HOLONOMY IN QUANTUM PHYSICS By Alexander R. Rutherford B. Sc. (Mathematical Physics) Queen's University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 1989 © Alexander R. Rutherford, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver, Canada Date October 4 1989 DE-6 (2/88) Abstract Holonomy in nonrelativistic quantum mechanics is examined in the context of the adiabatic theorem. This theorem is proven for sufficiently regular unbounded hamiltoni-ans. Then, simplifying to matrix hamiltonians, it is proven that the adiabatic theorem defines a connection on vector bundles constructed out of eigenspaces of the hamiltonian. Similar degeneracy regions, the natural base spaces for these bundles, are defined in terms of stratifications for the spaces of complex, hermitian matrices and real, symmetric ma-trices. The algebraic topology of similar degeneracy regions is studied in detail, and the results are used to classify and calculate all possible adiabatic phases for time-reversal invariant matrix hamiltonians in terms of the relevant topological data. It is shown how vector bundles may be used to impose transversality on the helicity vector of a photon. This is used to give a calculation, which is consistent with transver-sality, of quantum adiabatic phase for photons in a coiled optical fibre. As an additional application, the importance of quantum adiabatic in the dynamical Jahn-Teller effect is briefly explained. An introduction is given to some important aspects of algebraic topology, which are used herein. Moreover, a number of mathematical results for flag manifolds are obtained. These results are applied to quantum adiabatic holonomy. n Contents Abstract ii List of Figures v Acknowledgements vi Introduction 1 I Mathematical Preliminaries 4 1 Algebraic Topology 5 a Homotopy Theory 7 b Homology Theory 12 c Singular Homology 15 d Relation Between Homotopy and Homology 19 e Cohomology Theory 21 e Singular Cohomology 24 g De Rham Cohomology 32 2 Flag Manifolds 36 a Introduction to Flag Manifolds 37 b Algebraic Topology of Flag Manifolds 41 II Quantum Adiabatic Holonomy 52 1 Adiabatic Approximations in Quantum Mechanics 53 a The Adiabatic Theorem 53 b Matrix Hamiltonians 69 2 Regions of Similar Degeneracy 75 a Orbit Spaces 75 b Definition of Similar Degeneracy Regions 83 iii c Deformation Retracts of Similar Degeneracy Regions 87 3 Eigenspace Line Bundles . . 91 a Adiabatic Connection 92 b Holonomy and Stokes' Theorem 98 c Photons in an Optical Fibre 103 III Quantum Adiabatic Phase and Time-Reversal Invariance 114 1 Potentially Real Hamiltonians 115 a The Adiabatic Curvature in Potentially Real Subspaces . . . . 116 b Time-Reversal Invariance 119 2 Homotopy and Homology of Similar Degeneracy Regions 125 a Type I SD-Regions 127 b Type II SD-Regions 133 3 Computation of Adiabatic Phase 144 a Type I SD-Regions 146 b Type II SD-Regions 156 c Adiabatic Phase in Terms of Homotopy Classes 161 d Jahn-Teller Effect 164 References 168 iv List of Figures 1. The standard 1-simplex 15 2. The standard 2-simplex 15 3. Stratification of 5(4) C #erm(4, C) 81 4. Open star of VQ in A 2 86 v Acknowledgements I owe a debt of gratitude to many people for not only the work described in this thesis, but also many other projects, discussions, and musings, from which I learned many things. I thank my supervisor Gordon Semenoff for his support, a seemingly endless supply of interesting research problems, and many stimulating discussions. I thank Roy Douglas for teaching me algebraic topology, and for a productive and enjoyable collaboration resulting in the monograph [31], upon which this thesis is based. I am grateful to Lon Rosen and Joel Feldman for introducing me to the wonders of mathematical physics, and for their time and patience on many occasions. I benefitted greatly from the friendship and intellectual stimuli of my fellow gradu-ate students. I would like to thank my office mates Andre Roberge and Werner Keil; also, I thank Jim Booth and Chris Homes, who never let me forget that physics is an experimental science. During my tenure as a graduate student, I was fortunate to have the opportunity to visit the Institut fur Mathematik, E. T. H., and the Centre de Recherches Mathematiques, Universite de Montreal. I thank both of these institutions for their hospitality. At E. T. H., I profitted from discussions with Andrzej Lesniewski, Eugene Trubowitz, and Konrad Osterwalder, for which I thank them. I am grateful to many members of the Mathematics Department at U. B. C. for their patience in teaching me about mathematics. I thank Kee Lam, Dale Rolfsen, Richard Froese, and Tom Hurd. Especially, I would like to thank my wife Krisztina for her love and companionship. vi Introduction Simply put, holonomy is the geometrical concept which refers to the property that upon transport around a loop, an object may not return to its original state. Specifi-cally, holonomy pertains to the special form of transport, known as parallel transport. Quantum mechanics is not usually considered to be a geometrical theory, and therefore the recently discovered ubiquity of holonomy in quantum mechanics is surprising. We shall study holonomy arising from the quantum adiabatic theorem—hence the name, quantum adiabatic holonomy. The results described in this thesis appear in the monograph [31], by R. R. Douglas and this author. For quantum systems whose hamiltonians vary slowly with time, the adiabatic the-orem provides an approximate solution to the time-dependent Schrodinger equation in terms of eigenfunctions of the hamiltonian. The evolution of a quantum system is com-pletely determined by its hamiltonian, which is in general a hermitian operator on a separable Hilbert space: the state space of the system. Indeed, from a mathematical perspective, we can identify a quantum system with the hermitian operator that is its hamiltonian. A time-independent system is then represented as a point in the space of hermitian operators on the hilbert space. A time-dependent system is represented as path in this space of hermitian operators. The quantum adiabatic theorem defines a notion of parallel transport along this path. 1 Introduction 2 Implicit in the adiabatic theorem is the condition that some distinguished eigenvalue is bounded away from the rest of the spectrum of the hamiltonian. This eigenvalue is usually distinguished by the initial state of the system. When viewed in terms of the space of hermitian operators, such a "non-crossing constraint" is geometrically very complicated. The nontrivial geometry of regions defined by such constraints is at the heart of quantum adiabatic holonomy. These regions are termed similar degeneracy regions. In general, quantum adiabatic holonomy is difficult to compute because it depends on the details of the differential geometry of similar degeneracy regions. However, if the distinguished eigenvalue is nondegenerate, then the holonomy simplifies to a phase factor, the phase being called quantum adiabatic phase. Moreover, if the system is time-reversal invariant, then the adiabatic phase depends only on the most basic aspect of the topology of the similar degeneracy region. Specifically, a quantum system which is periodic in time is represented by a loop, and for time-reversal-invariant systems the adiabatic phase depends only on the homotopy class of this loop in the relevant similar degeneracy region. Quantum adiabatic holonomy was first noticed in analyses of the dynamical Jahn-Teller effect of molecular physics [60]. (For a more complete list of references on the dynamical Jahn-Teller effect, see Subsection 3.d in Chapter III, [2], and [50].) Indepen-dently, it was argued that adiabatic phases arise in the wave functions of particles with spin coupled to a slowly rotating magnetic field [1]. An important contribution was made by M . V. Berry [8], who demonstrated that adiabatic phase is a general phenomenon, Introduction 3 which occurs in many quantum systems. In commenting on Berry's paper, B. Simon [81] argued that adiabatic phase is an example of holonomy. Evidence of quantum adiabatic phase has been observed in a variety of experiments, many of which are reviewed in [2] and [50]. These include polarized photons in a coiled optical fibre [23], [88], nuclear quadrupole resonance spectra of rotating samples [89], neutron spin rotation [12], nuclear magnetic resonance [87], and half-odd integer quan-tization of pseudorotational levels in Na3, which results from the Jahn-Teller effect [27]. In Subsection 3.c of Chapter II, we examine the example of photons in an optical fibre. The Jahn-Teller effect is discussed in Subsection 3.d of Chapter III. Quantum adiabatic holonomy has figured prominently in recent theoretical work on the integer quantum hall effect [4], [5], [6], [81], the fractional quantum hall effect [3], [79], [78], vortices in 2-dimensional superfluids [43], superfiuid 3He [39], dipole-quadrupole interactions in spherical nuclei [58], optical pumping of atoms [76], and atoms in a slowly rotating electric field [64]. Also, it is now realized that holonomy is an important concept in gauge field theories. See [77] for a review of how chiral anomalies can be seen as holonomy in the hamiltonian formulation of chiral gauge theory. Chapter I Mathematical Preliminaries This chapter provides a brief introduction to some of the mathematical concepts and results which are used in this thesis. Unfortunately, it is impossible to make this introduction complete, because of length considerations. To offset this, references to mathematical background are supplied throughout the thesis. During this century, functional analysis and differential geometry have been exten-sively used in theoretical physics. However, until recently, algebraic topology has not been accorded the same attention. For this reason, many theoretical physicists are not as familiar with algebraic topology, and therefore the first section of this chapter is devoted to introducing some of the fundamental concepts of this field. For Chapters II and III, we require a number of results on the algebraic topology of flag manifolds. These results are easy to prove by standard techniques; however, a suitable reference for them could not be found in the literature. Therefore, flag manifolds are the topic of discussion in the second section of this chapter. 4 1.1 Algebraic Topology 5 §1 Algebraic Topology The central theme of algebraic topology is to express various properties of topological spaces in terms of algebraic data. The algebraic data usually takes the form of collec-tions of groups and homomorphisms between these groups. As might be expected, some information is lost when a topological space is analyzed algebraically. However, for some problems the essential features of a topological space may be encapsulated algebraically, and then it is possible to take advantage of the superior computational power of algebraic methods. Three very important concepts in algebraic topology are homotopy, homology, and cohomology. In this section, we shall introduce homotopy theory, singular homology the-ory, singular cohomology theory, and de Rham cohomology theory. In addition, we shall state some important theorems which express the relationship between these theories. Homotopy groups and singular homology groups are related by the Hurewicz homo-morphisms, which are the subject of the Hurewicz theorem. This theorem will be used a number of times in this thesis. An elementary discussion of homotopy theory, singular homology, and the Hurewicz theorem is given in [38]. For a more advanced treatment of these subjects, we suggest [83] or [92]. The relationship between singular homology groups and singular cohomology groups is expressed through the universal coefficient theorem for cohomology. This theorem is a special case of a more general universal coefficient formula for functors of complexes. An excellent discussion of functors and universal coefficient formulae is given in [29, Chapt. VI]. 1.1 Algebraic Topology 6 Homotopy, singular homology, and singular cohomology theories are defined for an arbitrary topological space. For the special case of a differentiable manifold, differential forms and the exterior derivative operator may be used to define de Rham cohomology. A good introduction to de Rham cohomology is Chapter 1 in [16]. The de Rham theorem provides a natural isomorphism between the singular cohomology of a manifold and the de Rham cohomology. This section does not cover all of the topics in algebraic topology which are prereq-uisite for a thorough reading of this thesis. Only the absolute versions of homotopy, singular homology, singular cohomology, and de Rham cohomology are presented, even though the relative versions of these theories are used in Chapter III. It is hoped that this elementary discussion will provide those physicists who are not versed in algebraic topology, with some feel for the subject. Relative homotopy theory is reviewed in [48], [83], and [92]. Relative singular homology and cohomology theory is presented in [29], [36], and [83]. We shall make extensive use of commutative diagrams, for which a good introduction is [61]. In particular, the five lemma will be used many times throughout the thesis, and the 3 x 3 lemma will be used in Chapter III. Fibre bundles are now used extensively in theoretical physics, and this thesis is no exception. A comprehensive reference on fibre bundles is [49]. Two excellent treatments of vector bundles and characteristic classes are [20] and [66]. 1.1 Algebraic Topology 7 (a) Homotopy Theory A pointed topological space is a nonempty topological space together with a distin-guished element, which is called its base point. Let X and Y be pointed topological spaces with base points XQ and yo, respectively. A map / from X to Y is said to preserve base points if /(zo) = yo- We shall always assume that maps between topological spaces are continuous maps. Two maps / : X —• Y and g: X —> Y are homotopic (written / ~ g) if there exists a map F: X x [0,1] —> Y such that F(x, 0) = f(x) for all xeX, F(x, 1) = g(x) for all x <E X, and F(x0,t) = y0 for all* € [0,1]. The relation ~ is an equivalence relation, partitioning the set of base-point-preserving maps from X to Y into disjoint equivalence classes, which are called homotopy classes. The set of all homotopy classes of base-point-preserving maps from X to Y is denoted by [X; Y], and the homotopy class of a map / is written [/]. If X and Y are path connected, then the following discussion is independent of which base points are chosen. However, if X and Y are not path connected, then the set of homotopy classes depends on which path components of X and Y contain the base points 1.1 Algebraic Topology 8 XQ and yo, respectively. For a discussion of the role of base points in homotopy theory, see [83, Sect. 7.3]. As an important example, consider the set of homotopy classes of maps from the circle Sl (with base point) to a topological space with base point XQ. The set [51; A] has a natural group structure, which is defined as follows. If S1 is parameterized by t 6 [0,1], then a base-point-preserving map from S1 to X may be written as a: [0,1] —• X , where cr(0) = cr(l) = XQ. The map a defines a loop in X, which is based at XQ. For two loops a and T , both based at XQ, the product loop a • r is defined to be the loop obtained by first going around a, and then going around r. In other words, J a(2t) if 0 < t < 1/2 a T I r(2t - 1) if 1/2 < * < 1 For two homotopy classes [a] and [r] in [5"1; A -], we define the product to be W)-[r] = [a.r}e[S1;X}. (1.1) It is easy to check that this definition is independent of which loops are chosen to represent the homotopy classes, and- hence we have a well-defined product on the elements of [S1; X]. This set, along with the multiplication defined in (1.1), forms the fundamental group of X, which is denoted by 7Ti(X). Thus far, we have taken the limited perspective that homotopy theory associates with two spaces the set of homotopy classes of maps from one space to the other. This is only part of the story. 1.1 Algebraic Topology 9 Take A to be a fixed topological space with a base point. We define a function from the collection of pointed topological spaces to the collection of sets by associating the set [A; X] with the pointed topological space X. Furthermore, if X and Y are two pointed topological spaces, and / : X —» Y is a base-point-preserving map, then we define an induced function fR.[A;X}-*[A;Y] by /#([#]) = [/ 0 g] f°r all [g] G [A; X). It is left to the reader to check that /# is well-defined. By associating the set [A] X] with every pointed topological space X, and the function /#: —> [A;Y] with every base-point-preserving function / : X —> Y, we have defined a covariant functor from the category of pointed topological spaces and base-point-preserving maps between them, to the category of sets and functions between them. Denoting this functor by IT A, we write TTA(X) = [A;X], and TTA(/) = /#• This is the proper context in which to understand homotopy theory. Functors and categories are discussed in Chapter 1 of nearly every book on algebraic topology. For example, see [29], [61], and [83]. It is instructive to reconsider in the context of category theory our example of homo-topy classes of maps from S1 to a pointed topological space X. The covariant functor nsi, which is given the special notation it\, assigns to each topological space the fundamental group TTI(X). In addition, if / is a map from X to Y, the induced function TTi(f) = /# is a group homomorphism from ni(X) to KI(Y). Therefore, we arrive at the remarkable LI Algebraic Topology 10 conclusion that n\ is a covariant functor from the category of pointed topological spaces and maps, to the category of groups and homomorphisms. It is by no means true that the functor TTA takes values in the category of groups and homomorphisms for all topological spaces, A. Indeed, the question, "For which topological spaces A does TTA take values in the category of groups and homomorphisms?" leads to the important topic of #-cogroups [47], [83, Sect. 1.6], [92, Sect. III.5]. To provide examples of if-cogroups, we introduce the concept of suspension. For a pointed topological space A with base point ao, the suspension S A is defined to be the pointed topological space obtained from A x [0,1] by identifying A x {0} U Ax {1} U {ao} x [0,1] to a single point, which is taken to be the basepoint of S A . Important examples of suspensions are the ra-dimensional spheres Sn, for n > 1. It is straightforward to show that Sn is homeomorphic to £ S 1 " - 1 , for all n > 1. This computation is carried out explicitly in [83, Lemma 1.66]. For any pointed topological space A, its suspension S A is an /f-cogroup. This is shown in most treatments of this subject, including [47, Chapt. 1], [83, Sect. 1.6], and [92, Sect. III.5]. Furthermore, the double suspension E 2 A is an abelian if-cogroup, which implies that the functor n^i A takes values in the category of abelian groups. Therefore, we conclude that the functors ITS* take values in the category of groups for n > 1, and in the subcategory of abelian groups for n > 2. Because of their importance, these functors are given the special notation 7rn, and the groups wn(X) are called the homotopy groups of the topological space X. 1.1 Algebraic Topology 11 It remains to consider the functor TTQ, which is obtained from the set of homotopy classes of base-point-preserving maps from the sphere S° to an arbitrary pointed space X. Clearly, the set ^o(-^) is in one-to-one correspondence with the path components of X. There is no natural group structure on 7TQ(X), and it should simply be regarded as a set with base point given by the homotopy class of the constant map. Extensive use will be made of the homotopy functors 7r„, for n > 0, and therefore we provide the following summary. The covariant functor irn from the category of pointed topological spaces and maps takes values in the category of ( sets with base point and base-point- .. £ ,. if n = 0 preserving functions groups and homomorphisms if n = 1 abelian groups and homomorphisms if n > 2 The basic philosophy remains the same for relative homotopy theory, which is defined on pairs of topological spaces with base points. However, there are important modifi-cations, which the reader should investigate in references such as [48] or [92]. A pair of topological spaces (X, A) is denned to be a topological space X, along with a subspace A C X} A base point for the pair (X, A) is a distinguished point in A. For the pointed pair (X, A), the relative homotopy group TrN(X, A) is a set with base point if n = 1 < a group, not necessarily abelian if n = 2 k an abelian group if n > 3 1 The term "subspace" will always mean topological subspace, rather than vector subspace. 1.1 Algebraic Topology 12 Usually, 7TQ(X,A) is not defined. (b) Homology Theory The most elegant formulation of homology theory is the axiomatic approach of Eilen-berg and Steenrod [36]. Consider (X, A) and (Y,B) to be pairs of topological spaces, without base points. A map / from the pair (X, A) to the pair (Y, B) is a continuous map from X to Y which satisfies f(A) C B. The class of all pairs of topological spaces, along with maps between them, form the category of topological pairs. The pair (X, 0) is identified with the topological space X, and therefore the category of topological pairs contains the category of topological spaces as a subcategory. Furthermore, the category of pointed topological spaces is obtained as a subcategory of the category of topological pairs by identifying the pair (X, {XQ}) with the pointed topological space X, which has base point xo. A homology theory consists of a collection of functors {Hq | q G Z}, where each Hq is a covariant functor from the category of topological pairs (or a subcategory) to the category of abelian groups. In addition, a homology theory contains a collection of homomorphisms 9Q{X,A): HG(X,A) — • Hq-i(A) , which are called the boundary homomorphisms. When there is no possibility for confu-sion, all of the homomorphisms dQ(X, A) will be denoted by <9+. Similarly, if / : (X, A) —> {Y,B) is a map of topological pairs, then often the induced homomorphisms Hq(f) will be collectively denoted by LI Algebraic Topology 13 A homology theory must satisfy the following five axioms. Axiom 1. (Naturality of 5*) For any map of pairs / : (X, A) —• {Y,B), let J\A • A —* B denote the restriction of f to the subspace A. Then, the diagram of groups and homomorphisms, HqU) Hq(X,A) • Hq(Y,B) dq{X,A) MY,B) (1.2) nq-i{fU) is commutative. Commutativity of the diagram (1.2) simply means that the compositions dq(Y, B) Hq(f) and Hq-\(f\A) dq(X, A) are identical homomorphisms from Hq(X, A) to Hq-i(B). In functorial language, Axiom 1 expresses the naturality of the boundary homomorphism. Axiom 2. (Exactness Axiom) For a topological pair (X, A), define the inclusion maps i: A t-* X and j: X «—>• (X, A) by i: a i—> a and j : x \-t x. These inclusions induce homomorphisms Hq(i) and Hq(j), and the sequence of groups and homomorphisms dq+l Hq{i) Hn(j) dq > Hq(A) > Hq(X) Hq(X, A) — Hq_x{A) • Hq^{X) • Hq_x{X,A) > ••• (1.3) is an exact sequence. The sequence (1.3) is called the homology exact sequence of the pair (X, A). 1.1 Algebraic Topology 14 Axiom 3. (Homotopy Axiom) Iff: (X,A) -> (Y,B) and g: (X,A) -* (Y,B) are nomotopic maps, then the induced homomorphisms Hg(f): Hq(X,A) — Hq(Y,B) and Hq{g): Hg(X,A) —+ Hq(Y,B) coincide for all q € Z. Axiom 4. (Excision Axiom) For any pair (X, A), let U be an open subset of X such that its closure is contained in the interior of A. Then, the inclusion k: (X -U,A-U)^(X,A) induces an isomorphism Hq{k): Hq{X -U,A-U)^ Hq{X, A), for all q G Z. Axiom 5. (Dimension Axiom) Let P be a one-point space, and recall that P is identified with the topological pair (P, 0). The homology groups of P are 0 ifq^Q It should now be demonstrated that homology theories exist. This will be established in the following subsection with the construction of singular homology theory. Also, it is important to understand the extent to which the homology theory is completely determined by the above five axioms. An extensive discussion of this question is given in [36]. LI Algebraic Topology 15 (c) Singular Homology The most cornmon homology theory, and the one that is used in this thesis, is singular homology theory. For simplicity, we define singular homology theory on the subcategory of topological spaces. Almost any text on algebraic topology, such as [29], [36], or [83], will give a definition of singular homology on the full category of topological pairs, and provide a verification of the axioms. The reader should also enjoy [34], one of the original papers on singular homology theory. Let (XQ, X\, ..., xg) denote the usual cartesian coordinates for the Euclidean space R ? + 1 . The standard g-simplex is defined to be the subspace d f 9 Ag = {(xo,zi, • • • | ^Px,- = 1 and Xi > 0} . i=0 Geometrically, A9 is the convex hull spanned by the vertices eo = (1,0, . . . ,0), t\ = (0,1, 0,. . . , 0), . . . , eq = (0,.. . , 0,1). For example, A 0 is a single point, A 1 is shown in Figure 1, and A 2 is shown in Figure 2. Xi x2 Figure 1. The standard 1-simplex. Figure 2. The standard 2-simplex. 1.1 Algebraic Topology 16 A map / : A p —> A 9 is said to be simplicial if it is the restriction of a linear function between Euclidean spaces, and maps vertices to vertices. Clearly, a simplicial map is completely determined by its action on vertices. We define a collection of simplicial maps f*: A 9 - 1 —• A 9 for i = 0,1, . . . , q by the vertex assignments ej if j < i Hence, /* maps A 9 1 isometrically onto the (q — l)-dimensional face of A 9 , which is opposite the vertex e,-. The boundary of A 9 , denoted by dAq, is constructed from the (q — l)-dimensional faces of A 9 . An orientation of A 9 is represented by an ordering of its vertices. The faces of A 9 inherit an orientation from the orientation of A 9 , and this orientation may possibly be different from the orientation inherited from A 9 - 1 through the simplicial map fg. Writing dA9 as a formal algebraic sum of the faces of A 9 , and using a minus sign to indicate a change of orientation, we have that S A « = ^ ( - l ) V i ( A * - 1 ) . t=0 For any topological space X, a singular ^-simplex is defined to be a continuous map a: A 9 —* X. The z t h face of the singular q-simplex a is the singular (q — l)-simplex LI Algebraic Topology 17 Following the formula for the boundary of A g , the boundary of a singular ^-simplex is defined to be the formal algebraic sum Our geometric intuition tells us that any reasonable definition of the boundary of a singular simplex should have the property that d(do~) = 0 when interpreted as a formal algebraic sum. Indeed this is true, and the proof is straightforward algebra. The first step in defining singular homology theory is to use the singular simplices to construct a chain complex. A chain complex C* is a sequence of abelian groups {Cq | q € Z}, and homomorphisms dg: Cg —> C g _ i , which have the property that the composition dqdq+\: Cg+\ —> Cq-\ is the zero homomorphism for all 5 6 Z. The elements of Cq are called ^-chains, and the homomorphisms dq are called boundary homomorphisms. The singular chain complex for a topological space X is denoted by S*(X). The group of singular ^-chains, Sq(X) is defined to be the free abelian group generated by all of the singular g-simplices of X. For q < 0, there are no singular g-simplices, and hence Sq(X) is defined to be the zero group. The elements of Sq(X) are written as formal sums and differences of singular g-simplices. The boundary homomorphism dq: Sq(X) —•» Sq^i(X) is defined by requiring that it map each generator of Sq(X) to its boundary as defined in equation (1.4). It follows directly from the result that d(da) = 0 for any singular simplex cr, that the composition dqdq+i is the zero homomorphism for all q G Z. (1.4) 1.1 Algebraic Topology 18 Consider a map / from a topological space X to a topological space Y. For each q, there is an induced homomorphism Sq(f): Sq(X) —* Sq(Y), which is denned by Sq(f): IT H (j o / , for all singular ^-simplices a in Sq(X). The collection of homo-morphisms {Sg(f)} is denoted by S*(f). It is easy to verify that dqSq(f) = Sq-i(f)dq for all q € Z. Any collection of homomorphisms defined on a chain complex and satisfy-ing this property is called a chain map. Therefore, we have denned a covariant functor 5* from the category of topological spaces and maps to the category of chain complexes and chain maps. Singular homology theory is denned in terms of the singular chain complex. The subgroup Zq(X) = kerdg is called the subgroup of ^-cycles, and the subgroup Bq = imdq+i is called the subgroup of ^-boundaries. Because dqdq+\ = 0, it follows that Bq(X) is a subgroup of Zq(X). The singular homology group Hq(X) is defined to be the quotient group Zq(X) / Bq(X), and the homology class of a cycle z € Zq(X) is denoted by {z}. Recall that if / : X —> Y is a map between two topological spaces, then the in-duced homomorphism on the singular chain complexes, S*(f): S*(X) —> ^ ( l^) is a chain map. This implies that for each q, there is a well-defined induced homomorphism from the quotient Zq(X) / Bq(X) to Zq(Y) / Bq(Y). This homomorphism is denoted by Hq(f): Hq{X) -> Hq(Y). Having constructed a collection of functors from the category of topological spaces to the category of abelian groups, and a collection of boundary homomorphisms, we have 1.1 Algebraic Topology 19 completed our definition of singular homology theory. It only remains to check the five axioms, for which a good reference is [36, Chapt. VII]. (d) Relation Between Homotopy and Homology For any topological space X with base point XQ, the homotopy functors provide a sequence of groups Kn(X), for n = 1,2,3, In addition, if we ignore the base point XQ and simply view X as a topological space, then the singular homology functors provide a sequence of abelian groups Hn(X), where n 6 Z. The homotopy and singular homology groups are related by the Hurewicz homomorphisms hn : 7Tn(X) —> Hn(X), for n > 1. The Hurewicz homomorphisms are defined as follows. Let Sn C R n + 1 denote the n-dimensional unit sphere, with base point. The sphere Sn is homeomorphic to dAn+1, the boundary of the standard (n + l)-simplex. Taking the orientation of Sn to be consis-tent with an outward pointing normal, we define e: 5 A n + 1 —• Sn to be an orientation-preserving homeomorphism. The homeomorphism e defines a singular n-cycle, and its homology class is denoted by {e}. The homology class {e} is a generator for Hn(Sn), which is isomorphic to Z, the abelian group of integers under addition. Consider any base-point-preserving continuous map f:Sn—*X. It induces on ho-mology a homomorphism Hn(f): Hn(Sn) —> Hn(X). The homotopy axiom implies that Hn(f) depends only on the homotopy class [/] in irn(X). Therefore, we define the Hurewicz homomorphism by hn: [/] ^ Hn(f)e. 1.1 Algebraic Topology 20 The Hurewicz homomorphisms are the subject of the Hurewicz theorem, one of the most important theorems in algebraic topology. A proof of the Hurewicz theorem for ab-solute homotopy and homology groups, and references to the original work of Hurewicz are provided in [34]. The Hurewicz theorem for topological pairs is covered in [48, Chapt. V], [83, Sect. 7.5], and [92, Sect. IV.7]. Before stating the absolute Hurewicz theorem, we introduce the concept of rc-con-nectedness. A topological space X is said to be n-connected if and only if Wk(X) = 0 for all integers 0 < k < n. Observe that O-connectedness corresponds to path connectedness, and 1-connectedness corresponds to simply connectedness. Theorem 1.5. (Absolute Hurewicz Theorem) (i) n > 2: If X is an (n — 1)-connected topological space, then hn: 7tn(X)Hn(X) is an isomorphism, and hn+i '• Kn+l{X) —* Hn+i(X) is an epimorphism. (ii) n = 1: If X is a path connected topological space, then fci:7n(X)—>#i(X) is an epimorphism, with kernel the commutator subgroup of TTI(X). 1.1 Algebraic Topology 21 The usual statement of the Hurewicz theorem does not contain the result in part (i) that hn+\ is an epimorphism. This is due to G. W. Whitehead [91]. Beware that if X is path connected and h\ is an isomorphism, it does not follow that hi is an epimor-phism. As a counterexample, consider the torus T2 = S1 x S1. The homotopy groups for the product of two topological spaces are given as the direct sum of the correspond-ing homotopy groups for the individual spaces,2 and therefore TTI(T 2 ) = Z © Z and 7T2(T2) = 0. The Hurewicz theorem then implies that hi: n\(T2) —+ H\(T2) is an iso-morphism. However, using a cellular decomposition,3 it may be shown that H2(T2) = Z. Hence, /12 cannot possibly be an epimorphism. (e) Cohomology Theory In a manner which will be made precise, the concept of cohomology theory is dual to that of homology theory. Here, we shall use two cohomology theories: singular coho-mology, which is defined in terms of homomorphisms on singular chains, and de Rham cohomology, which is defined in terms of differential forms. Our definition of cohomology theory is based upon the axiomatic definition given by Eilenberg and Steenrod [36]. It is necessary to go further then simply give the dual version of the axioms for homology, by defining cohomology with an arbitrary coefficient group G. 2 This follows from a special case of the homotopy'exact sequence for a fibration. The reader should become familiar with the homotopy exact sequence for a fibration, which is covered in most texts on homotopy theory, such as [48, Sect. V.6], [83, Sect. 7.2], or [92, Sect. IV.8]. 3 For a review of cellular decompositions, see [29, Chapt. V] . LI Algebraic Topology 22 Let G be an abelian group. A cohomology theory with coefficients G is a collection of functors {H9 \ q G Z}, where each Hq is a contravariant functor from the category of topological pairs (or some subcategory) to the category of abelian groups. For a topo-logical pair (X, A), the o t h cohomology group with coefficients G is written Hq(X, A; G). If / : (X, A) —> (Y, B) is a map of topological pairs, then the contravariant functors Hq induce homomorphisms Note that the direction of the arrow in H9(f; G) is reversed from that in / . This is what is meant by a contravariant functor. When there is no possibility for confusion, the homomorphisms Hq(f; G) will be denoted collectively by /*. In addition to the functors Hq, a cohomology theory has a collection of coboundary homomorphisms H9(f; G): Hq(Y, B; G) H9(X,A;G). 69(X, A; G): Hq(A; G) —-> Hq+1(X, A; G) . Often, these homomorphisms will simply be denoted by S*. A cohomology theory must satisfy the following five axioms. Axiom 1. (Naturality of 8*) For any map of topological pairs f: (X, A) —> (Y, B), let f\A: A —> B denote the restriction of f to the subspace A. Then, the diagram Hq(B\ G) Hq(f\A;G) H9(A; G) S"(Y,B;G) Sq(X,A;G) Hq+1(Y,B;G) (f;G) • Hq+1(X,A;G) 1.1 Algebraic Topology 23 is commutative. Axiom 2. (Exactness Axiom) For a topological pair (X, A) define the inclusion maps i: A <—> X and j: X <—>• (X, A) as in Axiom 2 for homology theory. Then, the sequence of groups and induced homomorphisms, • • • Hq-\A;G) ^ Hq{X,A;G) ^ Hq{X;G) Hq(A;G) ^ Hq+1(X,A;G) . . . ( 1 . 7 ) is an exact sequence. The exact sequence (1.7) is called the cohomology exact sequence of the pair (X, A). Axiom 3. (Homotopy Axiom) Iff: (X,A) -» (Y,B) and g: (X,A) (Y,B) are homotopic maps, then the induced homomorphisms Hq(f; G) and Hq(g; G) coincide for all qeZ. Axiom 4. (Excision Axiom) For any pair (X, A), let U be any open subset of X such that its closure is contained in the interior of A. Then, the homomorphisms Hq(k; G) induced by the inclusion k: (X -U,A-U)^(X,A) , are isomorphisms. LI Algebraic Topology 24 A x i o m 5. (Dimension Axiom) The cohomology groups of the one-point space, P are Hq(P; G) £ G if 0 = 0 0 ifq^O The most common cohomology theories are singular cohomology, Cech cohomology, and de Rham cohomology. Singular cohomology and de Rham cohomology will be denned in the remaining two subsections. . We remark that homology theory with an arbitrary coefficient group may be defined by generalizing in the obvious way the definition of homology theory in Subsection l.b. In the context of homology theories with general coefficient groups, the definition in Subsection l.b corresponds to homology theory with integer coefficients. (f) Singular Cohomology Just as singular homology theory is defined in terms of a chain complex, singular cohomology theory is defined in terms of a cochain complex. A cochain complex C* is a sequence of abelian groups {Cq \ q G Z} and homomorphisms 8q: Cq —• Cq+1, which have the property that the composition 6q+l8q is the zero homomorphism for all q G Z. A cochain complex may be obtained from a chain complex through the Horn functor. M C* is a chain complex and G is an abelian group, then Horn is a functor of two arguments, contravariant in the first and covariant in the second, which maps the pair (C*, G) to a cochain complex denoted by Hom(C*, G). 4 For each q G Z, the group Cq of 4 If B, C, and V are categories, and T is a functor from B x C to V which is contravariant in B and covariant in C, then T is called a bifunctor. Horn is an example of a bifunctor. 1.1 Algebraic Topology 25 g-cochains in the complex Hom(C*, G) is defined to be the group of all homomorphisms from Cq to G. The coboundary homomorphism 6q: C9 —> C 9 + 1 is defined by (S9f)(c) = f(dg+\c) for all / € Hom(C 9,G) and c € Cg+\. It is straightforward to verify that dq+idq+2 = 0 implies that S9+1S9 = 0 for all q € Z. If is the singular chain complex of the topological space X, then the singular cochain complex with coefficient group G is defined as S*(X;G)= Rom{S*(X),G) . Notice that it follows from the definition of the singular chain complex that S9(X; G) = 0 for all q < 0. The group of ^-cochains contains the subgroup of <7-c0cyd.es, Z9(X; G) = ker£ ? , and the subgroup of g-coboundaries, Bq(X; G) = im<S9-1. Because 6q8q~1 = 0, it follows that Bq(X;G) C Zq(X; G), for each q E Z. The singular cohomology group with coefficient group G is defined to be the quotient group Hq(X; G) = Zq(X; G) / Bq(X; G). If / is a continuous map from X to Y, the functors 5* and Horn provide an induced cochain homomorphism from G) to S*(X; G). This homomorphism of chain com-plexes induces a homomorphism H9(f;G): Hq(Y\G) —• Hq(X;G) on the cohomology groups. Again, the notation Hq(f; G) is usually abbreviated to /*. The relationship between singular homology and singular cohomology is given by the universal coefficient theorem for cohomology. Universal coefficient theorems are described in detail in [29], [61], and [83]. LI Algebraic Topology 26 Before stating the universal coefficient theorem for cohomology, we must first intro-duce short exact sequences and the derived functor Ext. For simplicity, we shall not discuss derived functors in general, but rather give a somewhat parochial construction of the Ext functor. An excellent introduction to derived functors is [61, Chapt. XII]. An exhaustive treatment is given in [18]. A sequence of abelian groups and homomorphisms a P A-+B-+C (1.7) is said to be exact at B if ima = ker/?. The exact sequence (1.7) is called a short exact sequence if moreover a is a monomorphism and /3 is an epimorphism. Sometimes this is indicated by writing the short exact sequence (1.7) as the sequence a 0 0—>A—*B—>C—*Q, which is exact at A, B, and C. A homomorphism 7 of short exact sequences is a triple of homomorphisms (71,72,73) such that A > B > C 71 72 73 A' > B' > C is a commutative diagram. If 71, 72, and 73 are isomorphisms, then the short exact sequences A —* B —> C and A' —• B' —> C' are said to be isomorphic. Note, that if 71 and 73 are isomorphisms, then it follows from the five lemma that 72 is also an isomorphism. 1.1 Algebraic Topology 27 An important example of a short exact sequence is the direct sum short exact sequence A A® B ^ B . The monomorphism i is defined by i: a I-+ (a,0), and the epimorphism 7r is defined by 7r: (a, b) H-> b. A short exact sequence A —> B —> C is said to be split if it is isomorphic, with the l IT identity on A and C, to the short exact sequence A —• A © C —* C. Some properties of split short exact sequences are given in a p Lemma 1.8. A short exact sequence A —• B —>• C is spiit iff any one of the following holds: (i) There exists a commutative diagram a p A > B • C 7 -A' A — A © C — C (ii) There exists a commutative diagram a P A • B • C | Jy | ^ — A © C — ^ c (iii) There exists a homomorphism ft': C —> B such that ft ft1 is the identity on C. The homomorphism ft' is called a right inverse for ft. (iv) There exists a homomorphism a': B —> A such that a1 a is the identity on A. The homomorphism a' is called a left inverse for a. 1.1 Algebraic Topology 28 Proof. Parts (i) and (ii) follow from the five lemma. Parts (iii) and (iv) are proven in Proposition 1.4.3 of [61]. D This lemma allows us to prove the following useful result. Proposition 1.9. If A —> B —> C is a short exact sequence of abelian groups, and C is free, then this short exact sequence is split. Proof. Let {c{} be a basis for C. For each i, choose an element 6,- € B such that 0bi = C{. Recall that /3 is an epimorphism, and therefore /? - 1 (CJ) is nonempty for all i. Define a homomorphism /?': C —»• B by requiring that ft'ci = b{. Because C is freely generated, it follows that /?' is well-defined. Obviously, /?' is a right inverse for fi, and therefore the proof follows from part (iii) of Lemma 1.8. D If A and C are fixed abelian groups, then an extension of A by C is defined to be a short exact sequence A —• B —> C. Two extensions of A by C are said to be equivalent, if they are isomorphic as short exact sequences with the identity on A and C. The set of equivalence classes of extensions of A by C is denoted by Ext(C, A). Let A\ —* B\ —> C\ be an element of the equivalence class E\ € Ext(Ci, A\), and A2 —* B2 —* C2 be an element of the equivalence class E2 € Ext(6^2,^2). The direct sum E\ © E2 € Ext(Ci © C2, A\ © A2) is defined to be the equivalence class of the short exact sequence Ai © A2 -* Bi © B2 - * C i © <72 . LI Algebraic Topology 29 It is easy to check that E\ © Ei is independent of the representative elements that are chosen for E\ and E2. We now show that extensions may be used to define a bifunctor from pairs of abelian groups to the category of sets and set homomorphisms. Let A —» B —* C be an extension of A by C, and denote its equivalence class by E € Ext(C, A). For an abelian group C', consider a homomorphism <f>: C —• C. Lemma III.1.2 in [61] states that there exists a commutative diagram A •* B' (1.10) B -* C for some abelian group B'. Furthermore, the equivalence class E' € Ext(C", A) of the extension A —+ B' —> C, is uniquely determined by the class E and the homomorphism (j>. This defines a morphism from Ext(C, A) to Ext(C',A). Now, consider a homomorphism if) from A to an abelian group A'. From Lemma III. 1.4 in [61], we know that there exists a commutative diagram B C (1.11) A1 • B" • C for some abelian group B". Furthermore, the equivalence class E" G Ext(C, A') of the extension A' —> B" —> C is uniquely determined by the class E, and the homomorphism if). This defines a morphism from Ext(C, A) to Ext(C, A'). Suppose that we have two pairs of abelian groups (C, A) and ( C , A'), and homomor-phisms (f>: C' —• C and if>: A —> A'. If the morphism induced by (1.10) is applied first, LI Algebraic Topology 30 and then the morphism induced by (1.11), we obtain the composition of morphisms Ext(C, A) -> Ext(C", A) -* Ext(C", A') . (1.12) However, if the induced morphisms are computed in the opposite order, then we obtain the composition Ext(C, A) -* Ext(C, A') -* Ext(C", A') . (1.13) It turns out that the compositions (1.12) and (1.13) coincide [61, Lemma III.1.6], and they define a morphism which is denoted by E x t ( ^ ) : Ext(C,A) —> Ext(C", A') . Therefore, Ext is a bifunctor from pairs of abelian groups to the category of sets and morphisms. The set Ext(C, A) has a natural group structure defined on it. In order to construct this group structure, we must first define the diagonal and codiagonal homomorphisms. For an abelian group G, the diagonal homomorphism A : G —> G@ G is Ag = (g,g), and the codiagonal homomorphism A ' : G © G —> G is A'(<7i,<72) = 9l + </2- For E\,E2 G Ext(C, A), the Bauer sum is denned by Ex + E2 d= Ext(A, A')(£i © E2) € Ext(C, A) . Under the Bauer sum, Ext(C, A) is an additive group [61, Thm. III.2.1]. The zero i element of this group is the equivalence class of the split short exact sequence A —* A © C —> C. Also, for group homomorphisms <f>: C' —> C and i/>: A —> A', the induced 1.1 Algebraic Topology 31 map Ext(^,^») is a group homomorphism. Therefore, if AQ is the category of abelian groups, then Ext is a bifunctor from AQ x AQ to AQ. If C is a free abelian group, then it follows from Proposition 1.9 that Ext(C, A) = 0 for all abelian groups A. Many of the Ext groups which arise in this thesis will fall under this example. Now that the Ext functor has been defined, we are able to state the universal coef-ficient theorem for cohomology. This theorem is proven in many textbooks on algebraic topology, such as [29, p. 153], [61, Thm. III.4.1], and [83, Thm. 5.5.3]. Theorem 1.14. (Universal Coefficient Theorem for Cohomology) If X is a topological space, and G is an abelian group, then there is a natural short exact sequence Ext(# g _ipO, G) -* H9(X; G) -* Eom(Hq(X), G) , which is split (although, not naturally) for all q 6 Z. This theorem implies that the singular cohomology group Hq(X; G) is isomorphic to Hom(# 9p0, G) ® Ext(#,_i(A-), G). Recall that it was remarked in Subsection l.e, that singular homology may be defined with an arbitrary coefficient group. For singular homology with arbitrary coefficients, there is a universal coefficient theorem for homology. We will not review this theorem, as it is not required in this thesis. However, interested readers will find it discussed in most texts on algebraic topology, including [29], [61], and [83]. LI Algebraic Topology 32 (g) De Rham Cohomology De Rham cohomology is a cohomology theory defined on the category of smooth manifolds and smooth maps between them. Throughout this thesis, smooth shall al-ways mean infinitely differentiable. De Rham cohomology satisfies the five axioms for cohomology in Subsection I.e. An elementary introduction to de Rham cohomology is given in Chapter 1 of [16]. Also suggested as references are de Rham's classic book [72], and [94]. References to many of the original papers on de Rham theory are given in [72]. Unlike the above references, we shall define de Rham cohomology with coefficients in the complex numbers C, rather than the real numbers R. For a smooth manifold M., let Vtp(M) denote the set of differential p-forms on M with complex coefficients. We shall allow p to range over Z, and define Q°(A/f) to be the set of C-valued smooth functions on M, and £l9(M) to be zero for q < 0. Under the operation of addition of differential p-forms, flp(M.) is an abelian group. Indeed, it has the further structure of a vector space over C . 5 The exterior differentiation operator is a group homomorphism for each p € Z. The collection of abelian groups {QP(M.)} and homomorphisms d form a cochain complex 0*(A/(), which is called the de Rham complex. 5 Abelian groups and vector spaces are examples of 72.-modules [61, Chapt. 1], where 1Z is a com-mutative ring with unit. Homology and cohomology may generally by defined as collections of functors from the category of topological pairs to the category of 7£-modules. 1.1 Algebraic Topology 33 The de Rham cohomology of M is defined to be the cohomology of the cochain complex fl*(M). Specifically, HPDR(M\C) is the quotient of the group kerd C Q.P(M) by the subgroup \md C £lp(M). Since both kerd and imd are vector spaces over C, it follows that HPDR{M; C) also has the structure of a vector space over C. To define Hp ft as a contravariant functor, consider two manifolds M.\ and M.2, and a smooth map / : Mi -»• M2- For w € flp(M2), the pullback6 f*u is an element of QP(Mi). Furthermore, /* commutes with d, which means that /* is a cochain homomorphism. This implies that /* induces a homomorphism HDRU)-- HPDR(M2]C)-,HpDR(Mi;C) . For a pair of manifolds (M,M) with J\f C M, it is possible to define the relative de Rham cohomology H*DR(M, Af; C). For a definition of relative de Rham cohomology, see [16, pp. 78-79]. On the category of smooth manifolds, we have defined two cohomology theories with coefficients in C: singular cohomology H*( •; C) and de Rham cohomology H*DR{ • ; C). The relationship between these cohomology theories is the subject of the de Rham the-orem [72, Chapt. IV], [94, Thm. IV.29A], which we shall now develop. Recall that a singular p-simplex a of a manifold M. is a continuous map a: Ap —• M. The singular chain complex constructed from all the singular simplices of M. is S*(A4). A smooth singular p-simplex a is defined to be a map cr: A p —> Ad, which can be extended to a smooth map defined on an open neighbourhood of Ap in R p + 1 . The 6 Pullbacks of differential forms are denned in [16, p. 19]. 1.1 Algebraic Topology 34 chain complex constructed from all of the smooth singular simplices of M. is denoted by At this point, it is necessary to describe some general properties of chain complexes. Consider two chain complexes C and C' with boundary homomorphisms d and d', re-spectively. Two chain maps r, 7/: C —> C are said to be chain homotopic if there exists a collection of homomorphisms {Dq \ q E Z} such that for all o, Chain homotopy, which defines an equivalence relation on the set of all chain maps between C and C', is reviewed in [83, Sect. 4.2]. The category with chain complexes as its objects and homotopy classes of chain maps as its morphisms is called the homotopy category of chain complexes. If a chain map r : C —> C is an equivalence in the homotopy category of chain complexes, then it is called a chain equivalence, and C and C' are said to be chain equivalent. The usefulness of this concept lies in the fact that free chain complexes are chain equivalent if and only if their homologies are isomorphic [83, As a special case of Theorem II in [35], S. Eilenberg proved the following about the chain map e. Sj!(M). Obviously, Sf{M) e: S?(M)l->S*(M). is a subcomplex of S*(M), and we denote the inclusion by Tq-T}q = + Dq-ldq . Thm. 4.6.10]. Lemma 1.15. The inclusion e:S^(M)^ S*(M) is a chain equivalence. 1.1 Algebraic Topology 35 We remark that for a chain complex constructed from singular cubes, the result corresponding to Lemma 1.15 is proven in [62, Appendix]. This proof is easily modified to give an alternate, and more modern proof of Lemma 1.15. It follows from Lemma 1.15 that the induced cochain map Hom(e): Hom(S*(A<), C) —•» Hom(5f (M), C) is a cochain equivalence. Denoting the cohomology obtained from S$(M;C) = Hom(^f(Af),C) by H*S{M;C), this implies Corollary 1.16. The induced homomorphism e*: H*(M;C) —> H*S(M;C) is an isomorphism. Q Given a p-form LO G Q?(M), we define a cochain k G Sg(M; C) by requiring that H*7) = I 0 0 ~ I <7*A, Jo JAP for all smooth singular simplices a G Sp(M). This defines a homomorphism rp: Vlp(M) —+ SVS(M). Stokes' theorem is just the statement that ij> is a cochain map. Therefore, ifr induces a homomorphism V>* '• H*DR(M; C) —»• H*S(M\ C) on cohomology. Theorem 1.17. (De Rham Theorem) ip* : H*DR(M; C) —> Hg(M; C) is an isomorphism. The de Rham homomorphism 1.2 Flag Manifolds 36 Notice that Corollary 1.16 combined with the de Rham theorem imply that the composition ( e * ) - V : H*DR(M;C) —+H*{MiC) (1.18) is also an isomorphism. This result will be used in Chapter III. The de Rham theorem also holds for relative cohomology groups. The relative version of the de Rham theorem is easily proven using the absolute de Rham theorem and the five lemma. For some hints on the proof, see [28, Chapt. 1], where the corresponding theorem is proven in the context of rational de Rham theory for arbitrary topological spaces. §2 Flag Manifolds In Chapters II and III, we shall need various results on the algebraic topology of flag manifolds. These results are not difficult to prove, and many of them represent a standard application of the concepts introduced in the preceding section. Indeed, with the exception of de Rham cohomology, this section should provide a good test of the reader's understanding of the previous section! Unfortunately, we have been unable to find a reference in the literature which provides a discussion of flag manifolds along the lines required here. 1.2 Flag Manifolds 37 (a) Introduction to Flag Manifolds To begin, we define complex flag manifolds. For positive integers n\,... , n p , which satisfy £2i=i «»' = n, the set F(ni,..., n p) is defined to be the collection of all sequences of vector subspaces V\ C V2 C • • • C Vp = Cn of the vector space C n such that their complex dimensions satisfy dimc(Vj) = X)j=i n«- Given the usual inner product on Cn, this set can be identified with the set of all ordered p-tuples (E\, E2,.. •, Ep) of mutually orthogonal subspaces of C n , which satisfy dime E{ = nj. This identification is constructed by taking E\ = Vi , and Ei to be the orthogonal complement of V,_i in Vi, for 2 < i < p. In terms of the usual orthonormal basis for C n , there is a one-to-one correspondence between ordered, orthonormal complex n-frames (ei, e2,. •., en) in C n , and elements in U(n), the group of n x n unitary matrices. Given an n-frame, we associate to it an element of F(n\,..., np) by setting the subspace E\ equal to the span of (e i , . . . , e n i), the subspace E2 equal to the span of (e n i +i, . . . , eni+n2), and so on. Let iVj = X)}=1 ni> and notice that two subframes, (e/v i +i, • • •, ^Ni+ni+i) a n ( i (ejv,-+l' • • •' eJV,+n,+1) & v e ^ n e same subspace En{+1 if and only if they are related by an element of U(rai+i). Therefore as a set, F(ni,... ,np) corresponds to the quotient U(n)/U(ni) x U(ri2) X • • • X U(n p ). We define U(ni) x • • • x XJ(np) to be the subgroup of all matrices which are block diagonal of the form " M i 0 . . . 0 ' 0 M 2 • : ••• 0 . 0 . . . 0 Mp. 1.2 Flag Manifolds 38 where Mi € U(n,) for all i — 1,... ,p. The set F(n\,... ,np) can be given a complex analytic structure to make it into a connected, compact complex manifold, which is called a complex flag manifold [85, Sect. 1]. Now, we define real flag manifolds. For positive integers ni,...np, satisfying IZjLi n« = n> the set F'(n\,..., np) is defined to be the collection of all sequences of subspaces V\ C V2 C • • • C Vp = R" of the vector space R n such that their real di-mensions satisfy dimji(Vj) = 2~2i=i n»- Let O(n) denote the group of n x n orthogonal matrices. Then, after fixing the usual orthonormal basis for R n , we see by the same rea-soning as for complex flag manifolds that as a set, F'(ni,... nv) can be identified with the quotient 0(n)/0(ni) x • • • x 0(n p), where 0(n\) x • • • x 0(np) is the subgroup consisting of all block diagonal matrices of the same form as (2.1), except that now M% E 0(n,), for all i = 1,... ,p. Because the quotient of a compact Lie group by a closed subgroup is a compact manifold, it follows that F'{n\,... ,np) is a connected, compact manifold, and it is called a real flag manifold. In this thesis, we are primarily concerned with two special cases of flag manifolds: short flag manifolds and Grassmann manifolds. The complex and real, short flag man-ifolds, denoted by F(p,q,r) and F'(p,q,r), respectively, are flag manifolds which have at most three nonzero arguments. The complex and real Grassmann manifolds, G(p,q) and G'(p,q), respectively, are flag manifolds which have two nonzero arguments. We remark that even though a Grassmann manifold may be viewed as a special case of a short flag manifold, we shall assume, unless otherwise stated, that the arguments of F, F', G, and G' are strictly positive. The distinct symbols F and G, will be used for short flag manifolds and Grassmann manifolds, respectively. This is done because there 1.2 Flag Manifolds 39 are important differences between short flag manifolds and Grassmann manifolds, which manifest themselves in their homotopy and homology groups. Since F(p, q, r) is diffeomorphic to U(p + q + r)/U(p) x 1%) x U(r), and F'(p, q,r) is diffeomorphic to 0(p + q + r)/0(p) x 0(q) x O(r), it follows that these manifolds appear naturally as the base spaces in the following two fibre bundles: U(p) x 1%) x U(r) • U(p + q + r) F(p, g , r) and 0(p) x 0(g) x O(r) ——> Q(p + q + r) (2.2) (2.3) Similarly, the complex and real Grassmann manifolds are the base spaces of the fibre bundles U(p) X V(q) — U U(p + <?) (2.4) and 0(p)xO(?) —-+ 0(p + g) (2.5) G'(P,9) respectively. The above four fibre bundles are specific examples of fibre bundles in which the fibre, base space, and total space are all homogeneous spaces. More generally, if G is a compact Lie group with closed subgroups G\ and G2 such that G2 C G\ C G, then the natural projection p: G/G2 —» GjG\ is the projection of a fibre bundle, and the fibre inclusion 1.2 Flag Manifolds 40 i: G1/G2 e—> G/G2 is the inclusion of the cosets. Further fibre bundles of this form, which are used in this thesis, are: GM -^-> F(p,q,r) G'(p,q) F'(p,q,r) Pi (2.6) and G(p + q,r) G'(p + q,r) G(q,r) F(p,q,r) G'(q,r) F'(p,q,r) P2 Pi (2.7) G(p,q + r) G'(p,q + r) It is useful for us to consider two sorts of imbed dings of flag manifolds. One of them imbeds a real flag manifold into the complex flag manifold with the same arguments. The other imbeds a complex Grassmann manifold into a larger complex Grassmann manifold, and similarly for real Grassmann manifolds. For any pair of flag manifolds (incuding Grassmann manifolds) F'(ni,... ,np) and F(ni,... ,np), the inclusion of O(k) in U(&) induces a smooth imbedding : F ' ( n i , . . . , np) ^ F(nu ..., np) . (2.8) Notice that the real dimension of F'(ni,..., np) is half of the real dimension of F(ni,..., np). Indeed, F(ni,..., np) is a complex algebraic variety defined over R [13, III.10.3], and j is simply the inclusion of its set of real points [13, V.15.3]. Now, for integers 1 < a < p and 1 < b < q, we define the inclusion G(a, b) <^-» G(p, q) as follows. Let 9: U(a+o) c—> V(p+q) be the smooth imbedding which maps W G U(a+6) 1.2 Flag Manifolds 41 to the (p + q) x (p + q) unitary matrix ' Ip-a 0 0 0 W 0 0 0 J g _ 6 where Ij is the j xj identity matrix. Since the image under 6 of the subgroup U(a) x U(6) is equal to the intersection of the image of U(a + b) with the subgroup U(p) x \J(q), it follows that 6 induces a smooth imbedding a: G(a,b)^G(p,q) . (2.9) The restriction of a to the submanifold of real points defines the smooth imbedding a': G'(a,b)^G'(p,q) . (2.10) (b) Algebraic Topology of Flag Manifolds We begin by computing the homotopy groups 7Ti and TT2 for complex Grassmann manifolds. This is done by examining the homotopy exact sequence for the fibre bundle (2.4). For integers 1 < a < p and 1 < b < q, the imbeddings $ and a induce the homomorphism from the homotopy exact sequence for G(a, b) to the homotopy exact sequence for G(p, q) 0 - 4 7r2(G(a,6)) iri(U(o) x U(6)) —• TTI(U(O + b)) h 6* *i(G(a,6)) 0 0 — K2(G(P,q)) — • 7n(U(p) x U ( 9 ) ) _> *i(U(p + 9)) (2.11) 1.2 Flag Manifolds 42 where 6 denotes the restriction of 9 to the subgroup U(a) x V(b). Notice that we have made use of the fact that 7T2 of a Lie group is zero. By the stability of the homotopy groups of U(n) [49, Thm. 7.4.1], 0# and 8# are both isomorphisms. It then follows by diagram chasing,7 that the homomorphisms a# : %i(G(a, b)) —> TTi(G(p, q)) and a#: n2(G(a, 6)) —• K2{G(p,q)) are also isomorphisms. Specifically, by setting a — b = 1 and noticing that G(l, 1) is the complex projective space CP(1), which is diffeomorphic to S 2 , we conclude that for all integers p,q> 1, *i(G(p,9)) = 0 and K2{G{p,q))*Z . (2.12) Furthermore, because a#: -K2{S2) —> K2{G(p,q)) is an isomorphism, the smooth imbed-ding a: S2 <—> G(p, q) represents a generator of Tr2(G(p,q)). We have shown that all of the complex Grassmann manifolds are simply connected. Therefore, the Hurewicz theorem and (2.12) imply that #i(<?(p,?)) = 0 and # 2(G(p, 9))S=Z, (2.13) for all integers p, q > 1. The universal coefficient theorem for cohomology implies that for all integers p,q > 1, the first and second cohomology groups with integer coefficients are H\G(p,q);Z) = 0 and H2(G(p, q); Z) £ Z . (2.14) 7 "Diagram chasing" is a standard technique used in algebraic topology to prove results about com-mutative diagrams. For some examples of diagram chasing, see Section 1.3 in [61]. 1.2 Flag Manifolds 43 For real Grassmann manifolds, it will only be necessary to compute the fundamental group, 7Ti. For integers 1 < a < p and 1 < b < q, let 9': 0(a + b) «—>• 0(p + q) denote the inclusion obtained by restricting 9 to the subgroup 0(a + 6) c U ( a + 6), and 8' the restriction of 0' to the subgroup O(a) x O(o). The inclusions a', 9', and 9' induce the following homomorphism between homotopy exact sequences obtained from the fibre bundle (2.5). *# p # 9 # H(0(a)x0(6)) —+ 7ri(0(a + 6)) —> ^(G'foft)) —> Z 2 © Z 2 0'. 7# (2.15) p # wi(0(p) x O(g)) —* 7ri(0(p-ro)) —- 7ri(G'(p,g)) — Z 2 © Z 2 In this commutative diagram, the last homomorphism in the top row is an epimorphism from 7r0(O(a) x 0(6)) = Z 2 © Z 2 onto 7r 0(O(a + &)) = Z 2 . Similarly, the last homomor-phism in the bottom row is an epimorphism from 7To(0(p) x O(q)) onto 7To(0(p -f q)). First, we shall assume that a -p 6 > 3. Then, by the stability of the homotopy groups of O(n) [49, Thm. 7.4.1], the homomorphisms i#: 7ri(0(a) x 0(6)) -+ 7ri(0(a + b)) and 7ri(0(p) x 0(q)) —»• 7ri(0(p-|-<?)) are both epimorphisms. In both cases, this implies that p'^ is the zero homomorphism. Therefore by diagram chasing, a'^. is an isomorphism. Furthermore, using (2.15), it is easy to compute that ni(G'(p, q)) is Z 2 if p + q > 3. We now consider the remaining case that a = b = 1 and p + q > 3. The homomor-phism O'jj.: 7Ti(0(2)) —> 7Ti(0(p + q)) is then an epimorphism [49, Thm. 7.4.1], and by diagram chasing, it follows that a'^ is also an epimorphism. Also, G'(l,l) is the real 1.2 Flag Manifolds 44 projective space RP(1), which is diffeomorphic to S1. Therefore, a'^: 7Ti(G'(l,l)) —> iri(G'(p, q)) is the epimorphism from Z to Z2 . 8 To conclude, the fundamental groups of real Grassmann manifolds are MG\VM*\1 " ' I 4 ; . 1 (2.16) I Z2 if p + q > 3 Since 7Ti(G'(p, q)) is abelian for all integers p,g > 1, the Hurewicz theorem implies that the first homology groups of real Grassmann manifolds are m<?M)*{z *py;\ (2.17) { Z2 11 p + q > 3 The universal coefficient theorem for cohomology implies that the first cohomology groups with integral coefficients are H\G>(P,q);Z)S{* (2.18) { 0 ifp + q>3 We now compute the first two homotopy groups of complex short flag manifolds. It follows immediately from the homotopy exact sequence for the fibre bundle G(p, q) F{Pi(lir) ~^ G(p + <7)r) m (2-6), that F(p,q,r) is simply connected. Indeed, by making an inductive argument using fibre bundles of this sort, we see that all of the complex flag manifolds are simply connected. 8 There are only two homomorphisms from Z to Z%. One is the zero homomorphism, and the other is an epimorphism. 1.2 Flag Manifolds 45 From the fibre bundle (2.2), we have the following short exact sequence for all integers p,q,r > 1. 0 > K2{F{p,q,r)) > 7ri(U(p) xU(o) xU(r)) > 7ri(U(p + q + r)) > 0 Z © Z © Z Z This short exact sequence is split because 7Ti(U(p+o+r)) is free abelian, and we conclude that 7C2(F(p,q,r))^Z®Z, (2.19) for all integers p,q,r > 1. The Hurewicz theorem implies that Hi(F(p,q,r)) = 0 and H2(F(p,q,r)) * Z © Z , (2.20) for all integers p,q,r >1. It follows from (2.20) and the universal coefficient theorem for cohomology that H1{F(p,q,r);Z) = d and H2(F(p, q, r); Z) S! Z © Z , (2.21) for all integers », g,r > 1. To compute the fundamental group of F'(p,q,r), we shall first assume that p + q + r > 4. From the fibre bundle (2.3), we have the homotopy exact sequence 7ri(0(p) x 0(q) x O(r)) -^7n(0(p + o + r)) -^7ri(F'(0,o,r)) —• *# TT0(O(P) x O(o) x O(r)) —•+ 7T0(O(» + o + r)) 1.2 Flag Manifolds 46 Since at least one of the integers p, q, or r is greater than or equal to 2, it follows that the homomorphism i'^: 7Ti(0(p) x 0(q) x O(r)) -» 7Ti(0(p + q -f- r)) is an epimorphism [49, Thm. 7.4.1]. Hence, p'^ is the zero homomorphism and we have the short exact sequence 0 — • iri(F'{p,q,r)) - 1 * 7r0(O(p) x O ( ? ) x O(r)) 7r0(O(p + ? + r)) • 0 z2 e z2 © z2 z2 This short exact sequence implies that 7Ti(.F'(p,q,r)) is isomorphic to Z 2 © Z 2 for all integers p, q, r > 1, satisfying /> + + r > 4. It now remains to compute the fundamental group of the closed 3-dimensional man-ifold F'( 1,1,1). By examining the homotopy exact sequence for the fibre bundle (2.3) with p = q = r = 1, we see at once that 7ri(F'( l , 1,1)) is a group of order eight. In Proposition III.2.23, it will be shown using a Mayer-Vietoris exact sequence9 that H\(F'(l, 1,1)) = Z 2 © Z 2 . Unfortunately, the proof of this result must be deferred until after similar degeneracy regions have been introduced in Section II.2. It follows from the Hurewicz theorem that 7ri(F '( l , 1,1)) must be a nonabelian group with commutator subgroup Z 2 . Up to isomorphism, the only two nonabelian groups with eight elements are the quaternion group and the dihedral group D% [44, p. 50]. Of these two groups, only Q% can be the fundamental group of a closed 3-dimensional manifold.10 This is because for any closed 3-manifold Ai, with finite fundamental group, the universal covering space 9 The Mayer-Vietoris exact sequence for homology may be found in [29, Sect. III.8], [36, Sect. 1.15], and [83, Sect. 4.6]. 1 0 The group Q$ is isomorphic to the multiplicative group of the eight quaternions usually denoted by {±l,±i,±j,±k}. 1.2 Flag Manifolds 47 M has the homotopy type of S3 [46, Thm. 3.6]. From this it follows that any element of order two in TTI(M) must belong to the centre of it\(M) [65, Cor. 1]. However, D% contains an element of order two which is not in the centre of the group [44, p. 50]. We summarize our results for real, short flag manifolds as follows: * i ( * W ) ) - ( g 8 i f ^ = r = 1 ( 2 . 2 2 ) " \ Z 2 © Z 2 if » + o + r > 4 V ; By the Hurewicz theorem, it follows that the first homology group is # i ( ^ W r ) ) = Z 2 0 Z 2 , (2.23) for all integers p, q, r > 1. The universal coefficient theorem for cohomology implies that H1(F'(p,q,r);Z) = 0, (2.24) for all integers p,q,r > 1. We now prove a proposition about homomorphisms induced by the inclusion of the real points defined in (2.8). This proposition saves us the trouble of computing the second homotopy and homology groups of real Grassmann manifolds and real, short flag manifolds. Proposition 2.25. Let j: G'(p,q) <^-> G(p,q) and j: F'(p,q,r) <^-> F(p,q,r) be the inclusion of the real points defined in (2.8). Then for all integers p,q,r > 1, each of the 1.2 Flag Manifolds 48 following four induced homomorphisms is trivial (i.e. the zero homomorphism): j # : ^2(G'(p,q)) — • 7r2(G(p,q)) (2.26) i#: *2{F'(p,q,r)) — > 7r 2 ( .F(p,r)) (2.27) j * : #2(G"(p,a)) —> #2(G(p,a)) (2.28) H2(F'(p,q,r)) —+H2(F(p,q,r)) (2.29) To prove Proposition 2.25, we require Lemma 2.30. For any n-fold covering, p: X —> X, the order of every element in the cokemel of the induced homomorphism p* : H2(X) —> H2(X), is a divisor of n. This lemma requires a knowledge of covering spaces. Two good reviews are [38, Sect. 1.5] and [83, Chapt. 2]. Proof. Let u> be any singular 2-cycle in X. By barycentrically subdividing [83, Sect. 3.3] to sufficiently often, we obtain a 2-cycle /3(u>) with the property that the image of each singular 2-simplex u{ in fl(io) is evenly covered [83, p. 62] by p. The 2-cycle f3(u>) is equal to the formal sum Yliaiaii where «i are integers. Let {Tij I j = 1) • • •) n} be the set of all singular simplices in X such that p o T{J = cr,-for all j. Then, we define u - - Y^ijaiTij- Since each o% is evenly covered, it 1.2 Flag Manifolds 49 follows that GJ is a 2-cycle in X, and furthermore p*{<2>} = {nP(u)} = n{p{u)} - n{u} . • Proof of Proposition 2.25. We will prove that the homomorphisms (2.27) and (2.29) are trivial. A similar argument proves that (2.26) and (2.28) are trivial. First, consider the homomorphism (2.27). We begin by noticing that the inclusion induced homomorphism ;#:*i(O(*0)-->7ri(U(&)) (2.31) is trivial for all integers k > 1. For k = 1, this is obvious because 7Ti(0(l)) = 0. Also, 7ri(0(fc)) £ Z 2 for all k > 3, and 7ri(U(&)) £ Z for all k > 1, which implies that (2.31) is trivial for k > 3. For k = 2, we require the trivial principal bundles11 SO(2) * 0(2) and SU(2) > U(2) det det O(l) U(l) which are associated with the determinant maps on 0(2) and U(2). The homo-topy exact sequences for these fibre bundles and the inclusion j: O(fc) c—> U(fc) 1 1 It is elementary to verify that both of these bundles are principal bundles. Obviously, the bundle over O ( l ) is trivial. Because the induced homomorphism d e t # : 7^(11(2)) —* 7Ti(U(l)) is an epimorphism, it follows that the bundle over U ( l ) admits a section. Any principal bundle admitting a section is trivial. 1.2 Flag Manifolds 50 provide the commutative diagram 7Ti(SO(2)) 0 = TTI(SU(2)) detj - *i(0(2)) Tl(0(l)) o det* •+ iri(U(2)) • *i(U(l)) 7T0(SO(2)) = 0 7T0(SU(2)) = 0 which implies that j#: 7Ti(0(2)) —> 7Ti(U(2)) is the zero homomorphism. The triviality of (2.27) follows from the triviality of (2.31), and the following commutative diagram whose rows are given by the homotopy exact sequences for the fibre bundles (2.2) and (2.3). 0 = ir2(Q(p + q + r)) 0 = TT2(U(p + q + r)) -> 7T2(F'(p,q,r)) 7Ti(0(p) X 0(q) X O(r)) 0 TTI(U(P) x V(q) x U(r)) We now show that the triviality of the homomorphism (2.27) implies the triviality of the homomorphism (2.29). Let p: F'(p,q,r) —» F'(p,q,r) denote the universal covering space of F'(p,q,r). Since ir\(F'(p,q,r)) is a finite group for all integers p, q, r > 1, it follows that p is a finite covering. Therefore, from Lemma 2.30 we conclude that the cokernel of the induced homomorphism p* : H2(F'(p, q, r)) —> H2(F'(p, q, r)) contains only elements of finite order. Consider the commutative square *2{F'{p,q,r)) • ir2(F(p,q,r)) H2(F'(p,q,r)) H2(F(p,q,r)) 1.2 Flag Manifolds 51 where the Hurewicz homomorphisms (vertical maps) are isomorphisms because both spaces are simply connected. Triviality of (2.27), and commutativity of this square imply that (j o p)+ : iJ2(-^'(p, q, r)) —> H2(F(p, q, r)) is the zero homomor-phism. The homomorphism (j op)* is equal to the composition of homomor-phisms p*. Since the cokernel of p* contains only elements of finite order, and H2(F(p, q,r)) is free abelian, it follows that is the zero homomorphism for all integers p, q, r > 1. D We remark that the universal covering space of F'(p,q,r) can be identified as a familiar space. If p + q + r > 4, then F'(p, q, r) is the oriented real flag manifold which is diffeomorphic to SO(p + o + r)/SO(p) x SO(o) x SO(r). If p = q = r = 1, then F'(l, 1,1) is diffeomorphic to S"3, which double covers SO(3) « RP(3), the corresponding oriented, real flag manifold. Chapter II Quantum Adiabatic Holonomy The quantum adiabatic theorem [14], [53] and its close relative the Born-Oppenheimer approximation [15], [24], [25], [82] are used extensively in nonrelativistic quantum me-chanics to find approximate solutions to the Schrodinger equation. An interesting and important feature of the adiabatic limit of solutions to the time-dependent Schrodinger equation is adiabatic phase. In this chapter, we shall show that the appearance of adia-batic phase can be traced to the twisting of certain eigenspace line bundles. Since the Born-Oppenheimer approximation is related to the adiabatic theorem, one should expect that nontriviality of the appropriate eigenspace line bundles will have important consequences for it. This is indeed the case, and the resulting phenomena have been observed in the Jahn-Teller effect, which arises because of the coupling between the vibrational modes of the nuclei and the electronic states of polyatomic molecules. References to this, and some of the other implications of adiabatic phase in physical systems are given in [50]. We shall be examining the examples of a particle with spin coupled to a magnetic field, and the molecular Jahn-Teller effect in more detail. Quantum adiabatic phase is introduced through the adiabatic theorem in the first section of this chapter. The second and third sections are on regions of similar degeneracy and eigenspace line bundles, respectively. Both of these concepts play an important role 52 II. 1 Adiabatic Approximations in Quantum Mechanics 53 in our analysis of quantum adiabatic phase. Also in the third section, some of our results are applied to the theory for a photon propagating in a coiled optical fibre. §1 Adiabatic Approximations in Quantum Mechanics Adiabatic phases arise in quantum mechanics when we consider solutions to the adiabatic limit of the Schrodinger equation. The adiabatic theorem is reviewed and then used to define the adiabatic phase. In Subsection b, we review M . V. Berry's computation of the adiabatic phase for a general periodic 2-level quantum system. These results are required in Chapter III, and it is useful to have them described in our notation. (a) The Adiabatic Theorem The state vector ip(i) for a time-dependent quantum mechanical system with hamil-tonian H(t) evolves according to the Schrodinger equation ijtm = H(twt), (i.i) where we have set Planck's constant % equal to 1. The hamiltonian H{t) shall be described as a continuous path of self-adjoint operators on a separable Hilbert space H. Of course, H(i) is not in general a bounded operator, which requires that we introduce a topology on the set of all self-adjoint operators on H. There is a standard procedure for defining a metric on the set of all closed operators on a Hilbert space [26], [56, Sect. IV.2], [69]. It exploits the fact that if T is a closed operator, then its graph G(T) is a a closed linear subspace of H x H. For two closed II.1 Adiabatic Approximations in Quantum Mechanics 54 operators S and T, let P(S) and P(T) denote the orthogonal projections onto G(S) and G(T), respectively. Then, denoting the operator norm for bounded operators on H x H by || • ||, the function g(S,T) d= \\P(S) - P(T)\\ is called the gap between S and T [26]. It is easy to verify that g(S, T) satisfies all of the properties of a metric, and the induced topology is called the gap topology on the space of closed operators on H . 1 The set of self-adjoint operators on a Hilbert space is a subset of the set of closed operators. We denote by Herm(H) the topological space of all self-adjoint operators on a Hilbert space H, with the gap topology. Beware that Herm(tl) is not a complete metric space. For an example of a Cauchy sequence which does not converge, see [56, Remark IV.2.10]. It is useful to know that the subset of bounded operators in Herm(H) is an open subset, and that the the relative topology defined on this subset is equivalent to the norm topology [56, Remark IV.2.16]. Before stating the adiabatic theorem, it is necessary to introduce a notion of differen-tiability for continuous paths T: [0,1] —> i7erm(H). Because Herm(H) is not a complete metric space, it is difficult to do this directly in terms of the gap metric. Instead, we define differentiability in terms of the resolvent of T(t). For a closed operator T, the resolvent set p(T) is defined to be the set of complex numbers ( for which T—(I is invertible, and R((, T) d= (T—(7) _ 1 is a bounded operator. Here and throughout the remainder of this thesis, the identity operator is denoted by I. The operator R((,T) is called the resolvent of T at (. Resolvents play an important role in the analysis of operators, and a good review is [56]. The resolvent set p(T) is an 1 This definition of g(S, T) makes use of the Hilbert-space structure of H. However, it is also possible to construct a metric for the set of closed operators on a Banach space. See [56] and [69]. II.1 Adiabatic Approximations in Quantum Mechanics 55 open subset of the complex plane, and a fundamental property of R((, T) is that it is a holomorphic function of ( on each connected component of p(T) [56, Thm. III.6.7]. The spectrum of T, denoted by a(T), is the complement in the complex plane of p(T). If T is a hermitian operator, then the spectrum is contained in the real line, R. We say that a path of self-adjoint operators T: [0,1] —> Herm(H.) is re-times differentiable in the norm resolvent sense, if the path of bounded operators R((, T(s)) is rc-times norm differentiable for all complex numbers ( with nonzero imaginary part. Norm differentiability is defined below. The space of bounded operators on H is denoted by B(H). If B is a function from the real line R to B(H), then the norm derivative is defined by ds V ' e—0 £ where the limit is taken in the norm topology on #(H). It is easy to verify that the norm derivative satisfies the following two properties. If B(s) and C(s) are two functions from R to #(H), which are at least once norm differentiable at s, then Ys (B(s)C(s)) - C(s) + B(s) ^(s) . (1.2) Also, if B(s) has a bounded inverse, then £*<•>-'--*(.)-'fMBM-1. (1-3) Using the identities (1.2) and (1.3), it is easy to prove II.1 Adiabatic Approximations in Quantum Mechanics 56 Lemma 1.4. A path B: [0,1] -» 0(H) D Herm(K) is k-times differentiate in the norm resolvent sense iff it is k-times norm differentiable. • This lemma suggests that norm resolvent differentiability is a sensible generalization of norm differentiability to unbounded self-adjoint operators. If a path of self-adjoint operators is &-times differentiable in the norm resolvent sense, then we shall simply say that it is Ck. If it is C°°, then it will be referred to as a smooth path. The definition of norm resolvent differentiability for a path T: R —• Herm(H) re-quires that R((, T(s)) be norm differentiable for all £ in the complex plane with the real line removed. The following lemma implies that it is necessary and sufficient for R((o,T(s)) to be norm differentiable for some fixed Co, with a nonzero imaginary part. Lemma 1.5. If R((Q,T(S)) is k-times norm differentiable for some Co £ C —R , then T(s) is Ck in the norm resolvent sense. Proof. If C, Co € p{T(s)), and ( ( 0, then from [56, Problem III.6.16] we have that (C — C o ) - 1 is contained in the resolvent set of it!(£o? T(s)), and R((, T(S)) = -(c - C o ) " 1 - (C - C o ) " 2 R( (C - C o ) " 1 , R«o, T(s))). This implies that if i?(Co, T(s)) is norm differentiable for some Co € C — R, then R((, T(s)) is norm differentiable for all ( G C - R. • II.1 Adiabatic Approximations in Quantum Mechanics 57 Having introduced the concept of differentiability in the norm resolvent sense, it is natural to say that a path T(s) in Herm(H) is continuous in the norm resolvent sense, or C°, if R((, T(s)) is continuous in the norm topology for all ( G C — R. For norm resolvent differentiability to be a sensible definition of differentiability on the topological space #erra(H), we would like continuity in the norm resolvent sense to be equivalent to continuity in the gap metric g on Herm(H.). It follows from [56, Thm. IV.2.25] that these two definitions of continuity are equivalent. Consider a path T: [0,1] —> -fferm(H), and assume that T(s) has an isolated eigen-value A(s) which remains bounded away from the remainder of the spectrum of T(s) for all s G [0,1]. Let P(s) denote the orthogonal projection onto the eigenspace E(s) of A(s). Lemma 1.6. If the path T is Ck, then the path of projection operators P is k-times norm differentiable. Proof. The projection P(s) is given by the contour integral where T is a closed contour in p(T(s)), encircling A(s). Because V is compact, and R((,T(s)) is rc-times norm differentiable in s, it follows that P(s) is &-times norm differentiable. • Differentiability of the projection P(s) may be used to prove that the isolated eigen-value A(^) is also a differentiable function of 5. II. 1 Adiabatic Approximations in Quantum Mechanics 58 Proposition 1.7. If the path T is Ck, then the function X(s) is Ck. Proof. For an arbitrary so € (0,1), let <f>o be an eigenvector of T(SQ) with eigenvalue A(so). There exists an e > 0 such that ||P(s)^>o|| > 0 for all s 6 (so — £,so + e). Then, <J)(s) = ||P(s)<^o||_1 P(s)(f>o is a fc-times differentiable normalized eigenvec-tor with eigenvalue A(s), for s E (so — e,so + e). The spectral theorem implies that for any fixed £ € C — R, the resolvent satisfies R((,T(s))<f>(s) = ((-\(s))-1<j>(s). Therefore, A(s) = ( - (<f>(s), R((,T(s))(f>(s))~1, and A(s) is fc-times differentiable at so- D The goal of the adiabatic theorem is to approximate solutions to the Schrodinger equation for a slowly varying, time-dependent hamiltonian, by using eigenvectors of the hamiltonian. To this end, we follow Kato [53] and construct the adiabatic transformation. Given a Ck path T: [0,1] —*• Herm(H) with k > 1, and an isolated eigenvalue A(s) which is bounded away from the remainder of the spectrum of T(s), there is an adiabatic transformation A(s) associated with A(s). The transformation A(s) will be defined so that it is a unitary operator which maps the eigenspace E(0) isometrically onto the eigenspace E(s). Denote the projection onto the eigenspace E(s) by P(s) and the norm derivative j^P(s) by P'(s). Then, the adiabatic transform A(s) is defined to be the unitary operator II. 1 Adiabatic Approximations in Quantum Mechanics 59 generated by the anti-self-adjoint commutator [P'(s),P(s)]. This means that A(s) is the unique solution to the differential equation ±A(s) = [P'(s),P(s)]A(s), A(0) = I. (1.8) Recall that I is the identity operator. Because [P'(s),P(s)] is a bounded operator, the Dyson expansion [71, Thm. X.69] may be used to write down a norm-convergent series solution to (1.8). From the Dyson expansion, it is evident that if T(s) is Ck, then A(s) is /c-times norm differentiable. If Q(s) = I — P(s), the adiabatic transformation satisfies the equivalent identities P(s)A{s) = A{s)P{Q) and Q{s)A{s) = A{s)Q{Q) . (1.9) These identities, which are verified below, imply that A(s) maps E(0) isometrically onto E(s), and £-L(0) isometrically onto £- L(s). In this computation, and throughout this thesis, the notation * is used to denote the adjoint of an operator. By differentiating P(s) = P 2 ( s ) , it is easy to verify that P'(s) = P'(s)P(s) + P(s)P'(s), which in turn implies that P(s)P'(s)P(s) = 0 . (1.10) Using (1.8) and these identities, compute the norm derivative j-s {A*(s) P(s) A{s)} = A*(s) {[P(s),P'(s)} P(s) + P'(s) + P(s) [P'(s),P(s)}} A(s) = 0 II.1 Adiabatic Approximations in Quantum Mechanics 60 This implies that A* (s) P(s) A{s) = A*(0) P(0) A(0) = P(0) which establishes (1.9). We return to considering solutions of the Schrodinger equation (1.1) as H(t) evolves continuously from H(Q) to H(T). Define the parameter s € [0,1] by 5 = t/r, and let T(s) = H(ST). Then, the Schrodinger equation can be rewritten in the form The map T: [0,1] —> Herm(H) defines a fixed path which is continuous in the gap topology. The time taken by the physical system to evolve along this path is r, and therefore the limit in which the time-evolution of the system is slow corresponds to r being large in (1.11). There is an extensive literature on the problem of existence and uniqueness of solu-tions to the Schrodinger equation (1.1). Because the hamiltonian is not usually a bounded operator, it is not possible to use a Dyson expansion to construct solutions to (1.1), or (1.11). However, there are a number of theorems which provide sufficient conditions on T(s) for (1.1), and hence (1.11), to have a unique solution for any specified initial data in the domain of T(0) [54], [55], [71, Sect. X.12], [80, Sect. II.7], [98, Sect. XIV.4]. For our purposes, the most useful theorem is B. Simon's statement [80, Thm. 11.21] of Yosida's theorem [98, Thm. XIV.4.2]. For simplicity, we state the theorem with slightly stronger hypotheses than did Simon. (1.11) II. 1 Adiabatic Approximations in Quantum Mechanics 61 Theorem 1.12. (Yosida) Let T: [0,1] Herm(H) be a C1 path satisfying the following conditions: (i) The domain of T(s) is independent of s. It is also assumed that this domain, denoted by T>, is dense in H . (ii) T(s) is bounded from below uniformly in s. This means that there exists a fixed Co € R such that T(s) > Co + 1 for all s G [0,1]. (in) The operator T(s) -^R{(o, T{s)) is bounded uniformly in s G [0,1]. Given conditions (i), (ii) and (in), the evolution equation d i—^{s)=.rT(s)i>{s), V(0) = ^ o (1.13) has a unique, strongly differentiable solution for all ip0 G T>. Moreover, i/>(s) G T) and | | ^ ) | | = | | ^ | | for allse [0,1]. The solution to (1.13) may be used to define an s-dependent operator function UT(s): V —• V by ^(s) = UT(s)il>o for all ipo G V. The subscript r reminds us of the parametric dependence of (1.13) on r. Because ||£/r(s) ^/>o|| = ||0o|| for all ipo G T), it follows that UT(s) has a unique extension to a unitary operator on H . Henceforth, the notation UT(s) will be used for this extension, which is called the propagator for (1.13). Theorem 1.12 implies that UT(s) is strongly continuous in s, and that Ut(S)I(}Q is strongly differentiable for all ^ o G V. We caution that the strong derivative of UT(s) need not be denned outside of T>. II.1 Adiabatic Approximations in Quantum Mechanics 62 Recall, that the limit of large r corresponds to the limit in which the time evolution of the system is slow. This limit is usually referred to as the adiabatic limit. The embodiment of the various adiabatic theorems is that under certain hypotheses, the appropriate adiabatic transformation may be used to approximate the propagator Ur, in the adiabatic limit. The first complete proof of an adiabatic theorem is due to Born and Fock [14]. In their theorem, it is assumed that the hamiltonian H(t) depends smoothly on t € [0,r], and that for all t it is a bounded, self-adjoint operator with a spectrum consisting only of discrete eigenvalues which are nondegenerate, except possibly for isolated crossings. Write T(s) = H(sr), and suppose that T(s) has a nondegenerate eigenvalue X(s), which is bounded away from the rest of the spectrum for all s G [0,1]. Associated with X(s) is an eigenspace E(s) and an adiabatic transformation A(s). Born and Fock proved that in the adiabatic limit (r —> co), the restriction of the propagator liT(s) to the subspace E(0) is approximated by the same restriction of the unitary operator exp[—ir fQs A(r) dr] A(s). In the physics literature, the phase factor exp[—ir fQs A(r) dr] is usually termed the dy-namical phase factor. A substantially more general adiabatic theorem was proven by Kato [53]. It applies to bounded hamiltonians which have a distinguished eigenvalue A(s) of arbitrary multiplic-ity. This distinguished eigenvalue must be bounded away from the rest of the spectrum, which may be of an arbitrary nature. Kato's theorem has been generalized by Avron, Seiler, and Yaffe [7], who showed that suitably modified results hold when the eigenvalue A(s) is replaced by a finite band, containing both discrete and essential spectrum. It is II.1 Adiabatic Approximations in Quantum Mechanics 63 required that this band be bordered by gaps in the spectrum, and that the width of these gaps be bounded away from zero for all s G [0,1]. We provide an adiabatic theorem generalizing Kato's theorem to sufficiently regular hamiltonians, which are no longer required to be bounded. Let T: [0,1] —» Herm(H.) be a C 2 path satisfying conditions (i), (ii), and (iii) in Theorem 1.12. This guarantees that the evolution equation (1.13) has a unique solution for each specified initial value in the domain V, and in addition provides enough regularity for the following adiabatic theorem. The notation a(T(s)) is used below for the spectrum of T(s). Theorem 1.14. (Adiabatic Theorem) Suppose that T(s) has an eigenvalue A(s) of arbitrary multiplicity, which satisfies for all s G [0,1], dist(A(s),a(r( 5 ))-{A( 5 )})>e, for some constant € > 0. Associated with the isolated eigenvalue A(s) is an eigenspace E($) and an adiabatic transformation A(s). Note that E(s) C T> for all s G [0,1]. Then for all <f>o G E(0), there is the estimate <-Uo\ T UT{\) <j)o - exp[-rr / A(x) dx] A(l) <f)0 Jo where K is a constant independent of r. The following proof is a simple generalization of Kato's proof in [53].2 2 We thank L . Rosen for suggesting a proof along these lines. II. 1 Adiabatic Approximations in Quantum Mechanics 64 Proof. Recall, that it follows from (1.8) that A(s) is twice norm differentiable, and that it follows from Theorem 1.12 that UT(s)ipo is strongly differentiable for all € T>. For any EV and 4>Q E -E(O), this allows us to consider (uT(s) tpo , exp[-ir J X(x) dx] A(s) cj>o = ^—irT(s) UT(s) , exp[—ir J X(x) dx] A(s) 4>o + ^UT(s) 4>Q , — irX(s)exp[—ir J X(x) dx] A(s) </>o^ + (uT{s) ipo , exp[-z'r J X(x) dx] A'(s) <^ = (ur(s)xpo , e x p [ - i r | X(x) dx] A'(s) <f>0^j , (1.15) We have used the fact that A(s) <f>o E -E(s), and also introduced the notation ' f o r i -Let P(s) be the projection onto the eigenspace E(s). Then, (1.8) and (1.10) imply that (1.15) is equal to UT(s) , exp[—ir J X(x) dx] P'(s) A(s) </)Q exp[ir X{x) dx] UT(s) i>0 , [I - P{s)] P'{s) A{s) ^ . (1.16) We now introduce the operator % ) = f Urn R(C,T(s))[I-P(s)] II. 1 Adiabatic Approximations in Quantum Mechanics 65 It is useful to observe that S(s) is easily recognized in terms of familiar operators. To this end, note that A(s) is contained in the resolvent set of the restriction of T(s) to the subspace E^^s), the orthogonal complement of E(s). The operator S(s) is the resolvent of evaluated at A(s) [56, Sect. V.3.5]. From this it follows that S(s) is a bounded operator with = [dist(A(s), a(T(s)) - {A(s)})]-1 . (1.17) More results for S(s) are proven in [56, Sect. III.6.5], where it is shown that SP = PS = 0 and [T(s)-\(s)}S(s) = I-P(s) . (1.18) Also, if T is any closed curve in p(T(s)), encircling only the isolated eigenvalue A(s), then C - A This formula implies that S(s) is a C 2 function of s. Substituting from (1.18) into (1.16) obtains exp[ir I X(x) dx] UT(s) *>0 , [T(s) - X(s)] S(s) P'(s) A(s) <j)Q Jo / = ^(J^{exp[ir J' X(x)dx]Ur(s)}^o , S{s) P'(s) A(s) <f>0 Therefore, the integral of (^r(s) , exp[—ir fQs \(x) dx] A(s) 4>Q) from 0 to 1 II. 1 Adiabatic Approximations in Quantum Mechanics 66 is equal to UT(l)ipQ , exp[-ir J X(x) dx] A(l) ^ - ^ o , ^ = ~7j0 (J;{eMir X(x) dx]Ur(s)^o , S(s)P'(s)A(s)<f>o^ds exp[irj^ X(x)dx]UT(s)ip0 , 5(a) J4(S) <^ 0 + - /Yexp[*r [' \{x)dx]UT{s)il>0, ^-{S(s) P'(s) A(s)} <f>0) , T Jo \ Jo ds J (119) after integrating by parts. It is easy to verify using (1.8), (1.10), and (1.18) that d —i T T 1 0 ds {S(s)P'(s)A(s)} <f>0 = {5'(«)P'( a ) + S'(a)P"(3)} A(s)<f>o . . By substituting this identity into (1.19), and making use of (1.17), we obtain the bound UT{l)ipo, exp[-zr J X(x)dx] A(l)<£0^ - , <f>i <*\\W\ \\M , (1.20) where * = 7(11^(1)11 + l|i"(0)ll+ ™P \\P"(s)\\)+ sup ||5'( 5)| | | |P'( 5)| |. 6 { *€[o,i] J *e[o;i] It is apparent from the form of K why we required that T(s) be C 2 , and that A(s) be bounded away from the rest of the spectrum. The bound (1.20) holds for II. 1 Adiabatic Approximations in Quantum Mechanics 67 all ipo e V. Therefore, it follows by a standard argument using the denseness of V that U~\l) exp[- i r / A (a?) dx] A(l) - <f>0 Jo • The adiabatic theorem has particularly interesting implications when the hamiltonian H(t) evolves periodically in time, with period r. In this case, T(0) = T(l) , and T is a loop in Herm(H.). The unitary operator UT{1) is called the monodromy operator for the periodic differential equation (1.11). The adiabatic theorem implies that for large r, the monodromy operator UT(1) is closely approximated by the dynamical phase factor times the monodromy operator A(l) of the differential equation (1.8). Let m denote the dimension of E(s) along the curve T, and now assume that m is finite. By hypothesis, m is a constant on T. If E(0) = E(l), then it follows that the restriction of A(l) to E(0) is a unitary operator on E(0). Hence, it may be represented as a matrix in U(m), the Lie group o f m x m unitary matrices. If X(s) is a nondegenerate eigenvalue for all s G [0,1], then m = 1 and A(l) may be written in the form e'7 for some real number 7, which is called the quantum adiabatic phase.3 Notice that 7 is only defined modulo 2n, and may be assumed to live on the interval [0,27r). The first systematic analysis of quantum adiabatic phase was conducted by M . V. Berry in [8]. He examined the hamiltonian for a particle of arbitrary spin coupled to a periodically varying magnetic field. This hamiltonian has only nondegenerate energy In the literature, 7 is also called Berry's phase. II. 1 Adiabatic Approximations in Quantum Mechanics 68 levels, and he computed 7 for each of these energy levels. For this class of hamiltonians, he made the important observation that 7 is essentially geometrical in nature. The differential equation (1.8) may be used to write down an integral formula for 7. Let </>(s) € H be a smooth family of normalized eigenvectors for the nondegenerate eigenvalue A(s). If T were a smooth open curve, then clearly it would always be possible to define <j>(s) smoothly everywhere on T. However, since T is a loop, it is not obvious that it is always possible to define <J>(s) such that it is single-valued and smooth everywhere on T. For now, we shall simply assume that such a </>(s) exists. This assumption will be justified in Section II.3. In terms of this basis, A(s) satisfies where 6(s) € R is a phase defined on the loop T. The adiabatic phase for the eigenvalue A(s) is now 7 = #(1)- Substituting (1.21) into the differential equation (1.8) gives the formula Using the notation of differential forms, the adiabatic phase on the loop T is then B. Simon observed that the 1-form (</>, dcj>) defines a connection on a hermitian line bundle, which when viewed as a subbundle of the trivial H-bundle has the eigenspace E for its fibre [81]. The adiabatic phase is then understood as the holonomy of this A(s)<l>(Q) = e"('V(s) (1.21) (1.22) II.1 Adiabatic Approximations in Quantum Mechanics 69 connection. An obvious question to ask is, "What should the base space of this line bundle be?" Roughly speaking, we would like the base space to be the largest region in Herm(H.) with the property that the distinguished eigenvalue A remains nondegenerate. Such a region will be called a similar degeneracy region. It was observed in [97] that for a degenerate eigenvalue, it is also interesting to look at the monodromy of equation (1.10). In this situation, e8 7 generalizes to a matrix in U(m). This "nonabelian adiabatic phase" is discussed in [4], [58], [64], [75], [78], and [97]. While we define the appropriate similar degeneracy regions, we do not attempt to compute nonabelian adiabatic holonomy in this thesis. (b) Matrix Hamiltonians In many applications, a hamiltonian may be represented by a matrix, at least for the purpose of computing adiabatic phase. For example, a particle with spin s and mass m, interacting through its magnetic dipole moment with a time-dependent magnetic field B(t) has H { t ) = ~ ^ { t ) " " as its hamiltonian. The coupling constant /i depends on the charge, mass, and gyromag-netic ratio of the particle; s is the vector spin matrix for spin s. The adiabatic phase is determined by the time-dependent (2s -f 1) x (2s + 1) matrix pB(t) • s, because it com-mutes with the time-independent kinetic term ^ V 2 . The hamiltonian H(t) is closely related to the hamiltonian for a photon in a coiled optical fibre. II. 1 Adiabatic Approximations in Quantum Mechanics 70 In the Born-Oppenheimer approximation [15], [24], [25], [82, Sect. 1.2] of molecular physics, the electronic part of the molecular hamiltonian is often approximated by a matrix. Adiabatic holonomy for eigenspaces of this matrix is important in Jahn-Teller theory, which describes the coupling between the vibrational modes of the nuclei and the quantum states of the electrons. The Jahn-Teller effect will be discussed further in Subsection III.3.d. The above examples of a particle with spin interacting with a magnetic field, and the Jahn-Teller effect illustrate that there is some physical value to analyzing adiabatic holonomy for matrix hamiltonians. In addition, studying matrix hamiltonians allows us to examine various topological problems without becoming mired in analysis difficulties. For these reasons, much of the remainder of this thesis is devoted to matrix hamiltonians. Of course, our ultimate goal is to obtain results for operator-valued hamiltonians. If a hamiltonian is represented as an n x n hermitian matrix, then the Hilbert space upon which it acts is C n , the vector space of complex n-vectors. A standard norm on the vector space of n x n matrices is the Hilbert-Schmidt norm, where * denotes conjugate transpose. We define Herm(n,C) to be the normed space of n x n hermitian matrices with complex entries. It is clear that the adiabatic theorem (Theorem 1.14) holds for matrix hamiltonians with the operator norm replaced by the Hilbert-Schmidt norm. Contained in the real vector space Herm(n,C) is i/"erm(n,R), the vector subspace of n x n symmetric matrices with real entries. This subspace plays an important role for physical systems which exhibit time-reversal invariance, and therefore the topology of its similar degeneracy regions will be important in Chapter III. II. 1 Adiabatic Approximations in Quantum Mechanics 71 The simpliest nontrivial example of a matrix hamiltonian is a 2 x 2 hamiltonian, which describes the dynamics of a 2-level quantum system. Adiabatic phases for 2-level systems may be computed in a straightforward manner, and this was done by Berry in [8]. His results are reviewed below. Let / denote the 2 x 2 identity matrix, and cr1, for i — 1,2,3, denote the Pauli matrices: a1 0 1" a 2 - "0 -x „ 3 _ 1 0 " u — 1 0 , a — i 0 , <7 — 0 -1 Then, any 2 x 2 hermitian matrix may be uniquely expressed in the form 3 H(x) = ^2xiai + xQI, (1.23) i = l where x = (xo, x\, x2, £ 3 ) is a coordinate vector in R 4 . From this we see that Herm(2, C) is isomorphic to R 4 , and (1.23) is used to identify these two vector spaces. The eigenvalues of H(x) are \± = dt\\x\\ + XQ, where x = ( x i , x 2 , £ 3 ) , and = (x\ + x\ + X 3 ) . Let R • / denote the 1-dimensional vector subspace of Herm(n,C) which is spanned by / . Then, the adiabatic hypothesis is satisfied for both eigenvalues on all loops in Herm(2, C) — R-J , the complement of R-7 in Herm(2, C). Furthermore, if U is any contractible open set in Htrm(2, C) — R - / , then it is possible to construct on U a smooth family of normalized eigenvectors for both of the eigenvalues of H(x). It should be emphasized, however, that it is not possible to define globally on Herm(2, C) — R • / , a smooth family of normalized eigenvectors. We return to these facts in Section 3, where they will be discussed in more detail for general hermitian n x n matrices. II. 1 Adiabatic Approximations in Quantum Mechanics 72 Denote by <^>+(x) and </>_(x) the normalized eigenvectors on U for the upper and lower eigenvalues, respectively. Notice that both </>+(x) and </>_(x) may be chosen to be independent of XQ, which simply shifts the spectrum of H(x). Therefore, the connection 1-form from the integrand in (1.22) takes the form for the upper eigenvalue on U. Similarly, the connection 1-form for the lower eigenvalue, A- may also be chosen so that it does not contain a dxn term. This implies that for any loop T in Herm(2, C) — R-7, the adiabatic phases for both of the eigenvalues are invariant under orthogonal projection of T onto Hermo(2, C), the subspace of 2 x 2 traceless, hermitian matrices. From the parameterization (1.23), it is clear that Hermo(2, C) is isomorphic to R 3 . Suppose that we now wish to compute the adiabatic phases of the upper and lower eigenvalues for any loop T in Herm(2,C) — R • I. By the above reasoning, it may be assumed without loss of generality that T is in Hermn(2, C) — {O}, where O is the origin in the coordinate system (x\, x2, xz). Furthermore, it is assumed for now that T is a simple loop which is not knotted in Hermo(2, C) — {0}. A loop is said to be simple, if it has no self-intersections. An extra subtlety occurs if T is knotted, and this will be dealt with in Subsection 3.b. Since T is not knotted, there exists an imbedding D of the 2-dimensional unit disc, such that D spans T in the sense that dD — T. (We will occasionally abuse the notation by using D to denote both the map and the image of the map in Hermn(2, C) — {0}.) II. 1 Adiabatic Approximations in Quantum Mechanics 73 Because D is contractible, it follows that we can define connection 1-forms A+ and A- everywhere on D. In Section 3, we shall examine how one goes about defining these connection 1-forms in general. For the upper eigenvalue, we define the 2-form /C+ = dA+. Then from Stokes' theorem and (1.22), the adiabatic phase for the upper eigenvalue is 7+(T) = i JJ JC+ (mod 2TT) . (1.24) We remark that /C+ may be interpreted as the curvature 2-form associated with the curvature of the line bundle mentioned at the end of the preceding subsection. This will be discussed in more detail in Section 3. For 2 x 2 hamiltonians, the curvature 2-form may be computed directly [8], and the result is = 2||a;||3 [ X 3 ^ X L ^ ^ X 2 + X 2 ^ 3 A dx\ + x\dx2 A dxs] (1.25) Also, it is easy to verify that for the lower eigenvalue, the curvature 2-form JC- = dA-. is equal to — )C+. Therefore, the adiabatic phase for the lower eigenvalue is 7-(T) = - 7 + ( n The coordinate system (rci,x2,x%) identifies D with a smooth disc in R 3 — {(0,0,0)}. Let $l(D) denote the solid angle subtended from the origin by this disc. Then, equations (1.24) and (1.25) imply that 7 +(r) = - |n (D) (mod27r), (1.26) Ill Adiabatic Approximations in Quantum Mechanics 74 which is the result found by Berry [8]. Contained in Herm{2, C) is Herm(2,H), the vector subspace of 2 x 2 real, symmetric matrices. In terms of the coordinate system (xo, x\, x2, £ 3 ) , this subspace is represented by {(XQ,XI,X2,23) | x2 = 0}. Now, suppose that T is a smooth loop in Hermn(2,R) — {O}, where Hermn(2, R) is the vector subspace of 2x2 traceless, real, symmetric matrices. In terms of the (x\, x2, ^-coordinate system, Hermn(2, R) is the a^a^-plane, and T is a loop in the punctured xia^-plane. If T is not simple, then it may be decomposed into simple loops, and (1.26) may be used to compute the adiabatic phase on each simple component. The total adiabatic phase is obtained by adding the adiabatic phases for all of the simple components (of course, taking "orientation into account). From this it follows that 7+(T) is simply TT times the number of times that T winds around the origin (modulo 2). The number of times that T winds around the origin represents the homotopy class of T in 7ri(i/ermo(2, R) — {O}), which is isomorphic to Z. This demonstrates that for any loop T in Herm{2, R) — R • I, the adiabatic phases 7+(T) and 7-(T) depend only on the homotopy class of T in iri(Herm(2, R) —R-7). As we shall see, this result generalizes t o n x n hamiltonians. Using the equation (1.25), observe that when restricted to Hermo(2,H) — {O}, the curvature 2-forms )C+ and /C_ are zero. More generally, /C+ and /C_ vanish when re-stricted to the direct sum of R • I with any plane through the origin in the (21, #2, X3)-coordinate system. Such subspaces are examples of potentially real subspaces of Herm(2,C), and these will be discussed in detail in Section III.l. We shall prove that II.2 Regions of Similar Degeneracy 75 it is generally true that the adiabatic phase is a homotopy invariant for loops in any potentially real subspace of n x n hermitian matrices. Moreover, such loops represent all periodic, time-dependent hamiltonians which are time-reversal invariant, and have a nondegenerate eigenvalue. §2 Regions of Similar Degeneracy In this section, we shall give a precise definition of the similar degeneracy regions, and describe their topology and the topology of their intersections with Herm(n,H). The finite-dimensional spectral theorem implies that a hermitian matrix may be specified by giving its spectrum and a unitary matrix, although of course, the latter is not unique. This simple idea can be used to describe similar degeneracy regions in terms of the orbit space associated with the conjugate action of the unitary group on Herm(n,C).4 Likewise, the similar degeneracy regions in Herm(n,'R) will be described in terms of the orbit space associated with the conjugate action of the orthogonal group on i7erm(ra,R). (a) Orbit Spaces First we shall discuss Herm(n,C), and then state the corresponding results for ii/"enn(n,R). The real vector space Herm(n,C) has real dimension n 2 , and as before, we endow it with the Hilbert-Schmidt norm, ||r|| = [ trT 2 ] 1 / 2 . Let Herm0(n, C) denote 4 This is essentially the well-known adjoint representation of U(n) on its Lie algebra. II.2 Regions of Similar Degeneracy 76 the (n 2 — l)-dimensional subspace of traceless matrices in Herm(n,C). With respect to the inner product (S,T) = tr[5T], this space is the orthogonal complement to the line R • I = {al | a 6 R}, which is the one-dimensional subspace spanned by the identity matrix, I. The compact Lie group U(n) acts by conjugation as a smooth transformation group acting on the right of Herm(n,C); that is, U «= U(n) takes T € R~erm(n,C) to Tu = U~^TU. The trace and the norm are well-defined, continuous functions on the orbit space Herm(n,C)/TJ(n), because. tr(r^) = tiT and \\TU\\ = ||T||. Orbit spaces are denned and discussed in [17, Sect. 1.3]. Under the action of U(rc), the vector subspace R • I is the set of all fixed points. Every matrix T 6 Herm(n, C) — R • J can be expressed uniquely as T = pB + rJ , with B € S(n) , the (n2 — 2)-dimensional unit sphere in Hermo(n,C). The parameters T , / J € R are defined by r = tv(T)/n and p = ||T — r/ | | > 0. Therefore, letting T correspond to (B, p, r) defines the diffeomorphism Herm(n, C) - R • 7 w <S(n) x R + x R , (2.1) where R + is the set of strictly positive real numbers. Furthermore, since for any U <= U(n), the equivariant product decomposition (2.1) is inherited by the orbit space, inducing the homeomorphism {Herm{n, C) - R J)/U(n) [5(n)/U(n)] x R+ x R II.2 Regions of Similar Degeneracy 77 From the spectral theorem, it follows that the U(n) orbits in Herm(n,C) are in one-to-one correspondence with sets of n real numbers, which are in fact just the eigen-values. This suggests that we may use the eigenvalues to parameterize the orbit space (Herm(n, C) — R • J)/U(n). However, we find that it is far better to use a canonical barycentric coordinate system which allows explicit control over the behavior of orbits as eigenspaces change dimension in Herm(n,C), and in particular near the set of fixed points R • I. Specifically for any T € Herm(n,C) — R • / , with eigenvalues denoted in non-decreasing order as A 1 < A 2 < - - - < A n _ 1 < A n , (2.2) we define the following parameters: T = T ( T ) = tr ( T)/n = i f > , i 1 / 2 , = ,(T) = ||r - r / | | = E ( f r 2 ) ' P = P(T) = (bo,b1,...,bn-3,bn-2), (2.3) where bk = (Xk+2 - Ajt+1)/(A„ - Ai) . Since X)fc=o = 1 a n d °fe ^ 0 f° r each k = 0,1, . . . , (n — 2), the barycentric coordinate vector /? ranges over all the elements of A™ - 2, the standard (n — 2)-simplex in R" - 1 . It is easy to see that the coordinate system (f3,p,r) uniquely labels each orbit which is not a fixed point. For each T € Herm(n, C) — R • / , we order the eigenspaces according to the ordering of eigenvalues in (2.2). Let nj denote the dimension of the jth eigenspace. Then, ny > 1 II.2 Regions of Similar Degeneracy 78 for each j = 1,... p and X/jf=i nj = n> where p > 2 is the number of eigenspaces or number of distinct eigenvalues of T. This associates with each orbit an ordered set of strictly positive integers (rii, «2> • • • > p^)> which we refer to as the degeneracy type of the orbit. If T is in an orbit of degeneracy type (n i ,n2, . . . , n p), then T is fixed under the action of a subgroup UT C U(n), which is isomorphic to YlPj=i U(n_j). This subgroup is called the isotropy subgroup [17, Sect. 1.2] at T. Note that the integers n i , n 2 , . . . , n p and p depend only on the barycentric coordinate (3 = (bo,..., 6 n-2)-If S and T are two points in the same orbit, it follows that the isotropy subgroups Us and UT are conjugate subgroups in U(n). In fact, the collection of isotropy subgroups for all of the points in an orbit form a complete conjugacy class of U(n). This conjugacy class is called the isotropy type of the orbit [17, p. 42]. We point out that two orbits with degeneracy types (n\,..., np) and (mi , . . . , mq) have the same isotropy type if and only if p = q and the set { n i , . . . , np} is equal to the set { m i , . . . , m ?}, without regard to order. The orbit space <S(n)/U(n) has the simple and natural structure of A n - 2 , with boundary faces in any subsimplex representing changes in the degeneracy type caused by eigenvalues becoming degenerate. Furthermore, the orbit space for all orbits whose isotropy type is the conjugacy class of the subgroup Ilj=i U(nj) m ^(n) 1S the union of at most p\ mutually disjoint, (p — 2)-dimensional open subsimplices5 of A n - 2 , where each 5 The standard g-dimensional open simplex A * C R ? + 1 is denned by A ? = {(a;o,a:i K j ) | ^ a ; , = l and a;,-> 0} . A n open subsimplex of A " - 2 is a simplicial map of Aq into A " - 2 . We caution that the image of such II.2 Regions of Similar Degeneracy 79 open subsimplex represents the orbit space for orbits of precisely one degeneracy type. If ft is a point in A™ - 2, then ft lies in a unique open subsimplex of A w _ 2 . This open subsimplex is denoted by x(ft)i a n d its closure in A " - 2 is the simplex determined by vertices associated to the positive barycentric coordinates of ft. Now define E(/?) to be the union of all orbits in S(n) with barycentric coordinates in x(ft)- Let p: E —• x be a restriction of the projection onto the orbit space S(n)/U(n). (When there is no ambiguity, we shall suppress the argument ft from our notation.) Lemma 2.4. E is a smooth submanifold of the (n 2 — 2)-dimensional unit sphere, S(n), and the projection p: E —* x is a trivial, smooth fibre bundle. Each fibre T(ft) = p~l(ft), is a single orbit diffeomorphic to the complex Bag manifold: 7{ft) « F(nun2, • • • , np) w U (n ) / J[ U(nj) . i = i Moreover, the inclusion F(ft) S(/3) is a homotopy equivalence. Proof. This follows immediately from Corollary VI.2.5 in [17]. The fibre bundle is trivial because the base is contractible. For the same reason, the inclusion of any fibre T(ft) E(/?) is a homotopy equivalence. D We remark that the real dimension of the fibre T(ft) is n 2 — X^j=iny> a n d the dimension of E is n 2 — 2 — J^j=i( n | — Moreover, since x(ft) 1S diffeomorphic to an open subsimplex is almost never an open set in A"-2. II.2 Regions of Similar Degeneracy 80 (p — 2)-dimensional Euclidean space, it follows that £(/?) &F(m,n2,--- ,np) x R p - 2 . In denning the diffeomorphism in Lemma 2.4, we have taken the convention that the arguments of F(ni,n2,..., np) are ordered according to (ni, n2,..., np), the degeneracy type of the orbit. Of course, the flag manifolds obtained by permuting the arguments of F are all diffeomorphic to each other. Recall from Section 1.2 that F(ni,n2,..., np) is nat-urally isomorphic to the set of ordered p-tuples (E\,E2,..., Ep) of mutually orthogonal vector subspaces in Cn with the property that dimc(Ei) = ni. Thus, the diffeomorphism in Lemma 2.4 identifies Ei as the eigenspace of the associated eigenvalue. By applying Lemma 2.4 to each open subsimplex of A n _ 2 , we obtain a stratification of S(n). For an example, consider Figure 3, which is a diagram of this stratification of «S(4), the unit sphere in #erm(4, C). It is interesting to note that since S(A) is diffeomorphic to the 14-dimensional sphere S 1 1 4, this stratification provides a decomposition of S14 into the seven subsets Et- for i = 1,2,..., 7. The straight arrows in this figure represent the projection maps p for the fibre bundles in Lemma 3.4, and the hooked arrows represent the inclusion of the fibres. We now consider the set of n x n real, symmetric matrices Herm(n, R) . It is a vector subspace of Herm(n,C), with real dimension (n 2 + n)/2. The subspace Herm(n, R) inherits its norm and inner product from Herm(n,C). Let #errao(n,R) denote the vector subspace of real, symmetric matrices with zero trace. II.2 Regions of Similar Degeneracy 81 E 5 — F(2,2) E 3 - F ( l , 2 , l ) Figure 3. Stratification of 5(4) C Herm(4,C). The compact Lie group of n x n orthogonal matrices O(n) acts by conjugation as a smooth transformation group acting on the right of i7erm(n,R). Let S'(n) denote the (n 2 + n — 4)/2-dimensional unit sphere in Hermn(n, R). Furthermore, (2.1) restricts to the diffeomorphism Herm(n, R) - R I w S'(n) x R+ x R . (2.5) As in our discussion of Herm(n, C), we find that the orbit space of Herm(n, R) — R • I II.2 Regions of Similar Degeneracy 82 has the direct product decomposition (tferm(n,R) - R • J)/0(n) w [S'(n)/0(n)] x R + x R , which allows us to parameterize the orbit space of i/erm(n,R) — R • I by the same coordinate system as is defined in (2.3). The orbit space S'(n)/0(n) is again the (n — 2)-dimensional standard simplex A™ - 2. Define £'(/?) to be the union of all orbits in S'(n) with barycentric coordinates in x(P)- Let p1: S' —* x he the projection to the orbit space. The same reasoning as in Lemma 2.4 obtains the following result. Lemma 2.6. £' is a smooth submanifold of the (n2 -f n — 4)/2-dimensionai unit sphere, S'(n), and the projection p': £'(/?) —»• x(P) i S a trivial, smooth fibre bundle. Each fibre — (p')_1(/?) is a single orbit diffeomorphic to the real flag manifold: Ptf) = (pT\P) « F'(nun2, • - . , * , ) « O(n)/ f[ 0(n3) . j=l Moreover, the inclusion T'(/3) <—> £ ' ( / ? ) is a homotopy equivalence. Q Observe that the fibre F((3) has real dimension ^ (n2 — Y?j=i nj2) > a n a ^ that this is exactly half the real dimension of the fibre J-(ft) in Lemma 2.4. Indeed, as we pointed out in Section 1.2, F'(n\,..., np) is the set of real points in F{n\,..., np), a complex variety defined over R. Also, the dimension of £' is \ {n2 — 4 — Y7j=i(nj2 ~ 2))> and the projection p' is the restriction of p to £'. II.2 Regions of Similar Degeneracy 83 We caution the reader that our notation uses the index j3 € A 7 1 2 in two different ways. That is, if fij 6 A n _ 2 for j = 1,2 are distinct barycentric coordinate vectors and xiPi) = X{P2), then £(&) = £(/? 2), and E'(/3i) = E'(/32). Also, there are dif-feomorphisms T{Pi) « ^(jh) and F'ifii) « Ffa). However, T(px) ^ ^(/32), and F'ifti) 4" Pith), since they are distinct fibres, or orbits. The results of this subsection provide a global generalization of the von Neumann-Wigner theorem [63], [68]. This theorem states that for an arbitrary hermitian matrix, it is only necessary to vary at most 3 parameters in order to cause two adjacent nonde-generate eigenvalues to cross. It also states that for real, symmetric matrices, it is only necessary to vary at most 2 parameters in order to cause two adjacent nondegenerate eigenvalues to cross. Using our stratifications for Herm(n, C ) and Herm(n, R) , it is easy to compute the relative codimension between regions with different degeneracy types. (b) Definition of Similar Degeneracy Regions Consider the adiabatic time evolution of an n-level quantum system, which initially has hamiltonian B £ Herm(n, C ) . In the definition of adiabatic phase in Section 1, we dis-tinguished some eigenspace of B. We are interested in the largest region in Herm(n, C ) , containing B and all points in Herm(n, C ) which can be reached from B by a path along which the dimension of the distinguished eigenspace is constant. We termed such a re-gion the similar degeneracy region (SD-region) of the distinguished eigenspace. Because such a region can be written as a union of orbits, it is most easily defined in terms of our parameterization of the orbit space. II.2 Regions of Similar Degeneracy 84 Definition 2 . 7 . Forl<k<k + d<n, with 0<d<n-2, define V(k, k + d) C A 7 1 - 2 to he the set of all (3 = (&o, 6 i , . . . , 6 N _ 2 ) winch satisfy bj = 0 i f k - l < j < k + d-2, & * _ 2 > 0 if k ^ 1 , and &jfc-Hi_i > 0 if k ^ n — d . It is easy to see that V(k, k + d) is the region in the orbit space for which the spectrum in ( 2 . 2 ) satisfies Afc_i < A* = Afc+1 = • • • = Xk+d < Xk+d+i • Of course, if either & = l ,or& = n — d, then one of the above inequalities is meaningless, and must be removed. The eigenspace associated with the eigenvalue A* = • • • = Xk+d has constant dimension, (d + 1 ) . Notice that V(k, k + d) is contained in the (n — d — 2)-dimensional subsimplex a C A n ~ 2 defined by requiring that bj = 0 for k — l<j<k + d — 2. Clearly, if d = 0 , then a = A™ - 2. We assume that the vertices of a inherit their ordering from the ordering of the vertices (WQ, . . . ,U>„_2) , of A n - 2 , which in turn is consistent with the ordering of the eigenvalues in equation ( 2 . 2 ) . If k = 1 , then cr is the ordered simplex whose vertices are the last n — d — 1 vertices of A™ - 2, and V ( l , l + d) is the open star6 in c of the first vertex t>o of a. Recall that VQ is wj, which is the (d + 1 ) vertex of A " - 2 . Similarly, if 6 Let Xi be the barycentric coordinate associated with the vertex Vi of A9. The open star of u,- in Aq is denned to be the set {(xo, x\,..., xq) G A 9 | X{ > 0}. Note that the open star of v,- is equal to the union all open subsimplices in A9, which contain V{ in their boundary. Unlike open subsimplices, open stars of vertices are indeed open subsets in A ' . II.2 Regions of Similar Degeneracy 85 k = n — d, then the vertices of a are the first n — d — 1 vertices of A n - 2 and V(n — </, n) is the open star in a of the last vertex vn_^_2 °f °"- The vertex u n_^_2 is also ww_<i_25 the (n — d — 1) vertex of A " - 2 . In the case 2 < k < n — d — 1, the region V(k, k + d) is the intersection of the open stars for the vertices v^_2 and vj--i, which are consecutive vertices in a. In A n ~ 2 , the vertices v^2 and are immediately before and after the d consecutive vertices of A n _ 2 which are not vertices of a. Note that V(k, k + d) is a contractible region in A n - 2 because it is always either an open star or the intersection of two open stars in a subsimplex of A " - 2 . For example, consider the 2-simplex A 2 which represents the orbit spaces of the unit spheres «S(4) and S1 (4) in Hermo(4, C) and Hermn{^, R) , respectively. The region in A 2 for which the first eigenvalue Ai is nondegenerate is V ( l , 1): the open star of VQ in A 2 . This region is shaded in Figure 4. Before writing down a careful definition of similar degeneracy regions, it is convenient to first define their projections onto the unit spheres S(n) and S'(n). We define the projected similar degeneracy regions, or PSD-regions, by W{k,k + d) = \J{X{P) | PeV(k,k + d)}cS(n) and W'(k, k + d) d^f |J{ £'(/?) | p € V(k, k + d)}c S'(n) . Similar degeneracy regions in Herm(n,C) and Herm(n,R) are defined in terms of PSD-regions by the diffeomorphisms (2.1) and (2.5), respectively. Let P denote the projection of Herm(n, C) — R-7 onto S(n) that is defined by (2.1), and P' the restriction II.2 Regions of Similar Degeneracy 86 V0 Vl Figure 4. Open star of vrj in A 2 . of P to Herm{n, R) — R • I. Then, the SD-region W(k, k + d) in Herm(n, C) is defined by W(fc, fc + d) d= i>_1(W r(ife, k + d)). Similarly, the SD-region W'(k, k + d) in i/erm(n,R) is defined by W'(Jfc, k + rf) =f ( P ' ) _ 1 ( ^ ' ( ^ ^ + ^)) • To keep our terminology concise, we shall refer to an SD-region in 7Term(n,R) as a real SD-region. Also, recall that the SD-regions W(k,k + d) and W'(k, k + d) are associated with an eigenvalue which has degeneracy (d + 1). Therefore, we will say that the SD-regions W(k, k + d) and W(k, k + d) have degeneracy (d + 1). If d = 0, then we shall call W(k, k) and W'(k,k) nondegenerate SD-regions. II.2 Regions of Similar Degeneracy 87 (c) Deformation Retracts of Similar Degeneracy Regions The existence of deformation retractions of the SD-regions W(fc, k + d) and W'(k, k + d), and the PSD-regions W(k, k + d) and W'(k,k + d) will play an impor-tant role in our analysis of SD-regions. For a review of deformation retractions see [83, Sect. 1.4]. The first deformation retraction is an immediate result of the definitions of VV{k,k + d) and W(k,k + d). Proposition 2.8. The pair (W(k,k -f d),W'(k,k + d)) is a strong deformation retract of the pair (W(k, k + d),W(k,k + d)). • We remark that this implies that W(k, k + d) ~ W(k, k + d) and W{k, k + d) ~ W'(k, k + d), where ~ denotes homotopy equivalence. Our second deformation retraction is a retraction of W(k, k + d) and W'(k, k + d) onto either a single fibre, or the cartesian product of a single fibre with the open interval. There are two types of PSD-regions, depending on whether the associated eigenvalue is at one of the extremes of the spectrum (i.e. the smallest or largest eigenvalue), or otherwise. This will be reflected in the deformation retracts that we obtain. For ease of reference, we refer to W(l, 1 + d) and W(n — d,n) as type I PSD-regions, while W(k, k + d) with 2<k<n — d — 1 will be called a type II PSD-region. The same terminology will also be applied to the real PSD-region W'(k, k + d), and to the SD-regions, W(fc, k + d) and W{k,k + d). II.2 Regions of Similar Degeneracy 88 Recall that the orbit space of a type I PSD-region is the open star associated with either the first vertex VQ, or the last vertex vn_d-2i 0 1 an ( n — d— 2)-subsimplex a C A n _ 2 . In the first case, when k = 1 the orbit at VQ is diffeomorphic to the complex Grassmann manifold G(d -f l , n — d — 1). Taking liberties with the notation, we shall write J-(vn) for the orbit at the vertex VQ. In the second case, when k = n — d the orbit ^ r(u„_ tj_2) at vn_d-2 is diffeomorphic to the complex Grassmann manifold G(n — d — 1, d + 1). Of course, G(d + l , n — d — 1) and G(n — d — l,d + 1) are diffeomorphic. We will show that J-{yo) and J-{vn_^_2) are deformation retracts of their corresponding PSD-regions W(l,d+ 1) and W(n — d, n), respectively. Notice that in this way, the orbit at each vertex in A n - 2 is a deformation retraction of exactly one type I PSD-region. The second type of PSD-region, W(k, k + d) with 2 < k < n — d — 1, has a more complicated deformation retract. Here the orbit space is the intersection of two open stars, which are associated with the adjacent vertices v^-2 and Ujt_i in the previously denned (n — d — 2)-subsimplex a C A n - 2 . Let x 1 be the open 1-simplex connecting Vk_2 and Vk-\. If S 1 = p - 1(x 1)> then by Lemma 2.4 the projection p: E 1 —> x1 1S a trivial, smooth fibre bundle and each fibre is diffeomorphic to F{k — l,d+l,n — d — k). We will show in the proof of Theorem 2.9 that E 1 is a deformation retract of its corresponding PSD-region. In this manner, every 1-simplex of A r a _ 2 corresponds to exactly one SD-region of type II. All of the above deformation retracts restrict naturally to the corresponding PSD-regions in S'(n), and the results are summarized in II.2 Regions of Similar Degeneracy 89 Theorem 2.9. First for type I PSD-regions, the pair (S(uo), S'(vo)) is an equi-variant, strong deformation retract of the pair (W(l,l + d),W'(l,l + d)), and the pair (£(t>n-<2-2)5 ^'(vn-d-2)) is a n equivariant, strong deformation retract of the pair (W(n — d,n),W'(n — d,n)). For type II PSD-regions, take any point ft £ x1* ^ n e n the pair (£(/3), £'(/?)) is an equivariant, strong deformation retract of the pair (W(k, k + d), W'(k, k + d)), where 2 < k < n - d - l . Proof. We first define our deformation retractions in the orbit space, and then equiv-ariantly extend them to the corresponding PSD-regions in S(n) and S'(n). For V(k, k + d), we consider the homotopy <f>: V(k,k + d) x [0,1] -+ V(k,k + d) , where <j>(ft,t) = (bo,..., bn-2) and ft = (b0,h,... ,bn-2) € V(k, k + d). The bj are defined as follows: (i) If k = 1, we define bj = tbj for 0 < j < n — 2 and j ^ d, and b^ = 1 — t + tbd. (ii) If k = n — d, we define bj = tbj for 0 < j < n — 2 and j ^ n — d — 2, and &n_d_2 = 1 - t + tbn_d-2-(iii) If 2 < k < n — d — l ,we define 6y = tiy for 0 < j < n — 2, but j ^ k — 2 and j ^ A: + <2 — 1, while ^ - 2 = I T T t 1 - * + * ( 6 * -2 + *fc+i-i)] Ok-2 + Ok+d-1 and Ojfc-2 + Ofc+d-l II.2 Regions of Similar Degeneracy 90 It is easy to check that for (i) and (ii), <f>((3,t) is a strong deformation retraction of V(k,k + d) onto the required vertex, and that for (iii), (f>(/3,t) is a strong deformation retraction of V(k, k+d) onto the interior of the required one-simplex. We now lift our deformation retraction <p to a deformation retraction $ of W(k, k + d). Consider any B € W(k, k + d), and let D be the unique traceless, unit-norm diagonal matrix in the orbit of B with the property that the diagonal entries of D are ordered in nondecreasing order. Denote by /? the coordinate in V(k, k+d) associated with D, and let (J>(/3, t) be the path in V(k, k+d) attached to /3 by the homotopy constructed above. Since there is a one-to-one correspondence between barycentric coordinates and traceless, unit-norm, diagonal matrices with nondecreasing entries along the diagonal, it follows that <f>(/3, t) defines a path of such diagonal matrices D(t). Let U be any unitary matrix satisfying B = U D U~l, and define the homo-topy $: W(k,k + d) x [0,1] -* W(k,k + d) by $(B,t) = U D(t) U~l. Note that U is not uniquely determined, but any such U will give the same result for $(B,t) because the isotropy subgroup for D(t) is the same for all 0 < t < 1, and this subgroup is contained in the isotropy subgroup for D(Q). Thus, $ is well defined. It is easy to check that in all three cases, the homotopy $ is the strong deformation retraction required. The homotopy $ is obviously equivariant by construction. In terms of the real PSD-regions W'(k, k + d) C W(k, k + d), the II.3 Eigenspace Line Bundles 91 homotopy $ is a relative homotopy since if B £ W'(k, k + d), then $(B,t) € W'(k, k + d) for all t £ [0,1]. • Theorem 2.9 establishes that the relative inclusion (£, £') <^-> (W, W') is a homotopy equivalence. With Lemma 2.4, Lemma 2.6, and Proposition 2.8, this proves Corollary 2.10. The inclusion of the orbit pair into the corresponding SD-region pair i: (F(k - 1, d +1, n - d - k), F'(k - 1, d + 1, n - d - k)) (VV(k, k + d),W(k, k + d)) is a homotopy equivalence. In the extreme case of k = 1 or k = n — d, it is understood that F and F' are replaced by the appropriate one of the two diffeomorphic Grassmann manifolds, G(d+l,n-d-l) w G(n-d-l,d+l) andG'(d+l,n-d-l) ss G'(n-d-l,d+l), respectively. • We remark that Corollary 2.10 illustrates the prominent role played by short flag manifolds and Grassmann manifolds in adiabatic holonomy. §3 Eigenspace Line Bundles On each similar degeneracy region W(&, k + d) in Herm(n, C) , we construct a vector bundle, which when viewed as a subbundle of the trivial product bundle W(&, k + d) x C", has fibre the (d + l)-dimensional eigenspace associated with the eigenvalue Afc. We refer II.3 Eigenspace Line Bundles 92 to this bundle as the eigenspace vector bundle. If d — 0, then the eigenspace bundle is a complex line bundle, which we denote by £. We will restrict our attention to nondegen-erate eigenvalues and their associated eigenspace line bundles, since in Chapter III, we are mostly concerned with adiabatic phase. (a) Adiabatic Connection The total space £(£) for the complex line bundle £ over the nondegenerate SD-region W(k, k), is the subspace of W(k, k) x C n given by {(B, <f>) \ B(j> = A ^ } . The projection map p: £(() -> W{k, k) is denned by p: (B, 4>) i-» B. For B G VV(k, k), the fibre F(B) = p~L(B) is the eigenspace associated with the eigenvalue A* of B. In the remainder of this section, we shall simplify our notation by writing W for W(fc,fc). To verify that £ is a complex line bundle, it only remains to check local triviality. Recall from Lemma 1.6, that the projection operator P(B) depends smoothly on B. Consider a point 5o G W, and let (f>$ be a normalized eigenvector for the eigenvalue Ajfc of BQ. Because P(B) is a continuous function of B, there exists a neighbourhood U of B0 such that | | P ( i ? ) ^ o | | > 0 for all B € U. Therefore, it is possible to set (j>(B) = H i ^ P ^ o H - 1 P(B)fo, and define a map h from { / x C t o p~l{U) by h: (B,z)^(B,z<j>(B)). It is easy to check that h is a diffeomorphism, and hence £ satisfies local triviality, as required. II.3 Eigenspace Line Bundles 93 The usual inner product on the vector space C n defines a natural hermitian structure on the line bundle £. We denote the vector space over C of smooth sections of ( by C°°((). For sections s\,S2 € C°°(£), the inner product, which is written as (si,S2), is a n element of C°°(>V,C), the space of all smooth functions from W to C. We suggest references [20] and [66] as excellent reviews on hermitian line bundles. Since the SD-region W is nondegenerate, it follows from Definition 2.7 that it is an open subset of the manifold Herm(n, C). This implies that W is an open manifold, and T* is defined to be the cotangent bundle of this manifold.7 A connection on the complex line bundle £ is a C-linear map V : C " ° ( 0 — C ° ° ( T * ® 0 which satisfies the Leibniz formula, V(/a) = df®s + fVs for every s € C°°(£) and every / € C°°(VV, C). Geometrically, V should be thought of as defining a notion of infinitesimal parallel displacement of the fibres of £. In this vein, a section s is called horizontal if Vs = 0. The differential equation (1.8) will be used to define a connection VA on £. This connection is first defined locally, and then extended to a global definition on £. On any m-manifold, there exists an open cover with the property that all finite, nonempty 7 Note that degenerate SD-regions are not open subsets of Herm(n,C). However, although it is significantly more difficult to prove, it is true that even degenerate SD-regions are submanifolds of Herm(n, C ) [32]. II.3 Eigenspace Line Bundles 94 intersections of sets in the Open cover are diffeomorphic to R m [16, p. 42]. Such an open cover is called a good cover. Let {Ua} be a good cover of W. For a particular open set Ua, the restriction £|j/ a defines a complex line bundle on Ua. However, because Ua is contractible, it follows that this line bundle is trivial and hence possesses a nonzero section, which when normalized with respect to the hermitian structure on (\ua is denoted by <j)a. Of course, <j)a is nothing more than a smooth family of normalized eigenvectors for the eigenvalue A^ of the hermitian matrices in Ua. The notation (si,d.S2) will be used to denote also the obvious product between a section s\, and the vector of differential 1-forms ds2 G C°°(r* ® £). The complex 1-form •Aa = ((f>a, d(f>a) from the integrand in (1.22) defines a smooth section in T*\ua. We define a map Va:C°°(t\Ua)—>C°°((T*®0\ua) by requiring that it satisfy the Leibniz formula, and map <f>a to Aa ® 4>a- Because any section s £ C°°(£\ua) may be written in the form s = f(f>a for some / € C°°(Ua,C), this uniquely defines a connection on £\ua-Using the above procedure, we construct a connection on the restriction of £ to every open set in the good cover {Ua}. For all of these connections to piece together to give a connection on £, we must check that they agree on the overlaps of the open sets. Suppose that Ua and Up are two open sets with a nonempty intersection, and consider normalized sections <j>a and <j>p of £\(/a and £\[/g, respectively. Then, in (\uar\U0i these sections are II.3 Eigenspace Line Bundles 95 related by <f)p = e <f>a, for some 6 € C°°(Ua n Up, R) . 8 We compute that Va(e''Va) = <8> e'V« + (<f>a,d(f)a) ® eie<f>a = V ^ ) (3.1) Therefore, the collection of connections {V a } piece together to give a connection for £, and we denote this connection by V ^ . Because of its relation to the adiabatic theorem, is referred to as the adiabatic connection. A connection V on a hermitian line bundle is said to be compatible9 with the her-mitian structure, if for any two sections si and s2, the relation d(si,s 2) = (s i ,Vs 2 ) + (Vsi ,s 2 ) (3-2) is satisfied. We remark that (3.2) is equivalent to requiring that the inner product (si, s2) be constant, whenever si and s2 are horizontal sections [20, p. 44]. For the adiabatic connection, (3.2) corresponds to the requirement that Aa + Aa = 0 on each open set Ua. It follows immediately from Aa = ((j)a,d(j)a) that this requirement is satisfied, and therefore, is a compatible connection on the hermitian line bundle £. The connection 1-form Aa represents the action of V a in terms of <j)a, a normalized basis for sections of £\ua- Consider a new normalized basis <f)'a, and let A'a be the connection 1-form relative to this basis. It is generally true that for a connection on a 8 For readers who are familiar with the language of gauge theory, this is simply a local gauge trans-formation. Specifying a section <j>a, with respect to which the action of V a is given, is called fixing a gauge. 9 The term compatible is used in [66]. The same property is referred to as admissible in [20]. II.3 Eigenspace Line Bundles 96 line bundle, the connection 1-form changes by an exact form under a change of basis. For V a , observe that if we write <j>'a = e%0<f>a, then A'a = Aa + id0. This implies that the 2-form tCa = dAa on Ua is independent of which basis is used to represent it. Also, if Ua and Up have a nonempty intersection, then it follows that the 2-forms fCa and ICp agree on Ua H Up. Therefore, all of the 2-forms )Ca piece together to give a smooth 2-form, which is globally defined on W and independent of the choice of local sections. This 2-form, which we denote by !C, is called the curvature 2-form for the connection VA-10 Let Wj denote a type I SD-region for a nondegenerate eigenvalue. This means that in Herm(n, C), the SD-region W/ is either W ( l , 1), or W(n,n). From Proposition 2.8 and Theorem 2.9, there is a strong deformation retract of VV/ to a subspace which is a single orbit diffeomorphic to the complex projective space CP(n — 1). Let i: CP(n — 1) «—• W/ be the inclusion of this subspace. If is the eigenspace line bundle over W/, then the pullback i*£j defines a complex line bundle over CP(n — 1). Also defined over CP(n — 1) is the canonical complex line bundle, 7„_i [66, p. 159]. It is constructed as a subbundle of the trivial complex vector bundle CP(n — 1) x Cn as follows. Recall from Section 1.2 that CP(n — 1) is the space of complex lines in C" . The total space of 7 n - i is defined as £ ( 7 n _ i ) = {(v, x)\v€ CP(n - 1) and x G v C C n } . The projection map p: £(7„_i) —» CP(n — 1) is defined by p: (v,x) n- v. 1 0 The concept of curvature in vector bundles is much richer than indicated above. We suggest [20] and [66] for some informative reading. II.3 Eigenspace Line Bundles 97 The canonical line bundles 7 n _ i are fundamental to the theory of complex line bun-dles. Indeed, they may be used to classify all isomorphism classes of line bundles over a given base space. More details on this construction are given in [20, Sect. 8], [49], and [66]. Given the importance of 7„_j, the following result is very useful. Proposition 3.3. The complex line bundle is isomorphic to 7 „ _ i . Proof. The proof of this theorem follows by inspection. Recall that Proposition 2.8 and Theorem 2.9 define a deformation retraction of Wj onto the subspace £, which consists of all matrices in Herm(n,C) with two fixed eigenvalues Ai and A 2 , such that Ai is the nondegenerate eigenvalue associated with Wj, and A 2 is an (n — l)-fold degenerate eigenvalue. From Lemma 2.4, it follows that £ is diffeomorphic to CP(n — 1), where CP(n — 1) is the subspace of all pairs (E\,E2) of orthogonal subspaces in C n , with the property that dime E\ = 1 and dime E2 = n — 1. Under this diffeomorphism, E\ is identified with the eigenspace of Ai , and E2 is identified with the eigenspace of X2. Therefore, £(i*£l) is diffeomorphic to £(7 n _i) . This diffeomorphism restricts to a vector-space isomorphism on fibres, and hence i*£j = 7 n _ i . D Consider now any nondegenerate, type II SD-region in Herm(n,C), and denote it hy Wrj. Let i: F(p,l,r) «—* W/j be the inclusion defined by Corollary 2.10. Of course, p and r satisfy p + 1 + r = n. The fibre bundles in (2.6) and (2.7) of Chapter I define II.3 Eigenspace Line Bundles 98 inclusions i\: CP(p) <-> F(p, 1, r) and z 2 : CP(r) <—> F(p, l , r ) . Then, the eigenspace line bundle £/j over W77 satisfies Proposition 3.4. The complex line bundle i\i*£n is isomorphic to the canonical bundle jp over CP(p), and i2i*£n is isomorphic to the canonical bundle 7 r over CP(r), Proof. This proof is similar to the proof of Proposition 3.3. In Section 1.2, we show that F(p, 1, r) is the space of all triples (E\, E2,E$) of mutually orthogonal subspaces of C n such that d i m c ^ i = p, dime-^2 = 1? and dime -^3 = f-The pullback is isomorphic to the complex line bundle with total space {((EuE2,E<i),v) <E F(pA,r) x C n | v e E2}. The proposition then follows immediately from the definitions of i\ and i2. D (b) Holonomy and Stokes' Theorem Recall from Section 1, that the adiabatic phase 7(T) for a nondegenerate eigenvalue A is defined by the U(l)-valued monodromy matrix for the differential equation (1.8). B. Simon observed [81] that exp[—i~f{T)] may be given a more geometrical interpretation: specifically, that of the holonomy of the connection on T. In the remainder of this section, we shall explain Simon's observation, and use Stokes' theorem to compute the adiabatic phase in terms of the curvature 2-form, K. The connection defines a notion of parallel transport in the eigenspace line bun-dle £ over W, the SD-region associated with the nondegenerate eigenvalue A. In general, II.3 Eigenspace Line Bundles 99 parallel transport defined by any connection satisfies the following properties [57, Propo-sition II.3.3]. If T is a C1 curve in the base space, then parallel transport along the inverse curve T - 1 is the inverse of parallel transport along T. Furthermore, if T and S are C 1 curves in the base space, then parallel transport along the composition S • T is the composition of parallel transport along S with parallel transport along T. Suppose now that T is a loop in W, with base point B. Then, the parallel transport generated by of the fibre ^ (B) around T defines an isomorphism from T(B) onto itself. It follows from the previous paragraph, that there is a group structure on the set of all such isomorphisms. This group, which is denoted by $(B), is called the holonomy group of Vyi with reference point B. Each isomorphism from ^(B) onto itself may be identified with an element of GL(1,C) , the structure group of £. Therefore, $(J9) is isomorphic with a subgroup of GL(1,C). We shall abuse the notation slightly, and write $(B) for this subgroup. Furthermore, because is compatible with the hermitian structure on it follows that $(B) is actually a subgroup of the reduced structure group U( l ) . This is simply a geometrical restatement of the fact that solutions to the adiabatic differential equation (1.8) are unitary. Let TR(T) € $(B) denote the holonomy of on the loop T, with base point B. If we choose another base point C for T, it follows from the discussion in [57, Sect. II.4] that Tg(T) € ®{B) and ^ c(T) £ ${C) are conjugate when viewed as elements of the reduced structure group of £. However, U(l) is abelian, and therefore rjj(T) = Tc{T). II.3 Eigenspace Line Bundles 100 This implies that the holonomy of defines a map T from the space of free loops in W t o U ( l ) . 1 1 The loop T is 1-dimensional, and therefore it has no cohomology in dimension 2. This implies that the first Chern class of the restriction of £ to T is zero. Line bundles are completely classified, up to isomorphism, by their first Chern class, and therefore we conclude that £\T is isomorphic to the trivial complex line bundle. It follows that there exists a smooth, normalized section <f> 6 C°°(£\T), which is defined everywhere over T. This section gives rise to a smooth connection 1-form A = (<f>, d<f>). Because A is defined globally on T, it follows that the holonomy of VA on T is given by the integral formula r(T) = exp[^ A] = exp[-»7(T)] . (3.5) This demonstrates the relationship between adiabatic phase and parallel displacement in the eigenspace line bundle. It is now shown how Stokes' theorem may be used to give a formula for the adi-abatic phase. Recall that a nondegenerate SD-region W C Herm(n,C) is an open n2-dimensional manifold, without boundary. Because the situation when n = 1 is unin-teresting, we shall assume that n > 2. Since the dimension of W is at least 4, it follows from the Whitney imbedding theorem [84, Sect. II.4], [93, Thm. 2] that any smooth loop T in W may be approximated arbitrarily closely by a simple loop. 1 1 Holonomy in general vector bundles and principal bundles is more complicated than in line bundles, and it is typically not possible to represent it as a map from the space of free loops in the base space to the structure group. For a review of holonomy in principal bundles, see [57]. II.3 Eigenspace Line Bundles 101 The holonomy T(T) varies smoothly under a smooth homotopy of T . 1 2 Hence, we can approximate T(T) arbitrarily closely by computing it on a simple loop which approx-imates T arbitrarily closely. It is therefore sufficient to compute the adiabatic phase only for simple, smooth loops in W. For the remainder of this section, we shall assume that T is any simple, smooth loop in W. In the Section 1.2, it was shown that all complex flag manifolds are simply connected. Along with Corollary 2.10, this implies that W is also simply connected, and hence T is null-homotopic. If dim W > 5 (i.e. n > 3), it follows from Theorem 7 in [93] that T may be extended to a smooth imbedding D of the 2-dimensional unit disc into W, such that T = 3D. If n = 2, recall from Section 2 that both of the SD-regions W ( l , 1) and W(2,2) are equal to Herm(2, C) — R • I, which is diffeomorphic to R 4 with a line removed. It follows from the corollary to Theorem 1 in [40] that T bounds a smooth imbedding D of the 2-dimensional unit disc in R 4 . However, a disc and a line do not generically intersect in R 4 , and therefore by at most an arbitrarily small perturbation of D, the image of D can be made to lie entirely in W. At this point, it is appropriate to remark on a subtlety which was not discussed when 2 x 2 hamiltonians were considered in Subsection l.b. It was argued that since the trace coordinate in Herm(2,C) does not appear in the curvature 2-form /C, the trace may be ignored for the purposes of computing 7. For this reason, only loops in Hermo(2,C) — {0} were considered. The space Hermo(2,C) — {0} is diffeomorphic to 1 2 This follows from the lemma on page 74 in [57]. II.3 Eigenspace Line Bundles 102 R 3 — {(0,0,0)}. Therefore, in general a simple loop T in Hermn(2,C) — {0} may be knotted. If this is the case, then T does not bound a smooth disc in Hermo(2, C ) — {0}, and it will not be possible to naively use Stokes' theorem in Hermn(2,C) — {0} to compute 7. However, we have shown that T bounds a disc D in Herm(2,C) — R • 7, and therefore the trace degree of freedom may be used to unknot T. Note, that if T is knotted in Hermn{2, C ) , then D must necessarily extend outside of Hermh(2, C ) . To conclude, we have shown that any smooth loop T in a nondegenerate SD-region W bounds a smoothly imbedded disc D in W. Because D is contractible, it follows that the restriction of the eigenspace line bundle ( to D is isomorphic to the trivial complex line bundle over D. This implies that there is a smooth, normalized section <f> € C°°(£|£>), which is defined everywhere over D. Using the section <f>, we obtain from a smooth connection 1-form A, defined everywhere on D. Note that the 1-form A depends on the disc that is used to span the loop T. In particular, suppose that we had chosen another disc D', with dD' — T. Let A' be the connection 1-form obtained from the construction of a normalized section °f £\D'- It need not be true that A and A' agree on T. Furthermore, although locally A — A' = da for some 0-form a, it does not follow that §j> A is equal to fjpA'. This is because it may not be possible to define a globally on T. Nevertheless, parallel transport generated by is uniquely determined. Hence, the holonomy of is independent of the connection 1-form. Therefore, we know that f^A — f^A' is equal to an integer multiple of 27r. Keeping this in mind, we rewrite (3.5) as II.3 Eigenspace Line Bundles 103 Recall that the curvature 2-form K satisfies K. — dA. Because A is defined smoothly everywhere on D, we may use Stokes' theorem to obtain the formula As an aside, note that although thus far D has been taken to be a disc with boundary T, we could more generally consider D to be an arbitrary, compact, connected, orientable surface with boundary T. To derive (3.6), we required that £\Q be a trivial line bundle. Since any connected 2-manifold with nonempty boundary has no cohomology in dimen-sion 2, 1 3 it follows that the first Chern class of £\D is zero. Therefore, £|x> is isomorphic to the trivial complex line bundle over D, and there exists a normalized section <j> de-fined everywhere over D. Using <j>, a connection 1-form, and a curvature 2-form may be constructed on D. Then, from Stoke's theorem, it follows that (3.6) holds if D is any compact, orientable surface with dD = T. (c) Photons in an Optical Fibre In [23], Chiao and Wu propose an experiment in which polarized light from a laser is injected into a helically wound optical fibre. They argued that because of the helical shape of the optical fibre, the wave function for a photon would acquire a quantum adiabatic phase. For a linearly polarized laser beam, this phase would cause a rotation of 1 3 Indeed for an arbitrary coefficient group, the 2-dimensional cohomology of any connected 2-manifold with nonempty boundary is zero. This is proven by considering a triangulation of the manifold, and not-ing that it is possible to contract the triangles bordering the boundary, without changing the homotopy type of the manifold. (3.6) II.3 Eigenspace Line Bundles 104 the plane of polarization. Subsequently, this rotation of polarization was experimentally verified by Tomita and Chiao [88]. The introduction of eigenspace line bundles in the previous two subsections provide the means for a careful computation of quantum adiabatic phase for photons in an optical fibre. As well, this example is an interesting application of line bundles to quantum physics. Following Chiao and Wu, we consider an individual photon propagating in an optical fibre which has been wound in a helix. Furthermore, it is assumed that the length of the fibre, the radius of curvature of the helix, and the radius of torsion of the helix are all much larger than the diameter of the fibre. With these assumptions, it is reasonable to assume that the direction of propagation of the photon is well-defined, and specified by the tangent vector to the helix. We shall take the point of view that this is a problem in 1-dimensional quantum mechanics, where the direction of propagation is an external parameter which may be specified by the experimenter through the positioning of the optical fibre. The two helicity states of a photon interact differently with a birefringent medium. We write the hamiltonian for the photon as H = - H f r + Hint , where H^t describes the interaction of the helicity states with the medium of the optical fibre, and H{t is the helicity-independent part of the hamiltonian. The helicity operator of a particle depends on the propagation direction, and therefore H[nt varies with the II.3 Eigenspace Line Bundles 105 tangent vector to the optical fibre. This suggests that Hint may give rise to a quantum adiabatic phase. However, H{t cannot contribute to such a phase, because it is a constant operator which commutes with H[nf In order to construct H[nt, note that the photon is a spin-1 particle. The helicity operator for a spin-1 particle is k • s, where s is the vector spin matrix for spin-1, and k is a unit vector in R 3 , giving the propagation direction of the particle. A 3 x 3 matrix representation for s = (51,52,53) is 51 -_L_ 71 0 1 0 1 0 1 0 1 0 52 = V2 0 -i 0 * 0 -i 0 i 0 and 53 = 1 0 0 0 0 0 0 0 - 1 (3.7) —* —* In terms of this representation, k • s is a 3 x 3 matrix parameterized by k, which takes values in the unit sphere S2. The eigenvalues of k • s are +1, 0, and —1; the associated eigenspaces are denoted by E+(k), En(k), and E_(k), respectively. Note that these eigenspaces depend on k G S2. The polarization vector of a photon must be perpendicular to the direction of propa-gation. This constraint, which is referred to as transversality, requires that the 0-helicity eigenvector must be projected out of the state space of the photon. Because the photon is massless, this constraint is relativistically invariant. Let T[k) denote the vector sub-space of C 3 defined by the span of E+(k) and E_(k). Then, the helicity operator for the photon is k-s\ (3.8) and i/i nt = AC k • s | j ^ , where K is related to the circular birefringence of the medium. II.3 Eigenspace Line Bundles 106 Observe that the transversality constraint applied in (3.8) depends on the parameter k, which indicates that it is most naturally imposed through fibre bundle theory. Con-structed over S2 from the eigenspaces E+(k), Eo(k), and E_(k) are the eigenspace line bundles £+, £o> and respectively. The coordinate representation of k • s, obtained by using (3.7) for s, defines an inclusion j of S2 into the type II SD-region W(2,2) of Herm(3,C). The line bundles £+, £o, and £_ are pullbacks over j of the appropriate eigenspace line bundles on W(2,2), which have been defined in Subsection 3.a. For £+, £o, and there are defined adiabatic connections, which we denote by V+, Vo, and V _ , respectively. Associated with V+, Vo, and V _ are curvature 2-forms /C+, /Co, and /C_, respectively. These 2-forms, which are globally defined on S2, were calculated by Berry in [8], and the results are fC± = ±i [k^dki A dk2 + k2dks A dk\ + k\dk2 A dk$] , (3.9) and /Co = 0 . (3.10) It follows from (3.9) that if T is a simple loop in S2, and D is a disc spanning T such that 3D = T, then the holonomy of V+ on T is exp[—iCl(D)], where fi(D) is the solid angle subtended from the origin by D. Similarly, the holonomy of V _ on T is exp[i£l(D)]. Also, equation (3.10) and the fact that S2 is simply connected imply that Vo exhibits no holonomy for all loops in S2. Before continuing with the problem at hand, we require the following general result for hermitian vector bundles. Suppose that a hermitian n-plane bundle n has a compatible II.3 Eigenspace Line Bundles 107 connection V , which has no holonomy for all loops in the base manifold. In the fibre Tx over some point x in the base manifold, choose an orthonormal n-frame (ei, e2,..., en). Because V has no holonomy, we can construct an orthonormal n-frame of sections for 77 by using V to parallel transport (ei, 62,..., en) to all other fibres of n. This implies that 77 is a trivial vector bundle. It should be remarked that the converse to this result is not true. In other words, trivial vector bundles may have connections which exhibit holonomy. In light of the above result, we reconsider the eigenspace line bundle £0 o v e r S2. The connection Vo has no holonomy, and therefore £0 is a trivial vector bundle. The first Chern class ci(£o) is an element of H2(S2;Z) = Z, and triviality of £0 implies that c i ( 6 ) = 0. We now return to the transversality constraint in (3.8). Taking the vector space T{k) as the fibre over each k 6 S2, we obtain a complex 2-plane bundle bundle over s2. This vector bundle is equal to the Whitney sum £ + © It is desirable to express the operator in (3.8) as a 2 x 2 matrix which depends smoothly on k £ S2. This requires a smoothly ^-dependent orthonormal basis for T(fc), which is of course nothing more than an orthonormal basis of sections for £+©£_. Therefore, H{nt can be expressed as a smoothly fc-dependent matrix if £+ © £_ is a trivial complex 2-plane bundle. The Chern classes of ©£_ are easily calculated using the product theorem for total Chern classes. Recall from the definition of eigenspace line bundles in Subsection 3.a, that £+ © £0 © £- is a trivial complex 3-plane bundle over S2. Therefore, the product theorem and ci(£o) = 0 imply that both the first Chern class c i ( £ + ©£_) and the second II.3 Eigenspace Line Bundles 108 Chern class c2((+ © £_) are zero. However, we caution that this does not necessarily imply that © £_ is trivial. Unlike with complex line bundles, the isomorphism class of a complex n-plane bundle is not completely determined by its total Chern class. We require some additional results.14 Lemma 3.11. Every complex n-plane bundle rf1 over S2 is isomorphic to the Whit-ney sum rj1 © e n _ 1 , where e n _ 1 is the trivial complex (n — l)-plane bundle on S2, and n1 is some complex line bundle, which is not necessarily trivial. Proof. For r)n, the fibre J-% over each point x G S2, is a complex re-plane. The collection of orthonormal (n—l)-frames in Tx form the Stieffel manifold Vn-\{F%)-All Stieffel manifolds are compact and connected. The manifolds Vn-iC^x) a r e the fibres of the fibre bundle Vn-i(rjn), which is one of the associated Stieffel bundles over S2. More details on associated Stieffel bundles are given in [66, Sect. 12]. The notation for Vn-i(r)n) is introduced through the diagram V»_i(C») — U £ p s2 The homotopy exact sequence for this bundle is i r i ( K _ i ( C B ) ) ) - > . - . (3.12) 1 4 We thank K . Y . Lam for bringing this lemma to our attention. II.3 Eigenspace Line Bundles 109 It is shown in [49, Thm. 7.5.1] that n(Vn-i(Cn)) = 0 for all integers n > 2. Therefore, it follows from the exact sequence (3.12) that the induced homomor-phism p#: 7T2(£) —* K2{S2) is a n epimorphism. This implies that there is no obstruction to constructing a section for the fibre bundle Vn-i(vn)-A section for Vn-i(nn) provides an orthonormal (n — l)-frame of sections for r)n. The span of this frame of sections gives the trivial vector bundle en~l as a subbundle of nn. The line bundle n1 is taken to be the orthogonal complement of e n _ 1 in rjn. Note that 7r 1 (y n (C n )) £ Z [49, Prop. 7.11.3], which implies that 771 need not be trivial. O Consider the Chern classes Ci(nn) for any complex n-plane bundle rjn over S2. For i > 2, the Chern classes ci(nn) must vanish, because S2 is 2-dimensional. Furthermore, it follows from Lemma 3.11 that the isomorphism class of rf1 is completely determined by cl( r/n)> which is equal to ci(rjl). Therefore, because c i ( £ + © £_) = 0, it follows that the Whitney sum of eigenspace line bundles £4. © £_ is a trivial complex 2-plane bundle.15 The triviality of £+ © £_ guarantees the existence of a frame of sections, with respect to which f/int can be represented as a 2 x 2 matrix. Indeed, there are infinitely many such frames of sections. It will become apparent that it is not strictly necessary to write 1 5 Although © £_ is trivial, the individual line bundles £ + and £_ are not trivial. The first Chern classes of £ + and £_ may be calculated by constructing normalized sections over the northern and southern hemispheres of S2, and then computing the homotopy classes of the transition functions on the equator. The result is that ci(£±) 6 H2(S2; Z) are represented by the integers ± 2 . Of course, which bundle is ascribed +2 and which bundle is ascribed —2 is simply a matter of convention. II.3 Eigenspace Line Bundles 110 down a specific matrix representation of -flint; however, for the sake of completeness, we do so. Observe that 1 yj2 + 2k\ l + k2 ^h(kx + ik2) (h + ik2)2 <f>2 = 1 yj2 + 2k\ 0 V2(h - ik2) -2h is an orthonormal basis of sections for With respect to this basis, ffint(fc) = l + k2 2k3 (h - ik2)2 ( fc i + ik2)2 -2k3 It is easy to verify that the eigenvalues of this matrix are ±K, as expected. Now that Hint(k) has a matrix representation, quantum adiabatic phase for the pho-ton is easily computed. Assume that the optical fibre is wound in a helix such that the —* « unit tangent vector k describes a loop T in S . This loop is parameterized by s, the length along the fibre. With suitable normalization, it is arranged that s € [0,1]. We shall assume that T is a simple loop. If T were not simple, then it may be decomposed into simple components, and the following analysis carried out for each component. On a disc D spanning T in S2, construct a family of normalized eigenvectors 4>+ a(k) b(k) for the eigenvalue +K of Hmt. Associated with <f>+ is an adiabatic connection 1-form *4+ = > d<f>+) = (a^i + b<j>2 , d[a<f>i + b<f>2}) II.3 Eigenspace Line Bundles 111 Observe that the complex 3-vector acj>\ +b<j)2 is an eigenvector for the 3x3 matrix k-s, with eigenvalue +1. Therefore, A+ is also an adiabatic connection 1-form for the eigenspace line bundle . It follows from (3.9) that the quantum adiabatic phase acquired by <f>+(k) under parallel transport around T is 7+(T) = — Cl(D) (mod 2TT). Similarly, if c/>_ is an eigenvector associated with the lower eigenvalue of i/hvt, then (j>- acquires a quantum adiabatic phase of —7+(T). It is perhaps useful to recapitulate the above computation. By showing that the vector bundle © £_ is trivial, we demonstrated that the transversality constraint in (3.8) is "topologically trivial". The result of this is that for the positive and negative helicity states of a photon, the quantum adiabatic phase agrees with the phase for the corresponding helicity state of an ordinary spin-1 particle. Therefore, the phases com-puted here agree with the results of Chiao and Wu in [23], where transversality was not imposed on the hamiltonian. It was shown in [23] that the adiabatic phases acquired by the helicity states of photons give rise to a rotation of the polarization plane for linearly polarized light. Let and ?/>_ be positive and negative helicity eigenstates, respectively, for the total hamiltonian H, such that i/fr0+ = eip+ and i/fr'i/'- = tip-. The wave function for linearly polarized light is a superposition of left-handed and right-handed polarization states. Therefore, the initial state of the photon is taken to be 0(0) = 2 - ? {0+ -f 0_}. The adiabatic theorem implies that at the end of the optical fibre, the photon's final state vector 0(1) is approximately parallel to 0(0). The phase difference between the II.3 Eigenspace Line Bundles 112 adiabatic approximation to 0(1), and 0(0) is the sum of the adiabatic and dynamical phases. If r is the optical length of the fibre, then 0(1) ~ 2~? {exp[-i(er + KT - 7+(T))] 0+ + exp[-i(er - KT + 7+(T))] 0_} . Squaring this gives ||0 ( l ) | | 2 ~cos 2(KT- 7 +(r)). (3.13) Malus' law states that for linearly polarized light, its amplitude along a given direction varies as the square of the cosine of the angle that this direction makes with the plane of polarization.16 Therefore, it follows from (3.13) that upon exiting the optical fibre, the photon's plane of polarization is rotated by 9 = KT — 7+(T). The contribution of the dynamical phase to 0 is proportional to the circular bire-fringence K. This is the usual optical activity which one would expect from a straight, birefringent optical fibre. However, the contribution from the adiabatic phase is inde-pendent of K. Therefore, if the birefringence of the fibre is small enough, we expect that the dominant contribution to the rotation of polarization would be the adiabatic phase 7+(T). By varying the geometry of the helix, the dependence of the polarization rotation on the solid angle Cl(D) can be experimentally tested. This was the experiment conducted by Tomita and Chiao [88]. They found that within experimental error, the rotation angle of the polarization vector was indeed equal to 7+(T). We remark that Ross [73] has performed an experiment similar to that of Tomita and Chiao, and his results agree with theirs. However, Ross and subsequently Haldane 1 6 Malus' law is easily proven. See [51, Sect. 24.5]. II.3 Eigenspace Line Bundles 113 [41] invoked a classical explanation for the rotation of polarization, which is based on a hypothesized law for parallel transport of polarization vectors. Berry [9] showed that in a certain approximation, this law may be derived from Maxwell's equations. This demonstrates that for large numbers of photons, the results obtained from quantum me-chanics survive the classical limit, and may be interpreted within the context of classical electromagnetism [9], [22], [42]. To conclude, we note that Pancharatnam [70] has considered another geometrical procedure for rotating the polarization vector of light. The relationship between Pan-charatnam's work and that presented here, is described in [10]. Some of the recent work on adiabatic phase in optics is reviewed in [21]. Chapter III Quantum Adiabatic Phase and Time-Reversal Invariance Isolated quantum systems may be described by a time-independent hamiltonian, and therefore they are invariant under time-reversal. Whether solutions of the Schrodinger equation for a time-dependent hamiltonian, or otherwise parameter-dependent hamil-tonian, exhibit time-reversal invariance in the sense of Wigner [95], [96, Chapt. 26], depends on the nature of the time dependence, or parameter dependence. For example, the hamiltonian of a charged particle with spin coupled to time-dependent magnetic field (c.f. Subsection I.l.b) is not in general time-reversal invariant. However, in the Jahn-Teller effect the electronic part of the molecular hamiltonian is time-reversal invariant. The electronic hamiltonian H(Q) depends on the configuration of the nuclei, which is specified by the vector parameter Q. The time-reversal invariance of the electronic hamil-tonian is expressed through the existence of a time-reversal operator 0 which commutes with H(Q) for all Q. Another time-reversal-invariant example is the Stark hamiltonian, which describes the coupling between atomic electron orbitals and a time-dependent electric field. This example has been examined by Mead [64] and Avron et al. [4]. 114 III.l Potentially Real Hamiltonians 115 In general, the adiabatic phase is difficult to compute because it depends on the details of the geometry of the similar degeneracy regions and the eigenspace line bun-dles over them. However, a remarkable simplification occurs for time-reversal invariant systems, where the adiabatic phase depends on the most basic aspect of the topology of the similar degeneracy regions. Specifically, recall that a periodic quantum system whose hamiltonian is represented by a matrix may be described by a loop in a space of hermitian matrices. For time-reversal invariant systems, the adiabatic phase depends on the homotopy class of this loop in the appropriate similar degeneracy region. These homotopy classes are given by the fundamental group of the similar degeneracy region. In the first section of this chapter, we will characterize time-reversal invariant sub-spaces in similar degeneracy regions, and explain the simplifications that occur for adia-batic phase in these subspaces. In Section 2, we develop those results on the homotopy and homology of similar degeneracy regions, which are required to compute the adiabatic phase of an arbitrary time-reversal invariant system in terms of its homotopy class. This computation is completed in Section 3. §1 Potentially Real Hamiltonians Recall that the space of real, symmetric matrices Herm(n, R) forms a vector subspace of Herm(n, C). Consider any pair of nondegenerate SD-regions (W, W ) , where as before, W' is the intersection of W with Herm(n,H). In the first part of this section, it is shown that the restriction of the curvature 2-form K, to W' is the zero form. Note that if T is a simple loop in W', then it does not follow from formula (II.3.6) and /C|>y = 0 that III.l Potentially Real Hamiltonians 116 7(T) = 0. This is because W' is not simply connected, unlike W. However, it does follow that j(T) depends only on the homotopy class of T in 7Ti(W'). The physical significance of Herm(n,H) is that time-dependent quantum systems which are represented by curves in .r7erm(n,R), are time-reversal invariant. In Subsec-tion (b), Wigner's definition of time-reversal invariance is reviewed. More details may be found in [74, Sect. 29] and [96, Chapt. 26]. Also, we demonstrate the relationship between 7ferm(n,R) and all other vector subspace in Herman, C), which have the property that all curves in them describe time-reversal invariant systems. This relationship will allow us to reduce the problem of computing the adiabatic phase for all time-reversal invariant, periodic, quantum systems to a computation of the adiabatic phase for all loops in real SD-regions. (a) The Adiabatic Curvature in Potentially Real Subspaces For a pair of nondegenerate SD-regions (W, W ) , let j: W' <—» W be the inclusion of W'. Then, the eigenspace line bundle over W' is the pullback £' = Of course, this means that (' is just the restriction of £ to W'. The adiabatic curvature 2-form for £' is denoted by K,'. It is simply the restriction of 1C to W. We now evaluate the curvature 2-form fC'. Let {Ua} be an open cover of W' such that £'\ua has a normalized section <f>a for each a. Obviously, such open covers exist (e.g., any good cover of W' has this property). Consider any Ua in this open cover. Because Ua is in 7/erm(n,R), it follows that the normalized section, or eigenvector <j>a may be III.l Potentially Real Hamiltonians 117 chosen to be a real vector. Recall that the curvature 2-form is independent of the choice of local section that is used to compute it. Therefore on Ua, i<f>a,d(f>a) + (d(j>aAa) = \d [d{(t>a,<t>a)\ = 0 This implies that although the connection VA on £ has nontrivial curvature, its restriction to £' is flat. Recall from Subsection Il.l.b, that if T is a loop in W 2 , the unique nondegenerate SD-region of Herm(2, R), then 7(T) depends only on the homotopy class of T in 7ri(W2). Suppose now that n > 3. A real, nondegenerate SD-region W' C Herm(nH) is an open, (n 2 + n)/2-dimensional manifold without boundary. If T\ and T 2 are two smooth loops in W', then by at most an arbitrarily small perturbation of T\ and T 2, we can ensure that T\ and T 2 are simple loops which do not intersect. If T\ and T 2 are homotopic, it follows from the Whitney imbedding theorem [84, Sect. II.4], [93] that there is an imbedding Y of a 2-dimensional cylinder into W' such that T\ is the boundary at one end of Y, and T2 is the boundary at the other end. Because a 2-dimensional cylinder has no cohomology in dimension 2, it follows that the first Chern class of the restriction of £ to Y is trivial. This implies that £|y is isomorphic to the trivial complex line bundle over Y, and therefore a 1-form A for the connection may be defined globally on Y. Applying Stokes' theorem on Y, and using flatness of V ^ , we conclude that 7(Ti) = 7(T2). This proves that the adiabatic phase is a homotopy invariant in W'. Another proof of this fact uses flatness of the connection V ^ , combined with the holonomy theorem of Ambrose and Singer [57, III.l Potentially Real Hamiltonians 118 Sect. 8] to prove that the holonomy group is discrete. It then follows that the adiabatic phase is a homotopy invariant [57, p. 93]. As it has been remarked, VV' is not simply connected. This fact will be proven in Section 2, where the fundamental groups for all of the SD-regions are computed. Our argument above suggests that once 7ri(W') has been computed for each nondegenerate, real SD-region W', then it should be possible to compute 7(T) in terms of the homotopy class of T in 7ri(W'). This is indeed the case, and these computations are carried out in Sections 2 and 3. A natural question to ask is, "Are there any subspaces in W, other than W', with the property that the adiabatic phase is a homotopy invariant for simple loops in that subspace?" Indeed, there are. Notice, that the conjugate action of a constant matrix U £ U(rc) on W leaves the curvature 2-form K invariant. This motivates Definition 1.1. For W, any SD-region in Herm(n, C), let W = W n Herm(n, R) . Then, a subspace VofVV is said to be potentially real if there exists a constant matrix U 6 U(n) such that V = UW'U*. We remark that neither SD-regions, nor potentially real subspaces of SD-regions are vector spaces. The restriction of K, to any potentially real subspace in W is the zero form, which implies lll.l Potentially Real Hamiltonians 119 Proposition 1.2. If V is any potentially real subspace of a nondegenerate SD-region, and T is a simple loop in V, then j(T) is determined by the homotopy class of T (b) Time-Reversal Invariance Before, continuing with our discussion of adiabatic phase, we shall give a brief in-troduction to time-reversal invariance in quantum mechanics, as it was first studied by E. P. Wigner in [95]. Consider a time-dependent hamiltonian represented by a path T: [0,1] —» Herm(n, C), and let U(s) be the propogator for the Schrodinger equation If it exists, the time-reversal operator 0 is defined to be an s-independent symmetry operator with the property that QUQ~l is the propogator for (1.3) with T(s) replaced by T(l — s). As a symmetry operator, 0 must preserve the transition probability between any two state vectors, which implies that 0 must be either a unitary or antiunitary operator. However, it is well-known that 0 cannot possibly be a unitary operator.1 Therefore, the hamiltonian T(s) is said to be time-reversal invariant if there exists an antiunitary operator 0 which meets the above requirements of a time-reversal operator. Clearly, the hamiltonian T(s) is time-reversal invariant if and only if there exists a time-reversal operator 0 such that T(s) commutes with 0 for all s. Since any antiunitary 1 For this result, and as well a general review of time-reversal invariance in quantum mechanics, see [74, Sect. 29] and [96, Chapt. 26] in 7Ti(V). • i±U(s) = T(s)U(s). (1.3) III. 1 Potentially Real Hamiltonians 120 operator may be written as the composition of a unitary operator and the complex conjugation operator,2 it follows that a time-reversal operator exists if and only if there exists a constant unitary matrix U such that UT(s)U* = T(s) for all s € [0,1]. Of course, the corresponding time-reversal operator is then given by U composed with the complex conjugation operator. Notice that taking U to be the identity provides a time-reversal operator for all paths in Herm(n,H). Indeed, since the defining equation is linear over R, it follows that time-reversal invariance is naturally associated, not with individual paths in Herm(n,C), but rather with real vector subspace of Herm(n,C). Definition 1.4. A subspace S of Herm(n, C) is said to be time-reversal invariant if there exists a fixed U € U(n) such that U AU* = A, for all A 6 S. The maximal time-reversal invariant subspaces in Herm(n, C) are vector subspace obtained by fixing U E U(n), and defining J(U) =f {A G Herm(n,C) \ U~AU* = A] . For adiabatic phase, we are interested in the intersection of maximal time-reversal-invariant subspaces J(U) with a nondegenerate SD-region W. Furthermore, it is prudent to require that J{U) be an irreducible subspace. By an irreducible set of matrices, 2 A detailed discussion of antiunitary operators is given in [96]. III.l Potentially Real Hamiltonians 121 we mean a set for which there does not exist a constant unitary matrix V such that conjugation by V puts every element of the set in the same block diagonal form. To see why it is reasonable to require that J(U) be irreducible, suppose that J(U) were instead reducible. Then, J(U) fl W would be a reducible subspace of W, and any loop T in J(U) D W could be reduced into block diagonal form. This would imply that the Schrodinger equation for T could be decoupled into irreducible components. The adiabatic phase could then be obtained by examining the adiabatic limit of the appropriate component, and the problem would be reduced to calculating the adiabatic phase in Herm(m, C), for some m < n. Requiring that J(U) be irreducible places the following restriction on U. Lemma 1.5. If J(U) is irreducible, then U must be either symmetric or skew-symmetric. The proof of this lemma follows from a routine application of Schur's lemma. For a review of Schur's lemma, see [37]. , Proof. For any A 6 J(U), note that UA = AU. Taking the complex conjugate of this gives UA = AU = U*AUU . Therefore, UUA=AUU, III.l Potentially Real Hamiltonians 122 for all A € J(U). Because J{U) is irreducible, it follows from Schur's lemma that U U = zl, where z is a complex number of modulus one. This implies that U = zUT. Furthermore, the unitarity of U requires that U U* = I, which in turn implies that z2 = 1. Hence, z = ± 1 , and the lemma follows. D Note that a complex SD-region W is invariant under the conjugate action of any unitary matrix. Therefore, any potentially real subspace of W is the intersection of W and a vector subspace conjugate to Herm(n,R). This suggests that Definition 1.1 should be extended to say that a vector subspace S in Herm(n,C) is potentially real if there exists a fixed V € U(n) such that S = V Herm(n, R) V*. In this language, the potentially real subspaces of W are obtained from the intersection of potentially real vector subspace with W. Lemma 1.6. • An irreducible J(U) is potentially real if and only if U is symmetric. Proof. First, assume that J(U) is a potentially real vector subspace Therefore, Herm(n,Il) — V* J(U) V for some V G U(n). It follows immediately from the definition of J(U) that V* J{U) V = J{V* UV). This implies that J(V* UV) = Herm(n,Il), and hence, V*UV commutes with every real, symmetric matrix. From Schur's lemma, we conclude that V* UV is equal to a complex number of modulus one times I. Therefore, U is symmetric. Conversely, assume that U is a symmetric matrix. Any symmetric, unitary matrix may be written in the form U — Q D QT, where Q is an orthogonal matrix, III.l Potentially Real Hamiltonians 123 QT is the transpose of Q, and D is a diagonal unitary matrix3 [96, p. 287]. Let D' be any one of the 2 n diagonal square roots of D. Then, J(U) = J((QD')(QD'f) = {QD>) J(I) (QD'Y = (QD')Herm(n,R) (QD1)* , which completes the proof. EH We remark that Lemma 1.6 implies that if U is symmetric, then an irreducible J(U) has a real structure defined on it. The remaining option that U be antisymmetric is also interesting, because then an irreducible J(U) has a quaternionic structure defined on it. We shall not investigate this situation here; however, it is discussed in [4] and [33]. Obviously, all potentially real subspaces of W are time-reversal invariant. If V = UW'U*, then a direct computation shows that V C J(UUT). We now prove that if W is a nondegenerate SD-region, then the converse is also true. Proposition 1.7. If W is a nondegenerate SD-region, then all irreducible, time-reversal invariant subspaces of W are potentially real. Proof. By Lemma 1.5, all irreducible, time-reversal invariant subspaces of W are of the form J(U) fl W for some U G U(n), which is either symmetric or skew-symmetric. Consider any matrix H G J(U) D W. Since >V is a nondegenerate 3 The author thanks R. Westwick for drawing this fact to his attention. III. I Potentially Real Hamiltonians 124 SD-region, the matrix H has a distinguished nondegenerate eigenvalue, which we label pL. We denote the remainder of the eigenvalues of H, not necessarily in any particular order, by A i , . . . , A„_i. The spectral theorem implies that there exists a unitary matrix V such that H = V DV*, where D is the diagonal matrix . 0 H 0 0 Ai 0 0 . . . 0 A„_i Because H G J(U), it follows that UVD VT U* — V DV*, which implies that U = VBVT (1.8) for some element B of the isotropy subgroup of U(n) at D. Since U is either symmetric or skew-symmetric, it follows from (1.8) that B must also be either symmetric or skew-symmetric, respectively. However, B is of the form where B\ G U(l) and B2 G U(n — 1). This implies that B is symmetric, and by Lemma 1.6, the subspace J(U) D W is potentially real. D Note that in the proof of Proposition 1.7, it was essential that the SD-region W be nondegenerate. There exist degenerate SD-regions which have a nonempty intersection with some irreducible J{U), where U is now taken to be skew-symmetric. For any III.2 Homotopy and Homology of Similar Degeneracy Regions 125 hamiltonian contained in such a J(U), it follows from the quaternionic structure on J(U) that the degeneracy of each of its eigenvalues is equal to an even integer [4]. Hamiltonians of this nature arise when considering Fermi systems with time-reversal invariance, and the aforementioned eveness of the degeneracies is called Kramers degeneracy [4], [64]. The appearance of quaternionic structure in Fermi systems with time-reversal invariance was first described by F. Dyson in [33]. The importance of this quaternionic structure to adiabatic holonomy in quadrupole hamiltonians is examined in [4]. The upshot of this section is that a computation of adiabatic phase for loops in each irreducible, time-reversal-invariant subspace of W may be reduced by a judicious choice of basis to a computation of adiabatic phase for loops in W'. The remaining two sections of this chapter are devoted to computing adiabatic phase for any loop in W', in terms of its homotopy class in 7r i(W') . §2 Homotopy and Homology of Similar Degeneracy Regions Consider an arbitrary SD-region W, and let W' be its intersection with Herman, R). For the computations in this section, there is no advantage to assuming that the pair (W,W') is nondegenerate. This section is devoted to studying the Hurewicz homomor-phisms from the homotopy exact sequence to the homology exact sequence for (W, W'). The groups and homomorphisms of interest appear in the following commutative dia-III.2 Homotopy and Homology of Similar Degeneracy Regions 126 gram, which has exact rows. 7r2(W) H2(W) 7T 2(W) h H2(W) -> 7r 2(W,W) h" H2(W,W) 7ri(W) h' ffl(W') 7Tl(W) #i(W) (2.1) Once the groups and homomorphisms are identified, this diagram will provide detailed information about the topology of the pair of SD-regions (W,W'). In particular, if (W, W') is a pair of nondegenerate SD-regions, then the diagram (2.1) will be used in Section 3 to compute the adiabatic phase 7(T) for any loop T in W'. If W and W' are type II SD-regions, then the isomorphisms induced by the inclusions in Corollary 2.10 of Chapter II form an isomorphism of commutative diagrams between (2.1) and TT 2 (F ' ) H2(F') 7r 2(F) h H2(F) - *2{F,F') h" H2(F,F') Tri(F') h' Hi(F') 7ri(F) (2.2) To simplify our notation, we shall often not write the arguments of F, F', G , and G ' explicitly. If W and W' are type I SD-regions, then we have an induced isomorphism between the commutative diagram (2.1) and the commutative diagram obtained by re-placing F by G and F' by G ' in (2.2). In Section 1.2, we showed that F' and G ' are connected manifolds and that F and G are simply connected manifolds. From the Hurewicz theorem, it follows that h is an isomorphism, and that h! is an epimorphism. Using (2.2), a diagram chasing argument shows that h" is also an epimorphism. This means that any relative homology class in III.2 Homotopy and Homology of Similar Degeneracy Regions 127 H2(F, F1) can be represented by the continuous image of a 2-cell in F, with the image of its boundary circle in F'. The same reasoning as above implies that h": x2(G, G') —> H2(G,G') is also an epimorphism. Therefore, any relative homology class in H2(G, G') can be represented as the continuous image of a 2-cell in G, with the image of its boundary circle in G'. We remind the reader, that by Corollary 2.10 of Chapter II, this same result holds for the pair of SD-regions (W, W ) . Using the formula (3.6) from Chapter II, a computation of the adiabatic phase for real hamiltonians, or more generally for hamiltonians with values in a potentially real subspace of hermitian matrices, requires us to integrate the adiabatic curvature 2-form over the relative 2-cycles described above. This motivates a detailed analysis of the commutative diagram (2.2), and the corresponding commutative diagram for Grassmann manifolds. (a) Type I SD-Regions If (W,W') is a pair of type I SD-regions, then by Corollary 2.10 of Chapter II, the commutative diagram (2.1) is isomorphic to the following commutative diagram in which the vertical maps are Hurewicz homomorphisms and the rows are exact. TT 2 (G') > H2(G') TT 2 (G) h H2(G) 7T 2 (G, G') > ffl(G') • 7Tl(G) = 0 h" ti (2.3) H2(G,G') Hi(G') • #i(G) = 0 III.2 Homotopy and Homology of Similar Degeneracy Regions 128 Recall that because G(p, q) is simply connected, it follows from the Hurewicz theorem that h is an isomorphism. We now summarize some results for this commutative diagram which were proven in Section 1.2. From (1.2.12) and (1.2.13), both 7r 2(G) and H2(G) are isomorphic to Z, for all integers p,q > 1. Furthermore, from (1.2.16) and (1.2.17), we have that h! is an isomorphism and Hence, we refer to the type I case as generic when p + q > 3. The special case that occurs when p = q = 1 is straightforward, and was discussed in Subsection Il.l.b. We leave it to the reader to translate that discussion into the language of this section. For the remainder of this subsection, we assume that p + q > 3. From (2.26) and (2.28) in Proposition 2.25 of Chapter I, the induced homomorphisms jft and j* in (2.3) are zero homomorphisms. Applying a generalization of the five lemma,4 we conclude that the relative Hurewicz homomorphism h" is an isomorphism. Since 7r 2 (G, G1) = H2(G,G'), it follows immediately that 7r2(Cr, G') is an abelian group, and furthermore must be isomorphic to either Z © Z 2 or Z. We shall now prove that it is the latter which is true. This will allow us to establish that with an appropriate choice of generators, the homomorphism k# from 7T2(G) to 7T 2(G,G') is multiplication 4 As usually stated, the five lemma applies to commutative diagrams of abelian groups, and a priori •K2(G,G') need not be abelian. However, with a little care it is easy to see that the standard diagram-chasing proof of the five lemma also applies to commutative diagrams which may possibly contain nonabelian groups. When quoting the five lemma in this chapter, it shall be this generalization which we have in mind. III.2 Homotopy and Homology of Simifar Degeneracy Regions 129 by 2. As we shall see in the next section, this fact is crucial to the existence of a nontrivial adiabatic phase for real SD-regions. We now show that in order to compute H2(G,G'), and hence 7r2(G, (?'), it suffices to compute H2(CP(2), RP(2)). For integers 1 < a < p and 1 < b < q, define the pair inclusion a": (G(a, 6), G'(a, b)) (G(p, q), G'(p, q)) (2.4) to be given by the inclusions a and a1 defined in (1.2.9) and (1.2.10), respectively. The homomorphisms a*, a[, and a", which are induced on homology by the respec-tive inclusions, give a homomorphism from the homology exact sequence for the pair (G(a,b),G'(a,b) to the homology exact sequence for the pair (G(p,q),G'(p,q)). When a = 1 and 6 = 2, this defines the commutative diagram, 0 Z Z 2 0 /f2(RP(2)) #2(CP(2)) — #2(CP(2),RP(2)) i #i(RP(2)) — #i(CP(2)) a* H2(G') H2{G) H2(G,G') a* Hi(G') Hi(G) Z Z 2 0 Because the induced homomorphisms a# and c/^ on the corresponding homotopy groups in (1.2.11) and (1.2.15) are isomorphisms, it follows from the Hurewicz theorem that a* and c/+ are isomorphisms. Hence, by the five lemma, we conclude that a" is an isomorphism and #2(CP(2), RP(2)) £ H2(G,G'). III.2 Homotopy and Homology of Similar Degeneracy Regions 130 To settle the issue of whether H2(CP(2), RP(2)) is isomorphic to Z © Z 2 or Z, we compute the relative cohomology with coefficient group Z 2 of the pair (CP(2), RP(2)), and then apply the universal coefficient theorem for cohomology. Lemma 2.5. Let j: RP(m) «—»• CP(ra) be the inclusion of the real points defined in (1.2.8). Then, the induced homomorphism j* : tf2(CP(m); Z 2 ) — * # 2(RP(m); Z 2 ) is an isomorphism for all integers m > 2. The result that H2(G, G') S Z follows from Corollary 2.6. Ifm>2, then H2(CP(m),RP(m)) £ Z . Proof of Lemma 2.5. First, note that # 2(CP(ra); Z 2 ) = Z 2 for all m > 1, and that # 2 (RP(m);Z 2 ) Z 2 for all m > 2. These results are well-known, and may be proven using either a CW-decomposition of CP(ra) and RP(m) [29, p. 102], or by a Thorn-Gysin exact sequence [83, p. 264]. To show that j* is an isomorphism, it now suffices to prove that j* is not the zero homomorphism. Let 7O T be the canonical complex line bundle over CP(m), and 7 m the real, oriented 2-plane bundle obtained by restricting 7 m (i.e. pulling back over the inclusion j), and then ignoring the complex structure. The basic fact that C is isomorphic to R 0 R , along with Lemma 3.2 in [66], implies that jm is isomorphic III.2 Homotopy and Homology of Similar Degeneracy Regions 131 to the Whitney sum j ' m ® "f'm , where j'm is the canonical real line bundle over RP(m). By naturality of the Stieffel-Whitney classes, it follows that j*(102(7™)) = W2(l'm © l'm)- The Whitney product theorem implies that ^2(7^ © 7^ ) = (wi(l'm))2- For m > 1, the first Stieffel-Whitney class, ^1(7^) represents the unique nonzero element in if1(RP(m); Z2). Furthermore, H*(RP(m); Z2) is an algebra with unit over Z2 , having 1^ 1(7^ ) as its only generator, and (wi(7m))m + 1 = O'as its only relation [66, p. 42]. Therefore, if ra > 2, we have that 1^ 2(7^ © j'm) 7^ 0, and hence j* is an isomorphism. CD Proof of Corollary 2.6. Recall that for m > 2 , we have already demonstrated that #2(CP(m),RP(m)) is isomorphic to either Z © Z 2 or Z. Since Hi(CP(m)) = 0 , it follows immediately from the reduced homology exact sequence for the pair (CP(m),RP(m)) that #i(CP(m), RP(m)) = 0. Therefore, the universal coefficient theorem implies that H2(CP(m), RP(m); Z 2 ) is isomorphic to either Z 2 © Z 2 or Z 2 . Now, consider the following cohomology exact sequence for the pair (CP(ra), RP(m)) with coefficient group Z 2 (coefficients are suppressed in the notation of this diagram), Hl(RP(m)) -i+ #2(CP(m),RP(m)) -?U #2(CP(m)) H2(RP(m)) z2 z2 z2 III.2 Homotopy and Homology of Similar Degeneracy Regions 132 From Lemma 2.5, we know that j* is an isomorphism. It follows that k* is the zero homomorphism, and 6* is an epimorphism from Z 2 to H2(CP(m),'RP(m); Z 2 ) . This eliminates the possibility that H2(CP(m), RP(ra); Z 2 ) is isomorphic to Z 2 © Z 2 - • We make use of a commutative diagram to summarize our results for generic, type I SD-regions. Denoting a pair of generic, type I SD-regions by (W 9 , Wp), we have estab-lished Z Z Z 2 0 x2 h" h' H2{Wg) - ! U H2(Wg) H2(Wg,Wg) HX{W9) T2(Wg) > 7 r 2 ( W 3 ) ^ 7 r 2 ( W g , w ; ) > 7 n ( w ; ) > T n C W , ) 0 (2.7) The rows in this commutative diagram are exact. To conclude, we compare (2.7) with the corresponding commutative diagram for the nongeneric case of 2 x 2 hamiltonians. The SD-regions W ( l , 1) and W(2,2) correspond to the same region in Herm(2, C), which we denote by W2- The intersection of W 2 with #erm(2,R) is denoted by W 2 . Recall that TTI(>V2) = # i ( W 2 ) = Z, which is not like the generic situation. However, as we showed in SubsectionTI.l.b, the computations for III.2 Homotopy and Homology of Similar Degeneracy Regions 133 2 x 2 hamiltonians can be done directly, and we obtain the following results. 0 Z Z © Z Z 0 T 2 ( W 2 ) #2(W 2 ) 7T2(W2) H2(W2) 7T2(W 2,W 2) h" z © Z A ' a. # i (W 2 ) T T I ( W 2 ) #i(W 2) (2.8) (b) Type II SD-Regions Now we turn our attention to the case when W and W' are type II SD-regions. Recall that by Corollary II.2.10, the commutative diagram (2.1) is now isomorphic to the commutative diagram (2.2). For convenience, we write this diagram again. 7T 2 (F ' ) > H2(F') *2{F) h H2{F) *2{F,F') h" 7Tl(F') V -+ 7Tl( JF)=0 (2.9) a* H2(F,F') > H^F') • #i(F) = 0 Recall that since F(p,q,r) is simply connected, the Hurewicz theorem implies that h is an isomorphism for all integers p, q, r > 1. Using various results from Section 1.2, we learn the following about the commutative diagram (2.9). From (2.27) and (2.29) in Proposition 1.2.25, both j# and j * are the zero homomorphism. Furthermore, from (1.2.19), we have that TT2(F(p, q, r)) = Z © Z for all integers p, q,r > 1. From (1.2.22) and (1.2.23), it follows that for p + q + r > 4, the Hurewicz homomorphism h' is an isomorphism, and 7Ti(F'(p,q,r)) = H\(F'(p,q,r)) = III.2 Homotopy and Homology of Similar Degeneracy Regions 134 Z 2 © Z 2 . By the generalized five lemma, this implies that the Hurewicz homomorphism h" is also an isomorphism for p + q + r > 4. For p = q = r = 1, we have from (1.2.22) that TTI(F'(1, 1,1)) = Q8, where Q8 is the quaternion group. Recall that Qs is a nonabelian group consisting of 8 elements. In this case, h!: 7Ti ( .F ' ( l , 1,1)) —» Hi(F'(l, 1,1)) is an epimorphism with kernel the commutator subgroup of Qs. Note that the commutator subgroup of Qs is Z 2 . It follows by diagram chasing that 7r 2 ( .F(l , 1,1), F'(l, 1,1)) is also nonabelian, and h": TT 2 (F (1 , 1 , 1 ) , F ' ( l , 1,1)) # 2 ( F ( 1 , 1 , 1 ) , F ' ( l , 1,1)) is an epimorphism. Without much difficulty, it can be shown directly that the kernel of h" is the commutator subgroup of 7r 2 ( .F(l , 1,1), ^ ' ( 1 , 1 , 1 ) ) . Thus, we are led to consider the type II case as generic when p + q + r > 4. As in our discussion of the type I case, we examine the generic situation first and then return to the special case p = q = r = 1. Since p + q + r > 4 for the generic type II case, it follows that at least one of the integers », q, or r is greater than or equal to 2. We shall assume that q > 2. However, if we had assumed that p > 2, or that r > 2, then there is a discussion which is almost identical to that which we formulate below. Indeed, we remind the reader that permuting the arguments of F(p, q, r) and F'(p, q, r) produce flag manifolds which are diffeomorphic to F(p,q,r) and F'(p,q,r), respectively. As the reader might suspect, the type II case may in some sense be thought of as the direct sum of two type I cases. To make this precise, we shall make use of two fibre III.2 Homotopy and Homology of Similar Degeneracy Regions 135 bundles, both of which have a short flag manifold fibred over a Grassmann manifold, with fibre another Grassmann manifold. First, using the homotopy exact sequence for the fibre bundles in (1.2.6), we obtain the following diagram. To simplify notation, we denote F(p,q,r) and F'(p,q,r) by F and F', respectively. 0 0 0 7r 2 (F ) x2 MG(P,q),G'(p,q)) i" *1# MG'(P,q)) TTl(F') *2(G(p + q,r)) - i ir2(G(p + q,r),G'(p + q,r)) TT^G'(p + q, r)) (2.10) The homomorphisms in this diagram are defined as follows. Each of the three rows are obtained from the homotopy exact sequences for the corresponding pair of spaces. The left and right columns are obtained from the homotopy exact sequences for the fibre bundles in (1.2.6). The homomorphism i " ^ is defined to be induced by the inclusion of the pair i'[: (G(p,q),G'(p,q)) c—• (F,F')), and p"^ is defined to be the homomorphism induced by the projection of the pair p'[^: (F,F') —> (G(p + q,r),G'(p + q,r)). With these definitions, (2.10) is obviously a commutative diagram. Exactness of the rows follows from Proposition 2.25 of Chapter I, and the fact that complex flag manifolds are simply connected. From the homotopy exact sequences for III.2 Homotopy and Homology of Similar Degeneracy Regions 136 the fibre bundles in (1.2.6), it follows that pi# and p'^ are epimorphisms. Therefore, to show exactness of the left and right columns, it suffices to show that ii# and i1^ are monomorphisms. From (1.2.12), we have that n2(G(p,q)) = TT2(G!(» + 4, r)) = Z, and from (1.2.19) that TT2(F) = Z © Z, for all integers p, q, r > 1. This implies that i\^ is monic, and hence the left column is a short exact sequence. For all integers p, q, r such that p + q > 3, and r > 1, we have from (1.2.16) and (1.2.22) that iri(G'(p,q)) £ 7ri(G'(p + q,r)) ^ Z 2 , and that 7ri ( F ' ) ^ Z 2 © Z 2 . This implies that i1^ is monic, and hence the right column is also a short exact sequence. We remark that here it is important that either p or q is greater than or equal to 2. In the previous subsection, it was computed that 7r2(G(», q), G'(p, q)) = 'K2{G(p + q,r),G'(p + q,r)) = Z. Therefore, commutativity of the diagram (2.10) im-plies that the composition p"^. i'^ is the zero homomorphism. From this it follows by a corollary of the 3 x 3 lemma [61, Exercise 1.5.2] that the middle column is a short exact sequence. Indeed, since 7r2(G(p + q, r), G'(p + q, r)) is free abelian, this short exact sequence must be split, and therefore 7r 2 ( i ? (p , q, r), F'(p, q, r)) is isomorphic to Z © Z for all integers p, q, r > 1, satisfying p + q + r > 4. At this point, we have computed all of the groups in the diagram (2.9). However, it will be useful to explore the relationship between the type I case and the type II case more fully. By examining how the assumption that q > 2 was used above, we realize that there is another commutative diagram of the form of (2.10) which we may construct. III.2 Homotopy and Homology of Similar Degeneracy Regions 137 The following diagram is constructed by making use of the fibre bundles in (1.2.7). 0 0 0 MG(q,r)) 7r2(G(q,r),G'(q,r)) 7n(G/( 9 >r)) '2# i" l2# *2{F,F') 4* »2# Tri(F') P' 2# *2(G(p,q + r)) 7r2(G(p,q + r),G'(p,q + r)) ^ ( G ' f o g + r)) 0 0 0 (2.11) The homomorphisms in this diagram are defined in an analogous fashion to those in (2.10). Notice that the diagrams (2.10) and (2.11) share the same middle row. This suggests that we should think of these two diagrams as being in perpendicular planes, meeting in their middle row, and thus comprising a 3-dimensional commutative diagram. Consider the left cross-section of this three-dimensional diagram. The fibre inclusion i2: G(q, r) <^-» F(p, q, r) is defined in terms of the appropriate fibre bundle in (1.2.7). By direct inspection, it is easy to see that the composition p\ o i2: G(q,r) —• G(p + q,r) is the same map as the inclusion a: G(q, r) c—> G(p + q, r) defined in (1.2.9). From the commutative diagram (1.2.11), it follows that for all integers p, q, r > 1, the induced composition »2# Pl# *2{G{q, r)) > 7r 2(F(p, q, r)) > 7r2(G(p + q, r)) (2.12) III.2 Homotopy and Homology of Similar Degeneracy Regions 138 is an isomorphism. Similarly, the composition »1# P2# *2(G{p, q)) • 7r 2(F(p, q, r)) • 7T2(G(p, q + r)) (2.13) is also an isomorphism. This implies that the left cross-section of the 3-dimensional commutative diagram formed by (2.10) and (2.11) provides a direct sum decomposition5 of 7r2(-F(p, q, r)) This decomposition is summarized by the commutative diagram *1# l2# *2(G(p,q)) > TT 2 (F ) « * 2(Gfa,r)) (2.14) P2# Pi# *2(G(p,q + r)).<—— irz(F) • TV2(G(P + q,r)) which is a direct sum diagram for all integers p, q,r > 1. We apply a similar argument to the right cross-section of the 3-dimensional diagram formed by (2.10) and (2.11), and conclude that the induced compositions L2# P\# MG'(q,r)) —+ ^{F'{p^r)) • *i{G'{p +q,r)) (2.15) and *l{G'{p,q)) — *i{F'(p,q,r)) —+ n{G\p,q + r)) (2.16) are isomorphisms as long as p, q, and r are integers satisfying p,r > 1, and q > 2. 5 Direct sum decompositions are reviewed in [61, Sect. 1.4] III.2 Homotopy and Homology of Similar Degeneracy Regions 139 Therefore, we have the direct sum diagram Kl(G'(p, q)) - ^ U iri(F')) r)) P2# P\# *l(G'(p,q + r)) « Tn(P') > ^ (G'(p + q,r)) (2.17) for all integers p, r > 1, and q > 2. The isomorphisms denoted by a'g are induced by the inclusion a', which is defined in (1.2.10). We remark that if p = q = r = 1, then (2.17) is not a direct sum diagram. We now show that the middle cross-section in the 3-dimensional diagram formed by (2.10) and (2.11) is also a direct sum decomposition for all integers p,r > 1 and q > 2. Because the compositions (2.12), (2.13), (2.15), and (2.16) are isomorphisms, the five lemma implies that the induced compositions -•" J' 2# Pl# K2(G(q, r), G'(q, r)) • 7T 2(F(p, q, r), F'(p, q, r)) • TT2(G(P + q, r), G'(p + q, r)) and i" P2# ir2(G(p, q), G'(p, q)) • *2(F(p, q, r),F'(p, q, r)) • n2(G(p, q + r), G'(p, q + r)) are also isomorphisms. Indeed, they are isomorphisms induced on homotopy by the relative inclusion a" defined in (2.4). Therefore, for all integers p, r > 1 and q > 2, the commutative diagram *2(G(p,q),G'{p,q)) - i *2(F,F') +± *2(G(q,r),G'(q,r)) (2.18) P2# Pl# *2(G(p,q + r),G'{p,q + r)) < ir2(F,F') > TT2(G(P + q, r), G'(p + q, r)) III.2 Homotopy and Homology of Similar Degeneracy Regions 140 is a direct sum diagram. The direct sum diagrams (2.14), (2.17), and (2.18) imply that for all integers p, r > 1 and q > 2, the commutative diagrams (2.10) and (2.11) provide direct sum splittings of each other. Proposition 2.19. Let F and F' denote F(p,q,r) and F'(p,q,r), respectively. Then, for all integers p,q,r > 1, the commutative diagram H2{G{p,q)) -^-> H2(F) ^ - H2(G(q,r)) 9* a* ^ a* (2.20) P2* Pi* H2(G(p,q + r)) < H2(F) • H2(G(p + q,r)) is a direct sum diagram. For all integers p, r > 1 and q > 2, the following two commutative diagrams are direct sum diagrams: Hi(G'(p,q)) - A * Hi(F')) /fi(G'( 9 ,r)) Si a't S a', (2.21) HiiG'faq + r)) J ^ - Hi(F') H^G'ip + q^)) and H2(G(p,q),G'(p,q)) H2(F,F') H2(G(q,r),G'(q,r)) Si ^ = a',' (2.22) „" Jl H2(G(p,q + r),G'(p,q + r)) H2{F,F>) —• H2(G(p + q,r), G'(p + q, r)) Proof. The Hurewicz theorem implies that (2.20) is isomorphic to the direct sum diagram (2.14). Similarly, (2.21) is isomorphic to the direct sum diagram (2.17). III.2 Homotopy and Homology of Similar Degeneracy Regions 141 It follows from the commutative diagrams (2.3) and (2.9) that (2.22) is isomorphic to the direct sum diagram (2.18). D lip = q = r = l, the middle and right columns of the 3 x 3 commutative diagrams (2.10) and (2.11) are no longer short exact sequences. Indeed, because the homotopy groups 7Ti( i ? ' ( l , 1,1)) and 7T2(-F(1,1,1),F'(1,1,1)) are nonabelian, they cannot possibly be a direct sum of abelian groups. However for homology, we have the following direct sum decompositions. Proposition 2.23. The diagrams P'I* p'i * #i(G'(l,2)) — #1(^(1,1,1)) —+#i(G'(2,l)) (2.24) and H2(G(1,2),G'(l, 2)) £ l H2(F(1,1,1),F'(l, 1,1)) ^ H2(G(2,1),G'(2,1)) (2.25) are both projective direct sum diagrams. Proof. We shall prove the proposition for diagram (2.25). A similar argument obtains the result for (2.24). Notice that the pairs of PSD-regions (W(3,3), W'(3,3)) and (W(l, 1), W'(l, 1)) are homeomorphic to the pairs of open-ended mapping cylin-ders for the relative projections P l : (F( l , 1,1), F'(l, 1,1)) — (G(2,1),G'(2,1)) III.2 Homotopy and Homology of Similar Degeneracy Regions 142 and P 2 : (F(l,l,l),F'(l,l,l)) —» (G(1,2),G'(1,2)) , respectively. Furthermore, the union (W(l, 1), W'(l, 1)) U (W(3,3), W'(3,3)) is homeomorphic to (S 1 7 , ^ ) . It follows from Lemma 2.4 in Chapter II, that W(l, 1) fl W(3,3) is homeomorphic to the product space F(l, 1,1) x (0,1), where (0,1) denotes the open, unit interval in R. Also, Lemma II.2.6 implies that W'(l,l) n W(3,3) is homeomorphic to the product space F ' ( l , l , l ) x (0,1). Choosing a pair of fibres, we define the relative inclusions i1: (F(l, 1,1),F'(l, 1,1)) - (W(l, 1), W'(l, 1)) and i2: (F(l, 1,1),F'(l, 1,1)) -> (W(3,3), W(3,3)) . Because (W(l, 1),W'(l, 1)) n (W(3,3),W'(3,3)) is homotopy equivalent to (F( l , 1,1), .F'(l, 1,1)), the relative Mayer-Vietoris exact sequence for homology6 gives H3(S7, S 4) - H2(F(1,1,1), F ' ( l , 1,1)) 1* H2(W(l, 1), W'(l, 1)) © H2{W{Z,3), W(3,3)) — #2(,S7, S 4) (2.26) The homomorphism y> is defined by </?: z i—> (i;];z,i2z) where and i2 are the homomorphisms induced by i1 and i 2 , respectively, on the 2-dimensional relative homology. 6 The relative Mayer-Vietoris exact sequence for homology is reviewed in [83, Sect. 4.6]. III.2 Homotopy and Homology of Similar Degeneracy Regions 143 From the homology exact sequence for the pair (S1, 5 4), it follows immedi-ately that H3(S7,54) = H2(S7, 5*4) = 0. Hence, (p is an isomorphism, and from the exact sequence (2.26) we obtain the projective direct sum diagram H2(W(1,1), W'(l, 1)) £ H2(F(1,1,1), F'(l, 1,1)) t H2(W(3,3), W(3,3)) (2.27) Recall from Lemma II.2.4, Lemma II.2.6, and Theorem II.2.9 that the pair (G(l,2), G'( l , 2)) is a deformation retract of the pair (W(l, 1), W'(l, 1)), and that (G(2,1),G"(2,1)) is a deformation retract of (W(3,3), W'(3,3)). This implies that the direct sum diagram (2.27) is isomorphic to the diagram (2.25). D We conclude this section by using two commutative diagrams to summarize the situ-ation for type II SD-regions. For n > 4, the complex, type II SD-regions in Herm(n, C), and the real, type II SD-regions in i7erra(ra,R) are all generic. We have shown that a pair of generic, type II SD-regions (W f f, W'g) satisfy the following commutative diagram, which has exact rows. Z © Z Z 0 Z Z 2 0 Z 2 0 • *i(W,) (2.28) • #i(W f f) Z 0 Z Z © Z Z 2 0 Z 2 0 *2(W f f) x2 *2 (W„W1) h" *l(W') h' III.3 Computation of Adiabatic Phase 144 There is only one type II SD-region in Herm(3, C), and it is the nongeneric, complex, SD-region which is homotopy equivalent to F ( l , l , l ) . Also, the unique type II SD-region in /7erm(3,R) is the nongeneric, real, SD-region which is homotopy equivalent to .F'(l, 1,1). We denote this pair of SD-regions by (W3, W3). The homotopy and homology in dimensions one and two for the pair (W^W^) are described by the commutative diagram with exact rows, Z © Z [nonabelian] Q8 0 7T 2(W 3) 7r 2 (W 3 ) h 7r2(W 3,W 3) *i(W8) epic epic h' 7Tl (W 3 ) Hi(m) ( 2 . 2 9 ) Z © Z ZfflZ z 2©z 2 §3 Computation of Adiabatic Phase In Section 1, it was shown that a physical system described by an irreducible, time-dependent matrix hamiltonian H(t), with values in a nondegenerate SD-region W of Herm(n, C), exhibits time-reversal invariance if and only if the path of hermitian matrices H(t) is in a potentially real subspace of W, for all t. Furthermore, we showed that to study paths in any potentially real subspaces of W, it suffices to consider paths in the nondegenerate, real SD-region W' = W Pi Jferra(n,R). The purpose of this section is to compute the adiabatic phase on all loops in each of the nondegenerate, real SD-regions of Herm(n, C). III. 3 Computation of Adiabatic Phase 145 Consider any pair of nondegenerate SD-regions (W,W'), and let T be a simple, smooth loop in W'. It was proven in Subsection II.3.b that there exists a smooth imbed-ding D of the 2-dimensional unit disc into W such that dD = T. Furthermore, viewing T as a smooth, oriented 1-cycle, the orientation on T induces an orientation on D. From equation (II.3.6), the adiabatic phase 7 (T) is given by the integral of the adiabatic cur-vature 2-form K over D. The disc D represents a relative homology class in ^ ( V V , W'). Our approach to computing f(T) is to relate fC to a relative cohomology class in H2(W, W'; Z), and then evaluate the cap product between this cohomology class and a choice of generator or generators for H2(VV,W'). Thus, we obtain 7(T) expressed in terms of the homology class of D in H2(W Our knowledge of the homomorphisms in the commutative diagram (2.1) is used to relate j(T) to the homotopy class of T in 7r i (W') . For a complex vector bundle, the relationship between curvature and cohomology is expressed by the Chern-Weil description of characteristic classes. A brief review, which is well suited to our needs, is given in [66, Appendix C]. For complex line bundles, Chern-Weil theory relies on the fact that if rj is a complex line bundle with connection V , defined over a base manifold Ai, then its curvature 2-form K is a glob ally-defined, closed 2-form on M.. This implies that K. represents a cohomology class {K,} G H2(M.;C). Furthermore, there is the remarkable fact that the cohomology class {£} is independent of the connection V , and depends only on the isomorphism class of the bundle rj [66, p. 298]. III.3 Computation of Adiabatic Phase 146 The computation of the adiabatic phase for simple, smooth loops is easily extended to all smooth loops, not necessarily simple. Indeed, because the adiabatic theorem (The-orem II.1.14) holds for twice continuously differentiable loops, we can also relax the requirement that the loops be smooth. However, this generalization is obvious, and we will not bother with it further. If T is an arbitrary, smooth loop in W', and n > 2, it follows from the Whitney imbedding theorem [84, Sect. II.4], [93, Thm. 2] that T may be approximated arbitrarily closely by a simple, smooth loop T', which also lies entirely in W'. Of course, T and T' are homotopic in W'. Furthermore, we can choose T' such that -y(T') approximates "f(T), arbitrarily closely. This allows us to obtain 7(T) in terms of the homotopy class of T in 7Ti(W'). A S in the previous section, we divide our analysis into first considering type I SD-regions, and then type II SD-regions. (a) Type I SD-Regions The two nondegenerate, type I SD-regions in Herm(n,Ii) are W' ( l , 1) and W'(n,n). We shall denote both of these regions by W', and their corresponding complex SD-region by W. By Proposition II.2.8 and Theorem II.2.9, there is a relative, strong deformation retraction of the pair (W,W') to the pair (CP(n —1), RP(n —1)). We denote the relative retraction mapping by r: ( W , W ) —• (CP(n - l ) ,RP(n - 1)) , and the corresponding relative inclusion by i: (CP(n - 1), RP(n - 1)) ^ (W, W') . III.3 Computation of Adiabatic Phase 147 In Subsection II.3.a, we defined the eigenspace line bundle £ over W. By representing the adiabatic connection in a basis of local sections, we obtained the adiabatic curvature 2-form JConW. The curvature form K. is closed, and hence represents a de Rham coho-mology class {JC} G H2DR(W; C). Furthermore, it follows from Chern-Weil theory that the cohomology class {/C} is independent of the connection VA- Perhaps surprisingly, this means that the cohomology class of any curvature 2-form on £ is identical to the cohomology class of the- adiabatic curvature 2-form. The de Rham isomorphism theorem (Theorem 1.1.17) implies that the de Rham cohomology group H2^R(W; C) is naturally isomorphic to the singular cohomology group # 2(W; C). Let <f>: H2(W; C) - • H2DR(W; C) be the inverse of the isomorphism defined in (1.1.18), and denote by (: Z <—> C the inclusion of the integers into the complex plane. Because # 2(W; Z) £ H2(CP(n - 1); Z), it follows from (1.2.14) that # 2(W; Z) is isomorphic to Z, and is therefore a free, finitely generated abelian group. This implies that the induced homomorphism (t: H2(W; Z) —> H2(W; C) is a monomorphism, which allows us to identify the integral, singular cohomology classes with their images under <f>(*. Under this identification, the cohomology classes in U 2 ( W ; Z ) are represented by closed 2-forms on W. Of course, not all closed 2-forms will represent an integral cohomology class. Consider the induced line bundle i*(, obtained by restricting £ to CP(n — 1). By Proposition II.3.3, the line bundle i*£ is isomorphic to the canonical line bundle 7 n _ i over CP(n — 1), and furthermore, I*VA is a connection on 7«_i. Thus, Chern-Weil theory implies that the cohomology class {^z*^} £ B^)R(CP(n — 1); C) is equal to the image of the first Chern class ci(7 n_i) G H2(QP{n — 1); Z) under the the monomorphism III. 3 Computation of Adiabatic Phase 148 [66, p. 306]. Moreover, the cohomology group H2(CP(n — 1); Z) = Z is freely generated by ci(7„_i) [66, Thm. 14.4]. By identifying integral cohomology classes with their images under we conclude that {-^i^fC} generates H2(CP(n — 1);Z). We have shown that ^ times the restriction of the adiabatic curvature 2-form to the canonical line bundle over CP(n — 1) gives a cohomology class which generates H2(CP(n — 1); Z). However, because CP(n — 1) is a strong deformation retract of W, it follows that the induced homomorphism r*: H2(CP(n - 1);Z) -+ H2(W;Z) is an isomorphism, and the homomorphism r*i* is equal to the identity on # 2(W; Z). Thus, the cohomology class r*{-^i*X} = {^FI^ C} generates H2(VV;Z). First, we consider a pair of generic, nondegenerate, type I SD-regions (W 5, W^), and examine the relative cohomology group H2(Wg, W'g; Z). Recall that type I SD-regions are generic when n > 3. Lemma 3.1. If (W<,,Wj) is a pair of generic, nondegenerate, type I SD-regions, then the closed 2-form i / C represents an integral, relative cohomology class {^j/C}, which generates H2(VVg, Wg; Z) ^ Z. Proof. The bottom row of the commutative diagram (2.7) gives the homology exact sequence H2(Wg) H2{yVg,Wg) HiCW'g) > Hx{Wg) I (3-2) z Z Z 2 0 III. 3 Computation of Adiabatic Phase 149 With an appropriate choice of generators, the homomorphism &+ is multiplication by 2. Applying the contravariant functor Hom( •, Z) to this exact sequence, we obtain the homomorphism Hom(A;*, Z): Rom(H2(Wg, Wg), Z) —> Hom(#2(W<,), Z) , (3.3) where rIom(H2(Wg, Wg), Z) and Hom(H2{Wg), Z) are both isomorphic to Z. It follows immediately from the definition of the functor Horn, that the homo-morphism (3.3) is also multiplication by 2. By extending the exact sequence (3.2) further to the right, we observe that Hi(yVg,VVg) = 0. Thus, the universal coefficient theorem for cohomology implies that H2(Wg; Z) =• Rom{H2(Wg), Z) and H2(Wg, Wg; Z) =• Hom(#2()%, VVJ), Z) . Since {^j/C} generates H2(Wg;Z), we conclude from the homomorphism (3.3) that {±K,} generates H2(Wg, Wg\ Z). • There are two possible homology classes which may be chosen as a generator of H2(yVg,yV'g)- We claim that one of these two generators, v is naturally dual to the gen-erator {^/C} for H2(Wg, WJ; Z), in the sense that , v) = 1. The product (•, •) between cohomology and homology, is the usual scalar product defined as evaluation of the cochain on the chain. III.3 Computation of Adiabatic Phase 150 It is crucial for the existence of a nontrivial adiabatic phase that one of the two possible generators of H2(VVg,Wg) be dual to {-^)C}. This claim may be proven in at least two ways. We present in detail a geometric argument that uses the imbedding a: CP(1) CP(n - 1), which is defined in (1.2.9). Poincare duality for CP(1) allows us to explicitly construct a dual generator for H2(Wg, W^). Alternatively, a more alge-braic argument using the universal coefficient theorem proves the same result. First, the geometric argument will be used to prove Theorem 3.5 and Corollary 3.6. Then, since the more algebraic argument clarifies some aspects of the proof, it will be briefly described in the remarks following Corollary 3.6. Consider the imbedding a: CP(1) <—> CP(n — 1), and recall that CP(1) is diffeo-morphic to the sphere S2. The 2-form a*i*K, is the restriction of K, to CP(1). From the definition of fC in Subsection II.3.a, it follows that K, is invariant under the conjugate action of a constant U £ U(n) on Wg. Hence, the 2-form a*i*JC is invariant under the usual action of U(2) on the homogeneous space CP(1). This implies that a*i*K is a nowhere vanishing 2-form on CP(1). For, if a*i*K vanished anywhere on CP(1), then it would follow that it must be identically zero on CP(1). This would mean that {a*i*JC} would be the trivial cohomology class in Hp^Wg] C), which we have shown not to be the case. In general, a choice of orientation for an orientable, n-dimensional manifold is uniquely determined by a nowhere vanishing n-form on the manifold.7 Therefore, the 2-form a*i*K. defines an orientation on CP(1) « S2. The cohomology class u = {{-^)o:*i*IC} € 7 This well-known fact is explained in [16, p. 29]. III.3 Computation of Adiabatic Phase 151 H2(S2; Z) is called the fundamental cohomology class of this oriented S2. Given a fun-damental cohomology class, there is a unique choice of generator p, G H2(S2) = Z, which is dual in the sense that (u,p) = 1. The homology class p, is called the fundamental homology class. More details on duality, and fundamental cohomology and homology classes are given in [38] and [66]. The homomorphism z*a*: H2(S2) —» H2(Wg) is an isomorphism, and hence i*a*p defines a preferred choice of generator for H2(Wg). The homology exact sequence for the pair (W s , W^) defines a homomorphism k +: H2(Wg) —> I/2(W f f, W^), which we have shown to be multiplication by 2. There are two possible homology classes which may be chosen as a generator for /^(Wg, W'g). However, a choice of generator for 772(W3) defines a preferred choice of generator, v G H2(Wg, W'g) by the condition that k*i*a*p, = 2v. Having fixed a choice of generator for /^(Wg, VVff), we have established a one-to-one correspondence between relative homology classes in H2(Wg,yVg), and integers. For a relative 2-cycle Z), we denote by {D} both the homology class and the integer obtained by this correspondence. The cap product8 defines a homomorphism from H2(Wg, Wg\ Z) ® i/"2(W f f, Wg) to H0(Wg). For c G H2(Wg, Wg\ Z) and a G H2(Wg, Wg), this homomorphism is denoted by c <8> a H-> c ^ a. If e is the augmentation on i7o(VV3), then the cap product satisfies e(c — a) = (c,a) . (3.4) Of course, since W 5 is path connected, the augmentation e is an isomorphism. 8 The cap product is reviewed in [29, Sect. VII.12] and [83, p. 254]. III.3 Computation of Adiabatic Phase 152 Let D be a smooth disc which represents the generator v 6 H2(Wg, VV^). Recall that because the Hurewicz homomorphism h" in the commutative diagram (2.1) is an epimorphism, any relative homology class in H2(VVg,yVg) may be represented by the smooth image of a disc in Wg, with its boundary in W'g. Also, the Whitney imbedding theorem implies that we can assume that the disc is imbedded in Wg. The de Rham isomorphism theorem for relative cohomology implies that / - U = /{i/c},iA, JD iri \ m j where {-^ j/C} denotes the cohomology class in H2(Wg, W^; Z) represented by ^fC. By (3.4), we have that / —X = e({i/C} - v) . JD 7™ 7™ Naturality of the cap product implies that e({—X} ^ v) = e({-^—X] ua*p) , where {^K,} € H2(Wg; Z) and i *a*/x G Pi2(yVg). Also by naturality, we obtain = e(u ^ u) = — 1 Let D now be a smooth imbedding of the unit disc which represents an arbitrary relative homology class in H2(Wg,W'g). Then, for the pair of generic, nondegenerate, type I SD-regions (Wg, W^), we have proven III.3 Computation of Adiabatic Phase 153 Theorem 3.5. Take v as the generator of ./Y2(W3, W p ) , and let {D} represent the integer corresponding to the relative homology class {D} 6 H2{Wg iW'g). Then, the integral of the adiabatic curvature 2-form K over D is given by I K, = {D}*i. JD • We are now able to compute the adiabatic phase for any simple loop T in the SD-region Wg. Any such loop is a 1-cycle in W'g, and hence represents a homology class {T} G H\(Wg). Recall from the commutative diagram (2.7) that Hi(yV'g) is equal to Z 2 . Therefore, each homology class {T} is uniquely represented by either 0 or 1, depending on whether it is trivial, or not, respectively. We use the notation {T} to also denote this integer (mod 2) in Z 2 . For the smooth, simple loop T in W f f, there exists a smooth imbedding D of the unit disc into Wg with the property that T = 3D. The disc D represents a relative homology class {D} G 7/2(Wg, W s ) . The homology classes {T} and {D} are related by the boundary homomorphism 5* in the short exact sequence 0 - * H2(Wg) H2(Wg,Wg) i HiiWg) -* 0 which is obtained from the bottom row of the commutative diagram (2.7). This short exact sequence, along with equation (II.3.6), and Theorem 3.5, provides f(T) for all simple, smooth loops in W g . By using the Whitney imbedding theorem to extend this result to all smooth loops in W'g, we prove III.3 Computation of Adiabatic Phase 154 Corollary 3.6. Let T be a smooth loop in a nondegenerate, type I SD-region in Herm(n, R) for n > 3. Then, the adiabatic phase is 7(T) = {T} ir. D We remark that it is not necessary to specify (mod 2ir) in the above formula for j(T), because {T} is an element of Z 2 . Also, the result in Corollary 3.6 is independent of the choice of generator for iJ^VV^, W^). This is not surprising, because the physics should not depend on the generators that are chosen for homology groups. The aforementioned algebraic proof for Corollary 3.6 is now sketched. Using the universal coefficient theorem, it illustrates that the central idea behind Corollary 3.6 is the existence of a basis for i f 2 (W 5 , W^), which is dual to {-^;fC}. Define the homomorphism / from H2(VVg, Wg) ^ Z to Z by /: o i—> ({—X},o~ Then, it is straightforward to verify that 7(r) = deg(/){r}7r (mod27r), where deg(/) is the degree of / . However, it follows from the universal coefficient theorem, that if (X, Y) is a pair of topological spaces, Hn(X, Y) is free abelian, and Hn_i(X, Y) is torsion-free, then there is an isomorphism between Hn(X, Y) and Hn(X, Y; Z), which allows us to identify dual bases for these two free abelian groups. This implies that deg(/) = 1, and Corollary 3.6 follows. III.3 Computation of Adiabatic Phase 155 Now, consider the case when n — 2. The single type I SD-region in Herm(2, C) is denoted by W 2 , and W 2 denotes its intersection with Herm(2, R). From the commutative diagram (2.8), we have the homology short exact sequence #2(w2) — • #2(w2) — • #2(w2,w2) — • /fi(w2) — • Hiim) 0 z z®z z 0 Because this short exact sequence is split, an analogue of Theorem 3.5 for nongeneric, type I SD-regions would be significantly different. However, in Subsection II.Lb, the adiabatic phase for any smooth loop T C W 2 was computed by directly evaluating the integral §D1C. The results of this computation are summarized in Proposition 3.7. Let T be any smooth loop in the SD-region W 2 C 7/erm(2,R). By choosing one of the two possible choices of generator for HiCW^) = Z, we assign an integer to each homology class in i7i(W 2). The integer associated with the homology class {T} € Jfi(W 2) is denoted by {T}. Then, the adiabatic phase is 7 (T) = {T} 7T (mod 2TT) . • We remark that in Proposition 3.7, the choice of generator for .ffi(W2) is irrelevant, because "f(T) is only defined modulo 27r. III.3 Computation of Adiabatic Phase 156 (b) Type II SD-Regions Let W denote a nondegenerate, type II SD-region in Herm(n,C), and W' its in-tersection with Herm(n,H). In this subsection, the pair of type II SD-regions (W, W ) may be either generic or nongeneric. This is because the commutative diagrams (2.28) and (2.29) differ only in homotopy, and it is homology which is used for computing the adiabatic phase. From Proposition 2.8 and Theorem 2.9 in Chapter II, we have a relative retraction mapping r:{WtW)-+(F(p,l1r),FJ(p,l,r)) and the corresponding relative inclusion i: (F(p, 1, r), F'(p, 1, r)) ^ (W, W) , where p, r > 1. Let i\ and i2 denote the fibre inclusions in the fibre bundles CP(p) — F ( p , l , r ) and CP(r) — F ( p , l , r ) PI P2 G(p+l , r ) G(p,r + 1) which are defined in (1.2.6) and (1.2.7), respectively. From the direct sum splitting in Proposition 2.19 and the universal coefficient theorem for cohomology, we obtain the direct sum diagram 2(CP(p);Z) ^— H2(F(p, 1, r); Z) # 2 (CP(r); Z) I T«2 (3.8) * * tf2(G(p,r + l);Z) > tf2(P(p,l,r);Z) ^— H\G(p + l , r) ;Z) III.3 Computation of Adiabatic Phase 157 where i\, i^, p*, and p 2 a r e the homomorphisms induced on cohomology by the maps i j , h-, Pi, and p 2 , respectively. The vertical homomorphisms a\ and are the isomorphisms induced on cohomology by the two appropriate inclusions defined by (2.9) in Chapter I. We define homomorphisms (p\ and <p2 by (p\ = / ^ ( ^ l ) - 1 a n a ^ V2 = P\{a2)~l• Because (3.8) is a direct sum diagram, it follows that the composition \ is equal to the identity on H2(CP(p); Z), the composition i\\p2 is equal to the identity on H2(CP(r); Z), and (p\i* + <p2i2- is equal to the identity on H2(F(p, l , r) ; Z). The adiabatic curvature 2-form on the eigenspace line bundle £ over W is denoted by K.. The induced line bundles i\i*£ and i2i*£ are the restrictions of the eigenspace line bundle to the subspaces CP(p) and CP(r) , respectively. It follows by Proposition II.3.4 that i\i*£ is isomorphic to the canonical line bundle *yp over CP(p), and i2i*£ is iso-morphic to the canonical line bundle 7 r over CP(r). The adiabatic curvature 2-form K restricts to give the curvature 2-forms K\ = i*i*K, and K2 = i\i*K for jp and 7 r , respectively. By the same reasoning as in the generic type I case, the cohomology class { 377^1} is equal to the first Chern class c i(7 p ), and therefore generates H2(CP(p); Z). Similarly, the cohomology class {217 -^2} generates H2{CP(r); Z). From the direct sum splitting (3.8), we have that the two cohomology classes Vli^fcl} a n d iP2{'2\i^2] generate H2(F(p, l ,r); Z). Furthermore, using the fact that the homomorphism y>\i\ + <£>2«2 is equal to the identity on H2(F(p, 1, r); Z), it follows that the de Rham cohomology class ^*{O4T^} is integral, and is given by ^h^-^h^+^h^- (3-9) III.3 Computation of Adiabatic Phase 158 The homomorphism r*, induced by the retraction mapping of (W,W') to (F(p, 1, r), F'(p, 1, r)), is an isomorphism, and hence the cohomology classes r*ip\{-^K,\] and r*tpii-^i^} g e n e r a t e -H"2(W; Z). Also because (r*) _ 1 = i*, it follows from (3.9) that is an integral cohomology class, and Our computation of -f(T) for a nondegenerate, real, type II SD-region W closely parallels the computation for type I SD-regions in the previous subsection. Generators for the relative cohomology group H2(W, W'; Z) are determined i n Lemma 3.11. Let (W, W ) be a pair of nondegenerate type II SD-regions. Then, the cohomology classes r*(pi{^tC\} and r*Lp2{-p%K<2} are integral, relative cohomology classes, and they generate # 2((W, W'); Z) ^ Z © Z. Proof. The proof of this lemma is almost identical to the proof of Lemma 3.1. How-ever, it uses the bottom rows of the commutative diagrams (2.28) and (2.29). • As in the type I case, generators for i ?2(W,W') , which are dual to r*<pi{^ICi} and r * ( / ? 2 { ^ ^ 2 } , m a y be constructed either geometrically using Poincare duality, or algebraically using the universal coefficient theorem. The arguments are similar to those used in the type I case, and to avoid being pedantic, only the geometric construction will be described. III.3 Computation of Adiabatic Phase 159 In order to construct generators for iz2(W,W'), consider the imbeddings c*3: S2 <—* CP(p) and cq: S2 <—> CP(r), which are two examples of the smooth imbedding a, defined in (1.2.9). Following the generic type I case, we observe that the cohomology classes ctH^ICi} e H2(S2;Z) and c^-f^A^} € H2(S2;Z) are both fundamental cohomology classes for S2. Fundamental homology classes p,\ € H2(S2) and \i2 6 H2(S2) are defined by the conditions that (^{^j /Ci} , /ii) = 1, and ( ^ { ^ j A ^ }, P2) = 1- The homology classes p,\ and p2 define preferred generators for ./^(W) by the homomorphisms in the diagram iJ 2(W) # 2 ( C P ( p ) ) / f 2 ( F ( p , l , r ) ) # 2 ( C P ( r ) ) Si <*3* <*4* # 2(S 2) # 2(S 2) This defines generators i^iua^pi and i* 12*0:4*^ 2 for /^ (W) . We remark that the non-homotopic imbeddings i\ 0 0 3 : S2 F{p,\,r) and i2 o a^: S2 <^-> F(p, l , r) represent generators for 7r 2(P(p, l,r))> which is isomorphic to Z © Z. From the homology exact sequence for the pair (W, W ) , we have the homomorphism '• H2(W) —» 7f 2 (W,W), which is multiplication by 2. This defines generators v\ and V2 for f/"2(W,>V') by the conditions that k^uiuot^pi — 2v\ and A;+i+i2+a4*//2 = 2*/2. Using (3.10), a computation which is similar to the proof of Theorem 3.5, proves Theorem 3.12. Consider a pair of nondegerate type II SD-regions (W, W'), and let D be any smooth, relative 2-cycle. Taking v\ and f2 as generators for ^ ( W , W'), the III.3 Computation of Adiabatic Phase 160 homology class {D} G H2(yV,W') is represented by a pair of integers ({D}i,{D}2) € Z © Z. Then, the integral of the adiabatic curvature 2-form K, over D is f )C = ({D}1 + {D}2)ni. JD • The homology group Hi(W') is related to H2(W, W') by the boundary homomor-phism in the short exact sequence 0 -> H2(VV) - i # 2(W, W) i # i (W) -> 0 . This defines the generators d*v\ and d*t>2 for H\{W') = Z 2 © Z 2 . Then, for any smooth loop T in W', the homology class {T} £ Hi(W') is represented by ({T}i , {T}2), a pair of integers (mod 2) in Z 2 © Z 2 . An immediate result of formula (II.3.6) and Theorem 3.12 is Corollary 3.13. For T a smooth loop in a nondegenerate, type II SD-region in Herm(n, R), where n > 3, tie adiabatic phase is given by 7 (T) = ({T} 1 + { T } 2 ) 7 r (mod27r). • III.3 Computation of Adiabatic Phase 161 (c) Adiabatic Phase in Terms of Homotopy Classes To recapitulate, the hamiltonian of a periodic quantum system with a finite number of energy levels is described by a loop T: S1 Herm(n,C). If T(s) has an eigenvalue A(s), which is nondegenerate for all s £ S1, then associated with this eigenvalue is an adiabatic phase 7(3°). Furthermore, if this system is also time-reversal invariant, then there exists a choice of basis for C n such that with respect to this basis, T is a loop in Herm(n, R). Corresponding to the eigenvalue A(s) is a nondegenerate, real SD-region W'. We have computed 7(T) in terms of the homology class {T} G Hi(VV'). However, from a physical standpoint, it may be more useful to obtain 7(T) in terms of the homotopy class [T] G 7Ti(W') . We remind the reader that two loops To and T\ in W' are in the same homotopy class if and only if there is a homotopy of loops $ : [0,1] x [0,1] —»• Herm{n, R) such that $(0,r) = T0(t), = Ti(r), $(a,0) = $(s, 1) for all s G [0,1], and the eigenvalue X(s,t) of $(s,t) remains nondegenerate for all (s,t) G [0,1] x [0,1]. Therefore, a homotopy of the loop T correspond to a continuous perturbation of the hamiltonian, which preserves the periodicity, the time-reversal invariance, and the nondegeneracy of A. For a specific time-reversal invariant quantum system, it may be possible to use such a homotopy to simplify the hamiltonian to the extent that its homotopy class in 7Ti(W') can be identified. We now summarize our results for the adiabatic phase 7 (T) in terms of the ho-motopy class of T in 7ri ( W ' ) . These result are obtained by using the the Hurewicz homomorphisms in the commutative diagrams (2.7), (2.8), (2.28), and (2.29) to relate III.3 Computation of Adiabatic Phase 162 the homotopy classes in 7ri(W') to the homology classes in i / i (W'), and thus obtain 7 (T) in terms of [T] € ir^VV'). If W 2 is the nondegenerate SD-region in .ff~erm(2,R), it follows from (2.8) that 7r i (W 2 ) = Z. Furthermore, the Hurewicz homomorphism h': 7Ti (W 2 ) —> i/"i(W2) is an isomorphism. By choosing one of the two possible generators for 7Ti(W 2 ) , we asso-ciate an integer, denoted by [T], with each homotopy class [T] G 7Ti (W 2). It follows from Proposition 3.7 that the adiabatic phase for a loop T C W 2 is 7(r) = [T] 7T (mod 2TT) . Of course, because 7 (T) is only denned modulo 2TT, this result is independent of the choice of generator for 7Ti (W 2). We now consider W'g to be a nondegenerate, type I SD-region in ^Term(ra,R) for n > 3. From the commutative diagram (2.7), we know that 7ri(W^) = Z 2 , and the Hurewicz homomorphism h!: 7Ti (W^) —> H\{yV'g) is an isomorphism. Therefore, it follows from Corollary 3.6 that 7 ( T ) = 0 if T belongs to the trivial class in %\(Wg). 7T if T belongs to the unique nontrivial class in 7ri(Wg). Now, consider W3, the nondegenerate, type II SD-region in i/erm(3,R). Notice that all three eigenvalues A i < A 2 < A 3 are nondegenerate in W 3 . For a loop T in W 3 , we denote by 7 i ( T ) the adiabatic phase associated the eigenvalue A ; . From the commutative III.3 Computation of Adiabatic Phase 163 diagram (2.29), recall that n\(yV3) is isomorphic to the nonabelian, 8-element quater-nion group, Q$. This implies that there are eight homotopy classes of loops in W 3 . Furthermore, the Hurewicz homomorphism h': 7ri(W3) —> Hi(W'3) is an epimorphism with kernel the two-element commutator subgroup of 0/8-The three adiabatic phases for each of the eight homotopy classes in 7ri(W3) are as follows. If T is in one of the two homotopy classes which comprise the commutator subgroup, then ~fi(T) = 0 (mod 27r) for i = 1,2,3. In Subsection III.3.b, we defined generators d*v\ and d+V2 for i / i(yV 3 ). If T is an element of one of the two homotopy classes in the preimage of d*v\ under the Hurewicz homomorphism h\ then 7i(T) = 72(T) = 7r (mod 27r), and 73(T) = 0 (mod 2TT). If T is an element of one of the two homotopy classes in (/V)_1<9*i% then 71 (T) = 0 (mod 27r), and 72 (T1) = 73(f) = 7r (mod 27r). There are only two remaining homotopy classes in 7r i (W3), and these lie in the preimage of d*v\ + d±vi. If T is in one of these two homotopy classes, then 7 l ( T ) = 73(T) = 7T (mod 2TT), and 72(T) = 0 (mod 2TT). Finally, consider a loop T in a nondegenerate, type II SD-region Wg in Herm(n,~R) for n > 4. The adiabatic phase associated with the eigenvalue A, which corresponds to W'g, is denoted by 7(T). Froni (2.28), the Hurewicz homomorphism h': 7ri(W^) —• Hi(W'g) is an isomorphism, and therefore (h')~ld*v\ and (h')~^d*V2 are generators of niiyV'g) = Z2 © Z2. Given this choice of generators, a homotopy class [T] £ Tri(VVg) is represented by ([T]i, [T]2), a pair of integers (mod 2) in Z2 © Z2. The adiabatic phase for A is then l(T) = {[T]1 + [T]2)v (mod27r). III.3 Computation of Adiabatic Phase 164 (d) John-Teller Effect Quantum adiabatic holonomy is important in the Jahn-Teller effect of molecular physics. Furthermore, it is a time-reversal-invariant manifestation of adiabatic holonomy, assuming that no external magnetic fields are applied to the molecular system. Therefore, the results of this section may be applied to the Jahn-Teller effect.9 This work is currently in progress, and only a brief outline of the methods will be presented here. Further details will appear in [31]. The motion of electrons within a molecule is described by a hamiltonian which de-pends on the positions of the nuclei. It follows that the electronic states must be coupled with the rotational and vibrational states of the nuclei. This coupling is the dynamical Jahn-Teller effect, and it has important consequences for the vibrational and rotational spectra of the molecule. Since the original investigation of the dynamical Jahn-Teller ef-fect by Longuet-Higgins, Opik, Pryce, and Sack in [60], there has developed an extensive literature on the subject. Some good reviews are [11], [52], [59], [86], and [90]. We consider a molecule with N nuclei, that are all assumed to have mass M, for the sake of simplicity. The positions of the nuclei are described by the coordinate vector Q € K3N. The hamiltonian is #mol = V2Q + H(Q) , where the operator H{Q) = — ^ V 2 , + V(Q,q) depends parametrically on the nuclear coordinates Q. The electronic mass and coordinate vectors are m and q, respectively. 9 We thank the referees of [31] for encouraging us to consider this problem. III.3 Computation of Adiabatic Phase 165 The total potential energy V(Q,q) contains the coupling between the electrons and the nuclei. Let (j>(Q,q) be a Q-dependent eigenfunction satisfying H(Q)<l>(Q,<i) = KQ)<KQ,q), for a l i o . It is assumed that X(Q) is a nondegenerate eigenvalue, that is bounded away from the rest of the spectrum of H(Q) for all Q.10 At least locally, <j>(Q) defines an adiabatic connection 1-form A = (<f>(Q), d(f)(Q)). If x(Q) 1S a wave function satisfying x(Q) = eX(Q), (3.14) then the Born-Oppenheimer approximation implies that i>(Q,q) = x(Q) ^{Q'? ?) approx-imates an eigenfunction of Hmo\, with eigenvalue e. In most applications, H(Q) is time-reversal invariant. Therefore, locally A may be chosen to be 0. For this reason, the usual statement of the Born-Oppenheimer approxima-tion does not include the 1-form A in (3.14). The importance of the adiabatic connection 1-form in the Born-Oppenheimer approximation was demonstrated in [67]. Recall from Section II.3, that the 1-form A is globally defined on Q-space only if the the eigenspace line bundle associated with (f>(Q) is trivial. This is not usually the case. Therefore, (3.14) is not really a single differential equation, but rather a collection of differential equations, defined on coordinate neighbourhoods in Q-space. 1 0 We have assumed that e(Q) is nondegenerate for simplicity. Degenerate eigenvalues may be handled by replacing the eigenvector <f>(Q) with an orthonormal frame for the associated eigenspace. ^ T ( V Q + A)2 + A(Q) III.3 Computation of Adiabatic Phase 166 Throughout this thesis, we have seen the importance of taking particular care when making expansions in terms of parameter-dependent eigenvectors. In general, the eigen-function <j>(Q,q) will have nontrivial holonomy when parallel transported around loops in Q-space. If H(Q) is time-reversal invariant and X(Q) is nondegenerate, then this holonomy will be restricted to changes in sign. However, the approximate molecular eigenfunction ij>(Q, q) = x{Q) 4>{Qi <?) must be a single-valued function of (Q, q). This is achieved by choosing appropriate boundary conditions for the differential equation (3.14) defined on the various coordinate neighbourhoods of Q-space. For example, if <f>(Q, q) were to change sign when parallel transported around a loop T, then it would be nec-essary to look for solutions to (3.14) which were odd functions on T. The spectrum of a differential operator depends on its boundary conditions, and therefore we see that quantum adiabatic phase will influence the molecular spectrum. Unfortunately, H(Q) is a hermitian operator on an infinite-dimensional Hilbert space, and not a matrix. Hence, it seems that the Jahn-Teller effect requires an analysis of adi-abatic holonomy for hermitian operators. However, the spectrum of a molecular hamil-tonian often consists of an infinite number of almost degenerate multiplets, separated by relatively large gaps. The usual procedure in Jahn-Teller analysis is to neglect mixing between eigenfunctions of different multiplets. In doing this, Q is fixed to some value Qo, usually an equilibrium position for the nuclei. Then a Q-mdependent orthonormal basis is chosen for the span of those eigenfunctions of H(Qo) which are associated with the multiplet of interest. In terms of this basis, H(Q) may then be represented as a Q-dependent matrix. Adiabatic holonomy is computed using this matrix. Therefore, many of the results in this chapter have direct application to Jahn-Teller theory. III.3 Computation of Adiabatic Phase 167 Implications of adiabatic holonomy have been worked out for rotating diatoms [67], and for the E®e Jahn-Teller effect, in which an electronic doublet E is coupled to a pair of vibrational modes e [45], [60]. For both of these problems, H(Q) may be represented as a 2 x 2 matrix. Hence, it is straightforward to compute adiabatic phases in terms of explicit eigenvectors, as was done in Subsection Il.l.b. For more complicated systems, such an approach becomes difficult, if not impossible.11 Experimental consequences of the E®e Jahn-Teller effect have been observed in the spectra of a number of molecular systems [2], [45]. The most compelling evidence is the half-odd integer quantization of pseudorotational levels in Na3 [27]. 1 1 Some explicit families of eigenvectors have been computed at great effort for Jahn-Teller systems involving electronic triplets. For example, see [52]. These families of eigenvectors have been used to compute adiabatic connection 1-forms in [19]. References [ 1] Y. Aharonov and L. Susskind, Observability of the Sign Change of Spinors under 2TT Rotations, Phys. Rev., 158, pp. 1237-1238 (1967). [ 2] I. J. R. Aitchison, Berry's Geometric Quantum Phase, Proc. 7th General Conf. of European Physical Society (Helsinki, Aug. 1987), Phys. Scr., T23, pp. 12-20 (1988). [ 3] D. Arovas, J. R. Schrieffer, and F. Wilcek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett, 53, pp. 722-723 (1984). [ 4] J. E. Avron, L. Sadun, J. Segert, and B. Simon, Topological Invariants in Fermi Systems with Time-Reversal Invariance, Phys. Rev. Lett., 61, pp. 1329-1332 (1988); Chern Numbers and Berry's Phase in Fermi Systems, CalTech. preprint (1989). [5] J. E. Avron and R. Seiler, Quantization of the Hall Conductance for General, Mul-tiparticle Schrodinger Hamiltonians, Phys. Rev. Lett., 54, pp. 259-262 (1985). [6] J. E. Avron, R. Seiler, and B. Simon, Homotopy and Quantization in Condensed Matter Physics, Phys. Rev. Lett., 51, pp. 51-53 (1983). (In their notation, some of the homotopy groups quoted in this reference for 9T34 are unfortunately incorrect.) [ 7] J. E. Avron, R. Seiler, and L. G. Yaffe, Adiabatic Theorems and Applications to the Quantum Hall Effect, Commun. Math. Phys., 110, pp. 33-49 (1987). [ 8] M . V. Berry, Quantal Phase Factors accompanying Adiabatic Changes, Proc. Roy. Soc. 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Amer. Math. Soc, 67, pp. 109-112 (1961). 41] F. Haldane, Path Dependence of the Geometric Rotation of Polarization in Optical Fibers, Optics Letters, 11, pp. 730-732 (1986). 42] F. Haldane, Comment on "Observation of Berry's Topological Phase by Use of an Optical Fiber", Phys. Rev. Lett, 59, p. 1788 (1987). 43] F. Haldane and Y.-S. Wu, Quantum Dynamics and Statistics of Vortices in Two-Dimensional Superfluids, Phys. Rev. Lett., 55, pp. 2887-2890 (1985). 44] M . Hall, The Theory of Groups, Macmillan (1959). 45] F. S. Ham, Berry's Geometrical Phase and the Sequence of States in the Jahn-Teller Effect, Phys. Rev. Lett., 58, pp. 725-728 (1987). 46] J. Hempel, 3-Manifolds, Ann. of Math. Studies, no. 86, Princeton University Press (1976). 47] P. Hilton, Homotopy Theory and Duality, Gordon and Breach (1965). 48] S.-T. Hu, Homotopy Theory, Academic Press (1959). 49] D. Husemoller, Fibre Bundles, McGraw-Hill (1966). 50] R. 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Holonomy in quantum physics Rutherford, Alexander R. 1989
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Title | Holonomy in quantum physics |
Creator |
Rutherford, Alexander R. |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | Holonomy in nonrelativistic quantum mechanics is examined in the context of the adiabatic theorem. This theorem is proven for sufficiently regular unbounded hamiltoni-ans. Then, simplifying to matrix hamiltonians, it is proven that the adiabatic theorem defines a connection on vector bundles constructed out of eigenspaces of the hamiltonian. Similar degeneracy regions, the natural base spaces for these bundles, are defined in terms of stratifications for the spaces of complex, hermitian matrices and real, symmetric matrices. The algebraic topology of similar degeneracy regions is studied in detail, and the results are used to classify and calculate all possible adiabatic phases for time-reversal invariant matrix hamiltonians in terms of the relevant topological data. It is shown how vector bundles may be used to impose transversality on the helicity vector of a photon. This is used to give a calculation, which is consistent with transversality, of quantum adiabatic phase for photons in a coiled optical fibre. As an additional application, the importance of quantum adiabatic in the dynamical Jahn-Teller effect is briefly explained. An introduction is given to some important aspects of algebraic topology, which are used herein. Moreover, a number of mathematical results for flag manifolds are obtained. These results are applied to quantum adiabatic holonomy. |
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Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-10-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085010 |
URI | http://hdl.handle.net/2429/29275 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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