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UBC Theses and Dissertations

Relativistic few-body quantum mechanics Monahan, Adam Hugh 1995

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RELATIVISTIC FEW-BODY Q U A N T U M MECHANICS By Adam Hugh Monahan B. Sc. (Physics) University of Calgary, 1993 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1995 © Adam Hugh Monahan, 1995 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 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract This thesis develops relativistic quantum mechanical models with a finite number of degrees of freedom and the scattering theories associated with these models. Starting from a consideration of the Poincare Group and its irreducible unitary rep-resentations, we develop such representations on Hilbert Spaces of physical states of one, two, and three particles. In the two- and three- particle cases, we consider systems in which the particles are non-interacting and in which the particles experience mutual in-teractions. We are also careful to ensure that for the three-body system, the formalism predicts that subsystems separated by infinite spatial distances behave independently. We next develop the Faddeev equations, which simplify the solution of multi-channel scattering equations. These are specialised to the three-body system introduced earlier and a series solution of the Faddeev Equations is obtained. A simple mechanical model is introduced to provide a heuristic understanding of this solution. The series solution is also expressed in a diagrammatic form complementary to this mechanical model. A system in which particle production and annihilation are allowed is then introduced by working on an Hilbert Space which is the direct sum of the two- and three-body Hilbert Spaces considered earlier. It is found that in this 2 - ^ 3 system, as the mass operator and the number operators do not commute, it is not possible for a system to simultaneously have a sharply defined mass and number of particles. The Faddeev Equations for this system are then considered, and a series solution of these equations is developed and discussed. It is also shown that the particle production and annihilation potential has a non-trivial effect on pure two-body and three-body scattering. In the last chapter we consider an attempt to derive from a more elementary field ii theory, using the dressing transformation, a form for the potential coupling the two- and three-body sectors of the Hilbert Space in the 2 o 3 system. It is found that this method is inherently ambiguous and is not, therefore, able to provide such information. i i i Table of Contents Abstract ii Table of Contents iv Acknowledgements vi 1 Introduction 1 2 The Poincare Group and its Irreducible Representations 5 3 Two-Particle Systems 20 4 Three-Particle Systems 29 5 The Relativistic Faddeev Equations 36 6 The 2 o 3 Body System 48 7 The Dressing Transformation: A Critique 64 8 Summary and Conclusion 73 References 77 Bibliography 79 A Lorentz Transformations of Spin Operators 81 iv B The Newton-Wigner-Pryce Position Operator 85 C Forms of Relativistic Dynamics, Lorentz Boosts, and Spin 91 D Normalisation Of the State U(\,a)\p\ > 96 E Clebsch-Gordan Coefficients of the Poincare Group 98 F Macroscopic Locality and Packing Operators 100 G Overview of Scattering Theory 104 v Acknowledgements I would first like to thank Dr. Malcolm McMillan for being so patient, interested, and supportive a supervisor. This work could not have been completed without his guidance and encouragement. Next, I would like to thank my friends and family for their their love and support during the last two years. Most of all, I appreciate their preternatural tolerance of my neuroses. Finally, I would like to thank NSERC for its financial support of this work. vi Chapter 1 Introduction Since the development of quantum field theory, most theoretical work in nuclear and particle physics has involved quantum mechanical models wi th an infinite number of degrees of freedom. These models, in which the number of particles comprising the system is a dynamic quantity, have been remarkably successful in describing physical phenomena at a wide range of energy scales. However, quantum field theories are in general extremely complicated and are beset with problems of divergent quantities. These problems can be overcome by using renormalisation theory, but this is again a rather complicated process. A t intermediate energy scales, where one must take into account relativistic kinematics and the possibility of particle creation or annihilation, but where for al l practical purposes the number of particles in the system is bounded above, one can use simpler quantum mechanical models in which the number of degrees of freedom is finite. These models, based on the representation theory of the Poincare Group, allow the number of particles in the system to be a dynamical variable, but keep this number wi thin a finite range. This thesis w i l l discuss such models, which are useful in both phenomenological and more fundamental theories of intermediate energy nuclear and particle physics. A l l relativistic quantum mechanical models are based on considerations of the Poincare Group and its representations. Chapter 2 w i l l introduce this group and discuss the con-struction of its irreducible unitary representations on the Hilbert Space of physical states of a single particle, which wi l l be shown to be characterised by the mass and spin of the 1 Chapter 1. Introduction 2 system. It will be seen that this construction leads naturally to the introduction of op-erators on this space which correspond to the familiar momentum, angular momentum, and Hamiltonian operators. In Chapter 3, we will discuss the extention of this construction to a system of two particles, for both the cases in which the particles are free and in which they interact. We will see that the Hilbert Space of this model can be decomposed as a direct integral of irreducible representation spaces of the Poincare Group, each characterized by different values of mass and spin. The consideration of interactions will lead to a consideration of the Bakamjian-Thomas construction, in which interactions are introduced in a Lorentz invariant manner by modifying the mass operator of the system. Chapter 4 describes the construction of the irreducible unitary representations of the Poincare Group on the space of physical states of three particles. This is for the most part similar to the construction of Chapter 3, but is complicated by the fact that one must be careful to ensure that the model has the property that independent subsystems behave independently when separated by an infinite spatial distance. These considerations lead to the introduction of Sokolov's packing operators, which are described in more detail in Appendix F. Most of the material in Chapters 2, 3, and 4 is summarized from a much lengthier discussion in the review article by Keister and Polyzou [Ke 91]. In Chapter 5, we discuss the relativistic Faddeev Equations, which can be used to simplify the determination of scattering T-matrices for systems of several interacting particles. In particular, we consider the case of 3-body scattering and describe in detail the manner in which the multi-channel 3-body T-matrices can be expressed in terms of the T-matrices of the individual 2-body interactions of the system, and we will develop series solutions of the Faddeev Equations and introduce a diagrammatic representation of these solutions. These solutions provide useful insight into the nature of the Faddeev Chapter 1. Introduction 3 Equations, and do not appear to be written down elsewhere in the literature. We consider in Chapter 6 a system in which particle creation and annihilation can occur. In this, the 2 «-» 3 system, particle number is a dynamical variable and the Hilbert Space of physical states of the system will contain both two-body and three-body sectors. We will examine in some detail the Faddeev Equations describing scattering in this model and shall see that the particle production and annihilation potential has a significant effect on pure two-body and three-body scattering. It seems that these results are original and do not appear elsewhere in the literature. Finally, Chapter 7 will critique the Dressing Transformation in simple field theoretical models, as formulated first by Schweber [Sc 64] and later by Hearn and McMillan [He 83]. It was hoped that this construction would allow the derivation of a form of the 2 ^ 3 interaction potential; however, it was found that the method is beset with ambiguities and consequently could not be used toward this end. This thesis also contains several Appendicies. The first of these, Appendix A, de-scribes in some detail the Wigner Rotation of spin operators under Lorentz boosts. Ap-pendix B contains a discussion of the Newton-Wigner-Pryce position operator and of its behaviour under Lorentz boosts. The components of the position operator do not transform as the spatial components of a 4-vector operator, and an interpretation of the transformation properties in the case of a system of spin zero is introduced. These results as well do not seem to appear elsewhere in the literature. Appendix C introduces Dirac's Forms of Dynamics and discusses the ambiguity as-sociated with the lack of a preferred initial-value surface in a relativistic Hamiltonian theory. Appendix D contains a short technical calculation involved in the construction of unitary irreducible representations of the Poincare Group. The Clebsch-Gordan coef-ficients of the Poincare Group are introduced in Appendix E. In Appendix F we discuss macroscopic locality and Sokolov's packing operators. Finally, Appendix G contains a Chapter 1. Introduction 4 short overview of the S-matrix formalism of scattering theory. The formalism developed in this thesis is ideal for a calculations of intermediate-energy scattering phenomena. In particular, the NN-NN7T system is naturally described by this formalism. In this thesis we will employ a system of natural units in which Planck's constant, h, and the speed of light, c, are set equal to one. Chapter 2 The Poincare Group and its Irreducible Representations Arguably, one of the most important fundamental developments of theoretical physics in this century has been the recognition of the importance of symmetry in physical theo-ries, with the consequent incorporation of group theory into the set of tools employed in mathematical physics. A physical system possesses a symmetry if a change in the con-figuration of the system leaves the result of physical experiments invariant. For example, a perfect sphere can be rotated through any angle about any of its diameters without changing the result of any measurement made on it. Similarly, a mathematical model possesses a symmetry if its predictions are invariant under transformations of the model. The set of points described in polar coordinates by: {{r,6,<P)\r = l,ee [0,TT],^G [0,2TT]} (2.1) is invariant under any rotation ip + a, 9 —> 8 + /?, so long as the new angles are mapped back into the appropriate angular ranges. In a quantum mechanical system, the physically observable quantities are the proba-bilities P^f/) of transformations between physical states, represented by vectors in a Hilbert Space ri. If the system is initially in the state \<j> >, then the probability of measuring it to be in the state \ip > is given by the standard formula 1<T/#>1 2 Any map from PL to itself which preserves the probabilities, ie, such that \ip >-)> \tp > 5 Chapter 2. The Poincare Group and its Irreducible Representations 6 and \4> >—> \4> > where for all normalizable vectors, is then a symmetry of the quantum mechanical model. Wigner [Wi 38] showed that all maps of this form may be expressed in the form \^>^U(G)^> (2.4) where U(G) is a unitary or an antiunitary operator. Thus, a l l symmetry transformations on the quantum mechanical model are induced by unitary or antiunitary operators on the Hilbert Space. As the subsequent application of two of these symmetry transformations is another such symmetry transformation, identity and inverse transformations exist, and multiplication of linear operators on PL is associative, these symmetry operators, U(G), possess a group structure. If the symmetry transformations considered are al l pathwise connected to the identity transformation (ie, if the transformations can al l be arrived at from the identity v i a pathwise continuous curves in the group manifold) then the symmetry operators w i l l be unitary. It is such a group that we wi l l consider. It is physically observed that the results of experiments performed on isolated systems, at scales less than astronomical, are unchanged i f the time, place, or orientation of the system is changed or if the experiment is performed in two different frames of reference moving with a constant relative velocity. Physical systems with these symmetries are said to be Poincare Invariant, and a mathematical model of such a system encodes these symmetries v i a the Poincare Group. This is a 10-parameter Lie Group, and much of the work in this thesis wi l l involve constructing its representations on Hilbert Spaces of physical states of few-body systems. Elements of the Poincare Group can be represented as ordered pairs (A, a), where A is a 4 x 4 Lotentz transformation matrix satisfying = (2.5) Chapter 2. The Poincare Group and its Irreducible Representations 7 w here 1 0 0 0 - 1 0 0 0 - 1 0 0 0 V (2.6) ) 0 0 0 -1 a n d a is a 4-vector . T h e group elements have the m u l t i p l i c a t i o n rule: (A2, a2) o (Ax, a i ) = (A 2 A 1 ; A 2ai + a2) (2.7) A n y L o r e n t z t r a n s f o r m a t i o n can be decomposed as the p r o d u c t o f a pure r o t a t i o n (which leaves the t i m e componen t of a l l 4-vectors i nva r i an t ) , charac ter i sed by the t r ip le t of real numbers 0, and a pure boost (which picks the coord ina te a l o n g a g iven spa t i a l d i r ec t i on of any 4-vector to m i x w i t h the t i m e coord ina te , l e av ing the other two spa t i a l coord ina tes unchanged) , un ique ly specified by the three parameters , u , a n d so can be pa ramete r i zed by s ix real numbers : A = A(0 ,u) (2.8) In te rp re t ing the t r ip le t of real numbers 0 as a 3-vector, 0 is pa ra l l e l to the axis of the r o t a t i o n and its m a g n i t u d e is the angle i n rad ians o f the r o t a t i o n abou t th i s axis , u is a spa t i a l 3-vector pa ra l l e l to the d i r ec t ion of the boos t and its m a g n i t u d e is the rap id i ty , u = t a n h w , where v is the re la t ive speed of the two frames re la ted by the boost . T h e set o f a l l L o r e n t z t r ans fo rmat ions is i t se l f a group , a n d forms a subgroup of the P o i n c a r e G r o u p . A s w i t h any L i e G r o u p , the un i t a ry operators o n TL represent ing the P o i n c a r e G r o u p m a y be expressed as the exponen t i a t i on o f the p roduc t of H e r m i t i a n opera tors and g roup parameters : C/(A(0,u),a) = e - ^ " < v e - i ( j . 0 + K . u ) (2.9) Chapter 2. The Poincare Group and its Irreducible Representations 8 The set of Hermitian operators (P^,J3,K3) are referred to as the generators of the Poincare Group. From any representation of the Poincare Group, these can be obained by the formal differentiations: J3 = i K3 d 8 _d_ dui U{A,a) U{A,a) U(A,a) af=Q3 =ui=0 (2.10) (2.11) (2.12) Of these ten generators, seven have familiar physical interpretations. P° = H is the Hamiltonian of the system, P3 is the j - th component of the total 3-momentum of the system, and J3 is the j-th component of the total angular momentum of the system. The Lorentz boosters K3 seem to have no simple physical interpretation. The ten generators of the Poincare Group satisfy a set of commutation relations among themselves, which define the Lie Algebra of the group. These can be determined by examining the results of successive applications of various transformations and their inverses in the neighbourhood of the identity (Keister and Polyzou [Ke 91] or Kalyniak [Ka 78]). The Lie Algebra of the Poincare Group, the Poincare Algebra, is given by: [Jj,Jk] = iejklJl [J\Kk] = -itjklKl [Jj,Pk] = kjklPl [P' \ P'y] = 0 [Jj,H]=0 [K3, H] = -iP3 (2.13) (2-14) (2.15) (2.16) (2.17) (2.18) Chapter 2. The Poincare Group and its Irreducible Representations 9 [Kj,Pk] =-iSjkH (2.19) [Kj,Kk] = -iejklJl (2.20) Equations (2.13), (2.14), and (2.15) ensure that the angular momentum, the Lorentz booster, and the 3-momentum, respectively, transform like 3-vectors under rotations; equations (2.16) and (2.17) imply conservation of system energy, total 3-momentum, and total angular momentum, and equations (2.18) and (2.19) lead to the familiar mixing of energy and 3-momentum under a Lorentz boost. Equation (2.20) has as a consequence the Wigner rotation of spin under Lorentz boosts. This will be discussed in some detail later. It is appropriate at this time to discuss what is meant when an operator is said to transform as a 4-vector operator, or in general as a tensor operator. If (A, a) is a Poincare transformation, with representation U(A,a) on ri , then A^ transforms as a 4-vector operator if U{A, a)A»U\A, a) = ( A - 1 ) ^ " (2.21) In general, an operator B^-a is a tensor operator if U(A, a)B^-aW{A, a) = ( A - 1 ) ^ " 1 ^ • • • {A~lYTBap-r (2.22) In the same sense, an operator Cj is said to transform as a 3-vector under rotations if R(e)cjR\e) = D[R(e)]rjcj' (2.23) where R(Q) is the representation of the rotation on ri and D[R(Q)] is its 50(3) repre-sentation. It is a well-known result that if one defines the tensor operator such that M0j = Kj (2.24) Mjk = tjklJl (2.25) Chapter 2. The Poincare Group and its Irreducible Representations 10 M^u = -Mvtx (2.26) then the Poincare Algebra can be expressed more compactly as: [P' \ P'y] = 0 (2.27) [P' \ MXa] i(Pxg»a - Pag»x) (2.28) i(M (2.29) It is not immediately obvious that the 4 x 4 object M ' " y should in fact transform as a Lorentz 2-tensor operator, but this can be shown to be the case. In discussions of the representations of a Lie Group, it is useful to determine the Casimir operators of the group; these are operators on Ti, constructed from the group generators, which commute with all of the group generators. In the case of the Poincare Group, there are two of these Casimir operators, both of which have important physical interpretations. The first is given by: M is defined to be the mass of the system. With some work, it can be shown that M commutes with all of P / l and M / t / y . It is clearly an Hermitian operator, and thus a potential physical observable; indeed, its eigenvalues correspond to the familiar notion of the mass of a physical system. We shall only consider physical systems for which the eigenvalues m of M are non-negative: m > 0. The boosts that we have so far considered take real numbers as arguments. One can also define boost operators which are functions of operators. It is natural to define the operator LC(Q)^ which takes as its argument the 4-velocity operator of the system: M 2 = (2.30) (2.31) Chapter 2. The Poincare Group and its Irreducible Representations 11 by its action on the arbitrary 4-vector operator A' ' : / y/TTWA° + Q A \ = (2.32) ^ A + + (1 + V / T T O ? ) - 1 Q ( Q • A) ) LC(Q) has the properties that it maps the 4-vector operator ( M , 0,0, 0) to (P°, P), where P° = VM2 + P 2 , and that LC{Q = (1,0,0,0))'; = g$. The inverse boost L~\Q) is obtained from (2.32) by replacing Q by —Q. While this seems the most natural definition of a boost operator which is a function of other operators on Pi , other definitions can be useful, and we refer the interested reader to Appendix C. The second Casimir operator for the Poincare Group is W ' W ^ , where is the pseudo-4-vector operator: W = -^a^PaM^ (2.33) tmM j s t n e totally antisymmetric tensor on Minkowski space, defined so that e 0 1 2 3 = 1. W ' is called the Pauli-Lubanski J^-Vector. One can show that the components of this operator are Hermitian and given by: W ° = P - J ; W = H J - P x K (2.34) and that and P^ are orthogonal: = 0 (2.35) The components of the Pauli-Lubanski 4-Vector satisfy the following commutation rela-tions among each other: [W, W] = ie^WpP* (2.36) and with the generators of the Poincare Group: [ P ^ W ^ = o (2.37) Chapter 2. The Poincare Group and its Irreducible Representations 12 [J\W°] = 0 (2.38) [P,Wk] = ujklW l (2.39) [Kj,W°] = -iWj (2.40) [Kj,Wk] = -iSjkW° (2.41) Equations (2.38) - (2.41) ensure that transforms as a 4-vector operator under Lorentz transformations. The Pauli-Lubanski 4-Vector has a straightforward physical interpretation. Consider those eigenstates |P = 0 > of P in % corresponding to the eigenvalue 0, ie, the rest states of the system. As K x P = —P x K and J • P = P • J hold as operator equations, we conclude that: W°|P = 0 > = 0 (2.42) W{\¥ = 0 >= MT\P = 0 > (2.43) The angular momentum of a system in its rest state is defined to be the spin of the system (up to a rotation, as is discussed in Appendix C), so we see that, acting on rest states, M _ 1 W corresponds to the spin of the system. This suggests that we define the spin operator S through the equation: T? = = jjL^QKW" (2.44) Note that because the argument of the boost is a 4-vector operator, the 4-component object S' 1 does not transform as a 4-vector. Expressed in terms of the generators of the Poincare Group, S is given by W W°P 1 P(P • S = M - M ( M + H) = M ( H 3 - P X K ) - M W T H ) ( 2 ' 4 5 ) Chapter 2. The Poincare Group and its Irreducible Representations 13 From definition (2.44), it is clear that S-S = ~ n (2.46) and from the commutation relation (2.36) one can show that M2[Sj,Sk] = iejklM2Sl (2.47) so, for a system in which the mass eigenvalues are strictly positive, we obtain the familiar commutation relations: [Sj,Sk] = iejklSl (2.48) which are formally identical to the commutation relation between the components of spin in nonrelativistic quantum mechanics (NRQM). The spectra of S 2 and the components of S are thus well-known. As mentioned above, £ is not a 4-vector operator. Using the fact that U\^a)L-c\QYvU{K,a) = L~l(AQY (2.49) it is straightforward to show that L7t(A, a)^U(A, a) = RC{A, Q ^ E " (2.50) where RC(A, QYV = L~l {AQYpA»aL{Q)l (2.51) The 4 x 4 matrix of operators RC(A, Q)% is in fact a rotation, as can be seen by noting that it leaves the 4-vector operator (M, 0) invariant. Thus, the spin operator transforms as U(A,a)SjUj{A,a) =R^(A,Q){Sk (2.52) This rotation of the spin under Poincare transformations is called the Wigner Rotation and is discussed in more detail in Appendix A. Chapter 2. The Poincare Group and its Irreducible Representations 14 In the discussion so far we have encountered a number of physical observables familiar from N R Q M : energy, momentum, angular momentum, and spin. The question naturally arises of whether this formalism contains observables that correspond to the position operators in N R Q M . The answer to this is a reserved yes. One can construct (and it can be shown that this construction is unique (Jordan [Jo 80])) an Hermitian 3-vector operator with components X1 canonically conjugate to the 3-momentum operator, that is, satisfying and transforming as a 3-vector operator under parity and time-reversal. This is referred to as the Newton-Wigner-Price Position Operator. It is given in terms of the Poincare Generators by: However, there are some problems with interpreting this object as a physical position operator. X as defined above is not the spatial part of a 4-vector operator, as we have not defined an operator to correspond to a time observable. In fact, the structure of quantum mechanics as a dynamical theory seems to preclude the definition of such an object; while the spatial position coordinates of a system are dynamical variables and represented by operators in the theory, time enters as a c-number parameter. Even in other forms of dynamics (see Appendix C) some combination of 4-position coordinates will, by construction, be a c-number. Some authors (Johnson [Jo 69], Fleming [Fl 64]) have tried to overcome these difficulties and construct a theory in which the spacetime position of the centre of mass of the system is given by a Lorentz covariant 4-vector operator, but these constructions are clumsy and difficult to work with. We will find it useful to use the Newton-Wigner-Pryce position operator frequently in the construction that follows, especially in writing down formally familiar expressions for the Poincare [X\ pi] = i5. (2.53) (2.54) Chapter 2. The Poincare Group and its Irreducible Representations 15 generators, but care must be taken in naively interpreting it in terms of the N R Q M position operator. For further discussion about this operator and its transformation properties, the interested reader is referred to Appendix B . The set {H, J J , K^} of Poincare generators can be expressed in terms of the set {Pi,Xi,S j,M} as follows: H = VP2 + M 2 (2.55) J = X x P + S (2.56) Thus, a knowledge of the set {P' ' , J j, K j } is equivalent to one of the set {P j, X j, S j, M}. The latter corresponds more closely to the set of operators familiar from N R Q M , and we will base our formalism in terms of this collection, not the first. We can now proceed to construct irreducible unitary representations of the Poincare Group on the Hilbert Space of physical states of the system. This is equivalent to solving the dynamics of the system: if the state of a system on a given 3-dimensional hypersurface in Minkowski space is known, then one has a well-posed initial value problem and one can use Poincare transformations to determine the state of the system at all other points in spacetime. Thus, if the representation of the Poincare Group on the Hilbert Space of physical states is known, and if the state of the system on some Minkowski hypersurface is also known, then one can determine the evolution of the system and the problem of determining its dynamics has been solved. Because the Poincare Group is non-compact, it can have no unitary representations on finite-dimensional vector spaces. Wigner's Theorem thus demands that the space of physical states be infinite dimensional. The basis we shall use on % is the set of simultaneous eigenstates of the complete set of commuting operators {P, 5 3 , M, S2} (this assumes that the system contains no internal degrees of freedom and thus contains just Chapter 2. The Poincare Group and its Irreducible Representations 16 a single particle). Now, each Casimir operator of a group will only have a single eigenvalue on any irreducible unitary representation of the group. This follows from Schur's First Lemma, the essential content of which is that if an operator, A, commutes with all elements in any irreducible representation of a group, then it must be of the form A = /.ii, where I is the identity operator and / i is a c-number. Irreducible representations of the Poincare Group can thus be uniquely labelled by a value of the mass and a value of the spin. The set {|pA >} of simultaneous eigenstates of our complete set of commuting oper-ators is defined such that P«|pA >=p*\p\ > (2.58) S 3 | pA >= A|pA > (2.59) M | p A >= m|pA > (2.60) S 2 | pA >= s(s + l ) |pA > (2.61) where for the basis states we shall use a notation in which the eigenvalues of M and S 2 are suppressed (as these are the same for all vectors in the space on which the irreducible representation is defined). As the momentum of the centre of mass of an isolated system can take on any value in K 3 , we know that these eigenvalues satisfy: p{ e ( - 0 0 , 0 0 ) (2.62) m > 0 (2.63) 5 = 0 , i 1 , | . . . (2.64) A € (—5, —s + 1,..., s — 1, 5) (2.65) These basis vectors are defined to be an orthonormal set, ie, to satisfy the normalization < p ' A ' | p A > = < J ( p - p % v (2.66) Chapter 2. The Poincare Group and its Irreducible Representations 17 and completeness condition 1 = E / ^ P I P A > < P A | (2.67) A=—s To construct irreducible representations of the Poincare Group, we will take the fol-lowing approach, known as the induced representation method (Tung [Tu 85], Keister and Polyzou [Ke 91]). The mapping between physical states of well-defined momentum and vectors in TL is known; this method constructs the representation of the group by defining how vectors in TL are connected by the representatives of the group elements based on the knowledge of the manner in which the group elements connect physical states. Consider a system of mass m and spin s. Then the 4-vector pf = ( m , 0), called the standard vector, is the momentum 4-vector of the system in a frame of reference in which it is at rest. The subspace in TL representing such a state is spanned by the vectors {|0A >}. Note that because the 3-vector 0 is mapped into itself by any rotation, this subspace is invariant under the group of rotations: U{R)\0X >= D^(R)xy\0\' > (2.68) where DXX,(R) is the 2s + 1 dimensional representative of the rotation, R. The rotation group is thus the little group of p[\ For this construction, two special cases of the Wigner Rotation are of particular importance. If A = R is a pure rotation and we are considering the action of the operators on the subspace spanned by the 4-velocity eigenvalues go = (1,0)) RC{R, g0) = L;1(Rq0)RLc{q0) = R (2.69) as Rqo = g 0 and £ c (g 0 ) i ' = 9v, s o the Wigner rotation is just the original rotation itself. If A = Lc(q), then considering the action on the same subspace of PL as above Rc(Lc(q),q0) = L-\q)Lc{q)Lc(q0) = 1 (2.70) Chapter 2. The Poincare Group and its Irreducible Representations 18 as q = Ag 0 , and Rc is simply the identity matrix. Let {|pA >} denote the subspace in ri representing the states of momentum p of the system. If q is the 4-velocity of the system corresponding to the 3-momentum p, then £ / (L c (g ) ) | 0A>=7V(p) |pA> (2.71) as the Wigner rotation is the identity in this case. Note that in the above we have taken Lc to be a function of the c-number, not the operator, 4-velocity. The c-number N(p) is needed to ensure that the transformation is unitary; it is determined by the normalization condition (2.66) and is calculated in Appendix D. Because the momentum and spin operators commute, spacetime translations do not couple eigenvectors of S3 corresponding to different eigenvalues, and so, defining T(a) = U(I,a) T(a)|PA >= e- l P" a" |pA >= e'ir^ |pA > (2.72) by definition of the states |pA >. This gives us the action of an arbitrary spacetime translation on an arbitrary basis vector in H . Using equation (2.51) and the fact that any Poincare transformation U(A, a) can be decomposed in the following manner (Tung [Tu. 85]): U(A,a)=T(a)U(A,0), (2.73) we have that U(\,a) = T(a)U[Lc(Aq)]U[Rc(A,q)}U[L;l(q)} (2.74) Consider the action of U(A,a) on the basis vector |pA >: U(A,a)\pX> = T(a)U[Lc(Aq)}U[Rc(A,q)]U[L;1(q)]\pX> (2.75) = N-1(p)T(a)U[Lc(Aq)}U[Rc(A,q)]\OX> (2.76) . = N-l(p)T(a)U[Lc(Aq)]\OX' > D^[Rc(A,q))xx, (2.77) Chapter 2. The Poincare Group and its Irreducible Representations 19 = ^^T(a)\pAA'> D^[Rc(A,q)}xx (2.78) = ^ p ^ ^ ^ ^ ) [ J R c ( A , g ) ] A A ' | P A A ' > (2.79) N{p) where p = (u;(p), p) and PA — Ap. It is shown in Appendix D that the normalisation N(p) is given by: so we have shown that the action of U(A, a) on |pA > is given by (7(A,a)|pA >= e~<a" X | 7 T ? ^ ^ W [ ^ ( A , 5 ) ] A A ' | P A A ' > (2.81) By construction, the basis chosen for Pi is irreducible under Poincare Transformations, so we have thus completed the construction of a unitary irreducible representation of the Poincare group on Pi . As a final point, note that on the basis chosen, the position operator will be repre-sented by a differentiation operator, ie, for any vector \ip >£pi : < p\\X*\ib >= i— < pA|V > (2.82) In the next chapter we will consider the problem of constructing representations of the Poincare group corresponding physically to a state with two particles. Chapter 3 Two-Particle Systems The previous chapter defined the Poincare Group, developed a number of results con-cerning its generators, and showed how to construct irreducible unitary representations of this group on the Hilbert Space of physical states. However, as we assumed that the system did not contain any internal degrees of freedom, the Hilbert Space considered in Chapter 2 was that of the physical states of a single particle. In this chapter we will extend the construction and develop irreducible unitary representations of the Poincare Group on the Hilbert Space of physical states of two particles. With the introduction of a second particle to our physical system, we can begin to consider the problem of how to include interactions in our formalism; this will lead to the introduction of the Bakamjian- Thomas construction. Let K ( m 1 , . S i ) be the Hilbert Space of physical states of a particle of mass m i and spin Si and 7i(m2, s2) be that of a particle of mass m 2 and spin s2- Then the 2-body Hilbert Space is given by the tensor product H(2)=H(m1,si)®H(m2,S2) (3.1) and is spanned by the basis | p i A i ; p 2 A 2 >= |p iA i ><8>|p2A2 > (3.2) If Ui(A,a) is a unitary irreducible representation of the Poincare Group on 7i(mi,Si), then a unitary representation of the Poincare Group on TC(2) is U(A,a) = U1(A,a)® U2(A,a) (3.3) 20 Chapter 3. Two-Particle Systems 21 That the two particles described are free can be seen from the fact that C/(A,a)|p1Ai;p2A2 >= t7i(A, a)|piAi > ®C/2(A, a)|p2A2 > (3.4) so no non-trivial dynamics have been introduced; the trajectories of the particles are independent. Because the two single particle spaces are independent, the Poincare Gen-erators on 7i(2) are given by While this representation of the Poincare Group is unitary, it is not irreducible. The construction of irreducible unitary representations is facilitated by a change of variables which leads us to a consideration of the Clebsch-Gordan coefficients of the Poincare Group. Instead of describing the system by the individual momentum of each particle, we will analyse the problem in terms of total and internal momenta. Now, the momentum operator on PC(2) is the sum of the momentum operators on %(mi,.si) and 7^(m2,.s2), and thus corresponds to the total momentum operator of the system, satisfying The total momentum of a system of two massive particles is spacelike, so there will exist a frame of reference in which the spatial part of the total momentum vanishes. In this frame, the 3-momenta of the individual particles will be equal in magnitude but opposite in direction; we will represent this physical state by the following subspace of vectors in P" = Pf ® J 2 + I, <g> P2" (3.5) AP" = Mf" <8> 72 + h ® M%v (3-6) ^ 1 P i A i ; p 2 A 2 >= ( K + P 2 ) | P i A i ; p 2 A 2 > (3.7) U{2): ^ c ( < 5 ) " 1 ] | P i A i ; p 2 A 2 > (3.8) (3-9) Chapter 3. Two-Particle Systems 22 The state vector |kAi; — kA2 > satisfies P"|kAi; -kA 2 >= #|kAi; -kA 2 > (3.10) where pr — (m 0 , 0) = (w m ] (k) + <x>m2(k), 0) is the total 4-momentum of the system in its rest frame. Note that mo, which is interpreted as the total mass of the system, is not fixed, but is functionally dependent on the internal momentum. In contrast to the one-particle case, the subspace of rest states is not closed under rotations: U(R)\kXi; -kA 2 >= D^[Rc(R,k1)]XlKD^Rc(R, fc2)]A2v2\RkX\- -i?kA 2 > (3.11) where k\ = (wm,(k),k) and k2 = (wm2(k), —k). Now, using the definition of the boost matrix Lc(q)%, with q = ^ L , it can be shown that L;1(Rq)RLc(q) = R (3.12) so [7(i?)|kAi; -kA 2 >= DM[R]MKD^[R]X2^\RkX[; -RkX'2 > (3.13) We now define the vector OA > as: \[la]ks;0X >= J dkr^(k)|kAi; -kA 2 >< A 1 A 2 (5is 2 )am c r >< mamt(al)sX > (3.14) where Y^(k) is the spherical harmonic with angles k, < AiA 2 ( .SiS 2 )am < 7 > and <mami(al)sX> are SO(3) Clebsch-Gordan coefficients, and k is the magnitude of the 3-vector k. The parameters I and o label the various degenerate states all of which trans-form with the same value of 5 and are not summed over in the above expression. Using the properties of the rotation matrices and the Clebsch-Gordan coefficients of SO(3), one can show that the subspace of states {|[/CT]£S; OA >} is irreducible under rotations: U{R)\[lo}ks;0X >= D{s\R)xx,\[lo]ks;0X' > (3.15) Chapter 3. Two-Particle Systems 23 and we have thus constructed a subspace of TC(2) of eigenstates of P ' 4 corresponding to the eigenvalue (mo,0) = (w m i (k) + w m 2 (k) ,0) which transforms irreducibly under rotations. As with the construction of the representation of the Poincare Group on the one-particle Hilbert Space, we define the subspace of total momentum eigenstates with eigen-value (w f n o(p),p) by \[lo]ks;pX >= J^—U(Lc{q))\[la]k.s-,OX > (3.16) V ^ mcAPj where = This definition (3.16) corresponds to the normalization: < [l'a']k's'- p'X'\[la\kS] pA >= <Jn W A V W ( p - p')^S(k - k') (3.17) and the resolution of unity I = [ d3P [°° k2dk\[lo]ks] pA >< [lo]ks; pX\ (3.18) We are thus now at the point where we can write down the irreducible unitary rep-resentation of the Poincare Group on %(2). A calculation identical to that in Chapter 2 shows that: U{A,a)\[la}ks;pX >= e'1^ \ ^ ^ / j S [ P c ( A , g ) ] | H ^ ; P A A ' > (3.19) The Clebsch-Gordan coefficients of the Poincare Group can now be calculated; the inter-ested reader is referred to Appendix E. This equation shows that the vector |[Z<r]/cs; pA > transforms under [/(A, a) in the same manner as would a single particle of mass "oio = ^mi(k) + w m 2 (k) and of spin .s. The states |[/cr]/cs;pA > are labelled by a set of six parameters; we have seen that |[/a]/:s; pA > is an eigenvector of the 3-momentum operator and that the parameter p labels the corresponding eigenvalue. Similarly, is is clear from (3.19) that |[Z«r]A;s; pA > satisfies S3|[/CT]A:S; pA >= X\[la}ks; pX > (3.20) Chapter 3. Two-Particle Systems 24 and S2\{lo]ks; pA >= s(s + l)\[lo}ks; pA > (3.21) where S is the total spin angular momentum operator on K(2). Similarly, the mass operator given by The Pauli-Lubanski 4-vector and Newton-Wigner-Pryce position operators of the system are defined in terms of the Poincare Generators on 7i(2) in exactly the same way as in Chapter 2. As well, on the basis used, the position operator acts in the familiar fashion as a derivative operator. The total momentum operator of the system is of great importance, as it is a Poincare Generator; however, also of importance is the internal momentum operator, k, given by: (3.22) satisfies M\[lo]ks; pA >= m0\[lo]ks] pA >= (u;nil(k) + u}m2(k))\[lo]ks] pA > (3.23) (Q)Pi (3.24) whose spatial components are given explicitly by (3.25) Defining the operator \K\ = VK2 we have that (3.26) and one can show, after some algebra, that 1 [M04 - 2M 2(m 2 + m22) + {m\ - ml)2} (3.27) Chapter 3. Two-Particle Systems 25 As with the operator £ ' ' defined in Chapter 1, it is clear from the definition of that it does not transform as a 4-vector operator but transforms instead with a Wigner rotation: U](A, a)kU(A, a) = RC(A, Q)k (3.28) Finally, one can write down operators 1 and o on Pi (2) such that |[Zcr]A;s;pA > is an eigenvector of l 2 and o2 with corresponding eigenvalues 1(1 + 1) and a(o + l) respectively. From the discussion of the Clebsch-Gordan coefficients of the Poincare Group in Appendix E, it is clear that the rotational Clebsch-Gordan coefficient coupling the I and s terms implies: S = l + a (3.29) and that coupling si and s2 that o = RC[L;1(Q),Q1]S1 + RC[L;1(Q),Q2]S2 (3.30) with Qi = jf. These equations define the operators 1 and o in terms of known operators. The problem of including interactions between the particles in this formalism is not as easy as the corresponding problem in N R Q M . The general problem of incorporating interactions into a relativistic Hamiltonian system was considered by Dirac [Di 49] and led to his development of the different forms of dynamics, as discussed in Appendix C. In this paper, Dirac made the point that it is not a trivial construction to modify the Poincare Generators of a system of non-interacting particles to include interactions and still perserve the Poincare Algebra. To see this, consider the commutation relation [K3, Pk] = —iSjkH. If to the non-interacting Hamiltonian is added an interaction poten-tial, V, then either the momentum or the boosters must also be modified to maintain the commutation relation. Thus, the construction of an interacting relativistic Hamilto-nian system involves the modification of a number of the Poincare Generators to include Chapter 3. Two-Particle Systems 26 interaction terms; Dirac used the name Hamiltonians to denote these. The other, un-modified, generators are said to be kinematic. In the body of this thesis, we will confine ourselves to the instant form of dynamics, in which the Hamiltonian and boost operators of the system are modified to include interactions, while the 3-momentum and angular momentum remain kinematic; here the kinematic subgroup is the set of transformations which leave the hyperplane x° = c invariant. This form of dynamics corresponds most closely to the familiar non-relativistic construction. Two other forms of dynamics, the point form and the front form, are discussed in Appendix C. The construction of a relativistic interacting quantum mechanical model can be car-ried out in a straightforward manner. It was shown by Bakamjian and Thomas [Ba 53] that this can be done through the modification of the mass operator of the system. Con-sider an irreducible unitary representation of the Poincare Group on the Hilbert Space of two non-interacting particles, as has been developed. The basis vectors are |[Za]/c.s; pA > and they transform as where the subscript 0 is now used to denote those operators which do not include inter-actions. As the representation of the Poincare Group is known, so too are the Poincare Generators and one can construct the set of operators { P 0 , X 0 , S 0 , M0}. Clearly, as M 0 commutes with all Poincare Generators, the Poincare Algebra is left unchanged if an interaction term added to M 0 commutes with all of the operators { P 0 , X 0 , S 0 } . If we then define an interaction U such that where M has a non-negative spectrum, and U is Hermitian and satisfies the commutation U0(A,a)\[lo]ks;p\ >= e~ip>» D[s>[Rc(A,Q0)}\[la}ks;pAX' > (3.31) M = M0 + U (3.32) relations: [Xo,U] = 0 ; [P0,U] =0 ; [S0,U] = 0 (3.33) Chapter 3. Two-Particle Systems 27 then we will have included interactions in our theory without violating the Poincare Algebra or the non-negativity of the mass spectrum. The commutation relations (3.33) are equivalent to the statement that the matrix elements of U must satisfy: < [l'o']k's';p'X'\U\[lo}ks;pX >= SslsSyx5{p' - p) < k'l'a'\\Us\\klo > (3.34) where the reduced matrix element is independent of A or p. As has been discussed, those sets of states which transform irreducibly under Poincare Transformations are sets of eigenstates of the mass operator corresponding to the same single eigenvalue. This is true in systems with interactions as well as in those without. Thus, to solve the dynamics of the system by constructing irreducible unitary represen-tations of the Poincare Group, one must determine the set of simultaneous eigenstates of the complete set of commuting operators: {M, P, S3, S2}. These eigenstates will be denoted by \ms; pX > and will satisfy the eigenvalue equation for M: M\ms]pX >= (y/m-l + K2 + \Jm\ + K2 + U)\ms;pX >= m\ms;pX > (3.35) where K is given by equation (3.25). Note that the spectrum of M may have both continuous and discrete components. Each subspace of H{2) spanned by all eigenvectors corresponding to a single mass eigenvalue m will be irreducible under the action of the Poincare Group: L7(A,a)|ms;pA >= e~ip>a \ f ^^I>? i [^c (A,p /m)] |m«;pAA' > (3.36) um\P) where the 4-momentum p has components (wm(p), p). This equation implies the normal-isation: < m's'] p'A'|ms; pA >= S[m'; m]6x'\Ss'sS(p' - p) (3.37) where 8[m'; m] is equal to S m i m in the discrete part of the mass spectrum and to S(m' — m) in the continuous part. Chapter 3. Two-Particle Systems 28 Note finally that the Poincare Generators {H, K, J} can be expressed as follows in terms of the set {M, P, K, S}: H = VM2 + P 2 (3.38) K = --{H,X\- fTX^ (3.39) 2X ' ; H + M v 1 J = X x P + S (3.40) As we are working in the instant form, note that only H and K contain mass terms and thus are modified by the interaction. This completes the construction of irreducible unitary representations of the Poincare Group on an Hilbert Space of the physical states of two non-interacting particles and on an Hilbert Space of the physical states of two interacting particles. The next chapter will extend this construction to Hilbert Spaces describing systems of three particles. Chapter 4 Three-Particle Systems In this chapter we will consider the Hilbert Space H(3) of state vectors of a system of three particles and the irreducible representations of the Poincare Group on this space. New difficulties are introduced in the construction of three-body representations from those of two bodies; in particular, special care must be taken to ensure that the system continues to demonstrate macroscopic locality (also known as cluster separability). A model is said to satisfy macroscopic locality if in the limit that disjoint sets of particles are separated by an infinite spatial distance, they behave independently. As will be shown, this physically necessary requirement is not encoded trivially into the theory, and must be put in later by hand using the method of packing operators introduced by Sokolov [So 78]. To construct representations we follow the methods described by Keister and Polyzou [Ke 91]. We will be considering the Hilbert Space ri(3) = n{mu si) <g> H(m2,s2) ® ri{m3, s 3) (4.1) which is spanned by the vectors I P 1 A 1 P 2 A 2 P 3 A 3 >= |p iAi > <g>|p2A2 > <8>|p3A3 > (4.2) Equivalently, we can consider H(3) to be given by li(2) ®rl{\) and consider the basis | [12,3]p 1 2 A 1 2 ;p 3 A3 >Tp= | [12]fc 1 2Si2 ;p 1 2 Ai2 > ®|p 3 A 3 > (4.3) In the above equation, we are using the new notation |[12]&4 2 si 2 ; p i 2 A i 2 > to label the vectors (3.16) that span irreducible subspaces of %{2), as introduced in Chapter 3. It is 29 Chapter 4. Three-Particle Systems 30 useful to start off with this basis if the system considered is one in which particles 1 and 2 mutually interact while 3 is free, a situation we will soon consider. Let us consider for the moment a system in which there are no interactions. The procedure for constructing irreducible unitary representations of the Poincare Group on the Hilbert Space of physical states of this system is essentially the same as that employed in the last chapter. Define the operators P" and Wu by P = Pl2 + P3 = f2Pi (4-4) i=l W = ( w m s ( W ) , W ) = L:\Q)(u,m3(p3),-p3) (4.5) where we define the total mass of the system as M 0 = \JPVPV so that Q = -^k The 4-component operator W'1 defined above is not related to the Pauli-Lubanski 4-vector. The spatial part of W is given explicity, then, by Note that W , M 0 , and H are functions of the operator K 1 2 , and that M 0 is given by: M 0 = JW* + Mf2fi + y / w 2 + m§ (4.7) where M 1 2 ) 0 = + + ^ 1 2 + ™! (4.8) is the mass operator on the (12)-system. For the non-interacting system, we interpret the 3-vector operators P and W as the total 3-momentum operator for the system and as the internal momentum of the (12)-system relative to particle 3, respectively. We can use this change of variables to define the basis states i I [12, 3 ]p 1 2 A : 2 ; p 3 A 3 >TP (4.9) I[12, 3]pwfci 2Ai 2; s 3 A 3 >BT= d ( P l 2 , P 3 ) | d(p,w) Chapter 4. Three-Particle Systems 31 where the square root of the Jacobian determinant of the coordinate transformation is included in the definition to ensure that both basis sets have delta-function normalisation. As in Chapter 3, where we built the irreducible basis set (3.16) we construct linear combinations of the basis vectors (4.9) irreducible under the action of Poincare Trans-formations: | [ 1 2 , % s ; p A > B r = £ / dwU{Lc(q))\[\2, 3]0wfc 1 2A : 2; s 3 A 3 >BT ^ ( W ) x < X[2^(si2S3)o-ma >< mLma(La)sX > (4-10) Each subspace of Tt(3) spanned by vectors of this form, with given values of fc12, |w|, and j will be irreducible under Poincare transformations, transforming with mass wi 0(|w|, k12) = ^ w 2 + mj2>0(ku) + v ^ 2 + mt (4-n) and spin s. We will now consider the situation in which particles 1 and 2 interact via a two-body interaction. Using the Bakamjian-Thomas construction as in Chapter 2, we modify the mass operator on M12 = M 1 2 ) 0 + V12 (4.12) where on vectors in ri{2) V\2 has matrix elements <[12] ; Pi 2 A' 1 2 |V 1 2 | [12]fc 1 2 s 1 2 ; P i 2 A 1 2 > = fya,iafyaAia*(p'i2 - P12) < [12]*i2||V£»||[12]A:1 2 > (4.13) This operator can be extended to one on TL(3) in one of two inequivalent ways: T P < [ ^ ^ I p ^ A ' ^ S p ^ l ^ l i ^ ^ l p ^ A ^ S P g A , > T P = Ss>12srA\2\iA>3x3S{p'12 ~ P12WP3 - Ps) < [12]fei2||^212||[12]fe12 > (4.14) Chapter 4. Three-Particle Systems 32 and BT < [l^p'w'feiaAiajs^A'glV^IIl^pw^aAiajssAs >BT = ^ , A ' 1 2 A 1 2 ^ ( W ' - w)<J(p' - p) < [12]A:i2||Vi'»||[12]A:12 > (4.15) The above matrix elements show that VBT commutes with P , S, and X , and so is a Bakamjian-Thomas interaction on PC(3). Now, the system binding energies and S-matrix are functions only of the reduced matrix elements < [12]/c 2^||V^211[12]A:i2 >• The potentials V£P and V ^ T both then yield the same scattering matrices and binding energies. They are not, however, identical: V^p satisfies the commutation relations [V™P,Pl2] = 0 ; [V#,p 3 ] = 0 (4.16) while Vg?r satisfies [V#,P] = 0 ; [VpT,W] = 0 (4.17) Therefore, as W = W ( P 1 2 , P 3 , K 1 2 ) and [V^p, « 1 2 ] ^ 0, we can conclude that in general • [ V # , , W 1 ^ 0 (4.18) so Vrfp and must be different operators on 7/(3). The Bakamjian-Thomas method of including interactions in the Poincare Algebra involves building the Poincare Generators from the set { X , P , S , M } , each of which are independent of the others but may depend on the internal variables. In the T P and B T representations, we have the mass operators MBT = / W 2 + M f 2 ) B r + y/w2 + Ml (4.19) MTP = \jl^P'ti + M\2~P + yTV + M 2 ] 2 - P 2 (4.20) Chapter 4. Three-Particle Systems 33 where M 1 2 , B r = M 1 2 ) 0 + VBT (4.21) and M12,TP = Mi2, 0 + VTP (4.22) This shows that we cannot write MTP as a function of internal variables only, so MBT is the correct interacting mass operator for use in the the Bakamjian-Thomas construction. However, as is discussed in Appendix F, while MTP satisfies the requirement of cluster-separability, this is not the case with MBT. To remedy this problem and construct a representation of the Poincare Group satisfying macroscopic locality for two interacting particles with a free spectator, we use the method of Sokolov and introduce packing oper-ators. These are unitary operators which map between the T P and B T representations. Denoting the packing operator by A, we have: [ / T P (A , a) = AUBT{A, a)A] (4.23) With a knowledge of the packing operators, then, we can use the Bakamjian-Thomas approach to construct the representation of the Poincare Group on the interacting Hilbert Space and then transform this into the T P representation which satisfies macroscopic locality. The packing operators are discussed in greater detail in Appendix F, and for the rest of the chapter we will assume they are known for the model being considered. We have so far included interactions between only two of the three, particles in the system. In general, of course, each pair of particles can experience mutual interactions, and we must also allow for the existence of 3-body interactions. In this case, the mass operator has the general form (in the T P representation) MTP = MTp + Mjp + MTp-2M0 + Vl2, (4.24) = M0 + V123 + (M?p - M 0 ) + ( M 2 T P - M 0 ) + ( M 3 T P - M 0 ) (4.25) Chapter 4. Three-Particle Systems 34 where MTP is the mass operator of the system if particle 1 is separated by an infinite spatial distance from particles 2 and 3, M{p — M2Z,TP, and similarly for Mlp and MTP', as described earlier in the chapter. V\2Z 1 S some potential which vanishes when any of the three bodies in the system is removed to in infinite distance from the other two; it is a pure 3-body interaction. From here, the construction of an irreducible unitary representation of the Poincare Group is straightforward. We have already considered irreducible representations of the Poincare Group on the space of physical states of 3 non-interacting particles. For each pair (ij) of particles, the B T basis \[ij, k]wks;p\ >BT can be constructed as detailed in equation (4.10). The B T pair interaction UkT between the pair (ij) is then defined on this basis to have the form < [ij, k]w'ks'.-p'\'\U?T\[ij, k]wks\ pA > k x ^a^l^m^k^ (4.26) as in equation (4.15). The potential UBT defined above modifies the 2-body mass oper-ator; the corresponding potential modifying the 3-body mass operator is where rriij = yfmi+kf + ^ nij + kf (4.28) Finally, then, the interacting mass operator in the B T representation is given by 3 MBT = M0 + £ + Vf2T (4.29) i=l To ensure the system satisfies the requirement of macroscopic locality, we use the packing operators A as described in Appendix F to obtain MTP = AMBTA] (4.30) Chapter 4. Three-Particle Systems 35 With MTF constructed, one can write H(3) as the direct integral of subspaces, each spanned by eigenvectors of MTP and S 2 corresponding to the same eigenvalue, and each of which is irreducible under the action of the Poincare Group. The construction of the irreducible representation of this group on the space of physical states of three interacting particles is thus concluded. This completes our discussion of the construction of irreducible unitary representa-tions of the Poincare Group on PL(2>). The next chapter will introduce the Faddeev equations for building the multichannel scattering T-matrices for a complex system in terms of simpler T-matrices, and will consider in some detail the scattering theory of a system of three particles. Chapter 5 The Relativistic Faddeev Equations In the last chapter we defined a three-body Hilbert Space on which we then constructed irreducible unitary representations of the Poincare Group, for both non-interacting and interacting systems. Having developed this machinery, it is still an extremely difficult task, in general, to solve exactly for the dynamics of the system. Consequently, it is useful to construct a scattering theory, in which one determines the evolution from states in the distant past to states in the distant future without calculating the precise details of the evolution for finite times. The general S-matrix scattering formalism is discussed in Appendix G. The scattering T-matrices for the scattering channels of the three-body system sat-isfy Lippman-Schwinger equations, but these are complicated integral equations and in general exact solutions cannot be obtained analytically. Numerical methods can be used to obtain solutions, but for the three-body system one encounters problems with delta function singularities in the kernel. These singularities, known in the literature as dis-connected diagrams, greatly increase the difficulty in obtaining numerical solutions to the Lippman-Schwinger equations. This difficulty can be avoided through the use of a method developed by Faddeev [Fa 65], in which the three-body T-matrix is given in terms of the T-matrices corresponding to the interactions between individual pairs of particles. This method will be discussed in this chapter, starting with a derivation of the Faddeev Equations for a general partition of an arbitrary mass operator and then considering a specialisation to the three-body system. 36 Chapter 5. The Relativistic Faddeev Equations 37 Non-relativistic S-matrix scattering formalisms are ususally formulated in terms of the Hamiltonian of the system. As is discussed in Appendix G, the Kato-Birman Invariance Principle allows us to build, without changing the S-matrix elements, a scattering theory dominated by a non-trivial function of the system Hamiltonian. Thus, as we desire to have a manifestly Lorentz-invariant scattering theory, we shall construct our theory using the mass operator of the system: M = VH2 - P 2 (5.1) This is also a useful scattering formalism to use in connection with the Bakamjian-Thomas construction, as there it is the mass operator which is modified to include interactions. Consider a system whose dynamics are governed by the mass operator M , which is partitioned as follows: M = M0 + JTVa (5.2) 0=1 where M 0 is the mass operator for the non-interacting system. This partition is in principle arbitrary, but the physics under consideration will usually suggest a certain form. Note that not all potentials Va need correspond to a scattering channel; that is, there will not necessarily be scattering channels in which all potentials but Va vanish. However, in general it is useful to partition the Hamiltonian so that each asymptotic channel corresponds to one of the Va. It is also useful to define Vo = 0. As described in Appendix G, one then defines the T-matrix for scattering from channel b to channel a as Tab(z) = Vb + VaG(z)Vb (5.3) where Chapter 5. The Relativistic Faddeev Equations 38 and defining V3 by V3 = £ V* (5-5) For each of the partition potentials Vj, one can define a T-matrix 7} (2) by TJ(z) = Vj + VjGJ(z)Vj (5.6) which can easily be shown to satisfy G0(z)Tj(z) = Gj(z)Vj (5.7) It follows then that Tab(z) = Vb + VaG(z)Vb = Vb + Y,ViG(z)Vb = Vb + J2VJGj(z)T3b(z) = Vb + Y,T^)Go(z)T3b(z) (5.8) where by definition T 0 = 0. Equation (5.8) describes a set of coupled integral equations for the scattering T-matrices, known collectively as the Faddeev Equations. These take as input the T-matrices corresponding to the partition potentials Vi and satisfying individual Lippman-Schwinger equations. With a judicious partition of the Hamiltonian, these Lippman-Schwinger equations are much easier to solve than that for Tab(z) directly, and consequently the solution of the problem is simplified. While these equations are formally identical to the familiar non-relativistic Faddeev Equations, they have the advantage that, as the T-matrices under consideration are built in terms of the mass operator, these equations are relativistically invariant. In the three-body scattering problem, there are 4 distinct scattering channels. As usual, denote by channel 0 the situation in which the particles are asymptotically free. Chapter 5. The Relativistic Faddeev Equations 39 One can then label by Channel j (j = 1,2,3) that asymptotic state in which particle j is free and the other two particles are bound. Complimentary to this notation is the partition of the mass operator: M = MQ + Y,Vj = M0 + V (5.9) where Vi is the two-body potential between particles 2 and 3 (and similarly for V2 and V 3 ) . This notation satisfies the requirement that when the system is in the asymptotic state corresponding to channel j , all interactions corresponding to potentials other than Vj vanish (with Vo = 0 by definition). In the three-body system, the Faddeev Equations can be cast in matrix form: \ (5.10) ' T°i(z) ) f ° T*(z) n(z)) ' T°i(z) Tli{z) v j 0 0 T2(z) Ts{z) G0(z) Tl3(z) T2i(z) + v j 0 0 Uz) T2i(z) 1° Ti(z) T2(z) 0 J v TV(z) which decouples into the two equations ' Tli(z) N ( 0 T2(z) Uz) ' ( Tli(z) N •T2\z) = Ti{z) 0 T3{z) G0(z) T2\z) V T*Kz) ) V Ti(z) ?2{Z) 0 J V T*>\z) j (5.11) and T°i(z) = V3 + J2 Tk(z)G0(z)Tk3(z) (5.12) k=l For the case in which the three particles are free in both the distant past and future, in which the scattering is governed by T00(z), the Faddeev Equations take on a particularly simple form. From (5.12), T00(z) = V° + ^Tj(z)G0(z)Ti°(z) J ¥ 0 (5.13) Chapter 5. The Relativistic Faddeev Equations 40 3 so, as V = V° = EJVJ, T°\z) = £ (Vj+T^Goiz^iz)) = J2Ti(z) where T^(z) is defined as T*{z) = VJ + Tj{z)GQ{z)T*\z) Now, the Ti(z) themselves can be shown to satisfy: T*{z) = VJ + TJWGMT^Z) = Vj + T^Goiz) (v+J2T^z)Go(^Tko(z) = Vj + T\z)G0(z)Vj + T^Goiz) £ (vk + Tk(z)G0(z)Tk0(z)) kfr = Tj(z) + '£Tj{z)G0{z)Tk(z) This equation has the matrix form T\z) T\z) \ T*(z) J + 0 Tx{z) Tx(z) T2(z) 0 T2(z) ' T\z) N G0(z) T\z) J v T ' 3 W ) (5.14) (5.15) (5.16) (5.17) The solution of this set of coupled operator equations yields the T J '(z) and thus T00(z) = ZUTJW-As with the Lippman-Schwinger equation, one can formally invert this matrix equa-tion to yield an explicit expression for T00(z) in the form of a series expansion. Equation (5.17) can be expressed as (5.18) ' 1 0 0 N ( 0 Ti(z) Ti(z) ) ' T\z) \ ' Tx{z) ^ 0 1 0 -- T2(z) 0 G0(z) T2(z) = T*(z) v0 0 1 y V Ts(z) n(z) 0 J v W ) Chapter 5. The Relativistic Faddeev Equations 41 So, formally, ' T\z) ^ Y i o °) ( 0 Ti(z) TM \ - i T\z) = 0 1 o -- T2(z) 0 G0(z) T2(z) (5.19) \ 0 0 1 J V T3{z) Uz) 0 ) K n(z)) Now, the series expansion (1 - A)~l = 1 + A + A2 + A3 + ... (5.20) holds formally for A an operator, provided that A is "small" in some sense. It is possible to define a metric on the space of operators on a vector space, so this notion of smallness can be made rigourous, and in the following we shall not concern ourselves with these details and shall assume that the expansion can be carried out. With this, we have that ' T\z) N ' 1 0 0 N f 0 T^z) T^z) ) ' Triz) ) T\z) = 0 1 0 -r T2(z) 0 T2(z) G0(z) + ... T2(z) (5.21) v T 3 W , 1° 0 1 / KT3(z) T3(z) 0 , V Uz) , so, to second order in the Tj(z), we have the Born series expansion for T00(z): T00(z)=Tl(z)+T2(z)+T3(z)+ £ Ti(z)G0(z)Tj(z) + ... (5.22) This expression has a naive but useful interpretation. As V\ is the two-body potential between particles 2 and 3, if particle 1 is sufficiently removed that its interaction with the other bodies vanishes, then all scattering information will be contained in T\. The scat-tering of particles 2 and 3 with 1 as a free spectator can be represented diagrammatically by 1 Chapter 5. The Relativistic Faddeev Equations 42 This describes a process in which the particles are originally free, particles 2 and 3 have an interaction described by the "black box" Ti(z) and emerge free, while particle 1 is free throughout. The operator G0(z) can be thought of as corresponding to the free propagation of all particles in the system. We then interpret equation (5.22) as describing the T-matrix T00(z) for three-body scattering between free channels as the sum of a set of successive two-body processes, represented by Tm(z) = 2 > > 3-3 > ^ > + 1-1 • 2-1 » ^ . 2-T", + 2 > > + 3-3 > 1-3-; &) • •+ »• ^—r *• + 2—*—^—*—7 ~~ — +... • 1 , If we try to understand the above diagrams in terms of a mechanical analogy, we must picture time as flowing from the left to the right (remembering that we must interpret time as flowing from right to left in an expression such as T^(z)G0(z)T2(z)). Note that this series contains no terms of the form W CO — » In terms of the simple mechanical picture described above, this is expected: the operator Chapter 5. The Relativistic Faddeev Equations 43 T\(z) describes the complete interaction of particles 2 and 3. It thus makes no sense physically to apply this operator twice in immediate succession. However, terms such as Ti(z)G0(z)T2(z)G0(z)Ti(z) are included in the series (5.22) and can be thought of as describing a process in which particles 2 and 3 interact then propagate freely, particles 1 and 3 interact and propagate freely, and finally 2 and 3 interact again. This picture should not be taken as an actual description of the physical processes involved in three-body scattering, but it does provide a useful paradigm allowing an understanding of the Born series solution of the Faddeev Equations for T00(z). This naively mechanical picture does contain some quantal elements. In this picture, the above processes, which are classically mutually exclusive, all occur simultaneously, in some sense, during the scattering. The above series expansion of Tm(z) is expressed in terms of powers of the Tt(z) operators. As described in Appendix G, each of these can be expressed as a Born series in the interaction potential: Ti(z) = Vt + ViGoizW + ViGoizWGoWVi + ... (5.23) We can use this expansion to express (5.22) as a series in powers of the interaction potentials: T 0 0 ( z ) = Vx + V2 + V3 + £ ViG0(z)Vj + ... (5.24) which can be represented diagramatically as Chapter 5. The Relativistic Faddeev Equations 44 Tm(z) = 2-3-1-1-+ 2-3-2-+ 3-1-3-+ 1-2-+ 3-1-+ . . . If we assume that each of these potentials are of about the same strength and range, then this series in powers of the potentials is more physically significant than that in powers of the T-matrices, in which each term contains potential terms to all orders. A similar analysis can be made of the T-matrices between the other scattering chan-nels. As an example, consider TVi(z). From equation (5.11), we know that ( T"(z) \ (V*\ ( 0 T2(z). T,(z) \ Ti(z) 0 T 3(z) + T23(z) | = | V3 V T33(z) ) \ V3 J which is formally solved to first order in the Tj by ( TVi{z) I V3^ V3 I + V3 T23(z) Ti(z) 0 T3(z) \ ' T13(z) N G0(z) T23(z) J v T 3 3 ( ^ ) i ( ' V3 \ G0(z) V3 +. \ KV3) (5.25) (5.26) Chapter 5. The Relativistic Faddeev Equations 45 so Tl\z) = V3 + T2{z)G0(z)V3 + T3G0(z)V3 + ... (5.27) To second order in the then, Tv\z) =V1+V2 + V2G0{z)V1 + V2G0(z)V2 + V3GQ(z)Vx + V3G0(z)V2 + ... (5.28) This T-matrix takes us from the channel in which particles 1 and 2 are asymptotically bound to that channel in which particles 2 and 3 are bound. It is represented diagram-mat i cai ly by and is equal to the series of diagrams to second order in the Vt. Note that none of the diagrams in this series have a V3 term to the extreme left or a Vi to the extreme right (except the first diagram, which will be discussed in the next paragraph). In terms of the mechanical model, we can understand Chapter 5. The Relativistic Faddeev Equations 46 this as arising from the fact that particles 1 and 2, interacting through potential V 3 , enter the scattering process bound. Before these two have interacted with particle 3, then, V3 is a binding and not a scattering potential. Similarly, in the out-channel particles 2 and 3 are bound and interact via Vi , so at the end of the scattering process this potential serves to bind these two particles and does not participate in the scattering. One feature of the series (5.28) which is somewhat unsatisfying is that it is asymmetric in the potentials Vi and V 3 . Neither the in- nor the out-channels are preferred in any way, and yet Vi appears on its own in a term in (5.28) while V3 does not. This problem is discussed in more generality in Appendix G, where the T-matrix fab(z) = Va + VaG0(z)Vb (5.29) is introduced. This T-matrix can be shown to satisfy the adjoint Faddeev Equations Tab(z) = Va + ^ r ' ( z ) G 0 ( z ) T i ( 2 ) (5.30) and it can be shown that the matrix elements of T 1 3 (z ) and Tn(z) are identical for z on shell. Let \ipi > be an eigenstate of Mi (the set of which span the Hilbert Space describing scattering channel 1) and \(p$ > be an eigenstate of M 3 . Then, as we can write T13(z) = \(Tn(z) + f 1 3(z)) + \(T'\z) - f'\z)) (5.31) we have that T13(z) has on-shell matrix elements < ^ | T 1 3 ( ^ 3 > = \<^\Tl\z)+f1\z)\^> = <y1\^L-± + v2 + V2GQ{z)V1 + +V3G0(z)V + V2G0(z)V2 + V3GQ{z)V2\(p3 > (5.32) This expression is then manifestly symmetric in the potentials Vi and V 3 . However, the asymmetry of equation (5.28) does not affect the physics and the result is symmetrized only so that the mechanical picture remains consistent. Chapter 5. The Relativistic Faddeev Equations 47 This completes the general discussion of the relativistic Faddeev Equations and the discussion of the special case of three-body scattering. In the next chapter, we consider a system in which the number of particles is not fixed (ie, in which the creation and annihilation of particles is permitted) and consider in some detail the Faddeev Equations describing scattering in this case. Chapter 6 The 2^>3 Body System In previous chapters, we have constructed Hilbert Spaces for two-body and three-body systems and discussed the Faddeev approach to the determination of the multichannel scattering T-matrix, looking in some detail at the case of three-body scattering. In all of these examples, the number of bodies comprising the system was fixed. One of the most important discoveries in modern particle physics is that the equivalence of mass and energy implies that the number of elementary particles in a physical system is not necessarily a constant of the motion; in a physical process, particles can be created and destroyed. This concept is of fundamental importance to quantum field theory, in which the Hilbert Space considered is one in which physical states can have any number of elementary excitations. However, in typical experimental situations, the energy range available to the system is bounded above and the number of particles that can exist in the system is thus limited. In such a case, instead of working in an Hilbert Space allowing an arbitrary number of excitations, one can work in a space in which only certain numbers of elementary particles can exist. This chapter will discuss such a model, in which the system can consist of two or three particles, and the dynamics of such a system will be considered. Note that in any such system, the particle which can be created or destroyed must be bosonic, as it a physical observation that fermion number is conserved in all systems. Consider the Hilbert Spaces H(2) and H(3), as constructed in Chapters 3 and 4, respectively. The Hilbert Space of physical states allowing either 2 or 3 particles is given 48 Chapter 6. The 2 o 3 Body System 49 by the direct sum H(2 <+3) = H{2) ®U{3) Vectors in this space can be represented by the column matrices (6.1) I* >= |V>2 > V 1^3 > where \ip2 > G %{2) and 1*03 > 6 TL{3). Physical states are defined to have unit norm (6.2) < > = < ip2\ijj2 > + < ip3\ips >= 1 Operators on TL{2 o 3) are 2 x 2 matrices: ^ ( 2 2 ) ^ ( 2 3 ) \ ( > (6.3) 4(32) 4(33) V | V > 3 > (6.4) In previous chapters, we constructed irreducible unitary representations of the Poincare Group on the spaces of physical states of systems of two and three interacting particles; these are given, respectively, by (71(A,a)(g)(7'2(A,a) (6.5) and t/ 1(A,a)<8»C/2(A,a)®f7 3(A,a) (6.6) We will assume in what follows that particles 1 and 2 are the same in the two and three-body sectors. In the absence of interactions coupling the two and three-body sectors, then, an irreducible unitary representation of the Poincare Group on 7i is Ux(A,a)®U2(r\,a) 0 \ 0 U1{A,a)®U2(A,a)®U3(A,a) ) U(A,a) = (6.7) Chapter 6. The 2 ^ 3 Body System 50 with irreducible basis ^ |[123]W. S;pA> J (6.8) on which the free mass operator is given by / M 0 = V JM? + K 2 + ^ M 2 + K2 0 0 xj ( ^ M 2 + K2 + y / M 2 + K2)2 + W 2 + y/' M2 + W 2 (6.9) We can also define the number operator N = (6.10) 2 0 such that any of its eigenstates have the zero vector in either the two- or three-body component. The inclusion of interactions that couple the two and three-body spaces is now straightforward. Define the interacting mass operator, M, by M = M 0 + V where the coupling potential, V, is given by ( o v ^ (6.11) V = V f 0 (6.12) V is a particle annihilation map from TL(Z) to TL(2) and V is Hermitian by construction. The form of V can be determined from a phenomenological theory or from some more fundamental field theory; an attempt to derive a form for V from the latter approach is discussed in the next chapter. Irreducible unitary representations of the Poincare Group for this interacting system are obtained by determining the spectrum of M and Chapter 6. The 2 ^ 3 Body System 51 the corresponding eigenstates. To each eigenvalue, m, of M will correspond a subspace of eigenstates satisfying M\ms] pA >= m\ms; pA > (6.13) and this subspace will transform irreducibly under the action of the Poincare Group. Note that as the mass and number operators do not commute, the eigenstates of M will have one of the components in H(2) or %(3) zero if and only if the map V is identically zero. This means that all states of the system that have a well-defined mass (and thus energy) will not have a sharp particle number, and that it no longer makes sense to speak of a two-particle state as having a precise mass. This is a familiar situation from quantum field theory, in which the eigenstates of the system Hamiltonian contain an indefinite number of particles, and one speaks of the bare particles as being "dressed" by a cloud of virtual bosons. This method of incorporating particle production and annihilation into a quantum mechanical model by taking a direct sum of two and three-body spaces and then coupling these by a 2 3 potential is not standard. More typically, this is done by considering an interacting field theory in which particle production is represented at the level of the primitive vertex. The method considered in this thesis was considered independently by McMillan [Mc 82] and by Keister and Polyzou [Ke 91]. It does not seem to appear else-where in the literature. Although unorthodox, it will be seen to provide some interesting results. If the energy of the system is less than J2f=i ftk, then it is kinematically forbidden for the 3-particle component of the state vector to be nonzero, and, as energy is conserved, this must remain true for the entire evolution of the system. Thus, the potential V cannot couple the two sectors of 7i(2 <B- 3) for energies less than the sum of the three particle masses. We can ensure that this is true for our interaction by defining the projection Chapter 6. The 2 ^ 3 Body System 52 operator JT: n = Jn nii+m2+m3 m\ +7T12 dmi 2 |mi2 >< m i 2 | where \m12 > is an eigenstate of the two-body mass operator on rl(2) M ( 2 ) | m i 2 >= m12\mi2 > (6.14) (6.15) The operator IT projects vectors in 7i(2) onto their components in the subspace of state vectors with masses in the range [mi + m 2 , mi + m2 + m.3]. The kinematic constraint on the interaction can be encoded into the theory through the imposition that the map V from rl(2>) to %(2) has the form v = w(i - n ) where W is some map from 7i(3) to TL(2). This form of V ensures that (6.16) vn = o (6.17) and the two and three-body spaces are not coupled below the threshold for the production of the third particle. We will now consider the scattering theory of this system. We will assume that the mass operator can be partitioned 5 M = M0 + J2Vi where M 0 = ( M , S 2 ) 0 V o is the direct sum of the free mass operators on %(2) and H(3), <0 0 ^ Vi = i = 1,2,3 (6.18) (6.19) (6.20) Chapter 6. The 2 o 3 Body System 53 where the V, are the two-body potentials on 7^(3), as described in Chapter 4, y 4 = (6.21) where V 4 is the two-body interaction in the 2-particle sector ofH(2f>3) , as considered in Chapter 3, and is the potential coupling the two- and three-body spaces. Thus, M has the form We have ignored the possibility of the existence of a purely three-body potential in the three-body sector of the interacting mass operator. As well, as in Chapter 4, for formal reasons it is useful to define Vo = 0. Although we have partitioned the mass operator using 5 different potentials, there are only 4 different well-defined scattering channels: not all of the partition potentials correspond to scattering channels. In the two-body sector, there can be no evolution between bound states and scattering states, so there is only one channel in this sector, and in the three-body sector we have the same set of scattering channels as described in Chapter 4. We thus have a 0 channel in which the particles are free in both sectors and channels 1, 2, and 3 in which two of the three particles in the three-body sector are bound. As before, in channel j (j = 0,1,2,3), the dynamics of the system are governed by the mass operator = M 0 + t^. Now, the T-matrices Tab(z) are defined as abstract mathematical objects for a,b € {0,1, 2, 3, 4, 5}, but only for a, b € {0,1, 2, 3} do these have physical significance as multi-channel scattering T-matrices. These objects satisfy the Faddeev Equations (6.22) (6.23) Chapter 6. The 2 ^ 3 Body System 54 derived in the previous chapter: fab(z) = Vb + J2 fj(z)G0(z)fib(z) (6.24) These differ from the Faddeev Equations considered in Chapter 5 in that in this equation the objects Tab(z) are 2 x 2 matrices of operators. This both complicates and enriches the calculations we will perform in this chapter. As was the case for purely three-body scattering, the consideration of these equations for particular scattering channels is quite instructive. Before we look at specific examples of multichannel scattering operators, let us con-sider the T-matrices and Green's operators corresponding to the partition potentials V^. By definition, the Green's operator Gi(z) is given by 1 1 Gi(z) z- Mi z- M0-Vi (6.25) Consider the Green's operator G\(z): Gi{z) = I z-M< Z-MQ-VX •(2) so Similarly, \ ( a - M ^ - V i ) " 1 ) - M 0 ( 3 ) - VY ( Gf{z) (6.26) \ V 0 Gf\z)) Gj(z) ( Gf{z) \ (6.27) V 0 Gf\z)J (6.28) for j = 2, 3, and G4(z) = ( <??>(z) \ V 0 Gf{z) J (6.29) Chapter 6. The 2 ^ 3 Body System 55 The calculation of G$ is more interesting. Now, -l G5(z) = z - M<2> M<3> ( Gf(z)-^ -V \ - V t G$\z)-' (6.30) — z — IVIQ The problem of inverting the above matrix is not a trivial one; its elements are not c-numbers, but non-commuting operators. However, it can be shown that the left and right inverses of this object exist and are equal, so an inverse does exist. It is given by: ( G$\z)+a (3 7 G$\z)+5 G5(z) = (6.31) V where (3 = G 0 2 ) ( z ) V G 0 3 ) ( z ) ( l - V t G 0 2 ) ( z ) V ^ 3 ) ( z ) ) - 1 a = f3V]G[2){z) 7 = G03\z)ViG?\z)(l-VG$\z)V'G02\z))-1 6 = iVG$\z) (6.32) This is a rather complicated object, but with its existence thus established, we can feel confident that the Faddeev Equations do in fact hold true for this system. Consider now the T-matrices of the partition potentials V,. The Ti(z) satisfy the Lippman-Schwinger equations fj{z) = Vi+VjG0(z)fj{z) For j G {1, 2,3}, the Lippman-Schwinger equations are 7K*)<22> 7 } ( z p ) \ (6.33) T3{z)W 7}(z)(33) ) 0 0 \ f + 0 0 ^ 1 G?\z) ( 0 Gf{z) Tj(z)(2 2) Tj{z)^ [T3(z)W Tj(z)W (6.34) Chapter 6. The 2 ^ 3 Body System 56 which after some matrix algebra becomes so Tj(z) has the form (6.35) (6.36) (° ° ) V 0 T3{z)W j and the Lippman-Schwinger equation for fj(z) just reduces to that for Tj(z) = T^z)* 3 3). Similarly, T 4(z) has the form f4(z) ( Uz) 0 X (6.37) where T 4(z) satisfies T 4(z) = y 4 + V 4C7( 2 )(z)T 4(z) (6.38) Again, T 5(z) is the most interesting in structure of the partition potential T-matrices. Starting with the Lippman-Schwinger equation t Uz)W T6(z)<23> T5(z)W T6(z)M V 0 V V* 0 0 V V* 0 / r»(2) \ V 0 <$\z) J T6(z)W Ts(z)W [T5(Z)W T 5 ( z ) ( 3 3 ) (6.39) V VG03\z)T5(z)W v + VG{03)(z)T5(z)W V* + V^G^iz^iz)^ ViG[02\z)T5(z)W so the components of T5(z) satisfy the coupled operator equations T5(z)W = VG$\z)T,{z)W T6(*) ( 3 2 ) = Vi + V^G{02\z)T5(z)W (6.40) (6.41) Chapter 6. The 2 ^  3 Body System 57 T5(zfV = V + VG$\z)Uz)W (6.42) Tb{z)W = V^\z)Tb{z)^ (6.43) The first two of the above equations pair to give T 5 ( z f 2 ) = VGt\z)V^ + VG?{z)V^G?\z)T5(zY22) (6.44) and the second two to give T 5(z)( 3 3) = V^G{2\z)V + V^G{02\z)VGi3\z)T5(z)W (6.45) If we define the new Hermitian operators U^=VGf\z)V] (6.46) and =V*G%\z)V (6.47) then the equations for T5(z)^ and for T5(z)^ are just Lippman-Schwinger equations for T-matrices associated with these new potentials: T5(z)W = W<2> + U^G{2\z)T5(zf2^ (6.48) T5(z)W = UW+UWGP(z)T5(z)W (6.49) The operator is a map from Ti(2) to TL(2) via 7^(3), and in terms of the mechanical model introduced in Chapter 5, can be thought of as describing a process in which the two-particle system sponteneously generates a boson which is subsequently absorbed. The formalism provides no way of determining which of the original two particles produces the boson and which of the two absorbs it, so the process could correspond to a boson exchange or a single-particle self-interaction. The process can be represented by the Chapter 6. The 2 ^ 3 Body System 58 diagram * ^ ^ > ^ * (6.50) Similarly, is a map from the three-particle part of the Hilbert space onto itself, corresponding to the situation in which three particles propagate freely until the boson is absorbed and subsequently re-emitted. The diagram representing this is )—»- = > ( V ) ( vt J (6.51) As will be seen, processes such as these do influence the dynamics even if there is no dif-ference between the number of particles in the distant past and distant future asymptotic states of the system. Equations (6.48) and (6.49) can be solved using standard techniques; in particular, approximate solutions can be obtained from Born series expansions: T5(z)W = + U^G02\z)U^ + ... (6.52) and T5(z)W = W 0 ) +U^Gf(z)U^.+ ... (6.53) The operators T5(z)^ and T5(z)^ are given in terms of T5(z)^ and T5(z)^ by simple expressions: T5{zfV = V + VG{03)(z)Tt\z) = V + VG03\z)U® + VG{3\z)U^G{3)(z)U^ + ... (6.54) and T5(z)W = ^ + V^G^(z)Tt\z) Chapter 6. The 2 ^  3 Body System 59 = Vt + V^\z)U^ + V^G{2\z)U^G02\z)U^ + ... (6.55) so the entire T-matrix Ts(z) is determined once one has calculated T^,(z)^ and T 5(z)( 3 3). Let us now return to the Faddeev Equations themselves. Consider the scattering process in which all particles are asymptotically free in both the distant past and in the distant future in both two and three-particle sectors. This is governed by the T-matrix f00(z). From equation (6.24), we know that f*\z) = V° + Y,fAz)Go(z)f>b(z) (6.56) so, in particular, f00(z) = V° + J2fj(z)Go{z)fi°(z) (6.57) A calculation formally identical to the one in Chapter 4 for the three-body T00(z) operator-shows that fO0{z) = Y/fj(z) (6.58) where the T3(z) are the solutions to the coupled implicit operator equations f'{z) = f3{z) + £ f , ( z ) G 0 ( z ) f *(z) (6.59) which take the partition potential T-matrices Tj(z) as input. These equations can be represented by the matrix equation: ( f\z) \ '' T,{z) ( 0 Ti(z) Ti{z) Ti(z) fi{z) } ( f\z) ^ T\z) %(z) Uz) 0 Uz) %{z) Uz) f\z) f\z) = Tz(z) + Uz) Ts(z) 0 %{z) %{z) Go(z) f\z) ' f\z) fA{z) f4(z) f4(z) f4(z) 0 f4(z) f\z) \ n z ) j ) K n(z) Uz) Uz) Uz) 0 J {nz) j (6.60) Again, as in Chapter 5, this equation can be solved if we assume that the matrix of operators: Chapter 6. The 2 ^  3 Body System 60 ^ 1 0 0 0 0 ^ 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ( 0 %{z) fx{z) f^z) fx{z) ^ %{z) 0 f2(z) f2{z) f2(z) %{z) %{z) 0 %{z) f3(z) f4(z) f4(z) f4(z) 0 f4{z) \t5(z) %(Z) t5(z) %{Z) o ; G0(z) can be formally inverted by a geometric series. Then we have that and so f00(z) = J2fi(z)+ J2 T i(z)G0(z)f i(z) + . . . After a little matrix algebra, one can show that T 0 0 ( z ) is given by the matrix f j > 0 0 ^ ( 2 2 ) j > 0 0 ^ ) ( 2 3 ) ^ T 0 0 (z)( 3 2 ) T 0 0 (z)( 3 3 ) where T°°{z)W T00(z)(23) (6.61) ' T\z) \ ( T,{z) \ ( 0 fi(z) Ti{z) fi(z) fi(z) } (f.(z)) f\z) %{z) 0 Uz) T2(z) n(z) T2{z) f\z) = %{z) + Ts(z) Uz) 0 n(z) n(z) G0(z) T3(z) + f\z) f4{z) f4(z) TA(Z) f4(z) 0 f4(z) f4(z) {n*) j v %{z) n(z) n(z) T5(z) 0 J V f5{z) J (6.62) (6.63) (6.64) T4(z) + Ts(z)W + T4(z)GP(z)T6(z)W + T5(*)<»>Gf (*)T4(*) + ... Ts(z)W + T4(z)G02\z)T6{z)W +J2T5(z)^GQ:3\z)Tl(z) + ... i=i T6(s)<32> + T B ( z ) < ^ i=i T°°{z)W Toow(33) = £ T . ( z ) +Tsrz)&) +^Tt(z)G$\z)T5(zfV +±T5(z)^G{3\z)Tl(z) i=l i=l Chapter 6. The 2 o 3 Body System 61 + J2 Ti(z)G03\z)Tj(z) + . . . (6.65) As was the case with the three-body Faddeev Equations in Chapter 4, a Born series expansion in terms of the partition potentials is more physical than one in terms of their T-matrices, but there is a complicating factor. Suppose that our interactions can be described in terms of an underlying field theory whose primitive vertex corresponds to the coupling constant, g. The potentials Vi, i € {1,2,3,4}, are all two-body potentials and thus arise from 2-vertex processes; they should then scale as g2. However, the particle annihilation and production potentials, V and V* respectively, are one-vertex processes and thus only scale as g. In a Born series expansion, then, it does not make physical sense to treat V and its Hermitian conjugate as being of the same order as the other-partition potentials. The potentials and on the other hand, scale as g2 and are thus of the same order as the potentials Vi, i € {1,2,3,4}. Taking this into account, we can expand the components of T00(z) as series in the potentials: T 0 0 ( z ) ( 2 2 ) = y 4 + U ( 2 ) + f V 4 + U ( 2 ) j t f 2 ) , z ) t V 4 + u { 2 ) } + _ Too ( z ) ( 23) = v + V i G f ) { z ) v + VG$\z) {ii® + X > i ) + • • • 7*0^(32) = V) + V \ G m { z ) ( U ( 2 ) + y\+J2ViG{*)(z)Vi + ... i=l 7*0(^(33) = j^V+U^ + (J2Vl+U^) G{03\z) (J2Vf+U®) + . . . (6.66) i=l \i=l / \i=i / Note that the diagonal elements of T00(z) are always sums of terms which scale as even powers of g and that the off-diagonal elements of terms which scale as odd powers of g. In the above series we have included terms to 0(gA) for the diagonal elements and 0{gi) for the off-diagonal. From the equations (6.66) we see that the T-matrix in the (22) sector is precisely that of a two-body system with interaction potential VA +U^2\ and that for the (33) sector of Chapter 6. The 2 ^ 3 Body System 62 a three-body system with potential Y$=\ +U^. Thus, the fact that particles can be created and annihilated affects two-body to two-body scattering, above the threshold for the creation of the boson, and three-body to three-body scattering. Thus, the potential V has a much more significant effect than just allowing asymptotic particle creation and annihilation; the opening of this new degree of freedom modifies all dynamics of the system. The off-diagonal components of T00(z) can be rewritten in a more symmetric manner by noting that V G 0 3 ) (z)U{3) = U™GQ2) (z) V (6.67) so that A l l of these equations can be represented diagrammatically as in Chapter 5. As well, as in Chapter 4, one can construct Faddeev Equations for all of the multi-channel T-matrices. For example, a calculation formally identical to that in Chapter 5 shows that the T-matrices Tj3,j ^ 0, satisfy the implicit operator matrix equation: ' fl3(z) \ ( v 3 ) ( 0 Uz) Uz) f4(z) %{z) \ ' fl3(z) N f23(z) V3 Ti(z) 0 n(z) f4(z) f5(z) f23(z) f33{z) = V3 + fx{z) Uz) 0 f4(z) %{z) Go(z) f33(z) f43(z) V3 fi{z) Uz) Uz) 0 Uz) •f43(z) V T53(z) , , v 3 ) V fi(z) Uz) n(z) f4(z) 0 J V T"3{z) , (6.69) Thus, while the operators f43(z) and f53(z) do not have the physical interpretation of multi-channel scattering T-matrices, they cannot be ignored, as the Faddeev Equations couple them to the physical T-matrices. Chapter 6. The 2 ^ 3 Body System 63 This completes the discussion of the 2 ^ 3 body system and its scattering theory. In this chapter, we assumed that the particle creation potential V* is known, but we made no specific assumptions about its form. In principle, one could use this formalism in a phenomenological theory, writing down a very general form of the particle production potential with several free parameters and then using experimental data to fit these parameters. Alternately, a form of the particle production potential can be derived from a more fundamental field theory. An attempt was made to do this using the dressing transformation, as will be discussed in the following chapter. Chapter 7 The Dressing Transformation: A Critique It was found last chapter that when the number of particles in a physical system is a dynamical quantity, the number operator and the mass operator of the quantum mechan-ical model describing the system do not commute. That a system cannot simultaneously have a sharp mass and a sharp number of particles is a general feature of quantum me-chanical models that allow particle creation and annihilation. In quantum field theory, this leads to the idea of renormalization, in which one finds that the original, or "bare", parameters such as particle mass or coupling constant that enter into the theory do not have the observed values of these parameters. Because of processes that can be pictured as arising from the exchange of "virtual particles", the bare and renormalized quantities differ by a (typically infinite) amount. One can then imagine that the physical particles of the theory are not the bare particles themselves with bare masses and whose interac-tions are governed by the bare coupling constant, but are the bare particles surrounded by a "cloud" of virtual bosons constantly being emitted and reabsorbed by the central particle. Considered in terms of these "dressed" particles, the system can then have si-multaneously sharply-defined mass and particle number. Schweber [Sc 61] discussed the issue of dressing field theories via unitary transformations and a general algorithm based on his method was developed by Hearn and McMillan [He 83]. The idea of this method is that one can write the Hamiltonian (for a non-relativistic system) in terms of creation and annihilation operators for the dressed particles, and from this extract the physical masses and interactions of the theory. It was hoped that 64 Chapter 7. The Dressing Transformation: A Critique 65 this algorithm would provide a form for the particle annihilation potential V discussed in Chapter 6. Instead, it was found that the method does not prescribe a unique dressing transformation and is thus ambiguous. This chapter will critique the dressing transfor-mation methods of Schweber, Hearn, and McMillan. Consider a simple non-relativistic field-theoretic model of a system of one species of spinless fermion with bare mass m^o and one species of spinless boson with bare mass mBfl described over the Fock space where TF and TB are the fermion and boson Fock spaces, respectively. We will denote the vacuum state of the system by the vector |0 >, and by F^(p) the creation operator whose action on the vacuum is to produce a one-particle bare fermion state of momentum p. F(p) will denote the corresponding annihilation operator. Similarly, B\p) and B(p) will denote the creation and annihilation operators for the bosons. The boson and fermion creation and annihilation operators satisfy the standard commutation and anticommutation relations: As well, the creation and annihilation operators of the fermions commute with those of the bosons. (7.1) {F(p),P(p)} = 0 ; {F{p),F\p)} = 5{p-p) [B(p),B(p)]=0 ; [5(p),B t (p)] = « J ( p - p ) (7.2) The number operator for the system is given by: N = NFy0 + NBfi = j d3pFi(p)F(p) + J d3pB^(p)B(p) (7.3) and the total momentum operator by: (7.4) Chapter 7. The Dressing Transformation: A Critique 66 We will assume that the basic interaction is a trilinear vertex with momentum-dependent coupling constant h(q). One can show that the most general coupling that produces a rotation, parity, and time-reversal invariant theory can depend only on the magnitude of the exchange momentum, q, and we shall consider such a coupling. The Hamiltonian of the system then has the form H = H0 + XHint = j d3peFfi(p)F^p)F(p) + j d3peBfi(p)B^p)B(p) +A ( | d3pd3q h(q)F^p)F(p - q)B(q) + h.c)j (7.5) where £Ffl(p) = V ^ F . O + P2 (7-6) and £Bfl(p) = \fmBfi+V2 (7-7) are the bare single-particle fermion and boson energies, respectively. A is an order pa-rameter characterising the strength of the interaction. Consider the bare one-fermion state of momentum p: F^(p)|0 >. While this is an eigenstate of the number operator: iVi ? t (p ) |0 >= 1 F f(p)|0 > (7.8) it is not an eigenstate of the Hamiltonian: H Ft(p)|0 >= i> F,oO) + e*,o(p)) ^ (P)|0 > +A / d3q h{q)F\p - ^)B\^)\Q > (7.9) Note that as the coupling h{q) goes to zero, the bare-one particle state approaches an eigenstate of the Hamiltonian. It is precisely the interaction term that prevents this state from being an eigenstate of H. A stable physical particle is an object with a well-defined Chapter 7. The Dressing Transformation: A Critique 67 energy; clearly, the bare particle is not then a physical particle. To determine the physical particles of the theory, and express the operators in terms of physical particle creators and annihilators, we employ the dressing transformation. The idea of the dressing transformation is to find a unitary operator, U, such that Vi leaves the vacuum invariant: W | 0 > = | 0 > (7.10) and such that UF^ (p)\0 > is an eigenstate of H: HUF\p)\0 >= eF{p)UF\p)\0 > (7.11) We can then define the physical, or dressed, fermion creation operator F^(p) as F\p)=UF\p)lfl (7.12) which, when acting on the vacuum, produces a physical single-particle state of sharply-defined energy. Similarly, we define the dressed fermion annihilator and boson creator and annihilator as: F(p) = UF{p)tf (7.13) B\p) = UB\p)tf (7.14) B{p) = UB{p)U] (7.15) As the transformation is unitary, the dressed creators and annihilators will satisfy the same commutation relations as the corresponding bare operators. One can also unitarily transform the Hamiltonian H(F, B) = U]H(F, B)U (7.16) where the transformed Hamiltonian H(F,B) is a new function of the bare creators and annihilators. This operator is such that -F^p^O > is an eigenstate. Alternately, as H is Chapter 7. The Dressing Transformation: A Critique 68 a polynomial function of the creators and annihilators, the unitary transformation H(F, B) - UH(F, B)U] (7.17) produces a Hamiltonian of the same functional form as the untransformed Hamiltonian but functionally dependent on the dressed creators and annihilators. One can combine these equations to show that H{F,B) = H(F,B) (7.18) The utility of this equation is that it expresses the Hamiltonian of the system in terms of the dressed creators and annihilators; from this, one can easily read off what are the interactions between the physical particles of the theory. It was shown by Schweber [Sc 61] that if the energy e>,o(p) of the fermion is approx-imated by the fermion mass m^o, then the normalized vector I* >= ^  t ^ I ft d%S(p-± k , -q) ft r i f f B t ( k l ) • • • 5t(k„)Ft(q)|0 > n=0 n ! J j=l i=l t=l e B , o ( « t j (7.19) where Z = e - x 2 L ; T - f ^ > -satisfies 77|* >= m F | * > (7.21) where the renormalized mass is given by r h2(a) mF = mFfi - X2 d3q y^- (7.22) J £Bfi(q) This then gives the dressed single particle state for this theory; as can be seen, it is a superposition of states with a single bare fermion but with all numbers of bosons. By definition, the dressed fermion creator ^(p) will satisfy Ft( p) |o >= | * > (7.23) Chapter 7. The Dressing Transformation: A Critique 69 This definition, however, does not uniquely specify -^(p). As the dressing operator is unitary, we can write it as the exponential U = eD (7.24) where D is an anti-Hermitian polynomial function of the fermion and boson creators and annihilators, D\F, B) = —D(F, B) (7.25) Note that because U = UUlfi (7.26) we have that D(F, B) = D(F, B) = D(F, B) (7.27) As the dressing transformation leaves the vacuum invariant, we have that D is specified by the equations: e^F^pJlO >= F f (p) |0 >= | * > (7.28) and eDB^{p)\0 >= B\p)\0 >= B\p)\0 > (7.29) Note that to any solution D of these equations we can add an antihermitian term which annihilates the vacuum and the one-particle states. A n example is a term of form iB^fitBB. The problem we are faced with is that we wish to uniquely determine an operator D defined on the whole Fock space, but the only information wi th which we are provided is its action on the vacuum and one-particle states, which form a set of measure zero in T. The situation is analogous to an attempt to construct a unique continuous function on $Rn given only its value at the origin. In either case, there are an infinite num-ber of distinct solutions that all satisfy the same condition, and we are provided with no other information with which to decide which is the correct solution. One cannot Chapter 7. The Dressing Transformation: A Critique 70 then use Schweber's construction to uniquely determine the unitary dressing operator U. This problem arises with any Hamiltonian if we try to use Schweber's method; it is not particular to this model. The dressing transformation method of Hearn and McMillan [He 83] is quite a different approach to solving the problem, employing a perturbation series in the order parameter A. We will assume that the operator D can be written as a power series in A: oo D = *nDn (7-30) ?l=l From previous considerations, we know that we can write H(F,B) =.tfH(F,B)U = e-DV&H(F,B)eD&& (7.31) It is a well-known result that we can expand such a unitary transformation as H = H + [H, D] + I [[H, D],D] + ... (7.32) where in the above equation all operators are functions of the dressed creators and an-nihilators. Remebering that H = H0 + A A and using the perturbation series expansion of D, this becomes H = H + \{H1 + [H0, Di]) + A 2 ([Hu A ] + i [[H0, A ] , A ] + [H0, A]) + • • • (7.33) By definition, the dressed single-particle state F^(p)\0 > is an eigenstate of H(F,B). If any term in H(F, B) contains just one fermion annihilator, other than the free term in F^F, then the single-particle vector will not be an eigenstate. This can be seen by explicit calculation. The idea of the method of Hearn and McMillan is to remove terms of this kind, order by order, by appropriately tuning the Dn. For example, Hx is just such a term, but if A is defined to satisfy -H1 = [H0,D1] (7.34) Chapter 7. The Dressing Transformation: A Critique 71 then the term linear in A vanishes and is no longer a problem. Then we have that H = H + \2(±[Hl, Di] + [Ho, D2\) + . . . (7.35) The operator D2 can be defined to remove any unwanted terms to 0 (A 2 ) , and, in principle, this process can be carried out to any order in A. There is, however, a problem with this method. Consider the equation (7.34) to which D\ is defined to be a solution. To any solution D\ of this equation, we can add any solution A of the homogeneous equation [77 0 ,A]=0 (7.36) Thus, with the information we have in hand, we can only determine Dx up to an arbitrary term which commutes with the free Hamiltonian. This ambiguity occurs at every order in A. As well, as the form of Dn is involved in the determination of the terms in the series for H(F, B) to all orders greater than n, these ambiguities propagate themselves along the series as well, with a contribution from each order. Note that this method takes as input information only the action of the Hamiltonian on the vacuum and dressed single particle states. As with Schweber's calculation, the ambiguity arises because the action of an operator on a set of states of measure zero in the Fock space does not uniquely define its action on the rest of the space. The method of Hearn and McMillan does not give a unique dressing operator D , but rather a continuous infinity of operators all of which satisfy the given constraints, and of which none is selected by the method as the correct dressing. These ambiguities seriously reduce the utility of this method. It was hoped that the dressing of a simple Hamiltonian such as the one considered above would provide information on the form of a 2 f ) 3 potential, as in general a dressed Hamiltonian would provide terms of the form (F^FB^B^B + h.c). However, the ambiguities in the Chapter 7. The Dressing Transformation: A Critique 72 dressing operator translate directly into ambiguities in the interaction terms of the dressed Hamiltonian. We can thus extract no useful information about the particle creation potential using this method. If we were provided with more constraints on the dressing operator, we would be able to at least reduce its solution space, if not select out a unique one. However, we have been unable to think of any other constraints on the operator that would provide such a reduction of the solution space. It is possible that such constraints do in fact exist. Not taking into account the ambiguities arising from the solutions to the homogeneous commutator equations, Hearn and McMillan were able to show that the dressing of the simple Hamiltonian above, in combination with some results from the MIT Cloudy Bag Model of the nucleon (Theberge, Thomas, and Miller [Th 80]), yielded an interaction potential between dressed fermions equal to the One-Pion Exchange Potential (OPEP) familiar from intermediate energy nuclear theory. As well, the algorithm produced a dressed theory from a simple field theoretic model in which the dressed fermions interacted via a Yukawa potential. If more constraints on the dressing operator could be determined, then, the method of Hearn and McMillan could be extremely useful in determining from the Hamiltonian expressed in terms of the bare excitations the nature of interactions between dressed particles of a field theory. This method is discussed in greater detail and a number of results are presented in the theses by Hearn [He 81] and James [Ja 82]. Thus, we were unable to determine a form of the particle-production potential using a dressing transformation approach. As mentioned in the previous paragraph, if one could learn to keep the ambiguities of this method under control, one could in principle calculate such a potential. There is still much work to be done towards this end. Chapter 8 Summary and Conclusion This completes our discussion of relativistic quantum mechanical models with a finite number of degrees of freedom. In Chapter 2, we introduced the Poincare Group, the set of flat spacetime symmetries, and considered its representation theory. We found that an irreducible representation of the Poincare Group on the Hilbert Space of states of a physical system is characterised by two real numbers, corresponding the mass and spin of the system. The model considered in Chapter 2 contained no internal degrees of freedom, so it was a description of a single particle. To describe a system of two free particles in Chapter 3, the tensor product of two spaces of the kind considered in Chapter 2 was taken. It was shown that the represen-tations of the Poincare Group on this space are fully reducible and that the space can be written as a direct integral of irreducible subspaces characterised by the pair of real numbers corresponding to system mass and spin, once one has determined the Clebsch-Gordan coefficients of the Poincare Group. This process involved a change of variables into internal and total variables for the system, and it was found that the mass and spin of each of these subspaces was determined by internal variables of the system. It was then desired to modify this construction so that we could consider systems in which the two particles interact. This led to the introduction of the Bakamjian-Thomas construction, in which interactions are included in a Poincare -invariant manner by modifying the mass operator. Chapter 4 described a similar construction on the Hilbert Space of physical states of a 73 Chapter 8. Summary and Conclusion 74 system of three bodies; it was found that this construction was similar to that of Chapter 3. The most important development of Chapter 4 was the introduction of the ideas of macroscopic locality and of packing operators, developed further in Appendix F. It was discussed that if one is not careful in including interactions among the three particles, then isolated systems will not behave independently as physical intuition would say they must. This problem was resolved using the packing operators, unitary operators which map from representations which do not obey the constraint of macroscopic locality to those that do. We introduced the Faddeev Equations in Chapter 5, first considering a mass operator partitioned in a very general manner, and then specializing to the case of three particle scattering in which the interaction was partitioned into three two-body potentials. It was found that the multi-channel T-matrices of the system can be determined from matrix operator equations which take as input the T-matrices corresponding to the partition potentials. These matrix equations were solved formally using a Born series solution, and a diagrammatic representation of these solutions was introduced. These solutions were discussed in terms of a naive mechanical model. Such a discussion of the Faddeev Equations does not seem to appear elsewhere in the literature. Chapter 6 introduced a model in.which the Hilbert Space contained both two- and three-body sectors, originally independent and then connected with a potential term that allowed the possibility of particle creation and annihilation in the system. It was pointed out that in general the mass operator and the number operator for the system will not commute; the system will not have, in general, simultaneously sharply-defined mass and particle number. We then discussed the Faddeev Equations for this system, again con-sidering the Born series solutions and interpreting these in terms of a simple mechanical model, and it was seen that the particle production and annihilation potential had an ef-fect on pure two-body and three-body scattering. This slightly unorthodox construction Chapter 8. Summary and Conclusion 75 seems original, and we were unable to find such an approach in other discussions of the subject. Finally, in Chapter 7, we discussed the dressing transformation, used to determine the physical single-particle states of an interacting field theory. It had originally been hoped that this transformation could be used to determine a form for the potential coupling the two- and three-body sectors of the Hilbert Space considered in Chapter 6. However, it was found that this method is fundamentally ambiguous. The method involves an attempt to determine an operator, defined on the entire Fock space, which satisfies certain conditions on the vacuum state and one-particle states. However, this condition is defined on a set of measure zero in the Fock space, and thus the desired operator is not uniquely defined. This lack of a uniquely defined dressing operator prevents this method from being useful for the purpose of determing the 2 f> 3 potential. The formalism developed in this thesis is of use primarily in the area of intermediate-energy nuclear and particle physics. In particular, the paradigmatic 2 f> 3 body system is that of two nucleons and a pion which can be created or annihilated. The NN-NN7T system is one which has been studied in some detail, both theoretically and experimentally. Betz and Coester [Be 80] and Betz and Lee [Be 81] have studied this system and produced a formalism that, while it bears some similarity to that considered in this thesis, is in several fundamental aspects quite different. We feel, however, that many of the constructions in this formalism are unnecessarily complicated, and that the construction developed in this thesis is much more transparent. The NN-NN-7T system was also considered by Hsieh [Hs 78], who studied an unortho-dox Hamiltonian in which the fundamental interaction was a pentalinear vertex of form F t i ? t F F 7 3 t + h.c. Although this method does not have fundamental predictive power, it could be useful in a phenomenological theory, and a comparison of its predictions with those of the formalism in this thesis would be valuable. Chapter 8. Summary and Conclusion 76 The work in this subject is by no means finished. This thesis simply establishes a formalism, the utility of which must be tested through an attempt to make concrete predictions for a given physical system and then to compare these to experimental data. There is much more work that can be-done in this direction. As well, much more work needs to be done on the formalism itself. We were unable to determine a form for the particle annihilation potential V from field-theoretic considerations; such an explicit mathematical expression for the potential would be of great use in understanding the 2 <r> 3 system. Perhaps this can be achieved by further considerations of the dressing transformation and the establishment of more constraints to eliminate the inherent am-biguities; perhaps an entirely different approach will need to be taken. Whichever is the case, the work presented in this thesis could easily serve as the springboard to much more work in the field of intermediate energy nuclear and particle physics. References [Ba 53] B. 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Polyzou, Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics, preprint (1991). [Mc 82] M . McMillan, Selected Topics From the Quantum Theory of Scattering, unpublished lecture notes, University of British Columbia (1982). [Sc 61] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, (New York; Harper and Row, 1961). [So 78] S.N. Sokolov, english trans.,Theor. Math. Phys. 3 6 , 682 (1979). 77 Chapter 8. Summary and Conclusion 78 Originally published 1978. [Th 80] S. Theberge, A . W . Thomas, and G. A. Miller, Phys. Rev. D22, 2838 (1980). [Tu 85] W . K . Tung, Group Theory in Physics, (Singapore; World Scientific, 1985). Bibliography [1] Bakamjian, B . , and Thomas, L . H . , 1953, Phys. Rev. 92 1300. [2] Betz, M . , and Coester, F. , 1980, Phys. Rev. C21 2505. [3] Betz, M . , and Lee, T.-S.H., 1981, Phys. Rev. C23, 375. [4] Birman, M . , 1962. Dokl. Akad. Nauk USSR 143, 506. [5] Coester, F . and Polyzou, W.N. , 1982, Phys. Rev. D26, 1348. [6] Dirac, P . A . M . , 1949, Rev. Mod. Phys. 21, 392. [7] Fleming, G.N. , 1966, Phys. Rev. B137, 188. [8] Hearn, D., 1981. M.Sc. Thesis, University of British Columbia. [9] Hearn, D., McMillan, M . , and Raskin, A. , 1983, Phys. Rev. C28, 2489. [10] Hsieh, W.W. , 1978, M.Sc. Thesis, University of British Columbia. [11] James, H.N. , 1982 M.Sc. Thesis, University of British Columbia. [12] Johnson, J.E:, 1969, Phys. Rev. 181, 1755. [13] Jordan, T.F. , 1969, Linear Operators for Quantum Mechanics, Robert E. Krieger Publishing Company, Malabar. [14] Jordan, T.F. , 1980, J. Math. Phys. 21, 2028. [15] Kalyniak, P.A., 1978, M.Sc. Thesis, University of British Columbia. [16] Kato, T., 1965. Pacific J. Math. 15, 171. [17] Keister, B .D. , and Polyzou, W . N . , 1991, Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics, preprint. [18] McMillan, M . , 1992, Quantum Leaps and Bounds: Elements of Intermediate Quan-tum Mechanics. Physics 500 Lecture Notes, University of British Columbia. [19] McMillan, M . , 1982, Selected Topics From the Quantum Theory of Scattering , un-published lecture notes, University of British Columbia. 79 Bibliography 80 [20] Pryce, M.H.L. , 1948, Proc. Roy. Soc. A150, 166. [21] Schweber S.S., 1961, An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York. [22] Sokolov, S.N., 1978. English Translation, 1979, Theor. Math. Phys 36, 682. [23] Taylor, J.R., 1972. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Robert E. Krieger Publishing Company, Malabar. [24] Theberge, S., Thomas, A.W. , and Miller, G.A. , 1980, Phys. Rev. D22, 2838. [25] Tung, W . K . , 1985. Group Theory in Physics, World Scientific, Singapore. [26] Wigner, E.P., 1939, Ann. Math. 40, 149. Appendix A Lorentz Transformations of Spin Operators C o n s i d e r the t r a n s f o r m a t i o n proper t ies o f the sp in opera to r i n t roduced i n C h a p t e r 2. W e showed tha t under a L o r e n t z boos t , the sp in rotates w i t h a W i g n e r r o t a t i o n , * 7 f ( A , a ) S ^ ( A , a) = i ? c ( A , Q)*V (A.l) T h i s resul t can also be seen i n another , more i l l u s t r a t i ve manner , w h i c h th is A p p e n d i x w i l l de t a i l . T h i s c a l c u l a t i o n is t aken i n par t f r o m K a l y n i a k [ K a 78]. In t e rms o f the P o i n c a r e Genera to r s , S is g iven by 1 p f p . x\ S = — (HJ - P x K) - I , J > ( A . 2 ) T h e c o m m u t a t i o n r e l a t ion (2.48) between the componen t s o f S guarantees t ha t i t t rans-forms as a 3-vector opera tor under ro ta t ions . W e w i l l now de te rmine the effect o f a L o r e n t z boos t on the sp in opera tor . F i r s t define S(u, 1) such t ha t S(u,l) = L1(u)SLlt(u) ( A . 3 ) T h e n we have tha t -^S{u,l) = -iLl(u)[K1,S]L1\u) ( A . 4 ) W i t h some work , it c an be shown tha t the c o m m u t a t o r is g iven by: [K\S] = w^-M(e1xP)xS . ( A . 5 ) 81 Appendix A. Lorentz Transformations of Spin Operators 82 where ei is the 3-vector operator (1,0,0). If we define P(u, 1) and H(u,l) analogously to S ( M , 1), SO that P{u, 1) = L^ujPL^iu) = (P 1 cosh(u) - / 7 s i n h ( « ) ) e i + P 2 e 2 + P 3 e 3 (A.6) and H(u, 1) = L^i^HL^iu) =Hcosh(u) - P 1 sinh(u) (A.7) where e 2 = (0,1,0) and e 3 = (0,0,1), then we have the differential equation for S ( u , 1): ^ S ' " - 1 » = g ( n , l ) + M ' e i X P ' X S < A - 8 » To obtain the above equation we have used the fact that e i x P ( « , 1) = d x P (A.9) which can be seen by considering the explicit form of P(u , 1) in (A.6). Now define the vector operator N = ^ | T (A.10) |e-ixP| v ; and the scalar operator n(u) defined through the differential equation dn lei x P l j{u) = r T ' ' A . l l duK ' H(u, 1) + M v 1 with the initial condition 77(0) = 0 (A.12) Then we have that | - S ( « , l ) = ^ x S ( U , l ) (A.,3) One can verify by explicit calculation that the solution to this equation with the boundary condition S ( 0 , 1 ) = S (A.14) Appendix A. Lorentz Transformations of Spin Operators 83 i s S ( « , l ) = Scos(?7) + (1 - cos(?7))(N-S)N + sin(?7)(NxS) (A.15) This expression has a simple interpretation: S(«, 1) is simply the vector obtained by rotating S around the axis parallel to N(u) through the angle n(u). Of course, both the axis and angle of the rotation are operator valued in the above equation, but as N(w) and rj(u) commute with each other and all of the components of S, this does not affect the interpretation. This rotation of the spin induced by a Lorentz boost is known as the Wigner Rotation, as introduced in Chapter 2. To finish the calculation, then, we must solve the differential equation for rj(u). Defin-ing the new variable z = eu, we can rewrite the equation for n as dn{z) | e ixP | dz \(H -P^zt + Mz + KH + P1) This can be simply integrated to give (A.16) where we have used the boundary condition n(0) = 0. With some work, and remembering the trigonometric identity . n . tan(a) + tan(/5) . . tan (a + B) = X—L— A.18 v ; 1 -tan(a)tan(/3) v ; one can show that r)(u) is given by the much simpler expression fl \ l e i x P l tan -77 = ! (A.19) V2 J coth(f ) ( / f+ M ) - e r P Thus, under a Lorentz boost, the spin operator of a system is rotated about an axis perpendicular to both the direction of the boost and the system momentum. If the boost is along the direction of motion, no rotation at all occurs. Appendix A. Lorentz Transformations of Spin Operators 84 The 50(3) representative of the Wigner rotation, ie, the matrix Rc(r](u)) such that S(u, 1) = Rc(r)(u))S, is given by: ^ cos(?7) P 2 s i n f a ) | e i x F | i P 2 s in(? j ) \ | e l X P | — P2 s in( t )) - P 2 P 3 ( l - c o s f a ) ) | e i X P | 2 — P 3 sin(?)) | e i x P | - P 2 P 3 ( l - c o s ( ? ) ) ) | e i x P | 2 \ * -i / / v - L I | e i X P | 2 / (A.20) Appendix B The Newton-Wigner-Pryce Position Operator To help make the connection with non-relativistic quantum mechanics, it is useful to introduce a 3-vector operator, formally canonically conjugate to the 3-vector momentum operator, which we will call the centre of mass position operator. This operator is also known as the Newton-Wigner-Pryce position operator. Denoting this operator by X\ it is defined to be Hermitian and to satisfy the commutation relation: It is also defined to be a 3-vector under spatial rotations, change sign under parity transformations, and be invariant under time reversal. As the fundamental operators on our Hilbert space are the generators of the Poincare Group, we wish to express X1 in terms of the set:{i7, P, J, K}. The unique triplet of operators that can be built from this collection which satisfies the above conditions is (Jordan [Jo 80]): Calculation shows that this these three operators are indeed Hermitian, and satisfy the commutation relations: [X\P>] = i5ij (B.l) (B.2) [J>\Xk] = iejUXl (B.3) We define Vj, the j-component of the velocity operator, by [X\H] = iVj (B.4) 85 Appendix B. The Newton-Wigner-Pryce Position Operator 86 When the operators are expressed in the Heisenberg representation, the above commu-tation relation is equivalent to the statement; Vj(t) = !*'•(<) (B.5) Evaluating the commutator explicitly, one finds that Pj V> = — (B.6) as one would expect for the centre of mass velocity for an isolated system. The commu-tation relations (B.3) guarantee that X transforms as a 3-vector operator under spatial rotations, eg: R\e)XlB}\6) = X1 (B.7) R\6)X2PJ\6) = X2 cos(0) + Xz sin(0) (B.8) Rl{e)X*R}\0) = -X2 sin(0) + X 3 cos(0) (B.9) where R1(6) is a rotation through the angle 9 about the 1-axis: R1(9) = exp(—iJl9). Because the velocity operator commutes with the Hamiltonian, the time evolution of the Heisenberg representation position operator is given by: Xj{t) = Xj(0) + VH (B.10) = X j + VH (B . l l ) The Lorentz boost operator along the direction j , IA(u), is defined by Lj{u) - e~iKJU (B.l2) where u = t a n h - 1 (^j is the rapidity, v being the relative speed of the frames related by the boost. The transformations of the 3-velocity components under a Lorentz boost in the 1-direction yield familiar results: Appendix B. The Newton-Wigner-Pryce Position Operator 87 VJ — v L 1 ( u ) y 1 L 1 t ( „ ) = _ _ - ( R 1 3 ) L i ( „ ) V ^ t ( u ) = 0 V 1 ) ( B . i4 ) where 7 = (1 — ?;2)~2 = cosh(zi). These are just the quantum analogues of the Einstein velocity transformations familiar from classical special relativity. Somewhat more complicated (and more interesting) is the behaviour of the position operators under Lorentz boosts. Consider the case of a system of spin zero and nonzero mass M. In this case, we know that WflW^ = 0 and so the spatial part of W ' is given by W = ff^M^W0 (B.15) Thus, as [W1*, P") = 0, we have that P x W = 0, so In the Heisenberg representation, these are the components of the position operator at t = 0. One can show, with a little work, that the transformations of the components of the position operator are: 7 27 I ' 1 - Vlv L\u)XW{v) = ^ ' + ^ { ^ 7 7 ^ ; } ti*l) (B.17) These results look surprising at first. In classical special relativity, the coordinate transformations associated with a boost of rapidity u along the 1-direction are: x° = cosh(w):r0 - s i n h ^ x 1 (B.18) Appendix B. The Newton-Wigner-Pryce Position Operator 88 xl = — sinh(?i)a;0 + cosh(u)xl "2 2 X = X x3 = x3 (B.19) (B.20) (B.21) so the time coordinate and the spatial coordinate along the 1-axis are mixed, while the coordinates perpendicular to the boost direction are unchanged. However, in the quantum case, while we can associate operators with the 3-vector position coordinates, we cannot do so with the time coordinate. In Hamiltonian quantum mechanics, time is just a c-number parameter, and not an observable in the same way as is the centre of mass position. Thus, the boost of a position operator cannot even be formally the same as that of a classical position coordinate, and this is indeed seen to be the case. Most striking is the fact that the components of the position operator perpendicular to the boost direction are not invariant under the boost. The physical interpretation of these results is straightforward, however. We will try to understand them in terms of a classical analogue. Consider the world-line, £, of a free classical particle of 4-velocity V 4 , and consider the 3-plane of constant coordinate time x° = t in some frame {x*}. Let r be the intersection of this plane with £. Consider now the coordinate system {x^} related to {x^} by: Ul,x" (B.22) u. = \ (B.23) where / 7 —wy 0 0 —wy 7 0 0 0 0 1 0 0 0 0 1 Define by r' the intersection of the plane x° — t with the worldline £, where t is the same coordinate time as for the first plane considered. In general, r and r ' will not coincide. V ) Appendix B. The Newton-Wigner-Pryce Position Operator 89 If we define by £' j(G:) the equation of f in the frame {£' '}, then in this frame the coordinates of the points r and r ' are related by: ev) = w + ^ V V e V ) ) (B.24) d^ = L^C(r) + ^ (t-LlC(r)) (B.25) d£v = KC(r) + ^ ( ( 1 - 7 ) ^ + 7 ^ ( 0 ) (B.26) The / i = 0 component of this equation just gives £ ° ( r ' ) = t, which we used as input. However, for I V ) = -TttfV) + 7*V) + Y Z ^ i i 1 - i)t + 7<V)) (B.27) where we have used the classical Einstein velocity transformation: V1 - v with dP VJ = ^7 (B.29) dt After some algebra, and remembering that xj{r) = xj{r0) + VH (B.30) where r 0 is the point where £ intersects the plane £° = 0, we obtain: U r } - — + 2^ | e ( T o ) ' T ^ ; ) + 7 3 7 ^ (B.31) But, from the above relations, we know that the action of a Lorentz boost along the 1-direction on Xl(t) is: Appendix B. The Newton-Wigner-Pryce Position Operator 90 Ll(u)Xl(t)Ll\u) = Ll{u){Xl + VH)Ll\u) (B.32) 7 27 \ ' 1 - VH + 1 - V 1 V1 - v v t (B.33) This equation is identical in form to equation (B.31), and allows us to give a physical interpretation to the Lorentz boost of the position operator. The action of the unitary transformation L1 on the position operator X is not to effect a Lorentz transformation on the coordinates of a point on the worldline of the centre of mass of the system. Instead, the result is the position of the centre of mass in the new frame at the transformed coordinate time equal to the coordinate time in the original frame. Note that the parametric role played by the time variable is maintained under the transformation, which is not surprising. The equivalent calculation for the case of a system with nonzero spin is much more complicated, and we have not been able to construct a classical analogue. Of course, a classical analogue should not be expected to exist: nonzero spin is a purely quantum phenomenon, and in general one will not be able to understand the behaviour of a quantum system in terms of a classical one. Attempts have been made to define a 4-vector position operator by defining a time op-erator and imposing the appropriate transformation properties (Johnson [Jo 69], Fleming [Fl 64]) but these are rather complicated and tend rather to obscure rather than clarify issues. The above calculations provide a useful and physical interpretation of the Lorentz transformation of the Newton-Wigner-Pryce position operator of a spinless particle. Appendix C Forms of Relativistic Dynamics, Lorentz Boosts, and Spin There is little ambiguity about the nature of dynamical evolution in a non-relativistic system. The inital conditions are specified on a surface of constant time and from this the equations of motion generate the configuration of the system on any other such surface. It was pointed out by Dirac [Di 49], however, that in a relativistic dynamical theory, the situation is more ambiguous. One is not compelled to specify the initial conditions on a surface of constant time; indeed, any smooth 3-dimensional surface, f(x^) = a, in Minkowski space can serve as a satisfactory initial-value surface, if the family of surfaces corresponding to different values of a do not intersect and if the family of surfaces fills spacetime as a sweeps through its allowed range of values. For any such surface, certain of the Poincare Transformations will leave the surface invariant; the generators associated with these are said to be kinematic. The other trans-formations will take points away from the surface to neighbouring surfaces in the family; the generators of these transformations are called generalised Hamiltonians. In an inter-acting system, the interaction terms will be contained in the generalised Hamiltonians. For surfaces of sufficient symmetry, most of the ten generators will be kinematic. Dirac discussed three such sets of surfaces and the generators associated with them. The most similar to non-relativistic theory of these initial-value surfaces is the constant-time hypersurface described by x° = const. Such an initial-value geometry defines the instant form of dynamics. Clearly, such hypersurfaces are left invariant by spatial ro-tations and translations; the components Pj of the 3-momentum and Jj of the angular 91 Appendix C. Forms of Relativistic Dynamics, Lorentz Boosts, and Spin 92 momentum are thus the kinematic variables. The Hamiltonian and Lorentz booster op-erators generate transformations which take points away from the constant time slices (the Hamiltonian maps each constant time surface into another such surface); they are the generalised Hamiltonians. The point form of dynamics corresponds to taking the initial value surface as the set of points equidistant from the origin x^x^ = const. This surface is clearly left invariant by Lorentz transformations; the kinematic variables are then the components of the Lorentz booster and the angular momentum vectors and the generalised Hamiltonians are the components of the 4-momentum. The third initial-value geometry considered by Dirac is the light front xQ + x3 — 0. This is in fact the most symmetric of the three geometries; the subgroup of the Poincare Group that leaves it invariant is a 7-parameter Lie Group, whereas the corresponding subgroups for the previous two geometries were 6-parameter Lie Groups. The following generators are kinematic in the front form of dynamics: K \ J 3 , E \ E 2 , P \ P 2 , andP+ (C.l) where El = Kl + J2 E2 = K 2 - J l P+ = P ° + P 3 (C.2) and the generalised Hamiltonians are the generators p i = K l - J 2 F 2 = K 2 + J l p - = p 0 _ p 3 (C.3) Appendix C. Forms of Relativistic Dynamics, Lorentz Boosts, and Spin 93 Working in the front form, instead of using the usual Minkowski coordinates a'' of a 4-vector, it is more useful to consider the front-form coordinates a + = a° + a 3 (a^al) = (a 1, a2) a- = a 0 - a 3 (C.4) Written in terms of these coordinates, the vector a has magnitude a^a^ = a + a ~ - sc\ (C.5) In principle, one could use any of these forms of dynamics to solve a physical problem; in practice, the problem under consideration will suggest one of the forms as the most useful. Consider now the canonical boost operator LC(Q) introduced in Chapter 2. It was defined so that it mapped the rest state 4-momentum operator ( M , 0,0,0) to the 4-momentum operator P = QM = ( V P 2 + M 2 , P ) . As was mentioned at the time, this definition is not unique; the vector (1,0,0,0) is invariant under spatial rotations, so any operator Lg(Q) = Lc(Q)R(a, (3,7) will produce the same mapping, where R(a, p, 7) is an arbitrary rotation through the operator-valued angles a,/3,and 7 . One can then define several different kinds of boosts; here we will discuss two. The action of the canonical boost LC(Q) is given explicitly by / AO \ / AO \ ( A M / A c = LC(Q) V A J { where A ' ' is an arbitrary 4-vector operator. This operator is useful as it corresponds to the familiar Lorentz boost from elementary special relativity. V I + Q 2 A° + Q • A , , I (C6) V A + QA° + (1 + V / T T C ? ) - 1 Q ( Q • A) Appendix C. Forms of Relativistic Dynamics, Lorentz Boosts, and Spin 94 One can also define a front-form boost Lf(Q) by its action on the arbitrary 4-vector operator (A+,A±,A~): ( A + \ ( A + \ ( A+ \ A ~ J = Lf(Q) AJL A~ Q+A+ (C.7) V (Q+)"1(Qi^ + + 2Q± • A x + A~) J This satisfies the condition that it maps the rest-state 4-momentum (in front form coor-dinates) ( M , 0, 0, M) into the 4-momentum (MQ+, MQ±, MQ~). It is denoted a front-form boost because it leaves invariant the light front characterized by x+ = 0. Unlike the canonical boost, its inverse is not obtained by replacing the argument (VI + Q2, Q) of the boost by + Q2, -Q). Rather, the inverse Lf1(Q) has the action on the arbitrary 4-vector operator A: < A+\ ( L7 (Q) A~ ( Q + ) " M + \ (C.8) A j . - (Q+)-1Q±A+ \ (Q+)-1Qi^ + - 2 Q ± • Aj_ + Q+A- ) Other Lorentz boosts may be defined; this is further discussed in Keister and Polyzou [Ke91]. The spin operator S of the system was defined in Chapter 2 as the result of the action of the inverse canonical boost on the Pauli-Lubanski 4-vector (C.9) However, as there are many different kinds of Lorentz boost operators that can be con-sidered, so there are as many different kinds of spin operators. The general spin operator Sg is defined as ( C I O ) Appendix C. Forms of Relativistic Dynamics, Lorentz Boosts, and Spin 95 The magnitude of the spin is the same for all forms: S 5 - S S = S-S = S 2 (C.ll) so the various spin operators are related among each other by rotations, known as Melosh Rotations. This is discussed in more detail in Keister and Polyzou [Ke 91]. Appendix D Normalisation Of the State U(A,a)\p\ > In Chapter 2, we showed that the irreducible unitary representation on our Hilbert Space of the Poincare Group, U(A,a), has the action on the basis states |pA >: U(A,a)\pX >= e - ^ « » i . ^ P ^ ) D W [ i 2 c ( A > g ) ] A V | p A A ' > (D.l) where the c-number normalisation N(p) is needed to ensure that U(A, a) acts unitarily. We will now calculate an explicit expression for N(p). To determine the normalisation N(p), we proceed as follows. If U(A,a) is in fact unitary, then: <pA |p 'A '> = < pA|(7 t(A,a)t/(A,a)|p 'A'> (D.2) =Dlf[Rc(A, q)}< P A a |p ' A a ' >£>« [RC(A, ^ N ^ N ^ f (D'3) Using the normalisation condition and the unitarity of the D^X\R), this becomes N(p)N(p')5(p - p') = N(pA)N(p'A)6(pA - pA) (D.4) As p° = Vp2 + rri2 = a>m(p), it can be shown with some work that Thus, § { p _ p l ) = ^ A S ( p A - p'A) (D.5) N2(pA) = N2(p)^± (D.6) Wm(p) V 7 96 Appendix D. Normalisation Of the State U(A, a)|pA > 97 and so we have A^(PA) W m(pA) N(p) \ l um(p) Since N(0) = 1, the above implies that (D.7) N(p) = (Wm(p) m (D.8) This completes the calculation of the normalisation N(p). Appendix E Clebsch-Gordan Coefficients of the Poincare Group Consider the group G and the irreducible unitary representations U\(G) and U2(G) de-fined on the Hilbert spaces Hi and 7i2, respectively. Then U\(G) <8> U2(G) will be a unitary representation of G on the Hilbert Space % = "Hi® H2. In general, this repre-sentation will not be irreducible on % ; that is, there will be proper subspaces of H which are invariant under the action of the group representatives. If H can be expressed as a direct sum of a number of such spaces, then it is said to be fully reducible. In this case, the restriction of the group representatives to any of these invariant subspaces is itself a group representative which is now irreducible. This process of expressing the product Hilbert Space as a direct sum of irreducible subspaces is equivalent to a diagonalization of the Casimir operators and can be effected by a unitary transformation of the basis used to span % . If {|^- >} is the tensor product basis set and {\<f>j >} is the basis set on which the Casimir operators are diagonal then the numbers < (f>j\ipk > a r e known as the Clebsch-Gordan coefficients of the group. Consider the two-body Hilbert space H{2) discussed in Chapter 3. It is a product space of two one-particle Hilbert Spaces: %(2) = H(mi,Si) ® H(m2, s2) and is spanned by the tensor product basis | p i A i ; p 2 A 2 >= |piAi > ®|p 2 A 2 > (E.l) However, it was shown in Chapter 3 that this basis is not irreducible, and that an 98 Appendix E. Clebsch-Gordan Coefficients of the Poincare Group 99 irreducible basis set is given by: \[lo]ks;pX> = J - ^ — r f d k y ^ ( k ) C / [ L c ( g ) ] | k A i ; - k A 2 > V w m 0 (Pj J x < XiX2{siS2)oma >< mamt(al)s\ > (E-2) Denote the irreducible subspace spanned by these vectors as ri(m(k), .$), where m(k) = u>im(k) + com.2(k). The Hilbert Space H(2) then is a direct integral of these irreducible subspaces %{2) =T dk ri(m(k), s) (E.3) The Clebsch-Gordan coefficients then are just the inner products < piA^; p'2A2|[7cr]A;.s; pA > and it can be shown that these have the explicit form <PiA,1;p2A2|[J0-]A:s;pA> = 6(p - p[ - p'2)^5(k - k[p[, p'2))^ um, {k)iom2{k)tom{k)(p) ^m 1 (pi)w m 2 (p 2 )m(k) In the above equation, x £ D§1 [Rc(Lc(q)i *i)Pg 2[Rc(Lc(q), ^ ) ]^ ( (k) x <. AiA 2 ( s i s 2 )am ( T >< mima(la)sX > (E.4) K = L; l(q)Pi , g = -77T , Q i = — (E.5) m(k) mi • P = K(k)(p),p) , k1 = (umi(k),k) ,and k2 = (com.2(k), - k ) (E.6) It can be shown that the factor (^)^m2(^)a;w(k)(p) (PiKn 2(p 2)m(k) ^ ' ' mi1 appearing in the Clebsch-Gordan coefficient (E.4) is the Jacobian of the transformation from the coordinates (p, k) to (p i ,p 2 ) . We have thus computed the Clebsch-Gordan coefficients for the irreducible unitary representations of the Poincare Group on the product Hilbert space H{2). Appendix F Macroscopic Locality and Packing Operators The concept of locality is of great importance to relativistic physical theories. That operators defined at spacelike-separated points should commute is taken as a postulate of axiomatic quantum field theory, and leads to the spin-statistics theorem of Pauli. However, this constraint cannot be tested directly, as the probing of arbitrarily small regions of space requires arbitrarily high energy experiments, and at any time the range of energies experimentally accessable will be bounded above. One can replace this strong requirement of locality, known as micro-locality, by the weaker, but physically necessary, constraint of macroscopic locality or cluster separability, which dictates that if a system is split up into two or more disjoint subsystems, each of which are then removed to infinite spatial distances from the others, then these subsystems will behave independently. The idea of macroscopic locality can be expressed in a concise mathematical form as follows. Considering a partition of the particles into two sets, labelled A and B , the Hilbert Space itself is partitioned U = HA®HB (F.l) If the two subsystems do not mutually interact, then a representation of the Poincare Group on H is U0(A,a) = UA(A,a)®UB(A,a) (F.2) where UA(A, a) and UB(A, a) are the group representations on the subspaces % A and 1iB, 100 Appendix F. Macroscopic Locality and Packing Operators 101 respectively. The operator which translates subsystem A by the 4-vector, b, is given by TA(b) = UA(I,b)®IB (F.3) and that which translates subsystem B by c is given by TB(c) = IA®UB(I,c) (FA) If U(A, a) is the representation of the Poincare Group on the Hilbert Space of physical states in which subsystems A and B interact, then the model will demonstrate macro-scopic locality if lim \\[U(A,a)-U0(A,a)]TA(b)TB(c)ty>\\ = 0 (F.5) (b—c)^->—oo for all \xb >£ U. Expressed in terms of any Poincare Generator G, macroscopic locality is described by lim \\{G-GA®IB-IA® GB)TA{b)TB(c)\iP > || = 0 (F.6) In the case of the Hamiltonian, for example, this states that in the limit that the two sub-systems are separated by an infinite distance, that the total Hamiltonian for the system just becomes the sum of the Hamiltonians of the subsystems, as would be expected. In Chapter 4 we considered the construction of representations of the Poincare Group on the Hilbert Space of states of three non-interacting particles, and then proceeded to include interactions using the method of Bakamjian and Thomas. Considering the case in which two of the three particles interact while the third is a free spectator, we found that the operation of embedding the 2-body mass operator interaction into the 3-body Hilbert Space was not uniquely defined, as the two interactions V^p and VB^, while they led to the same bound-state energies and on-shell scattering matrix, were not identical. Appendix F. Macroscopic Locality and Packing Operators 102 The interaction VTP is defined to commute with P 1 2 and P 3 and can depend on internal variables of the two-body system. The Hamiltonian defined with this potential is given by HTP = + MTP + y/Pl + Ml (F.7) which clearly then has the form HTP = H12>Tp ®h + I\2 ® H3 (F.8) The T P Hamiltonian thus trivially satisfies the requirement of macroscopic locality as expressed by equation (F.6). In fact, the matrix elements of HTP are exactly those to which the matrix elements of any representation of the Hamiltonian would have to converge in the limit that the third particle is separated by an infinite distance from the other two. However, it is shown in Keister and Polyzou [Ke 91] that, because of the behaviour of the off-shell matrix elemtents, this is not in fact the case for the matrix elements of the operator HBT, SO the interaction VBT fails to respect the requirement of macroscopic locality. This is a problematic situation, as the B T constructions are the generators needed for the Bakamjian-Thomas construction. To rectify the problem, Sokolov introduced packing operators which map between the T P and B T representations. Because these representations are scattering equivalent and have the same 2-body binding energies, the map between them is unitary. The packing operators, denoted by A are defined so that UTP{A, a) = AUBT(A, a)A] (F.9) Explicit expressions for these operators can be found in Sokolov [So 78] and Keister and Polyzou [Ke 92]. The situation is more complicated for a system in which all three particles mutu-ally interact. As is discussed in Chapter 4, the form of the mass operator (in the B T Appendix F. Macroscopic Locality and Packing Operators 103 representation) on H(3) is MBT = MfT + M2BT + MSBT — 2MQ + V123 (F.10) so that in the limit that one of the bodies in the system, for example particle 1, is separated by an infinite spatial distance from the other two, MBT —> M\. For each interacting pair (ij), we construct a packing operator Ak as described above. From these we can construct a total packing operator, A, on the interacting representation such that UTP{A,a) = AUBT(A,a)A^ (F.ll) is an irreducible unitary representation of the Poincare Group on the space of physical states of an interacting three-body system and satisfies the requirement of macroscopic locality. We require of A that in the limit that particle k is removed to an infinite spatial distance from particles i and j, A —> Ak- In the case that the three particles of the system are different so that the system does not have any symmetry under the interchange of particles, the simple product A = AzA2Ai (F.12) will satisfy these requirements, as it can be shown that in the limit that particle k is removed to an infinite distance from i and j, Ai —>• / and Aj —>• I. The situation is more complicated if there is a symmetry under interchange of identical particles; this is discussed by Sokolov [So 78] and by Keister and Polyzou [Ke 91]. Although the construction of the packing operators is complicated, once they have been obtained for the system the problem of constructing a theory that satisfies the phys-ically necessary requirement of macroscopic locality has been completed. The problem of macroscopic locality and the construction of packing operators for systems with more than three particles is discussed in great detail by Coester and Polyzou [Co 82] Appendix G Overview of Scattering Theory The exact determination of the time evolution of a physical system from a known initial state is in general quite difficult, because, in general, it is difficult to determine the exact eigenvalues and eigenstates of the system Hamiltonian. This problem can be simplified somewhat by concentrating instead on the question of how states of the system in the distant future, after the interactions in the system have ended, are related to those in the distant past, before the interactions began, and not trying to determine the exact behaviour of the system during the time of the interaction. This is known as the S-Matrix approach and will be discussed in this Appendix. The formalism is discussed in more complete detail in Taylor [Ta 72] Consider an N-particle physical system. One can define a partition, a, on the system that groups the particles into na clusters, denoted by <2j. One can then decompose the Hamiltonian as H = JTiHai + Va = Ha + Va (G.l) i=l where Ha. is the Hamiltonian restricted to the cluster a; with all couplings to other-clusters set to zero, and Va is an interaction between the clusters. Each partition a defines a scattering channel in that at an infinite time in the past or future, before or after the scattering has occured, Ha can be considered to be the exact Hamiltonian of the system, describing a situation in which the particles in each cluster interact among themselves, but in which there is no inter-cluster interaction. In particular, one defines 104 Appendix G. Overview of Scattering Theory 105 the asymptotically free channel, numbered 0, by the partition H = H0 + V° (G.2) where HQ is the sum of the free-particle Hamiltonians of each particle in the system H0 = £ ) ^ P ? + m\ (G.3) We then define the Moller operators Qa± by na± = lim eiHte-iHat (GA) These operators have the property that they map between asymptotic and scattering states; if the time evolution of \ip(t)fn > is governed by the Hamiltonian Ha, then &>+\ip(t)in > evolves with the Hamiltonian H in such a way that lim || \mtn > - * W ( * ) ? „ > || = 0 (G.5) We thus define the multichannel scattering operator, Sba, as Sba = ^1 (G.6) which takes asymptotic past states from channel a to asymptotic future states in channel b. Now consider the states \a > and \j3 >, eigenstates of Ha and Hb} respectively: Ha\a>=Eaa\a> Hb\P >= Ebp\(3 > (G.7) Between these states, it is a familiar calculation (Keister and Polyzou [Ke 91]) that Sba has matrix elements < P\Sba\a >=< p\a > 5ab - 2TTiS{Eb0 - Eaa) < p\Tba(Eaa + iO+)\a > (G.8) Appendix G. Overview of Scattering Theory 106 where the operator Tba(z), the T-matrix, is defined by Tba(z) = V° + VbG{z)Va (G.9) where G ^ = jzrji (G.IO) The first term in (G.8) describes forward scattering and is consequently not interesting. The physically important part of the S-matrix comes from the T-matrix term, and so the goal of scattering theory is the determination of the T-matrix. Definition (G.9) is asymmetric in the roles played by the potentials Va and Vb; clearly, as Va 7^  Vb in general, a T-matrix Tab(z) defined with V 6 as the first term will differ from that defined above. However, it can be shown (Taylor [Ta 72]), that the difference between matrix elements of the two operators vanishes on shell, that is, when the parameter z takes on the value of the energy of the system, < 4>a\Tab{E + z O + ) | 0 b >=< # 0 | f ab(E + i0+)\<j)b > ( G . l l ) so the two T-matrices yield the same physical predictions. It is not, in general, an easy task to solve for the T-matrix exactly, and one is usually constrained to find approximate solutions. This is facilitated by the consideration of a set of integral equations satisfied by the operators Tba(z), known as the Lippman-Schwinger equations. To construct these, we must define some notation. Suppose the Hamiltonian is partitioned H = HQ + J2Vt (G.12) i where the channel potentials V, are defined by Va = Ha- H0 (G.13) Appendix G. Overview of Scattering Theory 107 The form the partition will take will be determined by the physical system under con-sideration. From the above, it is clear that Va = J2 V3 (G.14) and that V0 = 0. Now define the operator Ga(z) by Ga(z) = (G.15) la It is clear then that Va = G~1{z) -G~\z) (G.16) Va = G-Q\z)-G-\z) (G.17) and thus that (where the summation convention does not apply to channel labels) G{z) = Ga(z) + Ga(z)VaG(z) (G.18) It is straightforward to show then that Ga(z)Tab(z) = G(z)Vb (G.19) and thus that Tab(z) = Vb + VaG{z)Vb (G.20) = Vb + VaGa{z)Tab(z) (G.21) This is known as the Lippman-Schwinger equation for the operator Tab(z). Once one has determined the resolvent (or Green's) operator Ga(z), Tab(z) can be calculated. One can obtain an explicit, although formal, expression for Tab(z) by noting that (I~VaGa(z))Tab(z) = Vb (G.22) Appendix G. Overview of Scattering Theory 108 so Tab(z) = (I - VaGa(z))-xVb (G.23) Formally inverting (J — VaGa(z)) as a power series, one obtains the Born series solution for the T-matrix: Tab(z) = Vb + VaGa(z)Vb + VaGa(z)VaGa(z)Vb + ... (G.24) One can easily make the connection between the T-matrix and physically observable quantities. It is a standard result that the differential cross section for scattering of a beam of energy E and momentum p from a target into the internal momenta phase space volume element d$N N d$Nd?pdE = FJ d3pi (G.25) i=i where the p; are the final momenta of the particles, is given by da = ^ - | < p i . . . pN\Tba(E + i0+)\ptp > \2d$N (G.26) where pt is the momentum of the target and vP-t is the relative speed of the incident particles and the target. More detail is given in Kiester and Polyzou [Ke 91] and Taylor [Ta 72]. Consider now relativistic Hamiltonian systems as considered in this thesis. The Bakamjian-Thomas construction is such that interactions are explicitly included in the mass operator of the system under consideration. It is thus convenient to construct the scattering theory in terms of this operator, which is a non-trivial function of the Hamil-tonian. It was shown by Birman [Bi 62] and Kato [Ka 66] that if / is some sufficiently smooth function of the exact system Hamiltonian H or of the channel Hamiltonian Ha, then, defining the generalised Moller operators naf. = lim e i f W t e~ i f { H a ) t (G.27) Appendix G. Overview of Scattering Theory 109 and the generalized S-matrix 57,6a = fi/L^"+ (G.28) then 57,6a = Sba (G.29) An heuristic derivation of this powerful and important result, the Kato-Birman invari-ance principle, is given in Keister and Polyzou [Ke 91]. It has the consequence that we can formulate our scattering theory in terms of the mass operator and not the Hamilto-nian. This has the added benefit that the scattering operator SM,ba can be characterised in a manifestly Lorentz-invariant manner, so we have thus also constructed a Lorentz-invariant scattering theory. 

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