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Hamiltonian treatment of the (2+1)-dimensional Yang-Mills theory 2007
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Title | Hamiltonian treatment of the (2+1)-dimensional Yang-Mills theory |
Creator |
Brits, Lionel |
Publisher | University of British Columbia |
Date Created | 2011-02-17 |
Date Issued | 2011-02-17 |
Date | 2007 |
Description | The (2+l)-dimensional Yang-Mills theory is studied in the functional Schrödinger formalism using the machinery laid out by Karabali and Nair. The low-lying spectrum of the theory is computed by analyzing correlators of the Leigh-Minic-Yelnikov ground-state wave-functional in the Abelian limit. The contribution of the WZW measure is treated by a controlled approximation and the resulting spectrum is shown to reduce to that obtained by Leigh et al., at large momentum. The inclusion of fundamental Fermions is done from first-principles, and it is found that the requirement of gauge invariance spoils the commutativity between gauge and matter fields. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-02-17 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084933 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/31435 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0084933/source |
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Hamiltonian Treatment of the (2+l)-dimensional Yang-Mills Theory by Lionel Brits B.Sc. (Hons), Memorial University of Newfoundland, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia September 6, 2007 © Lionel Brits 2007 Abstract The (2+l)-dimensional Yang-Mills theory is studied in the functional Schrodinger formalism using the machinery laid out by Karabali and Nair. The low-lying spectrum of the the- ory is computed by analyzing correlators of the Leigh-Minic-Yelnikov ground-state wave- functional in the Abelian limit. The contribution of the WZW measure is treated by a controlled approximation and the resulting spectrum is shown to reduce to that obtained by Leigh et al., at large momentum. The inclusion of fundamental Fermions is done from first-principles, and it is found that the requirement of gauge invariance spoils the comrnu- tativity between gauge and matter fields. ii Table of Contents Abstract 1 1 Table of Contents iii List of Tables vi List of Figures v " Acknowledgements v m Dedication 1 X 1 Background 1 1.1 Introduction 1 1.2 Yang-Mills theory 2 1.2.1 "Old" Canonical Quantization • • • 2 1.2.2 Gauge-independent degrees of freedom 4 1.3 Karabali-Nair formalism 7 1.3.1 2+1 Dimensions 7 1.3.2 Gauge Symmetry 9 1.3.3 "Holomorphic" Symmetry 10 1.3.4 Inner product 10 1.3.5 Evaluation of determinant 12 1.3.6 Hamiltonian 18 1.3.7 Vacuum wavefunctional 23 1.4 Spectrum in the planar limit 24 iii Table of Contents 1.4.1 Mass Spectrum 26 2 Controlled Approximation 28 2.1 Abelian Expansion 28 2.2 Analytic Structure 30 2.3 Discussion 33 3 Matter 34 3.1 Functional Schrodinger Picture 34 3.1.1 Bosonic case 34 3.1.2 Fermionic fields 37 3.2 Yang-Mills with Fermions 40 3.2.1 Holomorphic invariance 41 3.2.2 Non-canonical behaviour 42 4 Conclusion 44 4.1 Future work • 45 Appendices A The Adjoint Representations 46 A . l Group representation 46 A.2 Algebra representation 48 A. 3 Casimir Invariant 49 B SU(N) 51 B. l Completeness 51 B.2 Adjoint representation 52 B.3 Quadratic Casimir 52 Table of Contents C B e s s e l f u n c t i o n identi t ies 54 C l Proof of ̂ = i + 2 z £ £ i p r _ 3 r 54 C.2 Proof of J g = 2 , E r = i ^ ' 5 5 D F r e e B o s o n in ( 2 + l ) - D i m e n s i o n s 56 E Q u a n t u m n u m b e r s 57 B i b l i o g r a p h y 58 v List of Tables 2.1 Comparison of 0++ glueball masses given in units of string tension yfa ~ 32 vi List of Figures 1.1 Closed contour Px>x supporting Wilson loop 5 1.2 Complexified Wilson loop 8 1.3 Complexified Wilson loop H 9 2.1 Comparison of K(y) (dashed line) and IC(y) (solid line) 31 2.2 Zeros of J2 (grey) and Jo (black) 33 vii Acknowledgements I would like to thank Moshe Rozali for guidance, as well as insight and many useful dis- cussions. Also, the comments and clarifications made, amongst others, by Peter Orland, Oliver Rosten and Greg van Anders are greatly appreciated. Furthermore, without the wel- comed distraction of some of my peers, in particular James Charbonneau and Kory Stevens, I would most certainly have capitulated to the demand of reality. Finally, no amount of space can hold the gratitude I hold for my wife, Kelley, and for her support during the writing of this thesis. This work was supported by the National Science and Engineering Research Council of Canada. viii Dedication To my wife, Kelley. ix Chapter 1 Background 1.1 Introduction The study of non-Abelian gauge theories is complicated by the fact that by restricting one- self to states in the Hilbert space that are gauge invariant, or, alternatively, representatives of equivalence classes of states related by gauge transformations, one almost necessarily introduces interactions that complicate calculations. In perturbative treatments gauge in- variance is achieved at the expense of the form of the bare propagator of the gauge-boson, or the introduction of ghosts, while in functional treatments Gauss' law must either be imposed as a constraint, or solved implicitly through a change of variables [1], as is also the case in lattice calculations. Manifestly gauge invariant calculations [2-5] may also be done in the Effective Renormalization Group formalism [6-8] which necessitates the intro- duction of heavy regularization machinery. Finally, there exist studies of the anisotropic version of weakly-coupled (2+l)-dimensional Yang-Mills theory in which one dimension is descretized, replacing the continuum theory with infinitely many (integrable) chiral sigma models [9, 10]. One promising solution to the constraint problem is the use of Wilson-line variables [I] , for example in the work by Karabali and Nair on (2+l)-dimensional Yang-Mills theory [II] . The strength of their approach is that the field variables are encoded into a single variable along with a reality condition that allows it to be treated holomorphically, opening up the rich structure of complex analysis. Recent calculations by Leigh, Minic and Yelnikov based on this work have also yielded a candidate ground-state wave-functional as well as approximate analytical predictions for the JPC = 0 + + and Jpc = 0 glueball masses [12] 1 Chapter 1. Background which are in good agreement with lattice calculations. However, the effect of the non-trivial configuration space measure, i.e., the WZW action, was omitted from calculations without sufficient justification. In the following section we begin by reviewing the Hamiltonian quantization of Yang- Mills theory. We then discuss the use of Wilson-line variables to obtain manifestly gauge- invariant degrees of freedom. Subsequently, we review the formalism of Karabali and Nair, and outline the procedure used by Leigh et al. to obtain the vacuum wave-functional and glueball spectrum. In part II we attempt a conservative approximation of the glueball spectrum that incor- porates the WZW functional measure by expanding relevant operators about the Abelian limit. We then compare our results to that of [12] and analyze the robustness of the solution. In part III we develop the techniques necessary for including fundamental matter using the functional Schrodinger picture. We show that in order for gauge-invariant Fermionic fields to be consistent with the overall theory, the matter and gauge fields fail to commute (even classically), and therefore that the transformation is non-canonical. Finally, the feasibility of this approach is discussed. 1.2 Yang-Mills theory 1.2.1 "Old" Canonical Quantization We will begin with a review of the the "old"1 canonical treatment of Yang-Mills theory. Specifically, we will study SU(N) Yang-Mills theory in the Hamiltonian formalism, and denote the gauge potential by A = —iAfta, i — 1...D, where the N x N Hermitian matrices ta generate the su(N) Lie algebra [ta,tb] = ifabctc and we choose tr(tatb) = \5ah, while the gauge covariant derivative in the fundamental representation is D; = <9; + Ai. The classical 1 By "old" we mean that the modern BRST formalism is not used to implement first-class constraints 2 Chapter 1. Background action of the theory is then ~>YM 2 5 2 \ - fd3x trF^F^, YM J (1.1) where F^ — [Z)M,D„] = d^Av — dvA^ + [An,Au] is the field strength. The action can be written in a form that simplifies the process of obtaining the Hamiltonian, by by treating F^ as an auxiliary variable [13]: S[A,F]^—£- dzx tr -F^F^ - F^ (d^ - + [Ap, A„\) (1.2) "Integrating out" F^ shows that it equals [D^,DV], which can be substituted into the action (1.2) to obtain (1.1). More precisely, that what we are doing is quantum mechanically sound can be seen by considering the ordinary (Euclidean) integral JdFdye~^F2~zFY^ = V5ir fdye-±YM2 = V2^fdFdye~^F2-iFY^S(F + iY(y)). Defining E\ = F0i and B = ^eijFij (forshadowing the restriction to 2 dimensions), the action becomes S = — f ^ i t r [B2 + EiEi - 2Ei (d0Ai - 3iA0 + [A0, AA)] , 9YM J \B2 + EiEi - lEih - 2 / l 0 (diEi + [Ai, Ei}) 9YM dfx tr (1.3) after integration by parts. Taking the trace gives 9YM d3x _± (BaBa + EIEI) + El At + AaQ (A^<)° (1.4) This shows that Ei is the canonically conjugate momentum to Ai, and we shall take the equal-time Poisson bracket to be A?(x), EbAy) = SijS^S^ix - y). Canonical quanti- PB zation now proceeds by obtaining the canonical Hamiltonian and making the replacement .' iPB ih[, ]. The Hamiltonian is then / 2 1 1 d2x9j_KEaEa + _ ^ ^ B a B a _ (DiEi)" . (1.5) 2 2gYM gYM 3 Chapter 1. Background In the Hamiltonian formalism the component 4̂o is used up as a Lagrange multiplier in enforcing the Gauss' law constraint D • E = 0, and is subsequently ignored. Thus at this point we have yet to fix a gauge, so that the theory is invariant under (time independent) gauge transformations. 1.2.2 Gauge-independent degrees of freedom It is well known that from knowledge of all the holonomies of the gauge connection we can recover uniquely up to a gauge transformation [14]. Define the path-ordered phase U(Px>y) as the phase accumulated by parallel transporting some field ip along a smooth path from y —> x due to the connection A^. Infinitesimally, D^(x)e» = ty{x + e) - U(Px+£^(x)} (1.6) Therefore U(Px+t}X) = 1 - e^A^ + 0(e2), so that U{Px>y) =Ve-$ydxilA». (1.7) where the integral is taken along a curve joining y and x, and V denotes the path-ordering of the exponential [13, 15]. U(PXjy) depends on the path joining y to x, owing to the curvature of the connection. By definition, U(PXiy)ip(y) and ib(x) transform the same way under a gauge transformation, so that the path-ordered phase transforms bi-locally: U(Px,y) —¥ d(xW(^x,i/)5_1(y)- Additionally, the path-ordered phases give an N x N matrix representation of the holonomy group under multiplication, since U(PXly)U{Qylt) = U{PX>y O Qy,Z). (1.8) Given an open path-ordered phase (i.e., Wilson line), we can recover A^ as long dx11 does not vanish along the curve at x. Let Sx>y be a straight line segment joining y to x. 4 Chapter 1. Background Then An(x) = —-Q^U(Sy+X}y)\y = X. : i . 9 ) To reconstruct the connection from holonomies around closed loops (i.e., Wilson loops), first consider the effect of moving around a square spanned by ee1 and ee2, labeled Cx = PXfX (Figure 1.1) Figure 1.1: Closed contour PX;X supporting Wilson loop U(CX) = U(x,x + ee2)U(x + ee2,x + ee1 + ee2) x U(x + ee1 + ee2,x + eel)U(x + ee1 ,x), « exp x exp eA2(x) + — A2>2(x) exp *M{x) + —AlA(x) + e 2Ah2(x) -eA2(x) - e2A2,i(x) - ^A2t2(x) exp -eAi(x) - —Ai,i(x) = 1 + e2 [Al:2(x) - A2,i(x) - Ax(x)A2{x) + A2(x)Ai{x)] + 0 (e 3 ) . Thus, knowledge of the holonomy around an infinitesimal loop gives the curvature of the connection: U(Cx) = l-e2Fl2{x) + 0{e*). (1.10) Knowing the curvature allows us to reconstruct the connection, up to gauge transformations. If A^ is non-singular at the origin, then we can impose the coordinate gauge condition 5 Chapter 1. Background x^A^ = 0, from which x"F^ = x^d^Av - x*dvAp = (1 + x*dp)Av, so that [16, 17] ax^F^ax) = (1 + ax^d^A^ax) = -j^ [aAv(ctx)\, and Afl{x) = C daaxaF^{ax). (1.11) JO The phases U(CX) therefore encode all information about the field. It is conventional to define the object Wcix) = tr U(CX), also called a Wilson loop, which is gauge invariant because of the cyclicity of the trace. It is then possible to reconstruct U(CX) from Wc(x) modulo gauge transformations, although the construction is non-trivial [14]. For the time being, however, we will consider only open path-ordered phases. Note from equation (1.9) that we can also write A» = --^U(SZ+X,Z o SZiy o S~y)\z=X) (1.12) = -^-U(Sz+x,y)U-\SZiy)\z=x. (1.13) The path Sx,y is specific to the index being varied, so define Mfi(x) = U(S^X>V) where S(ii)x,y is a straight line segment directed along dx^. Then A^ = -(d^M^M'1. (no summation) (1-14) Note that we obtain a pure gauge if all the matrices are equal, corresponding to the vanishing curvature of the connection. Although the matrices transform bi-locally, the gauge transformations at the points y do not affect A^, as can be seen from equation (1.14). One may choose to set.y = oo, so that g(y) = 1; however, there is a symmetry that must be dealt with if the matrices € G are to be considered as variables in their own right, defined by (1.14): A^ is invariant under the replacement M M —> MllV(x) provided that 6 Chapter 1. Background d^V(x) = 0. 1.3 Karabali-Nair formalism 1.3.1 2-1-1 Dimensions In 2 spatial dimensions it is convenient to introduce the complex coordinates z = x — iy and z = x + iy. Correspondingly, 8 = £ = %^ = \ (b\ + id2) and 8 = £ = = \ (di —182). In the Hamiltonian formalism, Ao = 0, while the remaining two components of Ap become A = AZ = \ (A\ + iA2) and A = Az = \ {A\ — %A2). In two dimensions, the magnetic field has only one independent component, given by B = * {\Di,Dj]dx% Adx J ) , which becomes = (ib\A2 -id2Ai + [Ai, A2}), = 2^ (di - id2) (At + iA2) - 2 ^ (b\ + id2) (At - iA2) + 2[A, A] = 2(8A-8A) + 2[A,A] (1.15) We can again define matrices Mz and Mz that satisfy A = -(8MZ)M; -1 (1.16) A = -(8Mz)Mil. (1.17) 7 Chapter 1. Background However, solving equation (1.16) iteratively, we obtain Mz(r) = 1 - / dz'A(z', z)M{z', z) (1.18) J —OO / z rz rz1 dz'A(z',z)+ dz'A{z',z) dz"A{z",z)... (1.19) -oo J — o o J —CO = 1 - / Z dz'A(z',z) + ̂ V ̂ dz'j4(z',.z) / 2 d*"^*",*)... (1.20) J—OO J—oo J—oo so that instead Mz G G c , the complexification of G. The apparent over-abundance of degrees of freedom is resolved by the constraint that Az be related to Az by conjugation. i2 Figure 1.2: Complexified Wilson loop MZ The contour giving rise to MZ is shown in Figure 1.2, along with a schematic representation of the "lightcone coordinates" z and z. For "small" gauge configurations the contour may be taken to extend to the point at infinity as indicated. If G = SU(N) then the generators ta are Hermitian, so that A = -A^ = M\~1BMI = - 9 M ] _ 1 M J . Therefore we have the constraint that MZ = M J - 1 and with this understanding we can drop the subscript on MZ. The matrix M is an element of SL(N,C) = G c . 8 Chapter 1. Background Figure 1.3: Complexified Wilson loop H 1.3.2 Gauge Symmetry Under a gauge transformation, M —> M9 = gM, so that H = M^M is a gauge-invariant object. Any SL(N,C) matrix M can be uniquely written2 as M = hp, where h e,SU(N), P] = P, det p = 1. The interpretation of H as a closed Wilson loop is shown in Figure 1.3. Since p is obtained from H = M^M, which is gauge invariant, all gauge information is contained within h. The matrix M is related to p by a constant gauge transformation. Therefore matrices M with the same p lie on the gauge orbit through p, and all gauge-independent information can be extracted from H. It is reasonable to expect that wave-functionals of physical states (which are gauge-invariant) can be written in terms of H. 2Specificically, p = U^y/DU where D = U{MXM)U* is diagonal. 9 Chapter 1. Background 1.3.3 "Holomorphic" Symmetry In addition to the gauge symmetry there is the invariance of A under the replacement M —> MV(z) and Mt —> V(z)M\ discussed earlier. Here V(z) and V{z) are indepen- dent holomorphic and anti-holomorphic function, respectively, and this symmetry is loosely termed "holomorphic" invariance. Under such a transformation, H —• V(z)HV(z), but physical operators and wave-functionals corresponding to physical states should be "holo- morphically" invariant. 1.3.4 Inner product In order to define an inner product between states we need to determine the measure dp(C) on the configuration space C, i.e., the space of gauge potentials A modulo the (small) gauge group Qif. The metric on A is just the Euclidean one, i.e., ds2A = J dxSAaiSAai = -8 Jtr{5Az5Az). (1.22) Recall the parameterization, A = -dMM~1, (1.23) A = M]-ldM]. (1.24) Now, using <9(MM_1) = 0, SA = -(dSM)M-1 - (dM)5M~\ = -d(6MM-1) + 6MdM-1 +8MM-16MM~\ = -d (5MM-1) - 8MM-l{dM)M~l + (dM)M-16MM-\ = -d (6MM~X) + [A, 5MM~l] , = -D (6MM-1) . 10 Chapter 1. Background where D = d + [A, •] acts in the adjoint representation. Similarly, 5 A = SM^BM* + M^-ld5M\ = -M^-l5M^M^~ldM] + B (M^6M^ - (dM*" 1 ) 6M\ = -M^HM^M^BM^ + B{M^SM^ + Mt" 1 (dM^ Af t _ 1(JMt, = d(M^-16M^ - A,M]~l5M^ , = D (M]-15M^ , so that ds2A = 8 Jtr (D [6MM'1] D [ M T _ 1 JM*] ) . (1.25) On the other hand, the metric of SL(N,C) is asSL(N,C) 8 Jtr (6MM- 1) (Aft-^Mf) . (1.26) The Jacobian determinant is therefore det(DD), and M 0 - ^ ^ D D ) ^ t , (1.27) Finally, from the discussion in section (1.3.2), the space of Hermitian matrices (H) with unit determinant is given by H = SL(N,C)/SU(N), so that •wo - d^rv <L28) and dn(C) = det(DD)dfj,(H). (1.29) Up to complexification, det(DD) = det 0 where 0 = jlDi is the two-dimensional Dirac operator. This determinant was evaluated by Polyakov and Wiegmann [18] by investigat- ing the chiral anomaly that occurs when coupling the gauge potential to Fermions in 2 11 Chapter 1. Background dimensions. Instead, we may evaluate (and regularize) the determinant directly. 1.3.5 Evaluation of determinant Define e r = det(DD) where D and D are in the adjoint representation. Considered as matrices, these have the form D^ = Dab6W(x-x'), (1.30) D^^Dab5^(x-x'), (1.31) where Dab = Sabd + facbAc. (1.32) Then r = lndet(DZ)) = trln(DD). (1.33) One approach is to vary with respect to Aa and Aa separately, to obtain a functional differential equation for T. The solution to these equations gives T up to (divergent) Aa, Aa-independent terms. Since 6 (IndetM) = tr (M~l5M) , 5Aa(x) " V sAa{x) = j dr" dr' (D-l)bJ^rbc6W(x-a/)<5(V - x'), (1.36) = fabc(D-^\^x, (1.37) = t r (D^T") \^x, (1.38) 12 Chapter 1. Background where (Ta)bc == —ifabc are the generators in the adjoint representation. We are therefore tasked with determining (and regularizing) (-D_1)__ l ^ - ^ , i-e. solving ^ t " ^ " 1 ) ^ ^ (1.39) = Sac5^{x-x'), (1.40) where Aab = facbAc (it is to be understood that repeated spatial variables are to be inte- grated over). Suppose (£ ) _ 1 )* c x / = MbdGXiaj(M~l)dj for some yet to be determined matrix M. Then (5ahdx + Aab(x)) MbxdGx<x.{M-')<% = [(dM)^ + AfM^G^M'1)^ + 8M8^(x- x1)^ (1.41) which gives the requirement that dMab + AacMch = 0. (1.42) Thus Mah is the solution to (<9 + A)M = 0 in the adjoint representation. With our normal- ization convention for the Lie algebra, we obtain Mab = 2 tr (taMtbM-l^j , (1.43) ( M f ) a b = 2 tr ( t a A f t * 6 M t _ 1 ) • (1.44) Therefore D~1(x, x1) = M(x)G(x, a/)M _ 1(a/), where it is to be' understood that all objects are in the adjoint representation. 13 Chapter 1. Background Regular izat ion The propagator, D~1(x,x'), transforms bilocally, i.e., D~l(x,x') g(x)D~1(x,x')g^1(x'). (1.45) It is, however, holomorphically invariant. We want to consider the coincident limit x' —* x and write D~l(x, x) as a power series in e = x — x' (the limit x' —> x can be taken with- out difficulty). The coefficient D~l(x) of e™ transforms as D~l(x) —> g(x)D~l(x)g~1(x!) meaning that it is not gauge covariant. Therefore, introduce the gauge covariant regulator D-l(x,e) = D-\x,e)fl(x',x), (1.46) where n(a/,aj) = Vexp ^-J dx-A^J, (1.47) parallel-transports between the points x and x" in a guage-covariant manner. Then, let Dmlv{x,e) = M(x)G(x, o!)M-l(a!)e-*s'-s) + 0 (e 2 ) , = M{x)G{x,xl) [M-\x) - M-\x)dM(x)M-1(x)(x' - x)] x [l + A(x-x')] + 0 (e 2 ) , = G(x,a/) [l + dM{x)M~l(x'){x-x')] [l + A{x - x1)] + 0 (e 2 ) , = \ + -[A + aM(x)M-1(x)l + 0 (e 2 ) , f t t so that D-^ar, 6 ) = i [ i + BM(x)M-\x)] . (1.48) The regularization procedure used above is sufficient for determining D~^g, but holo- morphic invariance is not manifest. Therefore, following [11], we may instead regularize 14 Chapter 1. Background G(x, x1) by replacing it with a new function Q(x, x!) that is holomorphically covariant. Let Kab(x,x) = Hab(x) = M t a o ( x ) M c b ( x ' ) be the adjoint representation of H. To be certain, Kab(x,y) equals i J a b (x) only when y = x. Kab transforms holomorphically as Kab{x,x) -> Vac(x)Kcd(x,x)Vdb(x), (1.49) or more generally, Kab(x, y) -> Vac(x)Kcd(x,y)Vdb(y). (1.50) The above requirements are satisfied by the choice <?ab(x, y) = f G(x, w)a(w, y, e) [K~l (y, w)K(y, y)]ab , (1.51) J w £ a b ( x , y ) = / G(x,w)a(w,y, e ) [K(w, y)K~l{y, vj\ * , (1-52) J W | x _ y | 2 where cr(x, y,e) = e tends to the Dirac delta function <!>(2)(x — y) as e —> 0. Thus we damp out short-distance (UV) behaviour by keeping e strictly positive. Let its antiderivative be 5(x) = — l-e~^r + h(x), (1.53) •nx so that dS(x) = cr(x), and h(x) is arbitrary. Also, denote [K~1(y,w)K(y,y)]ab by fab(w). Then, on suppressing indices, £ ( x , y ) = / G ( x , w ) 6 ! U ) [ 5 ( w - y ) ] / ( t D ) ) :/w ^ [ G ( x , w ) 5 ( w - y ) / H ] - f dw[G(x,w)]S(w-y)f(w), i Jw | x - y | 2 -e « /(z). 'w G ( x , w ) ( — ^ e " ^ )/(«;) 7 T ( W - J / ) 7r(x - J/) There is a subtle point that must be considered: G(x) = lim7?^o+ xx+vz • ^n performing the regularization we keep n finite (along with e) and merely analyze the behaviour of the result as ry —-> 0. In the second term above, the derivative leads to a delta sequence, because 15 Chapter 1. Background if r] is small then dwG(x, w) only has support near w « x, so we can approximate the rest of the integrand (which is well-behaved) with its value at this point. Similarly, in the steps to follow, we would expand in powers of n2, in which case we would obtain terms that vanish as r) —+ 0. The first term of £ ( x , w ) is *2L 1 a. l(x -w)(w- y) 1 1 1 _ | w - y | 2 e * j(w) _(x-w-y) e ' f(w + y) (1.54) (1.55) Now dx Ady = — \dw A dw, so upon integration we obtain 2TT2 dw dw d„ 1 1 (x — w — y) w e * f(w + y) 27r2e dw dw 1 (x-w-y) e- — f{w + y). (1.56) We may rotate the integration contour as follows: w —> iw , so that the integral becomes and 7re J (x — w — y) V e 1 s(-)f(w + y) = -^f(y), •n x — y 1 1 7T x — y <? a 6 ( X ,y) = - G ( x , y ) 1 1 tab >-yr •n x — y e £ .Jx-y_ (^" 1 ( ? / ,a;)^( 2 / ,y)) c (l + â -1(y,y)/r(j/,y)(x-y) + ...) afe 16 Chapter 1. Background Then, as y —> x, since K — (K -l\ba After regularization, so that, gab(x>x) = ±-(K-18K)ab (x), Qah{*,K) = ~(dKK-1)ab (x G{r,r)Sab -> Gab{r,r) = i (H~1dH)ah ^IfM-1 (,4 + dMM~l) M]ab. ab (1.57) (1.58) (1.59) So (D- 1 ) r ^(r ,r) = - [ i + 9 M M - 1 ] a b ; ( £ > - % (r,r) = - 4-M^dM* aft (1.60) (1.61) Then, <L4a(r) <5r 2« 7T CAt 7T cAi Ifabcfdbc t r L d ^ + ( , g M ) M - i ) - ^ 2 tr [ta(A + (<9M)M-1)] , <5Aa(r) 2 tr 7T ia(,4 + M^-ldM]) (1.62) (1.63) where is the quadratic Casimir in the adjoint representation, i.e., (TcTc)a^ = —fcadfcdb = CASab, equal to N for SU(N). The solution to these differential equations is the Wess- Zumino-Witten action [19, 20], i.e., T = 2cASwzw(M^M), (1.64) 17 Chapter 1. Background where Swzw[H] = Jd2x tr(dHdH~l) + 777- [d?xe>iuatT(H-1dIJtHH-1b\/HH-1daH). (1.65) 127T J In the second line, the field H has been extended into the third dimension, where the boundary of the extended space is Coo, upon which the physical fields live. For Hermitian models, this can be written as an integral over 2-dimensional space only [11, 21]. Summarizing, the change of variables from (A,A) —> H results in a Jacobian determinant so that the measure on the configuration space C (the space of gauge potentials modulo allowable gauge transformations) is [11] dn(C) = det(DD)dn(H), (1.66) e2cASwzw^df4H). (1.67) det ' (99) n d i m G Jd2x Inner products and expectation values are calculated using this measure, as <tfi|0|#2} = J dp(H)e2cASwzw^*1[H}0^2[H}> (1.68) so that expectation values are essentially correlators of the Euclidean, Hermitian WZW theory. These, in turn, can be found (at least in principle) by solving the Knizhnik- Zamolodchikov equations for the unitary theory and performing an analytic continuation to the Hermitian case [11]. 1.3.6 Hamiltonian In order to quantize the theory in the H variables, the Yang-Mills Hamiltonian H = T+V= 2 0 i jE°;Ef + fBaBa, (1.69) 2 J 2gYM J 18 Chapter 1. Background must be written in terms of a single variable only. Therefore, we will use the fact that, when acting on physical (i.e. gauge invariant) states, the Gauss' law operator I = D • E = 2(DE + DE) vanishes. We can solve for E, as Ea(z) = £ [D-i(z,™)]ab Q / - D2J (w) where DD 1 ( z , w ) ] a b = SabS(z — w). The kinetic energy operator is then T = 2gYM / £ a ( x ) £ a ( x ) = ISYMJ [ ^ W ) (y) £ a (x ) . Since I vanishes on physical states, we move it toward the right, using (1.70) (1.71) l\y)Ea(x) = Ea(x)Ib(y) + [ / b (y) ,£ a (x) = £ a (x ) / 6 (y ) - iJ(x - y)fabcEc(y). (1.72) Therefore, T = - 2 g Y M [ [D-\-K,y)DE{y)]aEa(*)+gYM [ Dah-\^y)Ea{*)Ib{y) J x,y J x,y -2gYM j^-tr[D-l{^y)T\^Ec{^ (1.73) since T£b = —ifabc. Taking the coincident limit in the integrand leads to a divergence, so we must replace D~l with its regularized counterpart. Then 1 - L - tr [D-l(x,y)Ta] - ^ U - M^dM^Y , 2TT (1.74) 19 Chapter 1. Background where c^ab = famnfbmn, s o that the kinetic energy becomes T = - 2 9 y m [ [D-\^y)DE{y)]aEa{*)+gYM j Dab~\x,y)Ea{x)Ib{y) + 2im J [A-M^dM^YEa(yi), (1.75) where m = ^YnLCA is essentially the't Hooft coupling. The potential energy is simply V = ^— j BaBa, 2 9 YM (1.76) where Bata = 2 (OA- dA) + 2[A, A}. The Gauss' law operator is a generator for complex SL(N,C) gauge transformations because, properly, gauge transformations involve g~l, (which equals g^ if g is unitary), which gives rise to the commutator in the infinitesimal. Let g = e9 where 9 is complex and not antihermitian (as for SU(N)). Infinitesimally, g « 1 + 9, g~x & 1 — 0, then A^ =gAig-1-(dig)g-\ ~Ai + 9Ai-Ai9-di0 + O(92), = Ai- Did, = -i(Af-i{Di9)a)ta. (1.77) Now, dx [0 a(x)/ o(x),^(y)] = Jdx [9a(x)(6abdi + / a r f ^ ( x ) f ) ^ ( x ) , ^ ( y ) = - Jdx9a{*){5abdi + /a*4(x)?)i<J%<J(x - y), = -{(-6^ + fadcA(y)d)9a(y), = ^(Dj9y(y), = - H c ( y ) - (1.78) 20 Chapter 1. Background Furthermore. + 8Af] « *[A?] + Jdx5A)^)j^j-^[At} = *[A?] - i Jdx5A)(*)Eh^[At], = + jdx6\-x.)Ih(x)^[At}, (1.79) provided that surface terms vanish. For physical states, Ih(yi)^phys[At] = 0, expressing the fact that the wave-functional is gauge invariant. WZW current We wish to gather everything in terms of the WZW current J = ^dHH~l. In the Her- mitian WZW theory only correlators made up of integrable representations of the current algebra are well defined, so that all objects of interest can be written in terms of the WZW current J = ^dHH"1 [22]. First, note that dHH'1 = dM]M]~l + M^dMM-lM]~\ = dM]M*"1 - M+AM^-\ (1.80) so that A = M^{-dHH-l)M] + M]~ldM\ = M]-\-dHH^)M] - dM]-xM\ A = 9 M t " 1 M t . (1.81) Therefore (A, A) can be thought of as a field-dependent complex SL(N, C)-valued gauge transformation of (—dHH~l,0). In the ^-representation, the wave-functional ^f(A, A) may alternatively be written in terms of and J. Since the Gauss' law operator D • E is the.generator of infinitesimal gauge transformations (and vanishes on physical states), 21 Chapter 1. Background we can perform a sequence of such transformations Aft —* Aftgt until Aft = 1, with- out affecting ^(A,A). Hence, when acting on physical states, we may everywhere let (A, A) —> (—dHH^1,0). In terms of the generators, A = -iAata ——J = -—Jata, (1.82) so that Aa -> -i^-Ja, and 5aa 7v 6j - . In particular, we now have Dab = §abg _ ^jacbjc = fiabg + ^fabcjc^ £ a (x ) = ca $ 2Tv5Ja(x)' ca (1.83) The first term in the kinetic energy becomes rn / Ja(x) <5Ja(x) while the second term is -292YM I [D-\^y)DE{y)]aEa{x). Jx,y Using the substitution A —> 0, Aft —* 1 up to a holomorphic factor (recalling that D~l is holomorphically invariant), so that /3 - 1(x, y) = Aft~1(x)(5(x,y)Aft(y) —> ^•x^, the second term becomes 2 2 ~3YM 7T 1 m C y l /" 1 Sa% + i—rbcJc(y) ca 7 T ' mcyi / 1 7T" <rba^ - i—fbacjc(-x) ca E\yW(-x), S 5 6Ja{x) SJb{y)' 6 6 6Ja(x) 6Jb(y)' sj\yy (1.84) 22 Chapter 1. Background where H a b(x - y) = ^DfG{y - x), so that T = m J aU) fiab(x-y)- (1.85) '<JJ-(x) ' 4 / " v " JJ6J«(x)SJ»(y)_ Under the substitution B = 2 (dA — 8A) + 2[A, A] —> ^r-dJ, the potential energy becomes V mcA Jx 5Ja(x)5Ja(x) (1.86) The entire Hamiltonian is then H J a M ^ r - r + [ O a b(x-y) 6 8Ja(X.) j^y <5Ja(x) SJb(y) + rncA dJ a(x)<9Ja(x). (1.87) 1.3.7 Vacuum wavefunctional A remarkable side-effect of the non-trivial measure on the configuration space is that the wave-functional ^f[J] = 1 is both normalizable and an eigenstate of T, a solution to the Schrodinger equation can be obtained [22] by expanding around this infinite coupling limit. Writing *[J] = e p, we note that H$[J] = 0 implies e~pHep = V-[H,P] + ... = 0, which leads to a recursion relation for P in powers of m. The resulting wave-functional resembles that of [23], which was obtained by a perturbative resummation in the strong-coupling regime, as well as of [24, 25], obtained my Monte Carlo methods, and for example [26], though the solution is local in J rather than in B. Using this wave-functional, Karabali and Nair were able to exhibit the mass-gap of the theory as well as demonstrate area-law behaviour of the Wilson loop, giving evidence for confinement [22, 27]. However, while the solution can be resummed to second order in J , giving [22] P = 1 1 B(x) x,y [m + (m2 — V 2 ) 2] -B{y), (i- 23 Chapter 1. Background the solution cannot be invariant order-by-order in J , as this variable transforms as a connec- tion. Therefore the inclusion of terms of higher order in J is necessary for gauge invariance (or holomorphy), while terms of higher order in rn are needed for consistency with the Schrodinger equation. For computational purposes, some trade off is ultimately necessary. 1.4 Spectrum in the planar limit Leigh et al. instead proposed the following Ansatz for the vacuum wave-functional [12]: tt[J] = exp ( ~ 2 ^ 2 / 9 J K ( L ) B ? j + (1.89) where the kernel K is a power series in the holomorphic-covariant Laplacian A = = m2L. Here and onwards traces over adjoint indices are implicit. The general form of ^[J] was chosen to reflect the notion, as argued in [28], that the variables dJ somehow represent the correct degrees of freedom of the system (i.e., those in which the ground-state wave- functional is Gaussian). The use of the covariant Laplacian is then required by consistency with holomorphic invariance of the theory. Terms in the exponent of higher order in dJ that can't be absorbed into the kernel are denoted by ellipsis; therefore, the Ansatz is not the most general one. Note that dJK(L)dJ = dJK ( |f) dJ + 0(J 3 ) . Writing *[J] = e p, the action of kinetic energy term in (1.87) on \I/[J] becomes T*[J] = *[J]. (1.90) To second order in d J, the second term in brackets yields -A •Am J dzxdJ ddR2 fdd_ m2 \ m2 d.J + ... (1.91) The action of T on terms of the form On = J dJ(An)8J is less straight-forward. In reference [28] Leigh et al. argue that holomorphic invariance requires mixing between terms 24 Chapter 1. Background of different order in dJ in such a way that TOn = (2+n)mOn + ..., with the result explicitly demonstrated for Co and Q\. Whether this result acquires corrections for higher values of n is not known. Formally, then, the Schrodinger equation becomes H#[J] = TX I^ir^dX [L2^(L)]+lr2^+x)BJ] *=E*w- (i-92) cAm The eigenvalue equation -^[L2K(L)]+LK\L)+ 1 = 0 (1.93) is then solvable using the substitution K = — JTT which casts it into Bessel form. The unique, normalizable solution with correct UV asymptotics was found by Leigh et al., to be [12]: As mentioned earlier, the form of the Anscdz- is somewhat tautological once the variables 8 J are assumed to be the correct physical ones. However, as many of the resulting steps, for example, the requirement of a holomorphic covariant Laplacian, ruin the precise Gaussian form of the wave-functional, the Ansatz needs to be motivated further. Note that 8d = — | 2 L M T _ 1 S M T . so that in many expressions 8J simply reduces to the magnetic field. At large momentum (weak coupling) the field modes behave like free fields, and the well- known (Abelian) Maxwell-field ground-state wave-functional [29, 30] factors from the full wave- funct ional: ^A]uv = L * B { x ) T ^ T * B { y ) . (1.95) On the other hand, many have argued [31] or found [22] that the in the low-momentum (strong coupling) limit the wave-functional becomes •$[A}m = e~^J*B{x)2. (1.96) 25 Chapter 1. Background Thus the Ansatz may be seen as the minimal (but certainly not unique) way to interpolate between between these two limits while retaining the necessary holomorphic symmetry, and is analogous to the Ansatz proposed in [32]. As was pointed out by Greensite and Olejnik in [33], the Abelian wave-functional equation (1.95) can be made to satisfy the non-Abelian version of Gauss' law exactly whilst solving the Schrodinger equation to zeroth order in g by replacing V 2 with its gauge-covariant form D2. There it was found that this substitution leads to a confining state. But {D, 73} = ^D2 so that A = ^ {D, 9} is precisely the form of this operator where gauge covariance has been supplanted by holomorphic covariance. Therefore the Ansatz has many of the necessary properties, though these are certainly not sufficient. Indeed, the solution found by Leigh et al., does not agree with the exact (series) solution obtained by Karabali and Nair [22] at higher orders in the't Hooft coupling rn. However, when solving the Schrodinger equation to second order in dJ, only terms that are required for consistency were kept. Furthermore, the wave-functional obtained by Leigh et. al., is only motivated in the strict large-N limit, and so does not contain all leading corrections. Thus, while the Ansatz leads to an approximate, closed form wave-functional, from which correlation functions can be computed, it sacrifices corrections only seen in an exact, order- by-order treatment like that in [22]. 1.4.1 Mass Spectrum In order to extract meaningful information from the ground state \& it is necessary to compute correlation functions as in equation (1.68). Attempting to solve the Knizhnik- Zamolodchikov equations directly, one encounters (already at the four-point level) non- trivial expressions involving hypergeometric functions [34], whereas the calculation wish to attempt contains an infinite series of such correlators (from expanding the wave-functional), with the next-leading-order being a six-point function. The need for approximation is therefore evident. In [28] the operator trdJdJ was found to be even under parity and charge conjugation, 26 Chapter 1. Background and so creates 0 + + states (see Appendix E). A good starting point then is the calculation of ((dJdJ)x(dJdJ)y). As we have already noted, such a computation is presently intractable. However, in [12, 28] it is argued that in the large-A^ limit, the variables dJ represent the correct physical degrees of freedom so that integration over these variables can be done as in a free theory, in the sense that (dJxidJy)~5abK-1(\x-y\). (1.97) K~1(\x— y\)2 is then identified with a particular two-point function probing 0 + + glueball states, so that the mass spectrum of the vacuum can be read off from the analytic structure of K~l(k)2. Essentially, it is argued that in the dJ configuration space the WZW measure can be neglected. The interpretation of this statement will be explored further in this paper. The asymptotic form of K~l(\x — y\)2 was found to be [12] 1 oo K-\\x-y\)2^-— r V { M n M m f ' 2 e ^ M " + M ^ - y \ (1.98) 32^ -1 ,1 n ^ = 1 where Mn = 3 2 , n = 1,2,3... and ji,n are the zeros of Ji{z). Comparing to the two-point function of the free Boson, equation (D.l), we can see that the 0 + + states have masses given by various combinations of Mn. This result is in good agreement with lattice calculations presented in [35, 36] and in the following chapter we will attempt to give justification for this agreement in light of the approximations that have been made. 27 Chapter 2 Controlled Approximation 2.1 Abelian Expansion The statement that integration over the variables dJ can be done explicitly should not be interpreted literally, as the change of variables from H to dJ involves a further factor det(D<9)_1 where D is now the holomorphic covariant derivative with J as connection. Although the derivatives D and d are related to the original expressed in terms of (A, A) through conjugation by M t _ 1 , the determinant also suffers from a multiplicative anomaly which is expressed by the Polyakov-Wiegmann identity that relates Swzw[gh] to Swzw[g] and SVziy[^] (see [18, 37]). Therefore we shall approach the problem by exploring it in a particular limit where the WZW action at least allows for tractable calculations. We now attempt to calculate ((d~JdJ)x(dJdJ)y) by performing the path integral in equation (1.68) in the Abelian limit. Following [27] we write H = ev and expand terms of the form H~Xf(<p)H in powers of (p. In the adjoint representation, ipab — fabcipc, so that an expansion in <p is necessarily an expansion in the structure constants. In the limit of small ip (see e.g., [27, 38]), we can write the wave-functional and WZW factor in Gaussian form and calculate the four-point function as in a Euclidean field theory with action given by 2CAS[H] + In v̂ *[#]>&[//]. In all cases we expand terms inside the exponential, but not the exponential itself, i.e., we are performing a selective resummation [27]. The measure factor becomes the exponential of the free complex scalar field action, e2cAs\H\ ^ e - | £ J(Pzd<pad<pa_ r p n e c o m p i e t e measure factor can be integrated, so that the volume of the configuration space C is finite, which leads to the existence of a mass-gap [27]. By approximating the WZW factor in this way we hope to make the calculation tractable 28 Chapter 2. Controlled Approximation and at the same time capture non-trivial effects. Already we notice that in the Abelian limit the zero mode causes the volume of C to diverge - it was already noted in [27] that the WZW factor cannot be obtained in the Abelian limit; therefore the approximation may break down at low momentum. The correlation'function {(dJdJ)x(dJdJ)y) gives, up to a disconnected diagram, the square of the two-point function: ({dJdJ)X{dJdJ)y) = 2 (BjXdJy) (djjjy) . (2.1) To expand the wave-functional, note that [39] f ' d s e ^ i d ^ e - ^ , (2.2) * Jo * ^ > + ..., (2.3) 7T where we have dropped higher powers of <p, so that dJ ~ ^^-ip. That we are truly in the Abelian limit can be seen by allowing terms inside equation (2.2) to commute, thereby obtaining equation (2.3) exactly. Keeping in mind that k • z = kz + kz, we can Fourier transform the fields as ^(z) = i /̂ Wk-z, (2-4) so that the WZW term becomes Jd2zd<padya = - 7 ^ 2 / & z J d 2 k l J ^ 2 V a (k i ) fc ie i k r V a (k 2 ) fc 2 e i k 2 ' z , = J d 2 k ^ ) ^ ( - k ) , (2.5) Also, (Lip)(\s) « — ̂ jt^(k) , and we can write the exponent of the wave-functional as 29 Chapter 2. Controlled Approximation Finally, kz mL 4m2 J (2.7) To calculate (dJ(x)dJ(y)), it is sufficient to know (<pa(x)<pl'(y)), as the former can be obtained from the latter by repeated differentiation. Therefore we have to analyze the field theory with effective kernel given by IC - 4m* 14m2 ( k2 2~k~2~ V (2.8) 2.2 Analytic Structure Let us begin by writing the effective kernel in terms of the formal parameter y = 4yL: 11 1 J2(4VZ) '2L + JLJX(4^L) 2 | My) y [ y Mv)\ (2-9) (2.10) As a first step towards finding the inverse of the kernel, consider the identity [40] Jv-i(y) + Jv+\{y) = —Jv{y), y (2.H) from which we immediately see that /C(L) = 4 Mv) y -h (y)' (2.12) A comparison between the behaviour of K and K is shown in Figure 2.1; (note that all quantities here are dimensionless). Another identity of interest is Mv) My) oo (2.13) 30 Chapter 2. Controlled Approximation where jv>s denotes the s t h zero of Jv(y). This result can be derived using common Bessel function identities along with the infinite product representation, My) >\ \v oo n i (2.14) and the fact that Jv(y) is a meromorphic function. Figure 2.1: Comparison of K(y) (dashed line) and lC(y) (solid line) Then OO O V1 1 ic(L)-i = -y 2 _ o-2 • (2.15) We therefore take the inverse effective kernel to have the following analytical structure / C " 1 - 4m2 2 ' k2 + M 2 J ' (2.16) (2.17) where MS = 3 0 . Following the treatment of [12], the real space kernel has the following 31 Chapter 2. Controlled Approximation asymptotic behaviour: 1 0 0 M2 K~\\x-y\) = - 2 £ -^Ko {Ms\x-y\), (2 .18) 1 °° 3 1 -- MI . , e-M-M, (2 .19) 4 S V 2 7 r | x - y | while the four-point function is / C - ^ l a : - y\)2 = —-± r V ( A ^ M ^ f e " ^ ^ ^ . (2.20) 327r\x — y\ z-< 1 W | r ,s= l Comparing to the propagator of the free Boson evaluated at fixed time, equation (D.l), we see that a multitude of particles with masses Mr + Ms have been identified. A comparison between our masses and those obtained in [12] is given in table 2.2. Large-N lattice results [35, 36, 41] given for comparison in [12] have also been reproduced. We have omitted the spurious pole due to j'0,1 as it has no obvious value to compare to. As noted in [12], the discrepancy with the 0 + + * lattice result may suggest that this is in fact two states (corresponding to M\ + M2 and M2 + M2) which are not resolved by the lattice calculation, or it may indicate a low-momentum breakdown of our calculation. State Lattice0 Leigh, et., al. Our prediction 0 + + 4 . 0 6 5 ± 0 .055 4 . 1 0 4 .40 0++* 6 .18 ± 0 . 1 3 5 .41 5 .65 0++* 6.72 6 .90 7 .99 ± 0 . 2 2 7 .99 8 .15 QH—(-*** 9 .44 ± 0 . 3 8 9 .27 9.40 a S e e [35, 36, 41] Table 2 .1: Comparison of 0 + + glueball masses given in units of string tension ^fo « ^Jr^ 3 2 Chapter 2. Controlled Approximation 17 . 5 15 12 . 5 10 7 . 5 5 2 . 5 Figure 2.2: Zeros of J2 (grey) and Jo (black) 2.3 Discussion The calculation presented here shows that in the Abelian limit the mass spectrum probed by J pc = 0 + + operators comprises sums of pairs of zeros of Jo(y), in contrast with the result of Leigh et al., which expresses these masses in terms of zeros of J2(y). Thus we find a correction due to the inclusion of the WZW action to lowest order in fabc. Asymptotically Jv(y) \ f ^ c o s (y ~ T" ~ f ) s o t n a t -My) — ¥ ~Jo{y) as j / —> oo. Therefore our results reproduce those in [12] at large momentum and give justification for approximations made therein. That the results agree at large momentum suggests that the analytic structure of the wave-functional is rather robust against short-distance corrections. At low momentum (small y), however, an additional pole arises due to jo,i (see Figure 2.2), which is not seen in [12], and gives a constituent mass of M i « 0.96^/a. This correction can be seen in Figure 2.1. Combinations using M\ do not seem to appear in lattice calculations presented in [35, 36]. It is possible that this signals a breakdown of the Abelian approximation at low momentum that can be corrected by continuing the expansion to higher order. 33 Chapter 3 Matter 3.1 Functional Schrodinger Picture As the functional Schrodinger presentation of quantum field theory is not conventionally emphasized, it will be reviewed here briefly. 3.1.1 Bosonic case With ordinary Quantum mechanics, we are motivated to view the field </>(x,£) and it's con- jugate momentum 7r(x, t) as operators acting on a Hilbert space. In the Schrodinger pic- ture, these operators aren't explicitly time dependent and we simply write </>(x) and 7r(x). In canonical quantization we replace the fundamental Poisson bracket [<P{x),-K(y)]pg = (5(x —y) with the commutator [̂ >(x), 7r(y)] = iM(x —y). We then must find operators that obey this relation. This is normally done via the introduction of creation and annihilation operators that act on a Fock space. However, one can also consider operators that act on a space of wave functionals on the configuration space. The configuration space is the space of all (classical) fields </>(x), i.e., eigenvalues of the operator <j!>(x) in the same sense that the configuration space of a quantum mechanical point particle is the space of all positions x, i.e., eigenvalues of the position operator x. A wave functional is then a map \& : cf> —> ^[<p] £ C from field configurations onto complex numbers, and represents the probability of the system to be in the configuration (/>(x) [29]. It is just the functional analog of the wavefunction. In analogy 34 Chapter 3. Matter with ordinary Quantum mechanics, we then make the following algebraic substitution: <£(x) -» <£(x), 7r(x) —> —z/l <ty(x) Note that from functional differentiation, <ty(x) <ty(y) <*(x-y), so ^(x),7r(y) *[</»] = - ^ ( x <ty(y) i M ( x - y ^ ( y ) 0 ( x ) * M ) , (3-1) (3.2) (3.3) (3.4) (3.5) The Hamiltonian for the free real field is then (h —> 1) H = ljd3x [TT(X)2 + [V<Mx)]2 + m2</>(x)2] , + [V</>(x)]2-f-m2</>(x)2 + </>(x)(m2- V2)</>(x) 1 (5^(x)2 82 <^(x)2 (3.6) (3.7) (3.8) (3.9) The time-dependent Schrodinger equation is The (un-normalized) ground state of the real field is then (3.10) V j / ^ j _ & - \ Jd3x <j>(x)Vm2-V24>(x)-iE0t_ (3.11) 35 Chapter 3. Matter Using the expansion, (x)= dtk #k) = Id'xJ-^cfWe „ik-x i k x where w? = k 2 + rn?, we have / ^ ( k ) 6 ^(x ) y ^ (x) <ĵ (k)' = /cPJfe w fc „ - i k x 5 (2TT)3 <ty(k)' (3.12) (3.13) (3.14) (3.15) (3.16) The kinetic part of the Hamiltonian becomes 2 i 2 J J (2iry d<p(. 54>(k) 5cf>(k')' 1 f , 3 , <̂ ^ 2 y d / C " f e ^ ( k ) ^ ( - k ) The full Hamiltonian becomes H = - dfktok <ty(k) <ty(-k] + # k M - k ) = e - U r f 3 f c ^ ( k ) ^ - k ) - i e o * . Inserting into the Schrodinger equation gives £0 = I jd3kujk6(0), (3.17) (3.18) (3.19) (3.20) (3.21) which agrees with the vacuum energy one finds using the traditional (operator) method. 36 Chapter 3. Matter 3.1.2 Fermionic fields In analogy with the Bosonic case, we will develop a functional description of the Schrodinger picture of Fermionic fields. We will first describe the Hermitian case since a complex field can be described by a pair of real fields. Hermitian field To describe a real Fermionic field, i.e., a Majorana field, we need a representation of the following equal-time anticommutation algebra: {^a{x)^b{x')} = 5ab5{x-x'). (3.22) Since the field operator is it's own conjugate momentum, we cannot follow the Bosonic case where ir(x) — —ig^5^. Instead, we construct them in a way first proposed by Floreanini and Jackiw (F-J) [42], namely V>a(2) = a9a(x) + Pj^y (3-23) where 9a(x)* = 9a(x) is a real Grassmann number, i.e., {9a(x), 9b(x')} = 0. Then M*)Mt) = ( « * • ( * ) + fi^) (^(y, + / ^ r ^ ) , (3.24) a29a{x)9b{y) + aft6abS(x - y) + aft0a(x -aft6b($-/—+ft2-^-/— (3.25) so that {ijja{x),Mx")} = 2aftSabS{x-x'). (3,26) We can therefore choose complex numbers a and ft such that aft = For example, Mi) = ± ( « . < * ) + ^ ) , (3-27) 37 Chapter 3. Matter and Non-Hermitian field We can combine two Hermitian fields ipi and tp2 (dropping spinor indices) into a single complex field ip = ^= (ip\ -\-i1p2) which obeys {ip^tp} = \ {^i,V'i} + \ {^2,^2} = 5(x — y). For example, we may use the form * ( * ) _ ( „ , ( * , + i s y , (3.29) so that ^(f) = - ^ [ V ' i ( ^ + # 2 ( 5 ) ] , (3.30) = 7 f ( e ( f ) + «4>)' ( 3 3 1 ) = -i= -#2(f)], (3.32) - 7 1 + i ^ j ) • ( 3- 3 3 ) where 0(f) = [0i(£) + z02(z)],and 0*(x) = ^ - [6>i(x) - ^6l2(f)]. This is the Floreanini - Jackiw representation, and is suited to the description of neutral spin-^ particles [43]. Then {9(x),9*(x')} = {9(x),9(x>)} = 0. Note that 5 69 5 59* 5 1 / 5 5 \ ,„ _ and hence 50! 59i 59 S9i 59* y/2\59 59* 592 ~ y/2 \59 59* 1 ' ( 3 5 ) 1 / 5 5 \ 1 5 , 2\jfl+lW2) = ̂ ' (3-36) 38 Chapter 3. Matter There is an alternative, inequivalent choice for tpi and ip2 in terms of a single, real Grassmann variable 0a(x)* = 0a(x): *(x)=-L(%-, + _i_), (3.37) and Then w-Mw-m?1 ( 3 ' 3 8 ) 4>(x) = 9{x), (3.39) * ' ( « ) = j ^ , (3-40) This representation was proposed by Duncan, Meyer-Ortmanns and Roskies (DMR) [44]. Both representations are suitable for charged particles spin-^ particles [43]. In this rep- resentation the complication between canonical conjugation and Hermitian conjugation is evident. ^ is therefore the Hermitian conjugate of ip in the sense in which both are oper- ators on a Hilbert space. Vacuum wavefunctional Consider a Fermionic system with finitely many degrees of freedom and Hamiltonian H = hijipjipj. Let us work in the complex F-.J representation. This has the form of a har- monic oscillator, so we take as Ansatz the Gaussian ground state ^ = eGiiuiui where G is necessarily antisymmetric. = \ (< + hij (6jk + Gjk) uk^>, (3.42) 1 1 = 2 (hi + Gn) hii (5jk + Gjk) ujuh^ + -5ikhij (6jk + Gjk) (3.43) = \ [(I - G)h(I + G)]lku*lUk* + \ trh(I + G)tt. (3.44) 39 Chapter 3. Matter For this state to be an eigenstate, the first term must vanish, so that h — [G, h] — GhG must be symmetric or zero. Trivial solutions such as G = ±1 are excluded. Suppose G and h are simultaneously diagonalized. Then h(I — G2) = 0 so that G2 = I. The eigenvalues of G are therefore ±1. Since the exact form of G will not be needed here, the reader is invited to see [42] for a more detailed discussion. 3.2 Yang-Mills with Fermions It is inviting to try to extend the formalism of Karabali and Nair to something more closely resembling QCD, namely a gauge field interacting with Fermions. We will attempt to take the first steps towards such a description, and utilize the ideas of chapter 1 as far as possible. Let us then assume that the state of the system can be represented by a wavefunctional ^ [ ^ i , u] where u is the Grassmann variable representation of the Fermion field in either the F-J or DMR representation. Physical wave-functionals are subject to the constraint that the Gauss' law operator (generator of gauge transformations), that one may make the substitution ^[^4, 4̂] = ^f[— ̂ J, 0] throughout. When Fermions are introduced, this expression must be replaced with G^ = 0, which presumably means that *[A,u] = *[A»,u«]. The field-dependent "gauge transformation" by M l , i.e., O —> UOU~l sends the pure Yang-Mills part of the Hamiltonian into the Karabali-Nair form. It also sends ip —> x = M^tp and V f -> xH]M^~l. These are both gauge invariant quantities. Thus under (A, A) —> (—^-J, 0), the K K N Hamiltonian is obtained, plus a term proportional to j° (which was omitted in the treatment of Karabali and Nair), plus the gauged Dirac part. (3.45) vanishes. In the work by Karabali and Nair, it was required that (D • E ) ^ ^ = 0 so (3.46) 40 Chapter 3. Matter where V = + 7<9 — 7^- J(x). The variables x arise naturally from consistency with the field-dependent "gauge trans- formation". Evidently, these variables obey canonical anticommutation relations, so we may choose the Floreanini-Jackiw representation, as it is the easiest to implement. x = 7l(" +^)' *, = 7H",+i|- <3'47) 3.2.1 Holomorphic invariance Under a "holomorphic transformation", M -> Mh(z),Mi -> hM\ M^1 -» M T - 1 / i - 1 , we find that X -> h(z)x, X]^X]h-\z), (3.48) which is consistent with the fact that J transforms as a holomorphic connection. However, this would seem to require that u —» h(z)u v!-*uhrlfz), (3.49) which is not consistent. On the other hand, we recall that we u and are to be treated as independent variables, and so may transform independently. We may replace by another variable which transforms as —» v'[h~1(z). Then the F-J representation becomes 1 / 5 ^ = M v] + i ] (3- 50) 41 Chapter 3. Matter 3.2.2 Non-canonical behaviour We have thus far succeeded in finding a functional representation of the operators ip and i/>t that is both gauge and holomorphically invariant. The manner in which these variables arise is required for consistency with the functional manipulations done on the gauge theory sector. However, the price to be paid for this simplicity is that the gauge and matter field operators no longer commute (actually, the distinction between them is now a matter of viewpoint). Since M is a functional of A, it is no longer true that [E,x] ~ [A'^J = -̂ Instead, we now have [Ea,X} = 2 5Aa' i 5M^ 2 SAa (3.51) Therefore, the "pure" Yang-Mills Hamiltonian UKKN now acts on the fermionic fields in a rather non-trivial way. The additional terms coming from [Ea,x] are also of the same order in g, and it is not clear how successfully these can be treated perturbatively if man- ifest gauge-invariance is to be maintained. However, recent work by Agarwal, Karabali and Nair [45] involves a similar treatment of a massive scalar field in the adjoint repre- sentation, definining x = M ^ M ^ - 1 , which translates to xa = Ma^<f>a and is analogous to the Fermionic case presented here. There the contribution of [E, x] to the Hamiltonian is included, giving the highly non-trivial form for the Yang-Mills field Hamiltonian coupled to adjoint scalars. H m abc-a'w) SJa& +im f Acd(w,Z)rbcX J z,w i r s2 r ( i n ~2J 8XaSxa Q,(w, z) ba 5Ja(z) SJb(w) 6 8 SXb(w) 5Jd(z) + rncA dJ a8Ja + —dX a(Px) a + M2^- cA 2 (3.52) 42 Chapter 3. Matter where M is the mass of <j> and Urs(u, v) Ara(w, z) While the treatment is mostly qualitative, the authors analyze screening of the adjoint representations and give an estimate of the string-breaking energy which is within 8.8% of the latest lattice estimates. Gar(x, u)Kab(x)Gbs(x, v), [dzl\rs{w,z)\ K-a\(z) (3.53) (3.54) 43 Chapter 4 Conclusion The research presented in this thesis looked at the problem of calculating correlation func- tions in (2+l)-dimensional Yang-Mills theory in the Karabali-Nair formalism, specifically in the vacuum proposed by Leigh et al. The effect of the WZW non-linear sigma model action that arises anomalously from the configuration space measure was incorporated in a controlled manner by expanding the path integral around the Abelian limit; that is, in powers of the structure constants. In the Abelian limit it was found that the mass spectrum probed by Jpc = 0 + + oper- ators comprises sums of pairs of zeros of Jo{y), in contrast with the result of Leigh et al., which expresses these masses in terms of zeros of J2{y) (which approach those of Jo(y) as y —> oo). Thus, to lowest order in fahc, we find a correction due to the inclusion of the WZW action. Therefore our results reproduce those in [12] at large momentum and give justification for approximations made therein. That the results agree at large momentum suggests that a robustness of the analytic structure of the wave-functional is against short- distance corrections. On the other hand, the calculation presented here does not agree with that of Leigh et al. [12] at low momentum. This research also investigated the inclusion of Fermions in the fundamental represen- tation in a way that preserves as much of the machinery developed by Karabali and Nair while still maintaining gauge invariance. It was found that Fermions could be described in the functional Schrodinger picture using either the Floreanini-Jackiw or Duncan-Meyer- Ortmans-Roskies representations in a way that preserves holomorphic invariance of the theory. On the other hand, the restriction of gauge invariance (which translates into holo- morphic invariance) necessarily complicates the canonical commutation relations between 44 Chapter 4. Conclusion the gauge and matter fields. Presently it is hot known how to successfully deal with these complications, though other authors have been able to make some qualitative statements as well as preliminary strong-coupling numerical estimates [45]. 4.1 Future work If the theory is to be meaningfully expanded about the Abelian limit then the behaviour of the correlation function ((dJdJ)x(dJdJ)y) at low momentum needs to be understood. Whether the discrepancy in the mass spectrum in this regime signals a breakdown of the Abelian approximation and whether it survives at higher orders in fabc (equivalently, <p) is a possible direction for future investigation. These issues are independent of the fact that the vacuum wavefunctional was motivated in the large N, and it should be possible to answer these questions within the constraints of the Ansatz. The treatment of matter is indeed much more complicated than that of the pure Yang- Mills system, and many of the tools used thusfar are no longer available. On the other hand, it should be feasible to test the general approach by enlarging the theory to one that has a string theory or gravity dual. Thus far some relevant objects, e.g., Wilson loops, have been studied with some success[45]. 45 Appendix A The Adjoint Representations Since this thesis contains many calculations in the adjoint representation, the relationship between the adjoint representations of the Lie Group and its associated Lie Algebra will be reviewed-here. The adjoint representation of the Lie group G with Lie algebra Q is given by the deriva- tion of the conjugation action ghg~l at the identity: Ad{g){X)=gXg-\ (A.l) The adjoint representation ad(g) of the Lie algebra is given by the derivative of the expo- nential map. Since etYXe~tY = X + t[Y, X] + 0(t2), ad(Y) = [Y,-}. (A.2) In other words, writing g = etY, we say that ad(Y) is the derivative of Ad(g) = 1 • +t[Y, •} + 0(t2) at the identity (t = 0). A . l Group representation We can consider the adjoint action of the group G, on the algebra Q, i.e., X —> gXg~x, where g G G and X G TE(G) is a generator of the group. Recall that the generators form a representation of the algebra Q and are derivatives near the identity, and thus form a vector 46 Appendix A. The Adjoint Representations space. Let h(t) be a smooth curve through the identity h(0) = 1 such that h'{t) = X. Then h'(0) - gh'^g-1 € Te(G) (A.3) i.e. if X = Xata then X = gXg~l = Xata also for some Xa. Assuming a compact Lie group for clarity, so that we can write tr (tatb^j = X5ab, Xa = \~l tv{Xta). Xa = A" 1 tr [gXg-Ha) , - A" 1 tr (Xg-Hag) , = A" 1 tr (Xhthg-hag) , = A" 1 tr ( i W 1 ) Xb, = gabXb, where gab = A - 1 tr ( i a 5i b g _ 1 ) is the adjoint representation of g. Furthermore, let X = hXh~l = (hg)X(hg)-1. Then Xa = A" 1 tr (tahgtbg-lh-l^j Xb, = A - 1 tr (h-Hahgtbg-l^j Xb, but also, Xa = hacgcbXb, = A" 1 tr (h~xtahtc) A" 1 tr ( t ^ V 1 ) Xb. Therefore A " 1 tr (h-ltahtc) A" 1 tr (ifg^g'1) = A " 1 tr (h"Hahgtbg-1). UheG, and m = hg, then mab = hacgcb. Thus group multiplication is carried over naturally into the adjoint 47 Appendix A. The Adjoint Representations representation. Generally, for compact Lie groups, if A = Aata and B = Bata, then A tr(AB) = \2AaBa = tr(Ata) tr (taB). A . 2 A l g e b r a r e p r e s e n t a t i o n To obtain the representation of the Lie algebra we differentiate at the identity. If M = e u , t » = 1 + w a t a + 0(u>2), then Mab = A - 1 t r ^ M ^ M - 1 ) , = A" 1 tr(i a (l + toctc)tb(l - u>dtd)) + = Sab + uc\~l tr(tatctb - tatbtc) + 0(w2), = 6ab + c^A" 1 tr(ta[tc, tb}) + 0(UJ2), = 6ab + a ; c ( i / c W )A- 1 tv(tatd) + 0(u2), = 8ab + L u c ( i f c b a ) + O(oj2), = 8ab + o j c ( - i f c a b ) + 0(oj2), = 5ab + LocTZb + 0(u2). Thus T^b = —ifcab furnish the adjoint representation of the Lie algebra. Given [ta,tb] = i f a b c t c , it follows from the Jacobi identity that the matrices {Ta)bc = —ifabc satisfy These matrices form the adjoint representation of the Lie algebra. As in the fundamental representation, we take the inner product to be given by the trace, '] = ifabcTc. (AA) (A.5) 48 Appendix A. The Adjoint Representations In the case of non-compact Lie groups this inner product can be diagonalized, so that we may take tr (T a T b ) = ASab (A.6) for some positive A, and tr (TaTb) is known as the quadratic Casimir invariant of the adjoint representation. With a metric in place, we can recover the new structure constants fabc = t r ̂ [Ta) T6j T<A _ (A.7) A.3 Casimir Invariant Here we assume a compact Lie group, so that we can write tr (tatb) = dr5ab in some representation r. For any simple Lie algebra, the quantity t2 = tata) (A.8) commutes with all the other elements of the algebra: ta,tbtb\ = (ifabctc)tb + tb(ifabctc), = ifabc{tc,tb}, = 0, since fabc is antisymmetric in its last two indices. This object is an invariant of the algebra, known as the quadratic Casimir invariant. The (irreducible) matrix representation of t2 is therefore proportional to the identity matrix (since t2 commutes with all ta, and with 1, it commutes with all g G G, and by Schur's lemma, t2 oc 1): tata = c 2 (r)l , (A.9) Furthermore, c2(r) is sometimes labeled as cv, including in this thesis to be consistent with 49 Appendix A. The Adjoint Representations literature. In the adjoint representation, (TcTc)ah = -fadfdb = CA8ab. (A.10) If ta are suitably normalized, so that fabc is completely antisymmetric, then this can be written as jacdfbcd = rpj. (rparpb ̂ = ^ab_ {AAV) Now, in a particular irreducible representation r, tr(tata) = c rdim(r), - (A.12) but also tr(tata) = drSaa = dr dim(G). (A.13) Therefore we obtain the formula dr dim r cr dim G (A-14) 50 Appendix B SU(N) B. l Completeness Suppose we write the inner product in the Lie algebra as (̂ -'(Ŝ )-4"*- (B'1) Viewed as A^2-dimensional vectors in complex Euclidean space, this is just the Euclidean norm. We therefore have a subspace spanned by n orthonormal vectors. If these were a complete set, then we would have the completeness relation \ta) (ta\ = l N 2 x N 2 , (B.2) (summation implied). However, since the generators are traceless, they are all orthogonal to |1) the Ar2-dimensional vector corresponding to -^Srs. For su(iV), n = N2 — 1, so we simply project out the subspace |1): \ta)(ta\ = lNixN2-\l){l\. . (B.3) In matrix form, this becomes (keeping in mind that the generators are also hermitian) \W* = SprSgs - jj6pq6ra. (B.4) 51 Appendix B. SU(N) B . 2 Adjoint representation Consider the tensor product of the antifundamental representation and fundamental repre- sentations of SU(N), i.e., N®N. Now the trace (singlet) part is invariant, so transforms as the 1. The trace-free part stays trace-free under a transformation owing to the cyclic prop- erty of the trace. The trace-free part of X can be written as a linear combination of some basis of iV 2 — 1 traceless matrices. There happen to be iV 2 — 1 generators of SU(N), which are also traceless. For convenience, assume X is traceless (since the trace-part transforms trivially), so that X € su(N): X = Xata. (B.5) Therefore the transformation law X —> \JXU~1 is just the adjoint representation of SU(N), so N ® N = l 0 A d j . (B.6) B.3 Quadratic Casimir SU{2) is a subgroup of SU{N), since we can consider the subgroup of N x N matrices leaving all but the two indices fixed. Any complete representation of SU(N) is therefore a (possibly reducible) representation of SU(2). Three generators of SU(N) generate a (possibly reducible) representation of SU(2). Therefore tr(tatb) = d r S U ^ 5 a b = drtSU(2)dab where the indices a and b range over values which are defined for both groups. This assists in obtaining the Casimirs [15]. The N of SU(N) decomposes into N-2 singlets (1) and one doublet (2) SU{2). From the point of view of SU(2), dw,su(N) = (N - 2)dltSUi2) + d2jsu{2) = -z, (B.7) and , dimG l i V 2 - l N C N ^ N d h n N = 2-Hv— (R8) 52 Appendix B. SU(N) For N ® N, [N ® N] S t / ( J V ) - [2 @ (N - 2)1}SU{2) ® [2 © (TV - 2)1] w ( 2 ) , = [2 ® 2 + 2{N - 2)2 ® 1 + {N - 2)21 ® l ] S [ / ( 2 ) , - [3 + 1 + 2(/V - 2)2 + (TV - 2) 2l] w ( 2 ) , A d j 5 C / ( i V ) , 5t/(2) • = '3 + 2(/V - 2)2 + {N - 2) 21 It follows that cWc/ ( iv ) = [Rs + 2(N - 2)R2 + (N- 2)2i?i] = 2 + 2 ( / V - 2 ) ± + ( /V-2) 2 0, = N. SU(2) Consequently, since dimAdj = dimG, C N = 2 — ' ^ cA = TV. (B.10) 53 Appendix C Bessel function identities Since Jv{z) is merbmofphic, the following infinite product representation of Jv{z) exists in terms of the roots jViS [40]: c . i P r o o f o f ^ = f + 2 , E r = i F 4 : d 7(\ d ®* ft / l ^ = _ J 2 ( 2 ) + ^ ^ _ J 2 ( z ) ; - i 1 "^7 2 °° 1 - J 2 ( z ) + 2 z ^ - j r — - j J a ^ ) , 54 Appendix C. Bessel function identities C.2 'Proof of 444 = 2z E ^ i J0(z) ^s=l 5 = 1 7o,« CO ^ = 22 ? . 55 Appendix D Free Boson in (2+l)-Dimensions In order to make a connection between the correlators obtained in the Hamiltonian formal- ism, e.g., equation (2.20), wi th the (covariant) Kallen-Lehmann spectral representation, we need to recognize the analytic structure of a 1-particle state when expressed non-covariantly. To do this, we analyze the two-point function of the free Boson for purely spatial separation: A ^ - y ) = / ( 0 7 P T ^ e i k ' r ' kdk 2TTJo(kr) 0 (2TT)2 ^WTm^' 1 27r|a;- y\ re -m\x-y\ (D.l) 56 Appendix E Quantum numbers In 2 spatial dimensions, angular momentum is characterized by the quantum numbers as- sociated with the representations of the SO(2) subgroup of the 50(2,1) Lorentz group. Vectors (A, A) transform as J = (—1,1) whereas derivatives (<9, d) transform in the oppo- site way, as J = (1,-1). In this thesis we are chiefly interested in the operator dJ which has total JQJ = 0. Under parity, i.e. (xl,x2) —• (x1 — x2), we have that z —> z, leading to [28]: A -» A, M -» Aft - 1 , H -> H~\ dJ -> -H~ldJH, so that the trace tr dJdJ is even under parity. Finally, this operator is also even under charge conjugation, and so JPC = 0 + + . 57 Bibliography [1] K. G. Wilson, Phys. Rev. D 10, 2445 (1974). [2] T. R. Morris and 0. J. Rosten, Physical Review D 73, 065003 (2006). [3] S. 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