DOCSLIB.ORG

0461-0467.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
0461-0467.Pdf Pram~..na - J. Phys., Vol. 31, No. 6, December 1988, pp. 461-467. © Printed in India. Renormalization of a gauge theory in a nonlinear gauge SATISH D JOGLEKAR Department of Physics, Indian Institute of Technology, Kanpur 208 016, India MS received 1 March 1984; revised 29 August 1988 Abstract. We discuss renormalization of an 0(3) gauge model with the gauge fixing term given by ~.f.=-1/(l(O,-i,qA~)W+U[2-(1/2cO(OA3) 2. We utilize earlier results on the general theory of renormalization of gauge theories in quadratic gauges to prove multiplica- tire renormalizability of the theory together with a subtractive renormalization of gauge fixing and ghost terms. We show that this model has a double BRS invariance and that it is preserved under renormalization. Keywords. Renormalization; gauge theory; nonlinear gauge. PACS No. 11.15 1. Introduction Calculations in spontaneously broken theories are generally done in linear gauges Feynman rules and the renormalization procedure (Abers and Lee 1973; Fujikawa et al 1973; Zinn-Justin 1974; Lee 1975) in these gauges are simpler than in more complicated nonlinear gauges. As such nonlinear gauges have been more or less only of an academic interest (Joglekar 1974; Das 1981, 1982). Recently, however, many calculations in spontaneously broken theories have been performed(Deshpande and Nazerimonford 1983)in a particular nonlinear R~ gauge. In the context of an SU(2) × U(1) model described by the Lagrange density, (in the notations of Fugikawa et al 1973, Appendix C) Aao = Ous+ +iMwW~ +-~gwusi ""+° +i ( -eAu-t Gc°s202 Z~ ) s + 2 o Mz GZu o i 2 + O,s --i-~Z, - i~-~ +ToW;s+ , where q~= = sO v+q+i kc::/is,/2+ k / is a complex doublet. In these notations ~e~.f. in the nonlinear Re gauges is given by __.L#g.f.=j_~(O.A)Z + ~__~(OuZU+rtMzz) +.~l(Ou igA3) W+u- z z 1 _ i~Mws + l 6 I This choice for the gauge fixing has been made because it simplifies the Feynman rules 461 462 Satish D Joglekar of such theories and has in addition simpler electromagnetic WT identities. As such, renormalization of gauge theories in such gauges becomes an interesting problem. In an earlier paper (Joglekar 1988), we had developed the general theory of renormalization in nonlinear gauges. In this work we shall apply the results of this theory to the discussion of renormalization of an unbroken 0(3) model in a nonlinear R~-like gauge. This model in this gauge has an interesting property that it has a double BRS invariance as shown in § 2. We shall show that the theory can be renormalized to preserve the double BRS invariance and that the coupling constant 9 that appears in the gauge fixing term is renormalized the same way as the coupling constant that appears in the invariant action. 2. The model We shall consider the 0(3) Yang-Mills theory given by the Lagrange density LPo[A] = -~Fit~(x)F1 ~ ~itv(x) (1) with F~v(x) = ~?uA~ - ~?~A~ + ~,, JF~P~A~A It V~~ f,a~ _= e,a~' (2) We shall quantize the theory in a gauge whose gauge-fixing term is given by -~g.f.=~(OitA3u)2+~(Oit-igA3it)W + 2 = ~-f,E1 A], 2 (3) where W + = (A 1 +_ iA2)/x/2. ~o[-A] of (1) is invariant under the gauge transformations 6A~ = OitA ~ - ef~P~A~A ~, i.e. 6W~ = (?itA + -T- igA3A + + igA3W~, 6A3u = c3itA3 - ig(W+A - - W-A+), A ± = (A 1 -t- iA2)/x/2. (4) With the help of this form for the gauge transformations, one can write the ghost Lagrangian viz.: 5('g = 6+ {(Oit - igA 3) [(c~, - igA3)c + + ig W + c3] -igW+[Oitc3 -ia(W+c_ - W~-c+)] } + h.c. + 63{02c3 - igOit(W;c_ - W~-c+)} (5) with c1 +_ ic2 61 + ic2 c+= x/2 , 6+- x/2 Renormalization of a 9auge theory 463 The total effective action is La~ffE A, e, 6] = L,ao[A ] + 5°g.c[A] + Lag[A, c, ~]. (6) This effective action has an extra invariance. Omitting the term -- zr~un3~22~t~ rx~l in the Lag. f . [A], but keeping the rest of the terms in Lag.c and Lag., this c~,t t is invariant under the local "electromagnetic" gauge transformations viz: A~ ~ A~': Aus + Ou~(a W) ~ W+'= exp [+ i~l ,c)] W), c± ~e'+ = exp [+ io~(x)]c±, ~+ ~?'+ = exp [F i~(x)]6±, C 3 ~ C~ = C3~ C3 -"~ C'3 : C3~ (7) as can be seen from the form of Lag.c and Lag of (3) and (5). We shall express the two invariances differently. We add a pair of new ghost and antighost free fields C and C to the action. They do not couple to any other fields. Their use is in being able to express the two invariances as the BRS invariances of the single effective Lagrangian, LaCff[A, c, ~, C, C] = ~rr[A, c, c-] + C¢9~C. (8) Then £#err of (8) is invariant under the following two BRS transformations: (i) 6A] = D]#c#2, 6c~ = - ½9f'#¥#cy2, 66~ = - ~l- 1/2 f~[ A] 2 r/=~ for c¢=1,2 =0~ ---3, (9) 6C=6~=0. (ii). 6Aus = ~,C2, 6W + = +_ iaCW 52, 6c+ = +_ i9C2c+, 66+ = -~ igC26 ±, 6c3 = 0 = 663 = bC, 1 3 6C = --O"A,2. (10) Our aim is to renormalize the theory preserving both kinds of BRS invariances. We would also like to see if the coupling constant 9 that appears in gauge-fixing term is renormalized the same way as that appearing in the gauge-invariant Lagrangian. 464 Satish D Joglekar 3. Results on the renormalization in quadratic gauges In Joglekar (1988) we discussed the renormalization of gauge theories in quadratic gauges when the gauge fixing term of the form: 1 (I1) where the term r=sp~u,~ atlas. preserves the global invariance of the theory. It was shown there that the gauge fixing and the ghost terms are subtractively renormalized; so that it is convenient to start from the beginning with t ----- __ ~" L(O#A~ -4- ~Tn~ctYaA~aTtt or- l'~epCT) 2 = -- ~.f;2 [A, c, c-'] (12) with z~~ = - _~-P'.,'~_~depends on the loop expansion parameter a and are zero in the tree approximation. Then the ghost term must be modified as follows: o~g' = ,~g- ~gfi aPY c~c~z~K~- c~%. - (13) Let Sef f = .~o[a] + £ag.f. + £P,] d x. (14) d The structure of counterterms was established in Joglekar (1988). To state this structure, it is necessary to introduce sources for composite operators that enter BRS transformations. We consider, 3 Seffl'A , C, C, K, L, R] = So[A- ] _ ~ ~(f,1 , -- rl] ~/ZRo)2 + S o 0t=l + KiD~c~ + ½9f~ct3c~L~. (15) The structure of the counterterms is expressed conveniently in terms of the nilpotent operator 5S 5 5S 5 6S 6 + 4 (16) 6A i 3K i (~C~ 3L= 36~ 6R, and is given in terms of the divergent part of the generating functional for proper vertices in any given loop approximation as, {F} 'iiV = ~I(e.)So[A] + NX[A, c, ~, K, L,R; ~], (17) where X is a Lorentz invariant local functional of dimension 3 and ghost number (Joglekar and Lee 1976) minus one. From this it follows that X has the most general structure given below: 3 3 3 i=I p=l ~=i 3 O~6 (g,)"~c,)~ijAiA j 11= 1 (aq) + ~ c~7(,~) (e~pa1 ~, c~cr 7. (18) era Renormalization of a gauge theory 465 Then in terms of S'af, the following counterterms are needed (i) So[A] -= [~o[A] d"x, J (ii) f ~A~,(x)d,~S'~ff , . xa = 1,2,3 a not summed, (iii) f ~(x)C,~X)aSS'~ff .... x a = 1,2, 3 ~ not summed, (iv) fd"x{~'~a~cp + tl~l/zf'~8.A ~} not summed, (v) t_2,t~ J~'~u ~ --~ ~ u u t~3 dnx ~ not summed, (vi) j d "x[t h- 1/2f,p~ , ,t~- c~e~ + ¼gp~zgtjf'~c~c~] a not summed. (19) Further, our action has extra U(1) local invariance of (7). This has certain consequences on the structure of divergences. These will be derived in the next section in the form of an extra WT identity. This WT identity must be imposed on the structure of counterterms in (19) above and only those counterterms that satisfy this WT identity can be allowed counterterms. This, as we shall see restricts the counterterms such that the g in ~g.f. is renormalized the same way as that in 5%[A]. 4. Consequences of the U(I) invarianee To obtain the consequences of U(1) invariance of (7) in the form of the WT identity for the generating functional for proper vertices, we introduce the generating functionals and expectation values as usual W[J,~,~] = f [dAdedc-]expi f {~e'~ff[A,c,c-] + J](x)A~"(x) + ~-,c, + g,~,} d"x, (20) Z= -/In W, (21) (A'.) = ~z/~J'.; (e~) = ~z/C~; (e~) = - ~z/~. (22) Dropping brackets ( ) around expectation values, F[A, c, ~ = Z -- ~ [ J~uA"u + ~c~ + ~,~,] d"x. (23) 3 Now we consider transformations of (7) on the integration variables of (20) and, equating the change to zero, obtain: o=f[dAdcdc-]fd"x[-O"d3,+iJ-;W*"-iCW-"+i~_c+-i~+? + i6_ ~ + - i(+ ~_ - -1~?2(0"A 3)] exp i{... }, j:~ _ (j~ + iJ~)&/2. 466 Satish D Joglekar Translating in terms of F by the use of 6F/6A~ = - J",, 6F/6g, = - ~,; 6FI6c~= ~-~ (24) and using 3, (25) we obtain, , f ,fiFo + 6F0 W- 6Fo .6Fo f [ 3A/, 6W, ovv, oc+ 6F0 6Fo } -i~c_C--i(-~c - -= fir0 =0.
Load more

Recommended publications
  • Quantum Gravity and the Cosmological Constant Problem
    Quantum Gravity and the Cosmological Constant Problem J. W. Moffat Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada October 15, 2018 Abstract A finite and unitary nonlocal formulation of quantum gravity is applied to the cosmological constant problem. The entire functions in momentum space at the graviton-standard model particle loop vertices generate an exponential suppression of the vacuum density and the cosmological constant to produce agreement with their observational bounds. 1 Introduction A nonlocal quantum field theory and quantum gravity theory has been formulated that leads to a finite, unitary and locally gauge invariant theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For quantum gravity the finiteness of quantum loops avoids the problem of the non-renormalizabilty of local quantum gravity [15, 16]. The finiteness of the nonlocal quantum field theory draws from the fact that factors of exp[ (p2)/Λ2] are attached to propagators which suppress any ultraviolet divergences in Euclidean momentum space,K where Λ is an energy scale factor. An important feature of the field theory is that only the quantum loop graphs have nonlocal properties; the classical tree graph theory retains full causal and local behavior. Consider first the 4-dimensional spacetime to be approximately flat Minkowski spacetime. Let us denote by f a generic local field and write the standard local Lagrangian as [f]= [f]+ [f], (1) L LF LI where and denote the free part and the interaction part of the action, respectively, and LF LI arXiv:1407.2086v1 [gr-qc] 7 Jul 2014 1 [f]= f f .
    [Show full text]
  • Gauge-Fixing Degeneracies and Confinement in Non-Abelian Gauge Theories
    PH YSICAL RKVIE% 0 VOLUME 17, NUMBER 4 15 FEBRUARY 1978 Gauge-fixing degeneracies and confinement in non-Abelian gauge theories Carl M. Bender Department of I'hysics, 8'ashington University, St. Louis, Missouri 63130 Tohru Eguchi Stanford Linear Accelerator Center, Stanford, California 94305 Heinz Pagels The Rockefeller University, New York, N. Y. 10021 (Received 28 September 1977) Following several suggestions of Gribov we have examined the problem of gauge-fixing degeneracies in non- Abelian gauge theories. First we modify the usual Faddeev-Popov prescription to take gauge-fixing degeneracies into account. We obtain a formal expression for the generating functional which is invariant under finite gauge transformations and which counts gauge-equivalent orbits only once. Next we examine the instantaneous Coulomb interaction in the canonical formalism with the Coulomb-gauge condition. We find that the spectrum of the Coulomb Green's function in an external monopole-hke field configuration has an accumulation of negative-energy bound states at E = 0. Using semiclassical methods we show that this accumulation phenomenon, which is closely linked with gauge-fixing degeneracies, modifies the usual Coulomb propagator from (k) ' to Iki ' for small )k[. This confinement behavior depends only on the long-range behavior of the field configuration. We thereby demonstrate the conjectured confinement property of non-Abelian gauge theories in the Coulomb gauge. I. INTRODUCTION lution to the theory. The failure of the Borel sum to define the theory and gauge-fixing degeneracy It has recently been observed by Gribov' that in are correlated phenomena. non-Abelian gauge theories, in contrast with Abe- In this article we examine the gauge-fixing de- lian theories, standard gauge-fixing conditions of generacy further.
    [Show full text]
  • Gauge Symmetry Breaking in Gauge Theories---In Search of Clarification
    Gauge Symmetry Breaking in Gauge Theories—In Search of Clarification Simon Friederich [email protected] Universit¨at Wuppertal, Fachbereich C – Mathematik und Naturwissenschaften, Gaußstr. 20, D-42119 Wuppertal, Germany Abstract: The paper investigates the spontaneous breaking of gauge sym- metries in gauge theories from a philosophical angle, taking into account the fact that the notion of a spontaneously broken local gauge symmetry, though widely employed in textbook expositions of the Higgs mechanism, is not supported by our leading theoretical frameworks of gauge quantum the- ories. In the context of lattice gauge theory, the statement that local gauge symmetry cannot be spontaneously broken can even be made rigorous in the form of Elitzur’s theorem. Nevertheless, gauge symmetry breaking does occur in gauge quantum field theories in the form of the breaking of remnant subgroups of the original local gauge group under which the theories typi- cally remain invariant after gauge fixing. The paper discusses the relation between these instances of symmetry breaking and phase transitions and draws some more general conclusions for the philosophical interpretation of gauge symmetries and their breaking. 1 Introduction The interpretation of symmetries and symmetry breaking has been recog- nized as a central topic in the philosophy of science in recent years. Gauge symmetries, in particular, have attracted a considerable amount of inter- arXiv:1107.4664v2 [physics.hist-ph] 5 Aug 2012 est due to the central role they play in our most successful theories of the fundamental constituents of nature. The standard model of elementary par- ticle physics, for instance, is formulated in terms of gauge symmetry, and so are its most discussed extensions.
    [Show full text]
  • Operator Gauge Symmetry in QED
    Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 013, 7 pages Operator Gauge Symmetry in QED Siamak KHADEMI † and Sadollah NASIRI †‡ † Department of Physics, Zanjan University, P.O. Box 313, Zanjan, Iran E-mail: [email protected] URL: http://www.znu.ac.ir/Members/khademi.htm ‡ Institute for Advanced Studies in Basic Sciences, IASBS, Zanjan, Iran E-mail: [email protected] Received October 09, 2005, in final form January 17, 2006; Published online January 30, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper013/ Abstract. In this paper, operator gauge transformation, first introduced by Kobe, is applied to Maxwell’s equations and continuity equation in QED. The gauge invariance is satisfied after quantization of electromagnetic fields. Inherent nonlinearity in Maxwell’s equations is obtained as a direct result due to the nonlinearity of the operator gauge trans- formations. The operator gauge invariant Maxwell’s equations and corresponding charge conservation are obtained by defining the generalized derivatives of the f irst and second kinds. Conservation laws for the real and virtual charges are obtained too. The additional terms in the field strength tensor are interpreted as electric and magnetic polarization of the vacuum. Key words: gauge transformation; Maxwell’s equations; electromagnetic fields 2000 Mathematics Subject Classification: 81V80; 78A25 1 Introduction Conserved quantities of a system are direct consequences of symmetries inherent in the system. Therefore, symmetries are fundamental properties of any dynamical system [1, 2, 3]. For a sys- tem of electric charges an important quantity that is always expected to be conserved is the total charge.
    [Show full text]
  • Avoiding Gauge Ambiguities in Cavity Quantum Electrodynamics Dominic M
    www.nature.com/scientificreports OPEN Avoiding gauge ambiguities in cavity quantum electrodynamics Dominic M. Rouse1*, Brendon W. Lovett1, Erik M. Gauger2 & Niclas Westerberg2,3* Systems of interacting charges and felds are ubiquitous in physics. Recently, it has been shown that Hamiltonians derived using diferent gauges can yield diferent physical results when matter degrees of freedom are truncated to a few low-lying energy eigenstates. This efect is particularly prominent in the ultra-strong coupling regime. Such ambiguities arise because transformations reshufe the partition between light and matter degrees of freedom and so level truncation is a gauge dependent approximation. To avoid this gauge ambiguity, we redefne the electromagnetic felds in terms of potentials for which the resulting canonical momenta and Hamiltonian are explicitly unchanged by the gauge choice of this theory. Instead the light/matter partition is assigned by the intuitive choice of separating an electric feld between displacement and polarisation contributions. This approach is an attractive choice in typical cavity quantum electrodynamics situations. Te gauge invariance of quantum electrodynamics (QED) is fundamental to the theory and can be used to greatly simplify calculations1–8. Of course, gauge invariance implies that physical observables are the same in all gauges despite superfcial diferences in the mathematics. However, it has recently been shown that the invariance is lost in the strong light/matter coupling regime if the matter degrees of freedom are treated as quantum systems with a fxed number of energy levels8–14, including the commonly used two-level truncation (2LT). At the origin of this is the role of gauge transformations (GTs) in deciding the partition between the light and matter degrees of freedom, even if the primary role of gauge freedom is to enforce Gauss’s law.
    [Show full text]
  • 8 Non-Abelian Gauge Theory: Perturbative Quantization
    8 Non-Abelian Gauge Theory: Perturbative Quantization We’re now ready to consider the quantum theory of Yang–Mills. In the first few sec- tions, we’ll treat the path integral formally as an integral over infinite dimensional spaces, without worrying about imposing a regularization. We’ll turn to questions about using renormalization to make sense of these formal integrals in section 8.3. To specify Yang–Mills theory, we had to pick a principal G bundle P M together ! with a connection on P . So our first thought might be to try to define the Yang–Mills r partition function as ? SYM[ ]/~ Z [(M,g), g ] = A e− r (8.1) YM YM D ZA where 1 SYM[ ]= tr(F F ) (8.2) r −2g2 r ^⇤ r YM ZM as before, and is the space of all connections on P . A To understand what this integral might mean, first note that, given any two connections and , the 1-parameter family r r0 ⌧ = ⌧ +(1 ⌧) 0 (8.3) r r − r is also a connection for all ⌧ [0, 1]. For example, you can check that the rhs has the 2 behaviour expected of a connection under any gauge transformation. Thus we can find a 1 path in between any two connections. Since 0 ⌦M (g), we conclude that is an A r r2 1 A infinite dimensional affine space whose tangent space at any point is ⌦M (g), the infinite dimensional space of all g–valued covectors on M. In fact, it’s easy to write down a flat (L2-)metric on using the metric on M: A 2 1 µ⌫ a a d ds = tr(δA δA)= g δAµ δA⌫ pg d x.
    [Show full text]
  • Spontaneous Symmetry Breaking in the Higgs Mechanism
    Spontaneous symmetry breaking in the Higgs mechanism August 2012 Abstract The Higgs mechanism is very powerful: it furnishes a description of the elec- troweak theory in the Standard Model which has a convincing experimental ver- ification. But although the Higgs mechanism had been applied successfully, the conceptual background is not clear. The Higgs mechanism is often presented as spontaneous breaking of a local gauge symmetry. But a local gauge symmetry is rooted in redundancy of description: gauge transformations connect states that cannot be physically distinguished. A gauge symmetry is therefore not a sym- metry of nature, but of our description of nature. The spontaneous breaking of such a symmetry cannot be expected to have physical e↵ects since asymmetries are not reflected in the physics. If spontaneous gauge symmetry breaking cannot have physical e↵ects, this causes conceptual problems for the Higgs mechanism, if taken to be described as spontaneous gauge symmetry breaking. In a gauge invariant theory, gauge fixing is necessary to retrieve the physics from the theory. This means that also in a theory with spontaneous gauge sym- metry breaking, a gauge should be fixed. But gauge fixing itself breaks the gauge symmetry, and thereby obscures the spontaneous breaking of the symmetry. It suggests that spontaneous gauge symmetry breaking is not part of the physics, but an unphysical artifact of the redundancy in description. However, the Higgs mechanism can be formulated in a gauge independent way, without spontaneous symmetry breaking. The same outcome as in the account with spontaneous symmetry breaking is obtained. It is concluded that even though spontaneous gauge symmetry breaking cannot have physical consequences, the Higgs mechanism is not in conceptual danger.
    [Show full text]
  • Gauge-Independent Higgs Mechanism and the Implications for Quark Confinement
    EPJ Web of Conferences 137, 03009 (2017) DOI: 10.1051/ epjconf/201713703009 XIIth Quark Confinement & the Hadron Spectrum Gauge-independent Higgs mechanism and the implications for quark confinement Kei-Ichi Kondo1;a 1Department of Physics, Faculty of Science, Chiba University, Chiba 263-8522, Japan Abstract. We propose a gauge-invariant description for the Higgs mechanism by which a gauge boson acquires the mass. We do not need to assume spontaneous breakdown of gauge symmetry signaled by a non-vanishing vacuum expectation value of the scalar field. In fact, we give a manifestly gauge-invariant description of the Higgs mechanism in the operator level, which does not rely on spontaneous symmetry breaking. For concrete- ness, we discuss the gauge-Higgs models with U(1) and SU(2) gauge groups explicitly. This enables us to discuss the confinement-Higgs complementarity from a new perspec- tive. 1 Introduction Spontaneous symmetry breaking (SSB) is an important concept in physics. It is known that SSB occurs when the lowest energy state or the vacuum is degenerate [1]. First, we consider the SSB of the global continuous symmetry G = U(1). The complex scalar field theory described by the following Lagrangian density Lcs has the global U(1) symmetry: ! 2 2 ∗ µ ∗ ∗ λ ∗ µ L = @µϕ @ ϕ − V(ϕ ϕ); V(ϕ ϕ) = ϕ ϕ − ; ϕ 2 C; λ > 0; (1) cs 2 λ where ∗ denotes the complex conjugate and the classical stability of the theory requires λ > 0. Indeed, this theory has the global U(1) symmetry, since Lcs is invariant under the global U(1) transformation, i.e., the phase transformation: ϕ(x) ! eiθϕ(x): (2) If the potential has the unique minimum (µ2 ≤ 0) as shown in the left panel of Fig.
    [Show full text]
  • Gauge Fixing and Renormalization in Topological Quantum Field Theory*
    SLAC-PUB-4630 May 1988 P3 Gauge Fixing and Renormalization in Topological Quantum Field Theory* ROGER BROOKS, DAVID MONTANO, AND JACOB SONNENSCHEIN+ Stanford Linear Accelerator Center Stanford University, Stanford, California 94 309 ABSTRACT In this letter we show how Witten’s topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero. Submitted to Phys. Lett. * Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Work supported by Dr. Chaim Weismann Postdoctoral Fellowship. 1. INTRODUCTION Recently, there has been an effort to study the topology of four dimensional manifolds using the methods of quantum field theory!‘-‘] This was motivated by Donaldson’s use of self-dual Yang-Mills equations to investigate the topology of low dimensional manifolds!” The work of Floer on three manifolds, as interpreted by Atiyah!“can be understood as a modified non-relativistic field theory. Atiyah then conjectured that a relativistic quantum field theory could be used to study Donaldson’s invariants on four manifolds. This led Witten to work out a series of preprints in which just such topological quantum field theories (TQFT) were described.14-61 In the first paper a TQFT was proposed whose correlation func- tions were purely topological and reproduced Donaldson’s invariants!Q’ A later paper extended these ideas to a more complicated Lagrangian which could be used to study similar invariants in topological gravity!51These TQFT’s all pos- sess a fermionic symmetry analogous to a BRST symmetry.
    [Show full text]
  • Gauge-Underdetermination and Shades of Locality in the Aharonov-Bohm Effect*
    Gauge-underdetermination and shades of locality in the Aharonov-Bohm effect* Ruward A. Mulder Abstract I address the view that the classical electromagnetic potentials are shown by the Aharonov-Bohm effect to be physically real (which I dub: `the potentials view'). I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into `gauge-underdetermination': different gauge choices repre- sent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature (or in classrooms), neither by proponents nor by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellec- tually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e.
    [Show full text]
  • Gauge Theory
    Preprint typeset in JHEP style - HYPER VERSION 2018 Gauge Theory David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html [email protected] Contents 0. Introduction 1 1. Topics in Electromagnetism 3 1.1 Magnetic Monopoles 3 1.1.1 Dirac Quantisation 4 1.1.2 A Patchwork of Gauge Fields 6 1.1.3 Monopoles and Angular Momentum 8 1.2 The Theta Term 10 1.2.1 The Topological Insulator 11 1.2.2 A Mirage Monopole 14 1.2.3 The Witten Effect 16 1.2.4 Why θ is Periodic 18 1.2.5 Parity, Time-Reversal and θ = π 21 1.3 Further Reading 22 2. Yang-Mills Theory 26 2.1 Introducing Yang-Mills 26 2.1.1 The Action 29 2.1.2 Gauge Symmetry 31 2.1.3 Wilson Lines and Wilson Loops 33 2.2 The Theta Term 38 2.2.1 Canonical Quantisation of Yang-Mills 40 2.2.2 The Wavefunction and the Chern-Simons Functional 42 2.2.3 Analogies From Quantum Mechanics 47 2.3 Instantons 51 2.3.1 The Self-Dual Yang-Mills Equations 52 2.3.2 Tunnelling: Another Quantum Mechanics Analogy 56 2.3.3 Instanton Contributions to the Path Integral 58 2.4 The Flow to Strong Coupling 61 2.4.1 Anti-Screening and Paramagnetism 65 2.4.2 Computing the Beta Function 67 2.5 Electric Probes 74 2.5.1 Coulomb vs Confining 74 2.5.2 An Analogy: Flux Lines in a Superconductor 78 { 1 { 2.5.3 Wilson Loops Revisited 85 2.6 Magnetic Probes 88 2.6.1 't Hooft Lines 89 2.6.2 SU(N) vs SU(N)=ZN 92 2.6.3 What is the Gauge Group of the Standard Model? 97 2.7 Dynamical Matter 99 2.7.1 The Beta Function Revisited 100 2.7.2 The Infra-Red Phases of QCD-like Theories 102 2.7.3 The Higgs vs Confining Phase 105 2.8 't Hooft-Polyakov Monopoles 109 2.8.1 Monopole Solutions 112 2.8.2 The Witten Effect Again 114 2.9 Further Reading 115 3.
    [Show full text]
  • Path Integral and Gauge Fixing
    11.12.2020 Non-abelian gauge theories: path integral and gauge fixing (Lecture notes - a.a. 2020/21) Fiorenzo Bastianelli 1 Faddeev-Popov The quantization of gauge theories requires additional considerations for constructing a well- defined QFT and its perturbative expansion. The problem is clear in the path integral approach, where a naive path integral quantization gives a diverging result. Let us reconsider the case of the abelian U(1) theory (free electromagnetism) Z 1 S[A] = d4x − F F µν 4 µν Z (1) Z = DA eiS[A] ∼ 1 where Fµν = @µAν − @νAµ. Here the path integral diverges because one is summing over an infinite number of gauge equivalent configurations g −1 iα(x) Aµ(x) ! Aµ(x) = Aµ(x) + ig(x)@µg (x) ; g(x) = e 2 U(1) (2) which have the same value of the action, S[Ag] = S[A]. The field space decomposes into inequivalent gauge orbits, as in figure Figure 1: Gauge orbits One would like to define the path integral in such a way of getting a finite and gauge invariant result Z DA Z = eiS[A] ∼ finite : (3) Vol(Gauge) This can be done using a gauge fixing function `ala Faddeev-Popov, where unphysical ghost fields are introduced to exponentiate a measure factor. Ideally, the gauge fixing function should pick just one representative from each gauge orbits, as sketched in figure 2 1 Figure 2: Gauge fixing by the condition f(A) = 0 The gauge-fixed action thus obtained exhibits a global symmetry, called BRST symmetry, which is a remnant of the gauge symmetry and which is very powerful to prove many prop- erties of quantum gauge theories.
    [Show full text]