|Path-Integral Quantization of Gauge Theories|
In General > s.a. lattice
gauge theory; path integrals in quantum field theory.
* Advantages: It is the most convenient approach for gauge theories [@ Nash on quantum field theory].
* Gauge equivalence problem: Gauge variations in A don't produce any change in S, so the integral \(\int\cal D\)A won't converge if the gauge group is non-compact; Therefore, we must integrate over gauge equivalence classes of connections, i.e., choose some gauge representative, and apply the Faddeev-Popov procedure; This is not always possible (because of the Gribov problem), and should be done with care to ensure gauge invariance of the result; This prescription introduces ghosts in the theory.
* Evaluation of the integrand: Use the Steepest-Descent Approximation; This motivates the study of (anti)self-dual connections as extrema of the action; We now know, however, that this approximation is not enough [@ Uhlenbeck CMP(82)].
* Idea: To define the functional measure in gauge theory, one wants to integrate using only representatives of the gauge equivalence classes, appropriately weighted; This can be done by using \(\cal D\)Φ δ(F(Φ)) ΔFP, for some gauge fixing function F(Φ) = 0, where
ΔFP–1 = ∫ \(\cal D\)Φ δ(F(Φ)) .
@ References: Faddeev & Popov PLB(67); Ellicott et al MPLA(89) [geometrical]; Cabo et al PLB(91) [alternative derivation]; Vassilevich PLB(98)ht/97 [with boundaries]; Jaramillo et al RMF-ht/98; Akant JMP(08)ht/07 [as equivariant localization]; Limboonsong & Manoukian IJTP(06)-a0709; McKeon a1112 [with covariant gauge fixing]; Ornigotti & Aiello a1407 [pedagogical].
Ghosts > s.a. Ghost Fields.
* A: Extra fields introduced by the Faddeev-Popov procedure for path-integral quantization of gauge theories, in the procedure for making the path integral finite (gauge invariance would make the naive form infinite); They do not couple to the gauge field A, and are usually ignored in flat spacetime, but are important in curved spacetime.
* B: Negative norm states in quantum field theory; They seem to arise in higher derivative theories.
@ References: Faddeev & Popov PLB(67); Hawking & Hertog PRD(02)ht/01; van Tonder NPB(02) [as negative spinors]; Piguet CQG(00) [for diffeomorphisms, and vector supersymmetry]; Gibbons & Higuchi PRD(15)-a1410 [removing the Faddeev-Popov zero modes in spacetimes with compact spatial sections].
> Online resources: see Physics Stack Exchange page; Scholarpedia page; Wikipedia page.
References > s.a. BRST quantization.
@ General: Reshetikhin a1008-ln.
@ Techniques: Velo & Wightman ed-86, Bracken CQG(99) [measure]; Tomé 98 [on group manifolds]; Hüffel & Kelnhofer PLB(00)ht/99, ht/99-conf, NPPS(00)ht/99 [global path integral]; Shabanov PRP(00)ht [and phase-space geometry]; Conrady gq/05 [spin foams]; Jacquot PRD(09)-a0902 [regularization].
@ Hamiltonian formulation: Henneaux PRP(85); Dresse et al NPB(91); Su JPG(01)ht/00 [Lorentz-covariant]; Muslih HJ(02)mp/00.
@ Approaches: Whelan PRD(96)ht/95 [Hartle's generalized quantum mechanics]; Shabanov & Klauder PLB(99)ht [non-perturbative].
@ And Wilson loops: Faber et al PRD(00)ht/99.
@ Approximations: Hsu & Reeb IJMPA(10)-a0808 [path integral with positive-definite measure].
@ Related topics: Henneaux PLB(90) [antifield formalism, elimination of auxiliary fields]; Witten JGP(92) [2D, and Duistermaat-Heckman integration formula]; Moss & Silva PRD(97)gq/96 [BRST-invariant boundary conditions]; Tanimura ht/01-proc [symmetries and strata]; Reshetnyak a1412-proc [with soft BRST symmetry breaking]; Saito et al a1508 [Abelian gauge field, gauge freedom].
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