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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)].
Faddeev-Popov Procedure
* 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 CJP(12)-a1112 [with covariant gauge fixing]; Ornigotti & Aiello a1407 [pedagogical]; Chen et al a1712 [modification free from Gribov ambiguity].
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];
Etesi a1907
[partition function in the vicinity of the vacuum].
@ 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 PTEP(16)-a1508 [Abelian gauge field, gauge freedom];
Iraso & Mnev CMP(19)-a1806 [2D Yang-Mills theories with corners].
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