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)].

**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)) Δ_{FP},
for some gauge fixing function *F*(Φ) = 0, where

Δ_{FP}^{–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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jan 2015