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 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].

@ __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 a1806 [2D Yang-Mills theories with corners].

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