Path
Integral Quantization of Gauge
Theories| 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 
A won't
converge; 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 
(F()) 
FP,
for some gauge fixing function F()
= 0, where
FP–1 =
(F())
.
@ 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 ht/98/RMF; Akant ht/07 [as equivariant localization]; Limboonsong & Manoukian IJTP(06)-a0709.
Ghosts
* 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].
References > s.a. BRST quantization.
@ Techniques: Velo & Wightman ed-86, Bracken CQG(99)
[measure]; Tomé
98
[on group manifolds]; Hüffel & Kelnhofer PLB(00)ht/99, ht/99-in,
NPPS(00)ht/99 [global
path integral];
Shabanov
PRP(00)ht [and
phase space geometry]; Conrady gq/05 [spin
foams].
@ 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.
@ 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-in
[symmetries
and strata].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
8 jun 2008