Path Integral/Canonical Quantization of Gauge Theories

[TriTertButoxy]

I'm really getting frustrated right now, as I am unable to reproduce the two-point gauge-field correlation function (i.e. propagator) as derived from the path integral in an $R_\xi$ gauge using operators from canonical quantization. I believe the polarization 4-vectors of the gauge field ought to depend on the gauge parameter $\xi$ in order for this to work, but no references mention a set of $\xi$-dependent polarization vectors.

In mathematical notation, I am unable to derive the precise gauge-dependent ($\xi$-dependent) part of

$$\tilde{D}^{\mu\nu}(p)=\frac{-i}{p^2}\left(g^{\mu\nu}-(1-\xi)\frac{p^\mu p^\nu}{p^2}\right)$$​

by calculating the Fourier transform of

$$\langle0|T\Big(\hat{A}^\mu (x)\hat{A}^\nu (y)\Big)|0\rangle$$​

using the plane-wave expansion of the gauge field $\hat{A}^\mu(x)$

$$\hat{A}^\mu(x)=\int\frac{d^3 \mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \sum_\lambda\left(\epsilon_{[\lambda]}^\mu(\mathbf{p})\hat{a}_\mathbf{p}e^{-ipx}+\text{h.c.}\right)$$​

In order for me to be able to successfully get this to work, the commutation relations of the ladder operators and/or the polarization vectors need to carry a $\xi$-dependence. I would very much like any references that goes through the derivation of the canonical quantization of gauge fields in great detail, and also demonstrates how it is consistent with the path-integral quantization.

 

[tom.stoer]

Usually canonical quantization and PI quantization use different gauges; the propagator is gauge dependent, so it's hard to compare - and physically irrelevant b/c all what matters are gauge-invariant observables.

You are looking for a propagator $$D_{\mu\nu}(p)$$ in the Lorentz gauge $$\partial_\mu A^\mu=0$$ which is an unphysical gauge and which has several drawbacks in the canonical formalism (the latter one is not manifest Lorentz covariant so it doesn't make much sense to study a covariant but unphysical gauge; the Gupta-Bleuler formalism does not work for non-abelian gauge fields; so in many cases the canonical formalism comes with physical gauges, e.g. Weyl gauge or/plus Coulomb gauge).

The canonical formalism in QED with Lorentz gauge is called Gupta-Bleuler formalism; perhaps you can find some references in the internet.

http://en.wikipedia.org/wiki/Gupta–Bleuler_formalism

I am not sure but I think in Itzykson-Zuber "Quantum Field Theory" this formalism is explained in some detail.

 

[TriTertButoxy]

Well, yes; the propagator is gauge dependent. But when people calculate scattering amplitudes, they are using propagators inside their diagrams from the path integral quantization, and polarization vectors from canonical quantization, which seems totally inconsistent to me.

 

[vanhees71]

The reason might be that in most textbook treatments of the operator formalism are either done in the Coulomb gauge (radiation gauge for the free photon field), which is not manifestly gauge covariant, or in Feynman gauge, which means setting $\xi=1$ in your formula for $R_{\xi}$ gauge.

Of course, you can also take an arbitrary value for $\xi$ (except $\xi=0$, which is Landau gauge, which is a bit more complicated). Then the commutator relations contain the explicit dependence on $\xi$, and you can directly calculate the Feynman (time-ordered) or retarded propagator in arbitrary $R_{\xi}$ gauges within the covariant operator formalism, which in the case of abelian gauge theories like QED can be simplified to the Gupta-Bleuler formalism. For non-abelian gauge theories, the original papers by Kugo et al are very well written, better than most textbooks on the subject:

Kugo, T., and Ojima, I. Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I. Progress of Theoretical Physics 60, 6 (1978), 1869–1889.

Kugo, T., and Ojima, O. Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. II: SU (2) Higgs-Kibble Model with Spontaneous Symmetry Breaking. Progress of theoretical physics 61, 1 (1979), 294–314.

Kugo, T., and Ojima, I. Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. III—Pure Yang-Mills Theories without Spontaneous Symmetry Breaking. Progress of Theoretical Physics 61 (1979), 644–655.

 

[TriTertButoxy]

Kugo, T., and Ojima, I. Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I. Progress of Theoretical Physics 60, 6 (1978), 1869–1889.

Kugo, T., and Ojima, O. Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. II: SU (2) Higgs-Kibble Model with Spontaneous Symmetry Breaking. Progress of theoretical physics 61, 1 (1979), 294–314.

Kugo, T., and Ojima, I. Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. III—Pure Yang-Mills Theories without Spontaneous Symmetry Breaking. Progress of Theoretical Physics 61 (1979), 644–655.

This is awesome! Just what I have been looking for.
But in their papers they make references to "anti-commuting c-numbers" which don't make conceptual sense. What do they mean by that? Usually "c-number" means "commuting number"; so is "c" taken to mean "complex" instead of "commuting" in this case? (For example, see the line right below eqn 2.28 in their first paper.

 

[tom.stoer]

w/o having Kugo's paper at hand I guess that anti-commuting c-numbers are Grassmann numbers used to describe the fermion fields (anti-commuting for fermions, c-numbers b/c they are numbers and not operators); anyway - they shouldn't play a role for the boson propagator which is defined via the free, purely bosonic Lagrangian + gauge fixing terms

 

[Lapidus]

This is awesome! Just what I have been looking for.
But in their papers they make references to "anti-commuting c-numbers" which don't make conceptual sense. What do they mean by that? Usually "c-number" means "commuting number"; so is "c" taken to mean "complex" instead of "commuting" in this case? (For example, see the line right below eqn 2.28 in their first paper.

Have not read the paper, but maybe ghost fields, which are anticommuting scalars.

 

[vanhees71]

Yes, anti-commuting c-numbers are Grassmann numbers. In the description of BRST symmetry it's convenient to choose the group parameters as Grassmann numbers, commuting with bosonic and anti-commuting with fermionic field operators. You can, of course, also use usual c-numbers, but then you have to introduce the appropriate signs for the transformation of fermion-field operators.

 

[tom.stoer]

but maybe ghost fields, which are anticommuting scalars.

ghosts are not required in QED

 

[vanhees71]

Sure, that's what I meant when I said that for Abelian gauge models like QED the general covariant operator formalism simplifies to the Gupta-Bleuler method. I thought, the OP asked for the general case of non-abelian gauge theories. Of course, for QED it's much simpler to look in textbooks that use the operator formalism like Schweber or Weinberg.

 

[Lapidus]

ghosts are not required in QED

Yes, but all the three papers deal with Yang-Mills theories, so most surely Kugo refers here to what is also known as ghost fields.