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 [itex]R_\xi[/itex] gauge using operators from canonical quantization. I believe the polarization 4-vectors of the gauge field ought to depend on the gauge parameter [itex]\xi[/itex] in order for this to work, but no references mention a set of [itex]\xi[/itex]-dependent polarization vectors.(adsbygoogle = window.adsbygoogle || []).push({});

In mathematical notation, I am unable to derive theprecisegauge-dependent ([itex]\xi[/itex]-dependent) part of

[tex]\tilde{D}^{\mu\nu}(p)=\frac{-i}{p^2}\left(g^{\mu\nu}-(1-\xi)\frac{p^\mu p^\nu}{p^2}\right)[/tex]by calculating the Fourier transform of

[tex]\langle0|T\Big(\hat{A}^\mu (x)\hat{A}^\nu (y)\Big)|0\rangle[/tex]using the plane-wave expansion of the gauge field [itex]\hat{A}^\mu(x)[/itex]

[tex]\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)[/tex]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 [itex]\xi[/itex]-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.

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# Path Integral/Canonical Quantization of Gauge Theories

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