Lecture Notes for MATH4017 Quantum Field Theory — Feynman propagator for the photon

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6.3 Feynman propagator for the photon

The Feynman propagator for the electromagnetic potential/photon is easy to derive by following the same steps as for the Klein-Gordon field in Section 3.4. Let us first note that the $2$-point correlation function can be obtained using (6.32) and a short calculation

\begin{flalign} \nn \expect {0}{A_{\mu }(x)\,A_{\nu }(y)}{0} \,&=\, \sum _{\lambda ,\lambda ^\prime =0}^{d-1} \int _{\bbR ^{2(d-1)}}\frac {e^{\ii \,k\,x-\ii \,q\,y}}{\sqrt {2\,\vert \mathbf {k}\vert } \, \sqrt {2\,\vert \mathbf {q}\vert }}~ \expect {0}{a_{\lambda }(\mathbf {k})\,a^\dagger _{\lambda ^\prime }(\mathbf {q})}{0}~\epsilon ^\lambda _{\mu }(\mathbf {k})\,\epsilon ^{\lambda ^\prime }_{\nu }(\mathbf {q})~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\,\frac {\dd \mathbf {q}}{(2\pi )^{d-1}}\\ \nn \,&=\, \int _{\bbR ^{d-1}}\frac {e^{\ii \,k\,(x-y)}}{2\,\vert \mathbf {k}\vert }~ \sum _{\lambda =0}^{d-1}\eta _{\lambda \lambda }\,\epsilon ^\lambda _{\mu }(\mathbf {k})\,\epsilon ^{\lambda }_{\nu }(\mathbf {k})~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\\ \,&=\,\eta _{\mu \nu } \,\int _{\bbR ^{d-1}}\frac {e^{\ii \,k\,(x-y)}}{2\,\vert \mathbf {k}\vert }~\frac {\dd \mathbf {k}}{(2\pi )^{d-1}}\quad . \end{flalign} The second step uses the commutation relations (6.36) and the third step follows from the polarization sum identity (6.35b). Note that the scalar integral in the last line is the massless case $m^2=0$ of the $2$-point Wightman function (3.49b) for the Klein-Gordon field. This means that we can simply write

\begin{flalign} \expect {0}{A_{\mu }(x)\,A_{\nu }(y)}{0}\,=\, \eta _{\mu \nu }\,W_2(x,y)\big \vert _{m^2=0}^{}\quad . \end{flalign} The Feynman propagator for the photon is defined as the time-ordered $2$-point function

\begin{flalign} \nn \expect {0}{\TO \big ( A_{\mu }(x)\,A_{\nu }(y)\big )}{0} \, &=\,\begin{cases} \expect {0}{A_{\mu }(x)\,A_{\nu }(y)}{0}&,~\text {if }x^0\geq y^0\quad ,\\ \expect {0}{A_{\nu }(y)\,A_{\mu }(x)}{0}&,~\text {if }y^0\geq x^0\quad , \end {cases}\\[4pt] \nn \, &=\,\begin{cases} \eta _{\mu \nu }\,W_2(x,y)\big \vert _{m^2=0}^{}&,~\text {if }x^0\geq y^0\quad ,\\ \eta _{\nu \mu }\,W_2(y,x)\big \vert _{m^2=0}^{}&,~\text {if }y^0\geq x^0\quad , \end {cases}\\[5pt] ~&=\, \eta _{\mu \nu }~\Delta _{F}(x-y)\big \vert _{m^2=0}^{}\quad , \end{flalign} where $\Delta _{F}(x-y)\big \vert _{m^2=0}^{}$ is the massless case of the Feynman propagator for the Klein-Gordon field. Recalling the Fourier integral formula (3.57) for the latter, we obtain the following useful integral formula

\begin{equation} (D_F)_{\mu \nu }(x-y) \,:=\, \expect {0}{\TO \big (A_\mu (x)\,A_\nu (y)\big )}{0} = \lim _{\epsilon \to 0} \int _{\bbR ^d} \frac {-\ii \,\eta _{\mu \nu }}{k^2 -\ii \,\epsilon }~e^{\ii \,k\,(x-y)}\,\frac {\dd k}{(2\pi )^d} \quad . \end{equation}

The photon Feynman propagator is typically visualized in terms of a wiggly line

$(6.55) \{begin}{flalign} (D_F)_{\mu \nu }(x-y) ~=~ \expect {0}{\TO \big (A_\mu (x)\,A_\nu (y)\big )}{0} ~= \parbox {1.5cm}{ \begin{tikzpicture}[scale=1] \tikzset {decoration={snake,amplitude=.8mm,segment length=2.5mm, post length=0mm,pre length=0mm}} \draw [thick,decorate] (0.65,0) -- (-0.65,0); \draw (0.75,0) node{\footnotesize {$~~y$}}; \draw (-0.75,0) node{\footnotesize {$x~~$}}; \end {tikzpicture}} \{end}{flalign}$

in order to distinguish it from the Klein-Gordon and Dirac Feynman propagators that are visualized by a straight line or, respectively, a straight line with an arrow. (See Sections 4.3 and 5.4.)

6.3 Feynman propagator for the photon

Further reading