I have a question in Srednicki's book regarding path integrals, but first I'll set it up so that no familiarity of the book is required to answer the question.(adsbygoogle = window.adsbygoogle || []).push({});

The vacuum to vacuum transition amplitude for the photon field in the presence of a source is given by: [tex]<0|0>_J=\int \mathcal D A e^{iS+\int J_\mu A^\mu}[/tex] where [tex]S=\int -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} d^4x[/tex]. The result is [tex]e^{i\int \int J^\mu(x)\Delta(x-y)_{\mu \nu} J^\nu(y)} [/tex]. If you rewrite J, the current density, in terms of the line integral [tex] e \oint dx^\mu [/tex], and choose a path for the line integral, then this vacuum to vacuum transition amplitude, being equal to [tex]e^{-iE_{0}t [/tex], should give you the energy of the situation described by your line integral. In this way by choosing the path where an electron and anti-electron are just sitting a small distance R apart for a long time T, you actually recover the Coloumb force from QFT, [tex]V(R)=-\frac{\alpha}{R}[/tex]. This is equation (82.22) Srednicki.

My question is what does it mean to have the vacuum to vacuum amplitude in the presence of a source? If there is a source then is it really vacuum? Or does the vacuum mean it's a vacuum with respect to only photons, but you are allowed to have a fermion source [obviously J(x)]? If this is the case, then could you in principle use this same method but use the Dirac Lagrangian in place of the photon Lagrangian, and have the source J(x) be a photon source, and from this derive the energy between two photons? If one wants to derive Maxwell's equations (and not just Coloumb's law) from the path integral approach, how would you do this?

Also, this is the first time I've ever seen QFT used to describe something in position space. Is this the standard way to check if QFT gives the same results as classical physics - vacuum expectation values in presence of sources?

The coupling constant in QED gets big if the energy is high, and high energy is equivalent to short distances. Is this reflected in Coloumb's law with the 1/r^2 dependence? The beta function and renormalization seemed to have disappeared from this example, so this must mean that the 1/r^2 dependence is not due to changing coupling constants with energy?

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# Transition from QFT to Classical Physics using Path Integrals?

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