Comparing Path Integral Conventions in QED

[dman12]

Hello. I am doing some reading on QED and am getting a bit confused on the different conventions used. In Matthew Schwartz's book we have the Lagrangian given as:

L_QED = -¼ F_μνF^μν + iψ^*γ^μ(∂_μ + ieA_μ)ψ - V(ψ^*ψ)

And the path integral factor is exp(iS).

In another text, however, I see the QED lagrangian given as:

L_QED = ¼ F_μνF^μν + ψ^*γ^μ(∂_μ + eA_μ)ψ - V(ψ^*ψ)

And the exponential factor in the path integral is exp(-S).

How can I see that these two conventions are physically the same? In particular, what is the difference between using exp(iS) and exp(-S) in the path integral?

Thanks!

 

[Orodruin]

In another text

This, unfortunately, does not tell us very much. My guess would be that a Wick rotation has already been applied to the second case.

 

[vanhees71]

Yes, the first convention hints at that the book uses the west-coast convention and Minkowski-space path intgrals (that's indeed true for Schwartz's marvelous text which I tend to recommend as a first textbook on QFT instead of Ryder or Peskin/Schroeder). The other convention hints at that the author is writing down Euclidean path integrals with the positive definite metric.

 

[FieldTheorist]

How can I see that these two conventions are physically the same? In particular, what is the difference between using exp(iS) and exp(-S) in the path integral?

Thanks!

Your main question is easy --they have gone from the Lorentzian to the Euclidean (aka "Wick rotated") via a transformation of the kind:

t \to i \tau \, ,

which is allowed under many physical circumstances, so long as the contour integration and analytic continuation is appropriately taken into account. In the case of the path integral, it is almost exclusively done to make the path integral overtly convergent. (The minus sign in the spacetime metric makes Gaussian and similar approximations sketchy, but if you can convince yourself you can Wick rotate, these problems become manifestly convergent.)

HOWEVER: These two groups are using other conventional differences, too, if I'm not mistaken, including having separate fermion conventions. Note that there are various conventions for:

\eta = (\pm 1, \mp 1, \mp 1, \mp 1)
\varepsilon^{0123} = \pm 1
\{ \gamma^{\mu} , \gamma^{\nu} \} = \pm 2 \eta^{\mu\nu}
\bar{\psi} = \pm (\psi)^{\dagger} \gamma^0
G(x-y) = \int \frac{d^3 k}{(2\pi)^4} \frac{\pm e^{i k \cdot (x-y)}} {k^2 \mp m^2}
(\psi_{\alpha} \chi_{\beta})^{\dagger} = \pm \chi_{\beta}\,^{\dagger} \psi_{\alpha}\,^{\dagger}

and so forth. And they all lead to different i's, minus signs, $2\pi$'s etc. They all have to be checked against unitarity (e.g. vacuum persistence amplitude in the presence of a source is simple enough for most of it), that it has a positive-definite, canonically normalized Hamiltonian, etc. It is best to pick a convention (I recommend Srednicki's for its completeness and modernity) and derive the action for yourself. It's generally not easy to compute things from someone else's incomplete list of conventions (which often happens in papers). If you absolutely need to check between sources, scour the paper for their conventions, email the author's if necessary, and go back to the basics to see how the differences in conventions propagates through the fundamental formulas.