Euler-Heisenberg Lagrangian (2) - Schwinger Proper Time Method

In Schwinger's classic, On Gauge Invariance and Vacuum Polarization(Phys.Rev.82,664, 1951), One mysterious formula appeared.

\( {\mathcal{L}}^{(1)}(x)=\frac{1}{2}i\int_{0}^{\infty}ds s^{-1} \mathrm{exp}(-ims^{2})\mathrm{tr}\left \langle x|U(s)|x \right \rangle\),       (2.32) where \(U(s)=\mathrm{exp} \left[-i((p-eA)^{2}-\frac{1}{2}e\sigma_{\mu\nu}F^{\mu\nu})\right] \)

In the paper, you can derive this formula, but one cannot understand why we have to used expectation value of current '\(\left \langle j(x) \right \rangle\)' instead of current '\(j(x)\)'. For this reason, We will derive (2.32) by integrating out '\(\psi(x)\), Dirac field', the main ingredient of QED(Quantum Electrodynamics).

Keep in mind the concept of Effective theory(not 1PI Effective action).
$$
\int \mathcal{D}A\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathrm{exp}\left [i\int d^{4}x \left \{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i(\partial\!\!\!/-ieA\!\!\!/)-m)\psi \right \} \right ]\\
=\int \mathcal{D}A \mathrm{exp} \left[i\int d^{4}x S_{eff}[A_{\mu}]\right]\ \ \ \ \ \ \ \ \ \ -(1)
$$

If you are not familiar with path integral, here are some references. I. J.J.Sakurai, J.Napolitano, "Modern Quantum Mechanics", 122-129. II. A.Zee, "Quantum Field Theory in a Nutshell", 7-12. III. B.Desai, "Quantum Mechanics with Basic Field Theory", 473-478 IV. F.Mandl, G.Shaw, "Quantum Field Theory", 285-292 V. A.Altland, B.Simons, "Condensed Matter Field Theory", 95-155

I will make further assumption that \(F_{\mu\nu}\) is constant, \(i.e.\), constant \(\vec{E}\) & \(\vec{B}\).
Do not confuse: \(A_{\mu}\) is not constant because of the fact \(F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\).

Let's perform (1).
\(
\int \mathcal{D}\bar{\psi}\mathcal{D}\psi \mathrm{exp}\left[i\int d^{4}x \bar{\psi}(iD\!\!\!/-m)\psi \right]=\mathrm{Const.}\times \ \ \mathrm{Det}[iD\!\!\!/-m]
\)

Grassmann algebra was used. You can master(not in the mathematical precise way) Grassmann algebra by referring I. A.Zee, "Quantum Field Theory in a Nutshell" II. F.Mandl, G.Shaw, "Quantum Field Theory" III. A.Altland, B.Simons, "Condensed Matter Field Theory" IV. M.Nakahara, "Geometry, Topology and Physics"

\(
\therefore \ \ S_{eff}=\int d^{4}x \mathcal{L}_{eff}[A_{\mu}]=\int d^{4} x [-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}]-i \mathbf{Tr}[\mathrm{ln}(iD\!\!\!/-m)]
\) up to constant term.

The fact that \(\mathrm{Det}[A]=\mathrm{exp}[\mathrm{tr}(\mathrm{ln}A)]\) was exploited. Trace(\(\mathbf{Tr}\)) must be evaluated on states(\( \int d^{4}x \left \langle x| \ \ |x \right \rangle \)) and Dirac indices(\(tr\)): \(\mathbf{Tr}[f(x)]=\int d^{4}x \left \langle x|\mathrm{tr}[f(x)]|x \right \rangle \).

\(
\therefore \ \ \mathcal{L}_{eff}[A_{\mu}]=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-i\ \ \mathrm{tr} \left \langle x|\mathrm{ln}(iD\!\!\!/-m)]|x \right \rangle
\)

We can calculate trace term directly, but I will use trick by using property of logarithm.
\(
\frac{d}{d m} \mathcal{L}_{eff}=i \ \ \mathrm{tr}[\langle x |\frac{1}{i D\!\!/ \ \ -\ \ m} | x \rangle]=-i \ \ \mathrm{tr} \langle x| \frac{i D\!\!/ \ \ +\ \ m}{D\!\!/ ^2\ \ +\ \ m^2}|x \rangle\)

\(
=-i m \langle x|\frac{1}{D\!\!/ ^2\ \ +\ \ m^2}|x \rangle =m \int_{0}^{\infty} ds \ \ \mathrm{tr} \langle x| \mathrm{exp}\left[ -is m^{2}-i s D\!\!\!/ ^{2} \right]|x \rangle
\)

The fact that odd number of gamma matirces are traceless and Schwinger parametrization was used.

* Schwinger Parametrization$$ \frac{i}{A+i \epsilon}=\int_{0}^{\infty} ds \ \ e^{is (A+i \epsilon)} $$

Rest things are just integrate over \(m\) and follow (Schwinger, 1951) keeping your notation.

I postponed posting about 'applications of EH Langrangian' to the next(final) posting.