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.