Since the Green’s function of a Dirac particle in an external field, which is described by a potential A
μ(x), is given by
$$\displaystyle{ \left [\gamma \cdot \left (\frac{1} {i} \partial - eA\right ) + m\right ]G(x,x^{{\prime}}\vert A) =\delta (x - x^{{\prime}}) }$$
(37.1)
the Green operator G
+[A] is defined by
$$\displaystyle{ \left (\gamma \Pi + m\right )G_{+} = 1\,,\quad \Pi _{\mu } = p_{\mu } - eA_{\mu } }$$
or
$$\displaystyle\begin{array}{rcl} G_{+}& =& \frac{1} {\gamma \Pi + m - i\epsilon }\,,\quad \epsilon > 0 {}\\ & =& \frac{\gamma \Pi - m} {\left (\gamma \Pi \right )^{2} - m^{2} + i\epsilon } = \frac{-\gamma \Pi + m} {m^{2} -\left (\gamma \Pi \right )^{2} - i\epsilon } {}\\ & =& \left (-\gamma \Pi + m\right )i\int _{0}^{\infty }ds\exp \left [-is\left (m^{2} -\left (\gamma \Pi \right )^{2}\right ) -\epsilon s\right ]_{\epsilon \rightarrow 0}\quad. {}\\ \end{array}$$
This expression is needed in
$$\displaystyle{ i\frac{\delta W^{(1)}[A]} {\delta A_{\mu }(x)} = e\mathop{\mathrm{tr}}\nolimits _{\gamma }\left [\gamma ^{\mu }G_{+}\left (x,x\vert A\right )\right ]. }$$
(37.2)
One can show that the ansatz
$$\displaystyle{ iW^{(1)} = i\int (dx)\mathcal{L}^{(1)} = -\frac{1} {2}\int _{0}^{\infty }\frac{ds} {s} e^{-ism^{2} }\mathop{ \mathrm{Tr}}\nolimits \left [e^{is\left (\gamma \Pi \right )^{2} }\right ] }$$
(37.3)
fulfills equation (37.2). One can therefore write the unrenormalized Lagrangian
$$\displaystyle{ \mathcal{L}^{(1)} = \frac{i} {2}\mathop{\mathrm{tr}}\nolimits \int _{0}^{\infty }\frac{ds} {s} e^{-ism^{2} }\langle x\vert e^{is\left (\gamma \Pi \right )^{2} }\vert x\rangle + \text{const.} }$$
(37.4)
Let us write (37.3) in the form
$$\displaystyle\begin{array}{rcl} iW^{(1)}[A] = -\frac{1} {2}\int _{0}^{\infty }\frac{ds} {s} e^{-ism^{2} }\mathop{ \mathrm{Tr}}\nolimits \left \{\left [U(s) - U_{0}(s)\right ] + ct\right \}& &{}\end{array}$$
(37.5)
where
$$\displaystyle\begin{array}{rcl} U_{0}(s)& =& e^{is\partial ^{2} } {}\\ U(s)& =& e^{is\left (\gamma \Pi \right )^{2} } = e^{is\left (-\Pi ^{2}+\frac{e} {2} \sigma _{\mu \nu }F^{\mu \nu }\right ) } {}\\ V (s)& =& U_{0}^{-1}(s)U(s) = U_{ 0}(-s)U(s) = e^{-is\partial ^{2} }e^{is\left (-\Pi ^{2}+\frac{e} {2} \sigma _{\mu \nu }F^{\mu \nu }\right ) }. {}\\ \end{array}$$
V (s) satisfies the differential equation
$$\displaystyle\begin{array}{rcl} -i \frac{\partial } {\partial s}V (s)& =& e^{-is\partial ^{2} }\left [-\Pi ^{2} - \partial ^{2} + \frac{e} {2}\sigma F\right ]e^{is\partial ^{2} }V (s) {}\\ & =& e^{-is\partial ^{2} }\left [-\left [-\partial ^{2} - e\left (\frac{1} {i} \partial A + A\frac{1} {i} \partial \right ) + e^{2}A^{2}\right ] - \partial ^{2} + \frac{e} {2}\sigma F\right ]e^{is\partial ^{2} }V (s) {}\\ & =& e^{-is\partial ^{2} }\left [\partial ^{2} + e\left (\frac{1} {i} \partial A + A\frac{1} {i} \partial \right ) - e^{2}A^{2} - \partial ^{2} + \frac{e} {2}\sigma F\right ]e^{is\partial ^{2} }V (s) {}\\ & =& e^{-is\partial ^{2} }Qe^{is\partial ^{2} }V (s) = U_{0}^{-1}(s)QU_{ 0}(s)V (s) = U_{0}^{-1}(s)QU(s) {}\\ & & {}\\ \end{array}$$
where
$$\displaystyle\begin{array}{rcl} Q&:=& -e^{2}A^{2} + e\left (pA + Ap\right ) + \frac{e} {2}\sigma F\quad. {}\\ \end{array}$$
The corresponding integral equation, incorporating the boundary condition V (0) = 1, is
$$\displaystyle\begin{array}{rcl} V (s)& =& 1 + i\int _{0}^{s}ds^{{\prime}}U_{ 0}^{-1}(s^{{\prime}})QU_{ 0}(s^{{\prime}})V (s^{{\prime}}),\quad U = U_{ 0}V {}\\ U(s)& =& U_{0}(s) + iU_{0}(s)\int _{0}^{s}ds^{{\prime}}U_{ 0}^{-1}(s^{{\prime}})QU(s^{{\prime}})\quad. {}\\ & & {}\\ \end{array}$$
Iterating
$$\displaystyle\begin{array}{rcl} U(s)& =& U_{0}(s) + iU_{0}(s)\int _{0}^{s}ds^{{\prime}}U_{ 0}^{-1}(s^{{\prime}})QU_{ 0}(s^{{\prime}}) {}\\ & +& i^{2}U_{ 0}(s)\int _{0}^{s}ds^{{\prime}}U_{ 0}^{-1}(s^{{\prime}})Q\int _{ 0}^{s^{{\prime}} }ds^{{\prime\prime}}U_{ 0}(s^{{\prime}})U_{ 0}^{-1}(s^{{\prime\prime}})QU_{ 0}(s^{{\prime\prime}}) +\ldots. {}\\ \end{array}$$
Using U
0(s
1)U
0(s
2) = U
0(s
1 + s
2) and U
0(−s) = U
0
−1(s) we obtain
$$\displaystyle\begin{array}{rcl} & & \mathop{\mathrm{Tr}}\nolimits \left [U(s)\right ] =\mathop{ \mathrm{Tr}}\nolimits \left [U_{0}(s)\right ] + is\mathop{\mathrm{Tr}}\nolimits \left [U_{0}(s)Q\right ] {}\\ & & \phantom{\mathop{\mathrm{Tr}}\nolimits \left [U(s)\right ] =\mathop{ \mathrm{Tr}}\nolimits }+i^{2}\int _{ 0}^{s}ds_{ 1}\int _{0}^{s_{1} }ds_{2}\mathop{ \mathrm{Tr}}\nolimits \left [U_{0}(s + s_{2} - s_{1})QU_{0}(s_{1} - s_{2})Q\right ] +\ldots. {}\\ \end{array}$$
The contribution of \(\mathop{\mathrm{Tr}}\nolimits \left [U_{0}(s)\right ] =\mathop{ \mathrm{Tr}}\nolimits e^{is\partial ^{2} }\) is independent of A
μ
and hence to be dropped as the unwanted additive constant in (37.5).
