Solution
= Solution
Use units $\hbar=c=1$, the <Minkowski metric> $\eta_{ab}=\operatorname{diag}(1,-1,-1,-1)$, and the <Fourier transform> convention $f(x)=\int d^4p\,e^{-ip\cdot x}\widetilde f(p)/(2\pi)^4$. Here <gamma matrices> satisfy $\{\gamma^a,\gamma^b\}=2\eta^{ab}I_4$, and $\bar\psi=\psi^\dagger\gamma^0$ is the <Dirac adjoint>. A single <Dirac field> describes both <electrons> and <Positrons>; these are the particle and <antiparticle> sectors of that field.
The free <Maxwell Lagrangian> and <Dirac action>, with a linear <covariant gauge> condition for the <photon>, give
$$
S_0=\int d^4x\left[-\frac14F_{ab}F^{ab}-\frac1{2\alpha}(\partial_aA^a)^2+\bar\psi(i\gamma^a\partial_a-m)\psi\right],\qquad F_{ab}=\partial_aA_b-\partial_bA_a.
$$
Before <gauge fixing>, the <Maxwell Lagrangian> has <zero modes in field theory> along $A_a\mapsto A_a+\partial_a\lambda$, so its quadratic operator cannot be inverted on all potentials. In the linear <Lorenz gauge>, the <Faddeev-Popov determinant> is $\det\Box$. It is independent of $A$ and may be absorbed into the normalization; the corresponding <Faddeev-Popov ghost fields> have no interacting vertices in this Abelian linear gauge. A residual <gauge symmetry> is removed by the specified <boundary conditions>.
The free <generating functional> is
$$
Z_0[J,\bar\eta,\eta]=\mathcal N\int\mathcal DA\,\mathcal D\bar\psi\,\mathcal D\psi\;
\exp\!\left(iS_0+i\int d^4x\,[J_aA^a+\bar\eta\psi+\bar\psi\eta]\right),\qquad Z_0[0]=1.
$$
The <Dirac field> variables and their sources are independent <Grassmann fields> in this <path integral>. They anticommute; replacing them by ordinary commuting fields would give the wrong statistics and the wrong <functional determinant>. The bosonic <Gaussian functional integral> contributes an inverse square root of a determinant, and the <Grassmann Gaussian integral> contributes a determinant. If $K_A$ is the photon quadratic operator and $K_D=i\gamma^a\partial_a-m$, completing the square gives
$$
Z_0=\exp\!\left[-\frac i2\int J K_A^{-1}J-i\int\bar\eta K_D^{-1}\eta\right]
=\exp\!\left[-\frac12\int J D_F J-\int\bar\eta S_F\eta\right].
$$
In the last expression $D_F=iK_A^{-1}$ and $S_F=iK_D^{-1}$ are the <photon propagator> and <Dirac propagator>, with <Feynman i-epsilon prescription>. The products include the appropriate spacetime integrals and index contractions. <Functional derivatives> with respect to $J$, and consistently ordered left or right <Grassmann derivatives> with respect to the fermionic sources, generate the <time ordering> of the corresponding fields. In <Feynman gauge>, $\alpha=1$, the momentum-space two-point functions are
$$
\boxed{D_F^{ab}(p)=\frac{-i\eta^{ab}}{p^2+i0},\qquad
S_F(p)=\frac{i(\not p+m)}{p^2-m^2+i0}},\qquad \not p=\gamma^ap_a.
$$
The <Feynman i-epsilon prescription> specifies vacuum boundary conditions rather than an arbitrary inverse of the differential operator.
The covariant <photon propagator> uses four potential components. Its <operator formalism> counterpart is <Gupta-Bleuler quantization>: impose $(\partial_aA^a)^{(+)}|\mathrm{phys}\rangle=0$ and take the <Gupta-Bleuler null-state quotient>. This leaves a positive physical state space with two transverse <photon> <polarization vectors>. The temporal and longitudinal oscillator components occur in intermediate covariant expressions; the <Ward identity> removes their dependence from physical amplitudes. The free <Dirac field> uses the <canonical anticommutation relations>, producing the same <fermionic signs> as its <Grassmann fields> in the <path integral>.
The <operator-path-integral equivalence> can be seen directly with a regulator. Divide time into small intervals and insert complete sets of field-coordinate states for bosons, and resolutions in <fermionic coherent states> for fermions. The bosonic matrix elements produce the phase-space factor $\exp i\sum[\pi\Delta\phi-H\Delta t]$; integrating out the quadratic <canonical momentum> produces the bosonic <action>. The <fermionic coherent state> overlaps produce the first-order term $i\psi^\dagger\dot\psi$ and the <Berezin integral> measure. Multiplying the short-time kernels recovers the <path integral>. Projecting the remote endpoints onto the <Fock vacuum> with an infinitesimal damping selects the same <Feynman propagators> as the <operator formalism>. Field insertions become <time-ordered products> under this construction. Conversely, their quadratic <generating functional> obeys the canonical free-field equations and has precisely the oscillator two-point functions, so its higher free correlators agree by the <Wick theorem>. This establishes the equivalence for the regulated free theory and order by order in the <perturbation series>.
To couple the matter field electromagnetically, promote its global phase symmetry to the local transformation
$$
\psi\mapsto e^{-ie\lambda(x)}\psi,\qquad
\bar\psi\mapsto\bar\psi e^{ie\lambda(x)},\qquad
A_a\mapsto A_a+\partial_a\lambda.
$$
The <gauge covariant derivative> $D_a=\partial_a+ieA_a$ obeys $D'_a\psi'=e^{-ie\lambda}D_a\psi$. Replacing $\partial_a$ by $D_a$ in the <Dirac action> therefore gives the invariant matter density
$$
\bar\psi(i\gamma^aD_a-m)\psi
=\bar\psi(i\gamma^a\partial_a-m)\psi-e\bar\psi\gamma^a\psi A_a.
$$
The <electromagnetic field tensor> is unchanged by the local transformation. Thus the unfixed <quantum electrodynamics> action is <gauge-invariant>. The <Dirac current> $j^a=\bar\psi\gamma^a\psi$ is conserved by the matter equations, and the <electron> and <Positron> excitations carry opposite charges. The added <gauge fixing> density selects a representative and is not itself invariant under arbitrary local transformations; it does not change gauge-invariant observables. At a free fermion vertex,
$$
q_a\gamma^a=(\not p+\not q-m)-(\not p-m).
$$
For external on-shell <Dirac spinors> the corresponding current contraction vanishes. The quantum extension is the <Ward identity>, which makes physical amplitudes insensitive to adding a multiple of the photon <momentum> to its <polarization vector>. A gauge-compatible regularization preserves this vector-current identity.
With $\mathcal L_{\mathrm{int}}=-e\bar\psi\gamma^a\psi A_a$, expand $\exp(i\int\mathcal L_{\mathrm{int}})$ in powers of $e$. A term of order $n$ contains $n$ spacetime integrations and $1/n!$. Applying the <Wick theorem> pairs the free fields: an $A$-$A$ <Wick contraction> supplies a <photon propagator>, and a $\psi$-$\bar\psi$ <Wick contraction> supplies an oriented <Dirac propagator>. Each insertion supplies an <interaction vertex>. The permutations of contractions cancel the expansion factorials except for the <Feynman-diagram symmetry factor>. Interchanging <Grassmann fields> produces the <fermionic sign>, including a minus sign for every closed fermion loop. This is how <Feynman diagrams> arise from expectation values, rather than an extra dynamical assumption.
The <normalized vacuum generating functional> removes components with no external insertions. In the <operator formalism>, for an interacting-vacuum expectation value of an inserted product $\mathcal O$, the same cancellation appears as
$$
\langle\Omega|T\mathcal O|\Omega\rangle
=\frac{\langle0|T\{\mathcal O_I\exp(i\int d^4x\,\mathcal L_{\mathrm{int},I})\}|0\rangle}
{\langle0|T\exp(i\int d^4x\,\mathcal L_{\mathrm{int},I})|0\rangle}.
$$
Vacuum projection and the <Feynman i-epsilon prescription> are implicit. The <linked-cluster theorem> exponentiates all connected <vacuum bubbles> into the same factor in numerator and denominator, so it cancels. \b[Normalize by $Z[0]$ to remove every vacuum component.] This <cancellation of vacuum bubbles> still leaves products of disconnected diagrams that each contain external insertions. If only connected correlators are wanted, differentiate the <connected generating functional> $W=-i\log[Z[J,\bar\eta,\eta]/Z[0]]$.
The resulting momentum-space <QED Feynman rules> for the bare theory can be stated in <Feynman gauge> as follows:
* An internal oriented fermion line of <four-momentum> $p$ contributes $i(\not p+m)/(p^2-m^2+i0)$.
* An internal <photon> line contributes $-i\eta_{ab}/(p^2+i0)$.
* A one-photon two-fermion <interaction vertex> contributes $\boxed{-ie\gamma^a}$, with its spinor and photon indices attached to the incident lines.
* Each <interaction vertex> conserves <four-momentum>. With all incident <momenta> treated as incoming, include $(2\pi)^4\delta^4(\sum p)$; after using these constraints, integrate every independent loop <four-momentum> as $\int d^4\ell/(2\pi)^4$.
* Contract the <gamma matrix> and <Dirac propagator> factors in their order along the fermion line. A closed fermion loop has a <trace> and a factor $-1$. A relative odd permutation of external fermions also contributes $-1$.
* Divide each labelled diagram by its <Feynman-diagram symmetry factor> and sum the allowed diagrams. Omit vacuum components as explained above.
* For an amputated scattering amplitude, an incoming <electron> has $u_s(p)$ and an outgoing <electron> has $\bar u_s(p)$; an incoming <Positron> has $\bar v_s(p)$ and an outgoing <Positron> has $v_s(p)$. An incoming <photon> has $\epsilon_a(p)$ and an outgoing <photon> has $\epsilon_a^*(p)$. The external <polarization vectors> are physical and transverse, and external <four-momenta> are on shell. This last step follows from the <LSZ reduction formula> and <amputation of external propagators>; an unamputated correlator keeps its external <quantum field theory propagators>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-301-qed-vertex.png]
{title=An oriented electron line meets a photon at the QED interaction vertex}
The illustrated <interaction vertex> has incoming fermion <momentum> $p$, incoming photon <momentum> $k$, and outgoing fermion <momentum> $p'=p+k$. Its solid-line arrows indicate fermion flow. The <photon> wavy line has no fermion-flow arrow. \b[The propagators, the vertex $-ie\gamma^a$, momentum conservation, and the fermionic signs determine the perturbative amplitudes.]