The QED Path Integral

Introduction

The previous lecture introduced path integrals in a deliberately modest way. It did not claim that they solve interacting quantum mechanics exactly. It claimed something more useful: they provide a Lagrangian-based representation of quantum time evolution, and for weakly interacting systems they generate perturbation theory in a form that is often far cleaner than the Hamiltonian operator method.

This lecture now applies that machinery to quantum electrodynamics.

That is a big step, because QED contains both:

• bosonic degrees of freedom, namely the electromagnetic field,
• and fermionic degrees of freedom, namely electrons and positrons.

So the lecture’s task is not just to write “a path integral for QED” in one jump. It has to show how:

• bosonic coherent-state path integrals for photon modes,
• and fermionic Grassmann coherent-state path integrals for Dirac modes

fit together into one generating functional.

But that alone would not yet solve the real problem.

The deeper goal of the lecture is to recover manifest Lorentz invariance. In the Hamiltonian formulation, the theory is actually relativistic, but Lorentz invariance is concealed because the Hamiltonian chooses a preferred time slicing and a preferred frame. As the review slide states, that concealment makes every calculation much longer than it needs to be and makes it hard to see the structure of the whole theory. So the lecture wants to re-write QED in a genuinely Lagrangian form. :contentReference[oaicite:2]{index=2}

The strategy is:

1. start from the known bosonic and fermionic mode path integrals,
2. build the full QED source-dependent vacuum amplitude as one enormous product over momentum and polarization/spin labels,
3. change the fermionic variables from mode amplitudes to a 4-spinor field $\Psi$,
4. show that the free fermionic part of the exponent becomes the Dirac action,
5. rewrite the fermionic source terms as $\bar J\Psi + \bar\Psi J$,
6. deal with the photon sector by separating the real and imaginary parts of the coherent-state variables,
7. remove the auxiliary Gaussian part,
8. identify the remaining variables with the transverse vector potential,
9. then reintroduce $A_0$ through a Gaussian identity so that the Coulomb term is absorbed into the covariant Maxwell sector,
10. and thereby recover the Maxwell–Dirac action with sources.

That is the real content of the lecture.

The file ends just as the lecture is about to discuss the remaining gauge issue explicitly. So what is fully established here is not yet the final gauge-fixed covariant QED generating functional in textbook form, but the much more important structural fact that the Hamiltonian coherent-state path integral can be transformed into the Maxwell–Dirac Lagrangian form. :contentReference[oaicite:3]{index=3}

Learning Objectives

• Explain why manifestly Lorentz-invariant perturbation theory is the goal of the QED path integral.
• Recall the bosonic and fermionic coherent-state path integrals from the previous lecture.
• Understand how source terms generate arbitrary initial/final states and perturbative interaction insertions.
• Construct the combined QED coherent-state path integral over photon, electron, and positron mode variables.
• Change fermionic mode variables into a Dirac 4-spinor field $\Psi$ in momentum space.
• Show that the free fermionic action becomes the Dirac action in manifestly Lorentz-invariant form.
• Rewrite the fermionic source terms as $\bar J \Psi + \bar\Psi J$.
• Evaluate the noninteracting fermionic generating functional with sources.
• Reorganize the bosonic photon path integral by separating the auxiliary Gaussian variables from the physical transverse potential.
• Identify the remaining bosonic dynamical field with the transverse vector potential $A_\perp$.
• Reintroduce the scalar potential $A_0$ through a Gaussian identity.
• Understand how the Coulomb interaction is absorbed into the Maxwell part of the action.
• Show that the exponent becomes the Maxwell–Dirac action with sources.
• Understand that the explicit gauge treatment is only begun at the end of the uploaded file.

Prerequisite Knowledge

• Bosonic coherent-state path integrals
• Fermionic coherent-state path integrals and Grassmann variables
• Free Dirac field and Dirac spinors $u_\pm, v_\pm$
• Free photon modes in Coulomb gauge
• Maxwell–Dirac Lagrangian
• Fourier transforms in space and time
• Source generating functionals
• Coulomb-gauge Hamiltonian QED

1. Review: why QED needs a Lagrangian path integral

The lecture begins by repeating the central motivation. The theory is relativistic, but the Hamiltonian formulation conceals Lorentz invariance by choosing a particular frame. That makes calculations unnecessarily long and hides the overall structure. So the aim is to rewrite QED in a Lagrangian-based formalism. :contentReference[oaicite:4]{index=4}

The next review slide summarizes why path integrals are the right tool:

• they are equivalent to ordinary quantum time evolution,
• but they use the Lagrangian instead of the Hamiltonian,
• so every step is Lorentz-invariant,
• and for weakly interacting fields they generate manifestly covariant perturbation theory.

The lecture states three specific reasons:

1. without interactions, the path integrals are explicit Gaussian functions of initial and final states,
2. with arbitrary time-dependent linear classical sources added for each field, they become Gaussian functionals of the sources,
3. interactions can then be added perturbatively by variations with respect to those sources, followed by setting the sources to zero. :contentReference[oaicite:5]{index=5}

This is the whole conceptual foundation of the lecture.

2. Review: bosonic and fermionic mode path integrals

The lecture then briefly recalls the results of B5.

For a bosonic coherent-state path integral,

$$\langle \bar\alpha_f|\hat U(t_f,t_i)|\alpha_i\rangle = \int \mathcal D\alpha\,\mathcal D\bar\alpha\; e^{\frac12(\bar\alpha_f\alpha(t_f)+\bar\alpha(t_i)\alpha_i)} e^{\frac{i}{\hbar}\int dt\,L(\alpha,\bar\alpha,t)}.$$

For the driven harmonic oscillator, the result is a Gaussian functional of the source $J(t)$. :contentReference[oaicite:6]{index=6}

Likewise, for a fermionic coherent-state path integral,

$$\langle \bar\eta_f|\hat U(t_f,t_i)|\eta_i\rangle = \int \mathcal D\eta\,\mathcal D\bar\eta\; e^{\frac12(\bar\eta_f\eta(t_f)+\bar\eta(t_i)\eta_i)} e^{\frac{i}{\hbar}\int dt\,L(\eta,\bar\eta,t)},$$

and for the driven fermionic mode the result is again a Gaussian functional, now in Grassmann-valued sources. :contentReference[oaicite:7]{index=7}

The review also reminds us of the key perturbative identity: for a nonlinear Lagrangian $L=L_0+\epsilon L_1$, the full transition amplitude can be generated by source differentiation acting on the Gaussian generating functional of the quadratic theory. :contentReference[oaicite:8]{index=8}

This is exactly what the lecture now applies to QED.

3. Sources can generate initial and final states too

The lecture next adds an important technical point. The sources are not only for generating interaction insertions. They can also generate initial and final states.

For bosons, since

$$|\alpha\rangle = \sum_{n=0}^\infty \frac{\alpha^n}{n!}|n\rangle,$$

one has

$$\langle n|\hat U(t_f,t_i)|m\rangle = m!n!\, \frac{\partial^n}{\partial \bar\alpha_f^n} \frac{\partial^m}{\partial \alpha_i^m} \langle \bar\alpha_f|\hat U(t_f,t_i)|\alpha_i\rangle \Big|_{\bar\alpha_f=\alpha_i=0}.$$

Equivalently, those derivatives can be traded for source variations at the endpoints. :contentReference[oaicite:9]{index=9}

The lecture emphasizes that for photons in QED one usually needs at most 0 or 1 quanta in a given mode, so this endpoint-source trick is perfectly practical. For fermions the same logic applies, but the sources for electrons and positrons must remain distinct Grassmann sources because the interaction terms involve different combinations of $b,d,b^\dagger,d^\dagger$. For photons, only the combination $a+a^\dagger$ appears in the interaction term, so a real source is enough there. :contentReference[oaicite:10]{index=10}

This is the source technology that makes the final QED generating functional useful.

4. Fourier-space form of the Gaussian generating functionals

Before assembling QED, the lecture also rewrites the Gaussian source functionals in Fourier space. For the bosonic source,

$$J(t) = \frac{1}{2\pi}\int d\nu\, e^{i\nu t} j(\nu), \qquad \bar J(t) = \frac{1}{2\pi}\int d\nu\, e^{-i\nu t}\bar j(\nu),$$

the time-ordered double integral can be evaluated to give a propagator-like denominator. When $J$ is real, the lecture shows the result can be rearranged into the form

$$\int d\nu\, \frac{i\omega\, j(\nu)j(-\nu)} {\nu^2-\omega^2+i\epsilon}.$$

It explicitly remarks that this is helpful because $k_0+|k|$ is not Lorentz invariant, but $k_0^2-|k|^2$ is. :contentReference[oaicite:11]{index=11}

This is a very important hint of what is coming: the source-dependent Gaussian already wants to become a relativistic propagator.

5. Lecture outline for the QED path integral

The actual lecture B6 then gives its three-part structure:

1. The QED path integral 1: Fermions
2. The QED path integral 2: Photons
3. The QED path integral 3: Gauge :contentReference[oaicite:12]{index=12}

Only the first two are developed in the uploaded file.

6. The starting QED coherent-state path integral

The lecture now writes the combined QED source-dependent vacuum amplitude as one giant path integral over:

• bosonic coherent-state variables $\alpha_\pm(\vec k,t), \bar\alpha_\pm(\vec k,t)$ for photons,
• Grassmann variables $\eta_\pm,\bar\eta_\pm$ for electrons,
• Grassmann variables $\xi_\pm,\bar\xi_\pm$ for positrons.

The mode/operator replacements are

$$\hat a_\pm(\vec k)\to \alpha_\pm(\vec k,t), \qquad \hat a_\pm^\dagger(\vec k)\to \bar\alpha_\pm(\vec k,t),$$ $$\hat b_\pm(\vec k)\to \eta_\pm(\vec k,t), \qquad \hat b_\pm^\dagger(\vec k)\to \bar\eta_\pm(\vec k,t),$$ $$\hat d_\pm(\vec k)\to \xi_\pm(\vec k,t), \qquad \hat d_\pm^\dagger(\vec k)\to \bar\xi_\pm(\vec k,t).$$

The corresponding sources are named:

• $f_\pm,\bar f_\pm$ for photons,
• $r_\pm,\bar r_\pm$ for electrons,
• $s_\pm,\bar s_\pm$ for positrons. :contentReference[oaicite:13]{index=13}

The resulting object is the vacuum-to-vacuum amplitude ratio

$$\frac{\langle 0|\hat U_{f,r,s}(\infty,-\infty)|0\rangle} {\langle 0|\hat U_{0,0,0}(\infty,-\infty)|0\rangle},$$

written as a huge path integral with:

• the free bosonic action,
• the free fermionic action,
• source couplings,
• and the interaction Hamiltonian $H_I$. :contentReference[oaicite:14]{index=14}

The lecture then says: the next step is just to rewrite this big path integral to make it more obviously Lorentz invariant.

7. Fermionic change of variables to a Dirac spinor

The first substantial step is the fermionic sector.

The lecture defines a momentum-space Dirac spinor $\Psi(\vec k,t)$ from the electron and positron Grassmann variables:

$$\Psi(\vec k,t) = \sum_\pm \frac{M_0 c^2}{\hbar \omega(\vec k)} \Big( u_\pm(\vec k)\,\eta_\pm(\vec k,t) + v_\pm(-\vec k)\,\bar\xi_\pm(-\vec k,t) \Big),$$

with

$$\bar\Psi = \Psi^\dagger \gamma^0.$$

Using the eigenspinor identities

$$\gamma^0\left(c\vec\gamma\cdot \vec k + \frac{M_0 c^2}{\hbar}\right)u_\pm(\vec k)=\omega(\vec k)u_\pm(\vec k),$$ $$\gamma^0\left(c\vec\gamma\cdot \vec k + \frac{M_0 c^2}{\hbar}\right)v_\pm(-\vec k)=-\omega(\vec k)v_\pm(-\vec k),$$

the lecture shows that the free fermionic action becomes

$$\int d^3k \int dt\; \bar\Psi \left( i\gamma^0 \frac{d}{dt} - c\vec\gamma\cdot \vec k - \frac{M_0 c^2}{\hbar} \right)\Psi.$$

Then, after Fourier transforming in time,

$$\Psi(\vec k,t)=\frac{1}{2\pi}\int dk_0\, e^{ick_0 t}\Psi(\vec k,k_0),$$

this becomes

$$-\int d^4k\; \bar\Psi \left( c\gamma^\mu k_\mu + \frac{M_0 c^2}{\hbar} \right)\Psi,$$

which the lecture explicitly identifies as the Dirac action divided by $\hbar$, written in an obviously Lorentz-invariant form. :contentReference[oaicite:15]{index=15}

This is the key fermionic achievement of the lecture.

8. Fermionic source spinor

The lecture then shows that the electron and positron sources can also be packaged into a Dirac 4-spinor source:

$$J(\vec k,t) = \frac{M_0 c^2}{\hbar\omega(\vec k)} \sum_\pm \gamma^0 \Big( u_\pm(\vec k)\, r_\pm(\vec k,t) - v_\pm(-\vec k)\,\bar s_\pm(-\vec k,t) \Big),$$

so that

$$\bar J \Psi + \bar\Psi J = \sum_\pm ( \bar r_\pm \eta_\pm + \bar\eta_\pm r_\pm + \bar s_\pm \xi_\pm + \bar\xi_\pm s_\pm ).$$

This is exactly what is needed because later the interaction terms will be generated by functional differentiation with respect to $J$ and $\bar J$. :contentReference[oaicite:16]{index=16}

So now the fermionic part of the QED path integral looks like the Dirac action plus covariant source coupling.

9. The free fermionic Gaussian functional

The lecture pauses at this point to compute the noninteracting fermionic-plus-sources path integral explicitly.

It starts from the earlier driven-mode results and rewrites the sum over electron and positron modes into one expression involving step functions and the sign function $\operatorname{sgn}(t-t')$. Then it Fourier transforms in time, takes $t_i\to -\infty$, $t_f\to +\infty$, and evaluates the $\tau=t-t'$ integral. The result is

$$\exp\left[ \frac{i}{c\hbar^2} \int d^4k\; \bar J(k_\mu)\, \frac{\gamma^\mu k_\mu - M_0 c/\hbar} {(M_0 c/\hbar)^2 + k^2 - k_0^2 - i\epsilon}\, J(k_\mu) \right].$$

The lecture notes that this is now also obviously Lorentz invariant. :contentReference[oaicite:17]{index=17}

This is the fermionic Gaussian generating functional in covariant form. It is essentially the momentum-space Dirac propagator structure appearing already at the source-functional level.

10. Photon sector: separating real and imaginary parts

The lecture then turns to the bosonic photon part.

It writes

$$\alpha_\pm(\vec k,t)=\frac{k a_\pm(\vec k,t)+ i b_\pm(\vec k,t)}{2k},$$

with corresponding reality relations under $\vec k\to -\vec k$. The key observation is that the variable $b_\pm$ does not appear in the interaction Hamiltonian $H_I$, nor in the photon source couplings $f_\pm$, because only the combination

$$\alpha_\pm(\vec k,t)+\bar\alpha_\pm(-\vec k,t)=2k\, a_\pm(\vec k,t)$$

enters. :contentReference[oaicite:18]{index=18}

This is the bosonic analogue of identifying which coherent-state directions are physically relevant.

11. Integrating out the auxiliary bosonic Gaussian

After rewriting the free photon action in terms of $a_\pm$ and $b_\pm$, the lecture completes the square in $b_\pm$ and shifts to new variables $\tilde b_\pm$. The $b$-dependent part becomes a pure Gaussian factor independent of the sources and interactions. Therefore it cancels between numerator and denominator in the vacuum-amplitude ratio and can be discarded. The remaining bosonic action is

$$\frac{i}{2c} \sum_\pm \int d^3k\,dt \left( |\dot a_\pm|^2 - c^2 k^2 |a_\pm|^2 - 2k(f_\pm \bar a_\pm + \bar f_\pm a_\pm) \right).$$

The lecture is explicit that this cancellation is why the unwanted Gaussian can just be removed. :contentReference[oaicite:19]{index=19}

So the bosonic sector is reduced to the physically relevant dynamical variables.

12. Identifying the transverse vector potential

The lecture now defines

$$\vec A(\vec k,t)=\sum_\pm \vec\epsilon_\pm(\hat k)\, a_\pm(\vec k,t),$$

which is the vector potential (up to the lecture’s geometry/unit conventions). Because the polarization vectors are transverse,

$$\vec k \cdot \vec A = 0.$$

So at this stage the path integral still contains only the two transverse components $A_\perp$. The source is likewise assembled into a vector

$$\vec f(\vec k,t)= \sum_\pm \frac{f_\pm(\vec k,t)\vec\epsilon_\pm(\vec k)}{2k} + \frac{f_0(\vec k,t)\hat k}{2k},$$

and the lecture says we are free to add $f_0$ because it does not actually couple to the transverse $A$ yet. This brings the bosonic part much closer to a Lorentz-invariant form, but still without $A_0$. :contentReference[oaicite:20]{index=20}

This is the photon-side analogue of the covariant rewrite of the fermionic sector.

13. Why the Coulomb term is the obstacle

The lecture then looks directly at the interaction Hamiltonian:

$$H_I = H_{AJ} + H_{\text{Coulomb}}.$$

The $A\cdot j$ term is already part of the Maxwell–Dirac Lagrangian. But the Coulomb term,

$$H_{\text{Coulomb}} = \int d^3r\,d^3r'\, \frac{\rho(r)\rho(r')}{8\pi \varepsilon_0 |r-r'|},$$

is not manifestly relativistic. In momentum space it is

$$\int d^3k\, \frac{|\rho(\vec k,t)|^2}{2\varepsilon_0 k^2}.$$

The lecture states the key insight: we can remove the Coulomb term and at the same time restore the Maxwell Lagrangian. :contentReference[oaicite:21]{index=21}

That is the decisive step toward the covariant action.

14. Reintroducing $A_0$ by a Gaussian identity

To do this, the lecture uses the Gaussian identity

$$\int \mathcal D\Phi\; \exp\left[ \frac{ic}{2}\int d^3k\,dt \left( k^2 |\Phi|^2 + \frac{\Phi \rho^\ast + \Phi^\ast \rho}{\hbar c \varepsilon_0} \right) \right] \propto \exp\left[ -\frac{i}{2\hbar} \int d^3k\,dt\, \frac{|\rho(\vec k,t)|^2}{\varepsilon_0 k^2} \right].$$

Since the Gaussian factor not involving sources again cancels in the numerator/denominator ratio, the lecture simply replaces the Coulomb factor by a path integral over a new field and then renames

$$\Phi = A_0.$$

So the Coulomb interaction is traded for an integration over the scalar potential $A_0$. :contentReference[oaicite:22]{index=22}

This is the most important formal move in the whole lecture: the noncovariant Coulomb interaction is absorbed into a covariant-looking gauge-field sector by reintroducing $A_0$ as an integration variable.

15. Recovering the Maxwell–Dirac action

After this step, the path integral contains:

• the fermionic Dirac action with sources,
• the $A_\mu \bar\Psi \gamma^\mu \Psi$ interaction,
• and the bosonic quadratic form in $A_\perp$ plus the newly introduced $A_0$.

The lecture then writes the Maxwell Lagrangian density in Fourier space:

$$\mathcal L_M = -\frac{\hbar c}{4} (k_\mu A_\nu^\ast - k_\nu A_\mu^\ast) (k^\mu A^\nu - k^\nu A^\mu).$$

Because only the transverse components were originally present, one still has

$$\vec k\cdot \vec A = 0$$

at this stage. But the exponent is now exactly the Maxwell–Dirac action, except for the remaining gauge issue that has not yet been fully discussed in the uploaded pages. The lecture says this explicitly on page 24: we now have exactly the Maxwell–Dirac action — except that only transverse $A_\perp$ appears in the path integral, so $\vec k\cdot\vec A = 0$. :contentReference[oaicite:23]{index=23}

This is the central achievement of the lecture.

16. What the lecture has really shown by this point

By the end of the available pages, the lecture has already established the main structural fact:

Starting from the Coulomb-gauge Hamiltonian coherent-state path integral for QED, one can change variables so that the exponent becomes the Lorentz-covariant Maxwell–Dirac action with sources.

That is not a small cosmetic rewrite. It means:

• the free fermion sector becomes the Dirac action,
• the free photon sector becomes the Maxwell action,
• the current coupling becomes $A_\mu \bar\Psi \gamma^\mu \Psi$,
• and the noncovariant Coulomb term is absorbed by introducing $A_0$.

In other words, the manifestly relativistic Lagrangian form of QED is recovered directly from the Hamiltonian coherent-state formalism. :contentReference[oaicite:24]{index=24}

That is the real point of the lecture.

17. What is missing in the uploaded file

The file ends at “The QED Path Integral 3: Gauge …” without the continuation. So this PDF does not yet provide the final discussion of the gauge redundancy, gauge fixing, or whatever exact next step the lecturer intended to take. :contentReference[oaicite:25]{index=25}

So the honest boundary is:

• the lecture fully covers how the Maxwell–Dirac action with sources emerges,
• but the explicit final treatment of gauge in the path integral is only started here, not completed in the uploaded file.

Worked Examples

Example 1: Why the fermionic mode variables naturally combine into a Dirac spinor

The electron Grassmann modes $\eta_\pm$ and the positron modes $\bar\xi_\pm$ appear with exactly the same free frequency $\omega(k)$, but with opposite signs in the energy projector structure. Using the free Dirac eigenspinors $u_\pm(k)$ and $v_\pm(-k)$, the lecture combines them into

$$\Psi(\vec k,t) = \sum_\pm \frac{M_0 c^2}{\hbar \omega(k)} \left( u_\pm(\vec k)\eta_\pm + v_\pm(-\vec k)\bar\xi_\pm(-\vec k) \right).$$

Because the spinors satisfy the free Dirac eigenvalue identities, the entire fermionic quadratic form collapses into

$$\bar\Psi \left( i\gamma^0\partial_t - c\vec\gamma\cdot\vec k - \frac{M_0 c^2}{\hbar} \right)\Psi,$$

which is exactly the momentum-space Dirac action.

Example 2: Why introducing $A_0$ removes the Coulomb term

The Coulomb interaction is nonlocal in space:

$$\propto \int d^3k\, \frac{|\rho(\vec k,t)|^2}{k^2}.$$

A Gaussian identity shows that this factor is equivalent, up to a source-independent normalization, to integrating over a new scalar field $A_0$ with quadratic term $k^2 |A_0|^2$ and linear coupling $A_0\rho$. So the noncovariant instantaneous Coulomb interaction is replaced by a local-looking quadratic action for the scalar potential. This is exactly what is needed to reconstruct the Maxwell–Dirac Lagrangian form.

Intuition

The intuition of this lecture is that the Hamiltonian and Lagrangian formulations of QED are not different theories. They are two ways of packaging the same physics.

In the Hamiltonian picture, QED is built from:

• photon oscillators,
• electron modes,
• positron modes,
• and an ugly-looking Coulomb interaction.

In the path-integral picture, those same ingredients reorganize into:

• a Dirac spinor field $\Psi$,
• a 4-potential $A_\mu$,
• and the single Maxwell–Dirac action.

So what looked like many unrelated mode variables in the Hamiltonian basis becomes a small number of geometrically meaningful fields in the Lagrangian basis.

That is why the path integral is such a big improvement for relativistic field theory: it makes the Lorentz structure visible again.

Common Mistakes

• Thinking the QED path integral is introduced from nowhere. In the lecture it is built directly from the bosonic and fermionic coherent-state path integrals of the previous lecture.
• Forgetting that the purpose is not only to rewrite the theory, but to make perturbation theory manifestly Lorentz invariant.
• Confusing the mode variables $\eta,\xi,\alpha$ with the final field variables $\Psi$ and $A_\mu$.
• Missing why the fermionic source terms have to be reorganized into a Dirac source spinor $J$.
• Thinking the bosonic coherent-state variables directly are the vector potential. The lecture first has to separate and integrate out an auxiliary Gaussian part.
• Forgetting that only the transverse photon components are present initially, because the starting point is the Coulomb-gauge Hamiltonian theory.
• Missing the point of the Gaussian identity introducing $A_0$: it is what absorbs the noncovariant Coulomb interaction into the covariant gauge-field action.
• Claiming the uploaded PDF fully covers the gauge-fixing part. It does not; it only starts that discussion.

Short Summary

The lecture begins by restating the motivation for path integrals in QED: the Hamiltonian formulation is fully relativistic in content, but it conceals Lorentz invariance by choosing a particular frame, making calculations unnecessarily long and obscuring the structure of the theory. Path integrals provide an equivalent formulation of quantum evolution based on the Lagrangian, so every step can be made manifestly Lorentz invariant. After reviewing the bosonic and fermionic coherent-state path integrals and the source-differentiation method for generating perturbation theory, the lecture constructs the full source-dependent QED path integral over bosonic coherent-state variables for photons and Grassmann coherent-state variables for electrons and positrons. It then changes the fermionic variables into a Dirac 4-spinor field

$$\Psi(\vec k,t) = \sum_\pm \frac{M_0 c^2}{\hbar \omega(k)} \big(u_\pm(\vec k)\eta_\pm + v_\pm(-\vec k)\bar\xi_\pm(-\vec k)\big),$$

showing that the free fermionic action becomes

$$-\int d^4k\; \bar\Psi \left( c\gamma^\mu k_\mu + \frac{M_0 c^2}{\hbar} \right)\Psi,$$

which is the Dirac action in manifestly Lorentz-invariant momentum-space form. The electron and positron source variables are likewise combined into a covariant spinor source $J$, and the noninteracting fermionic generating functional is evaluated explicitly as a Gaussian functional of $J$ and $\bar J$. For the photon sector, the coherent-state variables are decomposed into real variables so that an auxiliary Gaussian part can be integrated out, leaving a path integral over the transverse vector potential $A_\perp$. The noncovariant Coulomb interaction is then replaced by a Gaussian path integral over a scalar field $A_0$, which allows the Coulomb term to be absorbed into the gauge-field quadratic action. The result is the Maxwell–Dirac action with sources:

• the Dirac action for $\Psi$,
• the Maxwell action for $A_\mu$,
• and the interaction $A_\mu\bar\Psi\gamma^\mu\Psi$, all in a manifestly Lorentz-covariant form. The uploaded file ends just as the lecture begins “The QED Path Integral 3: Gauge,” so the explicit final treatment of gauge redundancy is only started, not completed, in this PDF. :contentReference[oaicite:26]{index=26}

Practice Problems

1. Why does the lecture say the Hamiltonian formulation conceals Lorentz invariance?

2. Why are both bosonic and fermionic coherent-state path integrals needed for QED?

3. What is the purpose of introducing source fields $f,r,s$ in the QED path integral?

4. Why do the fermionic mode variables naturally combine into a Dirac spinor $\Psi$?

5. How do the free Dirac eigenspinor identities make the fermionic action collapse into the covariant Dirac form?

6. Why must the electron and positron source variables be reorganized into a 4-spinor source $J$?

7. Why is the noninteracting fermionic generating functional Gaussian?

8. What is the role of the auxiliary bosonic variables $b_\pm$, and why can they be integrated out?

9. Why does the path integral initially contain only the transverse vector potential $A_\perp$?

10. Why is the Coulomb interaction an obstacle to manifest covariance?

11. How does introducing $A_0$ through a Gaussian identity restore the Maxwell–Dirac structure?

12. What important part of the gauge discussion is only begun, but not completed, in the uploaded lecture file?