Covariant Quantization of the Electromagnetic Field

1. The Challenge of Quantizing the Maxwell Field

Our starting point is the classical Lagrangian for the free electromagnetic field:
${\mathcal{L}}_{\text{EM}}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }\text{where}{F}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$

To perform canonical quantization, we follow the standard procedure: first, we find the momentum ${\pi }^{\mu }$ conjugate to the field $A_{\mu }$.
${\pi }^{\mu }(x)=\frac{\partial \mathcal{L}}{\partial ({\partial }_0{A}_{\mu })}$
The components are:

• For $\mu =i$ (spatial components): ${\pi }^i=-F^{0i}=E^i$, the electric field. This is well-defined.
• For $\mu =0$ (time component): We find ${\pi }^0=-F^{00}=0$, because $F_{\mu \nu }$ is antisymmetric.

The fact that ${\pi }^0=0$ is a critical problem. The canonical quantization procedure requires us to impose non-zero equal-time commutation relations (ETCRs) between the fields and their conjugate momenta:
$[A_{\mu }(t,\mathbf{x}),{\pi }_{\nu }(t,\mathbf{y})]=i{\eta }_{\mu \nu }{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$
For $\mu =\nu =0$, this would require $[A_0,0]=i{\eta }_{00}{\delta }^{(3)}(\dots )=i{\delta }^{(3)}(\dots )$, which is impossible. The component $A_0$ is not a truly independent dynamical degree of freedom in the classical theory; its role is that of a Lagrange multiplier enforcing a constraint (Gauss's law). This prevents a straightforward canonical quantization.

2. A Solution: Covariant Gauge Fixing

To proceed, we need a method that makes all four components of $A_{\mu }$ appear dynamical, allowing us to quantize them in a manifestly Lorentz covariant way. The most common approach is the Gupta-Bleuler formalism.

The core idea is to modify the Lagrangian by adding a "gauge-fixing" term. This new term intentionally breaks the gauge invariance of the Lagrangian, but in a controlled way that we can deal with later. A widely used choice is:
${\mathcal{L}}_{\text{GF}}=-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2$
Here, $\xi$ (pronounced "ksi") is a dimensionless parameter. The full Lagrangian for the free gauge-fixed theory is:
$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2$

Different values of $\xi$ correspond to different "gauges". A particularly simple and common choice is $\xi =1$, known as the Feynman gauge. In this gauge, after an integration by parts, the Lagrangian simplifies significantly to:
${\mathcal{L}}_{\text{Feynman}}=-\frac{1}{2}({\partial }_{\nu }{A}_{\mu })({\partial }^{\nu }{A}^{\mu })$
The equations of motion derived from this Lagrangian are simply four uncoupled wave equations:
$\square A_{\mu }(x)=0$
This is a tremendous simplification! The theory now looks like four independent, massless Klein-Gordon fields. This form is ideal for quantization, although as we will see, the Minkowski metric signature in $A_{\mu }{A}^{\mu }=A_0^2-{\mathbf{A}}^2$ will introduce a crucial subtlety.

The following video provides an excellent introduction to this procedure, explaining why we need to break the gauge and how the gauge-fixing Lagrangian is introduced.

3. Quantization in Feynman Gauge

With our simplified Lagrangian, $\mathcal{L}=-\frac{1}{2}({\partial }_{\nu }{A}_{\mu })({\partial }^{\nu }{A}^{\mu })$, we can now quantize the theory.

1. Conjugate Momenta: The conjugate momenta are now non-zero for all components:
   ${\pi }_{\mu }=\frac{\partial \mathcal{L}}{\partial ({\dot{A}}^{\mu })}=-{\dot{A}}_{\mu }$

2. ETCRs: We can impose the canonical commutation relations:
   $[A_{\mu }(t,\mathbf{x}),{\pi }_{\nu }(t,\mathbf{y})]=i{\eta }_{\mu \nu }{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$
   $⟹[A_{\mu }(t,\mathbf{x}),{\dot{A}}_{\nu }(t,\mathbf{y})]=-i{\eta }_{\mu \nu }{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$

3. Mode Expansion: We expand the field operator $A_{\mu }(x)$ in terms of creation and annihilation operators, just as we did for the scalar field. However, we now have four types of polarization, labeled by $\lambda =0,1,2,3$.
   $A_{\mu }(x)={\sum }_{\lambda =0}^3\int \frac{d^{3}k}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}\left[{ϵ}_{\mu }(\mathbf{k},\lambda )a(\mathbf{k},\lambda )e^{-ik\cdot x}+{ϵ}_{\mu }^{\ast }(\mathbf{k},\lambda )a^{†}(\mathbf{k},\lambda )e^{ik\cdot x}\right]$
   Here, ${ϵ}_{\mu }(\mathbf{k},\lambda )$ are the four polarization vectors. For a photon moving in the $z$-direction ($k^{\mu }=(\omega ,0,0,\omega )$), a standard choice is:

   • ${ϵ}^{\mu }(\lambda =0)=(1,0,0,0)$ (Timelike/Scalar polarization)
   • ${ϵ}^{\mu }(\lambda =1)=(0,1,0,0)$ (Transverse polarization)
   • ${ϵ}^{\mu }(\lambda =2)=(0,0,1,0)$ (Transverse polarization)
   • ${ϵ}^{\mu }(\lambda =3)=(0,0,0,1)$ (Longitudinal polarization)

4. Operator Commutation Relations: The field ETCRs translate into the following commutation relations for the creation and annihilation operators:
   $[a(\mathbf{k},\lambda ),a^{†}(\mathbf{p},{\lambda }^{\prime })]=-{\eta }^{\lambda {\lambda }^{\prime }}(2\pi {)}^{3}2{\omega }_{\mathbf{k}}{\delta }^{(3)}(\mathbf{k}-\mathbf{p})$
   (Note: Here we treat $\lambda$ like a Lorentz index, so ${\eta }^{00}=1,{\eta }^{11}={\eta }^{22}={\eta }^{33}=-1$).

The Cost of Covariance: Negative Norm States

Look closely at the commutation relation for the timelike polarization, $\lambda ={\lambda }^{\prime }=0$:
$[a(\mathbf{k},0),a^{†}(\mathbf{p},0)]=-(1)(2\pi {)}^{3}2{\omega }_{\mathbf{k}}{\delta }^{(3)}(\mathbf{k}-\mathbf{p})$
This has the wrong sign! For a normal harmonic oscillator, this commutator is positive. Let's see the catastrophic consequence. Consider a one-photon state with timelike polarization, created by acting on the vacuum: $|\psi ⟩=a^{†}(\mathbf{p},0)|0⟩$. Now let's calculate its norm:
$⟨\psi |\psi ⟩=⟨0|a(\mathbf{p},0)a^{†}(\mathbf{p},0)|0⟩=⟨0|[a(\mathbf{p},0),a^{†}(\mathbf{p},0)]|0⟩$
The norm is negative! This is a disaster for the probabilistic interpretation of quantum mechanics. These unphysical, negative-norm states are often called ghosts. We have traded a quantization problem (${\pi }^0=0$) for a physical one (negative probabilities).

For a detailed explanation of this procedure and its pitfalls, the following reading from David Tong's notes is invaluable.

4. The Gupta-Bleuler Physical State Condition

The resolution lies in remembering that we started with a gauge theory. The physical states of the theory should not contain the unphysical scalar ($\lambda =0$) and longitudinal ($\lambda =3$) photons. We need a condition to project our vast, unphysical Fock space onto a smaller, physical Hilbert space.

The original gauge choice that motivated our formalism was the Lorenz gauge, ${\partial }_{\mu }{A}^{\mu }=0$. One might be tempted to impose this as an operator equation on the physical states $|\psi {⟩}_{\text{phys}}$:
${\partial }_{\mu }{A}^{\mu }|\psi {⟩}_{\text{phys}}\stackrel{?}{=}0$
However, this condition is too strong. It can be shown that this would imply that even the vacuum state is not a physical state!

The correct, weaker condition, proposed by Gupta and Bleuler, is to demand that only the annihilation part (positive frequency part) of the operator annihilates physical states:
$({\partial }_{\mu }{A}^{\mu }{)}^{(+)}|\psi {⟩}_{\text{phys}}=0$
where $({\partial }_{\mu }{A}^{\mu }{)}^{(+)}$ contains only annihilation operators $a(\mathbf{k},\lambda )$. This condition ensures that for any two physical states $|\psi ⟩$ and $|\varphi ⟩$, the matrix element of the full operator vanishes:
$⟨\varphi |{\partial }_{\mu }{A}^{\mu }|\psi ⟩=0$
This is sufficient to recover all the physical predictions of electromagnetism. The Gupta-Bleuler condition elegantly ensures that the contributions from the unphysical timelike and longitudinal photons to any physical observable (like the Hamiltonian or S-matrix elements) exactly cancel. The negative-norm "ghosts" are removed from the physical spectrum, and unitarity is restored.

The video you watched earlier continues to explain exactly how this works.

5. The Photon Propagator

We are finally ready to derive the photon propagator, $D_{F\mu \nu }(x-y)=⟨0|T\{A_{\mu }(x)A_{\nu }(y)\}|0⟩$. The calculation is analogous to the one for the Klein-Gordon field, but we must carry the Lorentz indices $\mu ,\nu$ and sum over all four polarizations $\lambda$. Since the vacuum is a physical state, we can compute this expectation value in our full Fock space.

The result of this calculation in Feynman gauge ($\xi =1$) is remarkably simple. In momentum space, the propagator is:
${\tilde{D}}_{F\mu \nu }(k)=\frac{-i{\eta }_{\mu \nu }}{k^2+iϵ}$
This beautifully covariant expression is the final piece we need for tree-level QED. It describes the propagation of a virtual photon between two points in spacetime.

For the more general case of an arbitrary gauge parameter $\xi$, the propagator is:
${\tilde{D}}_{F\mu \nu }(k)=\frac{-i}{k^2+iϵ}\left({\eta }_{\mu \nu }-(1-\xi )\frac{k_{\mu }{k}_{\nu }}{k^2}\right)$
While expressions look different for different $\xi$, any physical quantity we calculate (like a scattering cross-section) must be independent of our choice of $\xi$. Feynman gauge ($\xi =1$) is often preferred for its simplicity.

The following resource provides a direct derivation of this result.

Conclusion

In this lesson, we navigated the complex but elegant procedure of quantizing the electromagnetic field in a covariant gauge. We saw how the naive canonical approach fails and how modifying the Lagrangian with a gauge-fixing term allows us to proceed, at the cost of introducing unphysical states with negative norms. We then learned how the Gupta-Bleuler condition provides the crucial final step, defining a physical Hilbert space where these unphysical states have no effect, restoring unitarity and a sensible physical theory.

Key Takeaways:

• The standard Maxwell Lagrangian cannot be canonically quantized because the momentum conjugate to $A_0$ is zero.
• Adding a gauge-fixing term, such as $-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2$, makes the theory quantizable in a covariant manner but introduces four photon polarizations instead of two.
• The timelike polarization ($\lambda =0$) corresponds to states with negative norm, threatening the probabilistic interpretation of QM.
• The Gupta-Bleuler condition, $({\partial }_{\mu }{A}^{\mu }{)}^{(+)}|\psi {⟩}_{\text{phys}}=0$, defines the subspace of physical states. Within this subspace, the unphysical longitudinal and timelike polarizations do not contribute to physical observables.
• The Feynman propagator for the photon in Feynman gauge ($\xi =1$) is ${\tilde{D}}_{F\mu \nu }(k)=\frac{-i{\eta }_{\mu \nu }}{k^2+iϵ}$.

Preview of the Next Lesson:
With the propagators for both the Dirac field and the photon field in hand, along with the QED interaction vertex, we have assembled all the necessary components to build our Feynman rules for QED. In the next lesson, we will put everything together to construct the full QED Lagrangian and state the complete set of Feynman rules in momentum space, preparing us to calculate our first QED scattering processes.