Questions: Electromagnetic Field Quantization (QED)

Question 1

The photon has spin 1, which naively allows three polarization states (m = -1, 0, +1). Why does a physical photon have only two polarization states?

• A. The third polarization state has negative energy and is excluded
• B. Gauge invariance eliminates the longitudinal and timelike polarizations, leaving only two transverse physical states
• C. The photon is massless, so it cannot be at rest and therefore spin projections are meaningless
• D. Experimental evidence shows only two polarizations, but the theory actually predicts three

Answer: (B)

A massive spin-1 particle has three polarization states. The photon has only two because gauge invariance (the freedom to shift A^mu -> A^mu + partial^mu Lambda without changing the physics) makes the longitudinal and timelike components unphysical. In Coulomb gauge, this is manifest: only the two transverse components of A are dynamical. In covariant gauges, all four components appear in intermediate calculations but the unphysical ones cancel in any physical observable (this cancellation is guaranteed by Ward identities). Masslessness alone is not sufficient — it is gauge invariance that reduces the polarization count.

Question 2

In Coulomb gauge (div A = 0), the quantized electromagnetic field has a clear physical interpretation but breaks manifest Lorentz covariance. In Lorentz gauge (partial_mu A^mu = 0), Lorentz covariance is manifest but unphysical ghost states appear. How is this resolved?

• A. Ghost states are real particles that have been observed in accelerator experiments
• B. The Gupta-Bleuler condition restricts the physical Hilbert space to states where the unphysical polarizations have zero expectation value, ensuring that only transverse photons contribute to physical processes
• C. The Lorentz gauge is abandoned in favor of Coulomb gauge for all practical calculations
• D. The ghost states cancel each other exactly due to supersymmetry

Answer: (B)

The Gupta-Bleuler quantization method works in Lorentz gauge by allowing all four polarization states in the full Hilbert space but imposing a subsidiary condition that defines the physical subspace. In this subspace, the contributions of timelike and longitudinal photons cancel in all matrix elements between physical states. The physical photon count is two transverse polarizations, consistent with Coulomb gauge. This tension between manifest Lorentz covariance and manifest unitarity (physical states only) is a recurring theme in gauge theory quantization.

Question 3

The quantized electromagnetic field in the vacuum has zero electric and magnetic fields everywhere — it is completely empty and inert.

• T. True
• F. False

Answer: False

The vacuum expectation values of E and B are zero: <0|E|0> = <0|B|0> = 0. But the expectation values of E^2 and B^2 are not zero — the vacuum has nonzero field fluctuations. These vacuum fluctuations are physically real: they cause the Lamb shift (a measurable energy difference between the 2S_{1/2} and 2P_{1/2} levels of hydrogen), the anomalous magnetic moment of the electron, and the Casimir effect (an attractive force between conducting plates in vacuum). The vacuum is not empty — it is the ground state of a quantum field, with zero average field but nonzero field fluctuations.

Question 4

Explain why the masslessness of the photon is intimately connected to gauge invariance, and what would go wrong if you added a mass term (1/2)m^2 A_mu A^mu to the electromagnetic Lagrangian.

Model answer: The electromagnetic Lagrangian L = -(1/4)F_{mu nu}F^{mu nu} is invariant under gauge transformations A^mu -> A^mu + partial^mu Lambda. A mass term (1/2)m^2 A_mu A^mu is NOT gauge invariant because it changes under A^mu -> A^mu + partial^mu Lambda. Therefore, gauge invariance forbids the photon mass. Conversely, adding a mass term explicitly breaks gauge invariance, which would allow three polarization states instead of two (a massive spin-1 particle has a longitudinal mode) and would make the theory non-renormalizable in its naive form. The photon is massless because gauge invariance demands it.

This connection between gauge invariance and masslessness is central to the Standard Model. The W and Z bosons are massive spin-1 particles, which would seem to violate gauge invariance. The Higgs mechanism resolves this by breaking the gauge symmetry spontaneously rather than explicitly, generating mass while preserving the underlying gauge structure needed for renormalizability.