Photons: Quantizing the Electromagnetic Field

Introduction

This lecture is where the electromagnetic field finally becomes a genuine quantum field in the same concrete way that the scalar and Dirac fields did earlier in the course.

Up to now, the Maxwell field has been brought into Hamiltonian form in Coulomb gauge. That already gave a clear classical picture:

• two transverse dynamical degrees of freedom,
• transverse electric and magnetic radiation fields,
• plus the instantaneous Coulomb electric field tied directly to the charge density.

But that was still classical field theory.

This lecture takes the next step: quantize the free electromagnetic field.

The first half is conceptually very clean. Once the sources are removed, the Coulomb-gauge Maxwell Hamiltonian becomes an infinite collection of independent harmonic oscillators, one for each wave vector $\vec k$ and each transverse polarization. Quantizing those oscillators gives creation and annihilation operators, and their quanta have exactly the properties Einstein guessed in 1905:

• energy $E=\hbar ck$,
• momentum $\vec p = \hbar \vec k$,
• transverse polarization,
• zero rest mass,
• Bose–Einstein statistics.

Those quanta are photons. :contentReference[oaicite:1]{index=1}

But the lecture does not stop there. Once the photon field is quantized, one naturally asks about causality. In Coulomb gauge the vector potential operator $\hat A_i(x,t)$ has a commutator that does not vanish outside the light cone. In fact, the lecture emphasizes this very strongly on page 8. That looks alarming. If the basic field operator itself is acausal, has something gone wrong?

The lecture’s answer is no. The acausal part is a pure gauge term, and the physically gauge-invariant fields $\hat E$ and $\hat B$ do commute at spacelike separations. So physical electrodynamics remains causal even though the gauge-fixed $A$-field itself does not look manifestly causal.

Then the lecture goes one step further. It presents a Glauber-style photon-detection model with two localized two-state atoms coupled to the electromagnetic field. The question is whether detector 1 can influence detector 2 faster than light. The lecture shows that for times shorter than the light-travel time $L/c$, detector 2’s excitation probability is not necessarily zero — because of vacuum noise and dark counts — but crucially it does not depend on the initial state of detector 1. Therefore there is no way to signal faster than light. :contentReference[oaicite:2]{index=2}

That is the real conceptual arc of the lecture:

1. photons arise from quantizing the free electromagnetic field,
2. Coulomb gauge makes locality look strange at the level of $\hat A$,
3. but gauge-invariant observables and actual signalling remain relativistically causal.

Learning Objectives

• Recall the Coulomb-gauge Hamiltonian of the electromagnetic field.
• Understand why only the transverse electromagnetic modes are dynamical.
• Quantize the source-free electromagnetic field as an infinite set of harmonic oscillators.
• Define the photon creation and annihilation operators in Coulomb gauge.
• Diagonalize the free electromagnetic Hamiltonian in terms of photon number operators.
• Derive the energy and momentum carried by free electromagnetic quanta.
• Explain why the quanta of the free electromagnetic field are massless bosons with transverse polarization.
• Understand why the Coulomb-gauge vector potential commutator does not vanish outside the light cone.
• Explain why the acausal part of the $A$-field commutator is only a pure gauge artifact.
• Show that gauge-invariant fields $E$ and $B$ commute at spacelike separations.
• Understand the Glauber-style detector model introduced in the lecture.
• Explain why detector dark counts do not imply faster-than-light signalling.
• Distinguish operator nonlocality in a gauge-fixed description from genuine violation of relativistic causality.

Prerequisite Knowledge

• Maxwell Hamiltonian in Coulomb gauge
• Canonical quantization of bosonic fields
• Harmonic oscillator quantization
• Fourier expansion and polarization vectors
• Noether momentum for fields
• Equal-time commutators
• Basic relativistic causality in QFT
• Interaction picture and time-ordered evolution

1. Review: the Maxwell Hamiltonian in Coulomb gauge

The lecture starts from the Coulomb-gauge Hamiltonian derived previously. In this formulation, the vector potential is expanded in transverse modes,

$$\vec A(\vec x,t) = \frac{1}{(2\pi)^{3/2}} \int d^3k\, e^{i\vec k\cdot \vec x}\sum_\pm A_\pm(\vec k,t)\,\vec\epsilon_\pm(\hat k),$$

with

$$\vec k\cdot \vec\epsilon_\pm(\hat k)=0.$$

The Hamiltonian contains:

• free oscillator-like terms for the transverse modes,
• coupling to the transverse current,
• and the instantaneous Coulomb term from the charge density.

The lecture also recalls the canonical equations of motion for the mode amplitudes $A_\pm(\vec k,t)$ and their conjugate momenta $P_\pm(\vec k,t)$. It emphasizes that the second-order equation for the transverse modes is the only actual dynamical equation for the electromagnetic field in this Hamiltonian description. The Coulomb part is instantaneous and non-propagating. :contentReference[oaicite:3]{index=3}

That is the classical starting point for photon quantization.

2. The Coulomb term and Poisson’s Green function

Before quantizing, the lecture briefly reviews the Fourier representation of the Poisson Green function. The goal is to justify the momentum-space form of the Coulomb potential

$$\frac{1}{4\pi}\int d^3r'\,\frac{j^0(\vec r',t)}{|\vec r-\vec r'|}.$$

The lecture derives the standard result that the Fourier transform of the Coulomb kernel is proportional to $1/k^2$, up to the usual rapidly oscillating terms that vanish in Fourier integrals. :contentReference[oaicite:4]{index=4}

This matters because the Coulomb-gauge Hamiltonian contains the term

$$\propto \frac{|J^0(\vec k,t)|^2}{k^2},$$

and the lecture wants to show this is exactly the Fourier-space form of the Poisson solution.

3. The free electromagnetic field: no sources

Now the lecture specializes to the pure electromagnetic field without sources, effectively setting the charge and current to zero. Then the Coulomb term disappears, and the Hamiltonian reduces to the source-free form

$$H_A^0 = \sum_\pm \int d^3k\, \frac12 \left[ \frac{1}{\hbar c}|P_\pm(\vec k)|^2 + \hbar c\, k^2 |A_\pm(\vec k)|^2 \right],$$

up to the lecture’s conventions. The slide explicitly says this is: an infinite set of quantum harmonic oscillators. :contentReference[oaicite:5]{index=5}

That is the central simplification. Each transverse polarization mode at each $\vec k$ behaves like a harmonic oscillator.

4. Canonical quantization of the transverse modes

The lecture then promotes the transverse mode amplitudes and their conjugate momenta to operators:

$$A_\pm(\vec k,t)\to \hat A_\pm(\vec k,t), \qquad P_\pm(\vec k,t)\to \hat P_\pm(\vec k,t),$$

with canonical commutators

$$[\hat A_\pm(\vec k,t),\hat P_{\pm'}(\vec k',t)] = i\hbar\,\delta_{\pm\pm'}\delta^3(\vec k-\vec k').$$

This is just the usual bosonic field quantization, but applied only to the two transverse electromagnetic degrees of freedom. :contentReference[oaicite:6]{index=6}

So the quantization of the free electromagnetic field is now literally the quantization of infinitely many harmonic oscillators.

5. Creation and annihilation operators for the electromagnetic modes

The lecture then diagonalizes each electromagnetic harmonic oscillator by defining annihilation and creation operators. In its conventions the ladder operators are combinations of $\hat A_\pm(\vec k)$ and $\hat P_\pm(\vec k)$ with coefficients chosen to match the oscillator frequency $ck$. Schematically,

$$\hat a_{\pm \vec k} \sim \sqrt{k}\,\hat A_\pm(\vec k) + \frac{i}{\sqrt{k}}\,\hat P_\pm(\vec k),$$

and

$$\hat a^\dagger_{\pm \vec k} \sim \sqrt{k}\,\hat A_\pm(-\vec k) - \frac{i}{\sqrt{k}}\,\hat P_\pm(-\vec k).$$

The lecture then inverts these relations to write $\hat A_\pm$ and $\hat P_\pm$ in terms of $\hat a$ and $\hat a^\dagger$. :contentReference[oaicite:7]{index=7}

This is exactly parallel to the harmonic-oscillator quantization of scalar fields.

6. Diagonalizing the source-free Hamiltonian

Substituting the ladder operators into the free electromagnetic Hamiltonian gives

$$\hat H_A^0 = \sum_\pm \int d^3k\, \hbar ck \left( \hat a^\dagger_{\pm \vec k}\hat a_{\pm \vec k} + \frac12 \right).$$

The lecture explicitly says: Diagonalize the source-free E-M Hamiltonian = diagonalize the HO! and immediately adds that this agrees with Einstein’s 1905 hypothesis that electromagnetic energy is quantized. :contentReference[oaicite:8]{index=8}

This is the field-theoretic derivation of photon energy quantization.

7. Momentum of the free electromagnetic field

The lecture next computes the linear momentum of the free electromagnetic field as the Noether charge associated with spatial translations. After rewriting it in terms of the ladder operators and using symmetry under $\vec k\to -\vec k$, it gets

$$\hat{\vec P} = \sum_\pm \int d^3k\, \hbar \vec k\, \hat n_{\pm \vec k},$$

where

$$\hat n_{\pm \vec k} = \hat a^\dagger_{\pm \vec k}\hat a_{\pm \vec k}.$$

So each quantum created by $\hat a^\dagger_{\pm \vec k}$ carries momentum $\hbar \vec k$. :contentReference[oaicite:9]{index=9}

The lecture then lists the physical properties of these quanta:

• transverse polarization,
• wave number $k$,
• Bose–Einstein statistics,
• energy $E=\hbar ck$,
• linear momentum $p=\hbar k$,
• and

$$E^2 = c^2 p^2,$$

so zero mass.

Then it states the conclusion in a boxed sentence: The quantized excitations of the free EM field are PHOTONS. :contentReference[oaicite:10]{index=10}

8. Photon field operator in Coulomb gauge

The lecture then writes the Coulomb-gauge vector potential operator in terms of the photon ladder operators. In its notation,

$$\hat A(\vec x,t) = \frac{1}{(2\pi)^{3/2}} \int \frac{d^3k}{\sqrt{2k}}\, e^{i\vec k\cdot \vec x} \sum_\pm \Big( \hat a_\pm(\vec k)e^{-ickt} + \hat a^\dagger_\pm(-\vec k)e^{ickt} \Big)\vec\epsilon_\pm(\hat k).$$

This is the photon analogue of the free scalar field expansion. :contentReference[oaicite:11]{index=11}

So the vector potential operator is the field whose quanta are photons.

9. The surprising $A$-field commutator

The lecture then examines the commutator of the Coulomb-gauge vector potential:

$$[\hat A_i(\vec x,t), \hat A_j(\vec x',t')].$$

After evaluating the momentum integrals, it finds a striking result: the commutator is not zero outside the light cone. In fact, the slide emphasizes this very dramatically. It even remarks that the commutator vanishes inside the light cone except for the expected lightlike contribution, which is exactly what one expects for light-speed propagation, but the concerning feature is that it does not vanish at spacelike separation. :contentReference[oaicite:12]{index=12}

At first glance this looks like a violation of relativistic causality.

10. Why this does not violate causality

The next slides explain why the alarming result is not actually a problem.

The lecture shows that the only acausal part of the commutator is a pure gauge term. More concretely, when one applies the curl operator to form the magnetic field operator,

$$\hat B = \nabla \times \hat A,$$

the problematic piece drops out exactly. The lecture shows

$$[\hat B_i(\vec x,t), \hat A_j(\vec x',t')] = 0$$

at spacelike separation, and similarly for the magnetic and electric field commutators. It also notes that

$$[\hat E_i(\vec x,t), \hat E_j(\vec x',t')] = 0$$

outside the light cone. The slide explicitly concludes:

Gauge-invariant E-M operator fields, which can appear in physical Hamiltonians, do commute at spacelike separations. Even with interactions turned on, QED is relativistically causal. :contentReference[oaicite:13]{index=13}

That is the key conceptual resolution:

• the gauge-fixed potential $A$ can look nonlocal,
• but the gauge-invariant physical fields remain causal.

11. Glauber and photon detection

The lecture then shifts to a physically motivated causality test, citing Roy J. Glauber and quantum optics. It introduces a simple model with:

• the quantized electromagnetic field,
• plus two localized two-state detector/emitters of light, modeled as two-state atoms.

The free Hamiltonian contains the two-state atoms plus the free photon field. The interaction term couples the electric field $E_x(\vec r,t)$ to Pauli-matrix operators localized near the detectors. The lecture notes that the coupling is not rotationally invariant, but it is gauge invariant. :contentReference[oaicite:14]{index=14}

The question is then: how do the probabilities of the detector states evolve in time?

This is a much more operational notion of causality than just field commutators.

12. Eliminating polarization modes that do not couple

To simplify the detector problem, the lecture chooses polarization vectors so that one polarization is orthogonal to the detector axis. That polarization does not couple to the detectors and can be eliminated. The electric field operator component relevant for the detectors is then written in terms of only the coupling polarization modes. :contentReference[oaicite:15]{index=15}

This simplifies the interaction-picture evolution substantially.

13. Interaction-picture evolution

In the interaction picture, the state evolves according to

$$i\hbar \frac{d}{dt}|\Psi(t)\rangle = \hat H_I(t)|\Psi(t)\rangle,$$

so formally

$$|\Psi(t)\rangle = \mathcal T \exp\left( -\frac{i}{\hbar}\int_0^t dt'\, \hat H_I(t') \right) |\Psi(0)\rangle.$$

The lecture splits the interaction Hamiltonian into the contributions from the two detectors,

$$\hat H_I(t) = \hat H_1(t) + \hat H_2(t).$$

Then it states the crucial commutator property:

$$[E_x(\vec x,t),E_x(\vec x',t')] = 0 \qquad \text{as long as}\qquad |t-t'| < \frac{|\vec x-\vec x'|}{c}.$$

This is the light-cone condition. Therefore, if the two detectors are separated by distance $L$, then for

$$|t-t'| < \frac{L}{c},$$

one has

$$[\hat H_1(t),\hat H_2(t')] = 0.$$

So before a light signal has time to travel between them, the two detector interactions commute. :contentReference[oaicite:16]{index=16}

That is the operator statement behind causal independence.

14. Factorization for short times

Because the two detector interaction Hamiltonians commute for times shorter than $L/c$, the time-evolution operator factorizes:

$$\mathcal T e^{-\frac{i}{\hbar}\int_0^t dt' \hat H_I(t')} = \hat U_1(t)\hat U_2(t) = \hat U_2(t)\hat U_1(t), \qquad t<\frac{L}{c}.$$

The lecture emphasizes that $\hat U_1(t)$ does not act at all in the Hilbert space of detector 2 for such short times. :contentReference[oaicite:17]{index=17}

This is the clean causality statement in the detector model.

15. Probability that detector 2 is excited

The lecture then asks for the probability that detector 2 is in its excited state at time $t$, given some arbitrary initial state of detector 1 and the electromagnetic field. It writes the general expression by summing over a complete basis of photon states together with the detector states. For $t<L/c$, the local factorization allows the photon sector to be rewritten in a locally factorized basis that diagonalizes the relevant commuting operators for the two detector regions. The lecture notes that this basis gives an entangled representation of the electromagnetic vacuum, not a simple number-state basis. :contentReference[oaicite:18]{index=18}

That detail is important: locality is not simple when expressed in terms of definite photon-number states.

16. No faster-than-light signalling

After the algebra, the lecture reaches the key result: for

$$t < \frac{L}{c},$$

the probability $P_{2+}(t)$ that detector 2 is excited is not necessarily zero, but it does not depend on the initial state of detector 1. The slide states this explicitly and dramatically: there is no way to send signals from detector 1 to detector 2 faster than light. :contentReference[oaicite:19]{index=19}

This is the operational version of relativistic causality in the detection model.

17. Dark counts and vacuum noise

The lecture then explains why $P_{2+}(t)$ need not vanish even when $t<L/c$: there can be dark counts. Even when no signal is present, the detector can be excited by vacuum noise. So a detector click by itself is not enough to conclude that a signal arrived from the other detector. The important question is whether detector 2’s statistics depend on what detector 1 did. For $t<L/c$, they do not. :contentReference[oaicite:20]{index=20}

That is a subtle but crucial point:

• vacuum fluctuations can cause excitations,
• but they cannot be used for superluminal communication.

18. Why explicit photon calculations can be surprising

The final slide says that this is a rather formal proof. It depends critically on introducing a locally factorized basis for the photon sector, but those states are extremely complicated to write out explicitly and are very difficult to use in practical calculations. In particular, they do not have any simple expression in terms of states with definite photon numbers. The lecture closes by saying that causality, when expressed in terms of photons and explicit calculations, can be surprising. :contentReference[oaicite:21]{index=21}

That is an honest summary of the subtlety:

• QED is causal,
• but locality is not always obvious in the photon-number language.

Worked Examples

Example 1: Why the free electromagnetic field is an infinite set of oscillators

In Coulomb gauge, the free electromagnetic Hamiltonian is a sum over wave vectors $\vec k$ and transverse polarizations $\pm$:

$$H_A^0 = \sum_\pm \int d^3k\, \frac12 \left[ \frac{1}{\hbar c}|P_\pm(\vec k)|^2 + \hbar c\, k^2 |A_\pm(\vec k)|^2 \right].$$

This has exactly the harmonic-oscillator form for each independent mode. Quantizing each mode gives bosonic ladder operators $\hat a_{\pm\vec k}$, and therefore the field is an infinite tensor product of harmonic oscillators.

Example 2: Why photons are massless

From the diagonalized Hamiltonian,

$$\hat H_A^0 = \sum_\pm\int d^3k\, \hbar ck \left( \hat a^\dagger_{\pm\vec k}\hat a_{\pm\vec k}+\frac12 \right),$$

and the momentum operator,

$$\hat{\vec P} = \sum_\pm \int d^3k\, \hbar \vec k\, \hat n_{\pm\vec k},$$

each single quantum has

$$E=\hbar ck,\qquad p=\hbar k.$$

Therefore

$$E^2 = c^2 p^2,$$

which is the relativistic dispersion relation for zero rest mass. So the field quanta are massless.

Intuition

The first half of the lecture is the easy part conceptually: the free electromagnetic field is just like every other free bosonic field once you reduce it to its true dynamical degrees of freedom. Each transverse mode is a harmonic oscillator. Quantize it, and the oscillator quanta are photons.

The second half is the subtle part: gauge-fixed fields can look nonlocal even when the underlying physics is perfectly causal. In Coulomb gauge, the vector potential operator is not the best object to judge causality by, because part of it is pure gauge. The true physical observables are the gauge-invariant electric and magnetic fields, and those do behave causally.

Then the detector model adds the operational statement: even though vacuum noise can cause local clicks, one detector cannot influence the statistics of a spacelike separated detector faster than light. So QED stays causal both formally and operationally.

Common Mistakes

• Thinking the electromagnetic field has four independent oscillator modes after quantization. In Coulomb gauge only two transverse modes are dynamical.
• Forgetting that photons come from quantizing the free transverse electromagnetic field, not the instantaneous Coulomb piece.
• Confusing the Coulomb-gauge vector potential with a directly observable local field.
• Thinking a nonzero spacelike commutator of $\hat A$ automatically means acausality. It does not.
• Forgetting that the acausal part of the $A$-field commutator is pure gauge.
• Missing that physical gauge-invariant fields $E$ and $B$ still commute at spacelike separation.
• Interpreting dark counts as faster-than-light signals. Dark counts are vacuum noise, not communication.
• Assuming photon-number states are the most natural basis for local causality questions. The lecture explicitly says they are not.

Short Summary

The lecture begins from the Coulomb-gauge Hamiltonian of electromagnetism and considers the source-free case, in which the electromagnetic field reduces to two transverse dynamical modes for each wave vector $\vec k$. After canonical quantization of the transverse mode amplitudes and momenta, the free electromagnetic Hamiltonian becomes an infinite set of harmonic oscillators. Introducing ladder operators diagonalizes the Hamiltonian:

$$\hat H_A^0 = \sum_\pm \int d^3k\, \hbar ck \left( \hat a^\dagger_{\pm\vec k}\hat a_{\pm\vec k} + \frac12 \right).$$

The corresponding Noether momentum operator is

$$\hat{\vec P} = \sum_\pm \int d^3k\, \hbar \vec k\, \hat n_{\pm\vec k}.$$

Therefore the quanta created by $\hat a^\dagger_{\pm\vec k}$ carry transverse polarization, momentum $\hbar \vec k$, energy $\hbar ck$, obey Bose–Einstein statistics, and satisfy $E^2=c^2p^2$, so they are massless. These quanta are photons. The lecture then studies the Coulomb-gauge vector potential operator and shows that its commutator does not vanish outside the light cone. However, the acausal part is a pure gauge term, and the gauge-invariant electromagnetic field operators $E$ and $B$ do commute at spacelike separations, so QED remains relativistically causal. Finally, using a Glauber-style model of two localized two-state detectors coupled to the electric field, the lecture shows that for times shorter than the light-travel time $L/c$, detector 2’s excitation probability may be nonzero because of vacuum dark counts, but it does not depend on the initial state of detector 1. Therefore no faster-than-light signalling is possible. The lecture closes by stressing that locality is subtle in a photon-number description and can look surprising in explicit calculations. :contentReference[oaicite:22]{index=22}

Practice Problems

1. Why does the source-free electromagnetic field in Coulomb gauge reduce to an infinite set of harmonic oscillators?

2. Why are there only two dynamical polarization states for the free electromagnetic field?

3. Show conceptually why diagonalizing the free electromagnetic Hamiltonian gives quanta with energy $E=\hbar ck$.

4. Why do the photon quanta satisfy $E^2=c^2p^2$?

5. What makes photons bosons in this construction?

6. Why is the Coulomb-gauge vector potential $\hat A$ not the best object for judging physical causality?

7. What does it mean that the acausal part of the $\hat A$-field commutator is a pure gauge term?

8. Why do the gauge-invariant fields $E$ and $B$ still commute at spacelike separation?

9. In the detector model, why can detector 2 click even before a signal from detector 1 could arrive?

10. Why do such dark counts not imply faster-than-light signalling?

11. Why does the lecture need a locally factorized photon basis to make the causality proof work?

12. Why can locality look surprising when expressed purely in terms of definite photon-number states?