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Photons

At this point we have a non-interacting Fock space and we have Dirac particles. We want to introduce interactions between particles, such that the interaction operator has an invariant form. Since the current density observable, j^a(x), is a vector, a covariant theory can be found by contracting it with another Hermitian vector operator, A^a(x). The possibilities are severely restricted. The natural and simplest thing to try is to introduce a particle with a spin index which transforms as a vector, and which is its own antiparticle, i.e. its creation and annihilation operators appear in the same field operator. Vector particles may have non-zero mass, but empirical evidence is that this is not so for the photon» at the limit of experimental accuracy. Zero mass is assumed.


With this Hamiltonian density photons are always either created or destroyed in interaction. We cannot, therefore, talk of the measurements of the position of a photon, but only of measurement of the position at which it was annihilated, or the position at which it was created. x is not the position of a photon, but rather the position at which a charged particle would be found to have emitted or absorbed a photon, if a measurement were carried out.

Plane Wave Photon States

Since momentum is a conserved quantity, it is possible to talk about the measured momentum of a photon state. A photon created with a given momentum will be annihilated with the same momentum. So, it will be required that plane wave states are an orthogonal basis. First define a basis for spin states.


The scalar, λ, is required because the states refer to the hypothetical measurement of position of the electron which emits a photon, not the position at which a photon can be measured. Photons are always created or annihilated in interaction, and cannot be in eigenstates of a position operator. It is not meaningful to annihilate a photon at the instant of its creation. We do not require that states, , are orthogonal. Direction is determined by the distribution of matter, not by fundamental assumption, so λ depends only on the magnitude of p.

We require that is a delta function,
where η(0) = −1 and η(r) = 1 for r = 1, 2, 3. The minus sign from η(0) does not alter the expansion of the inner product for an orthonormal basis. The braket for the photon is,
The resolution of unity takes the form,
We do not have ; the braket is not positive definite, in conflict with the calculation of probabilities. In practice, we only need to generate probabilities for observations. We impose the condition that, in observations on the photon, there is no polarisation between time-like and longitudinal states,
With this restriction, probabilities for the observation time-like and longitudinal states are zero, and the braket reduces to
which is positive semidefinite». It will be seen that all four polarisation states are required for the derivation of Maxwell’s equations». We can conclude that the unobservable states have a real effect, and represent real particles, but the probability interpretation allows only the observation of a subspace containing the two transverse polarisation states, on which the inner product is positive definite. The braket is invariant under the addition of a light-like polarisation state, from which it follows that light-like polarisation cannot be determined from experimental results.

We require that the probability for the creation of a photon at x and its annihilation at y is covariant. Observe that
Then, setting
gives
which is covariant, as required.

Evolution of Photon States

We may expand using a basis of plane wave states,
Then the wave function for the state is
Since p is the momentum vector for a zero mass particle, the wave function satisfies a Klein-Gordon Equation,
Conservation of probability applies to the creation and annihilation of particles. Differentiating gives a first order equation as required by Stone’s theorem,

The Photon Field Operator

The creation operators for a plane wave state is given by . Substituting gives the photon field operator,


Proof: Differentiate directly.


Proof: Differentiate and use absence of polarisation between light-like and longitudinal states.

Photons are Bosons, obeying commutation relations,
Substituting p → −p in the second term gives the equal time commutator,
Since the integral is invariant, the commutator is zero outside the light cone, satisfying locality.

The Locality Condition for Photons

Because the photon commutator vanishes, the time evolution of the expectation of the photon field is trivial. Physical laws depend on derivatives of the photon field, not directly on A. It is required that locality is obeyed.


Proof:   Differentiating,
and
Substitute p → −p at x⁰ = y⁰. Then, for i = 1, 2, 3,
and, for the time component,
The integrals are invariant, so they are zero outside the light cone.

The Photon Field Operator ↑ Classical Electromagnetism →