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The Classical Field

A(x) has the required property of an observable, that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law and Maxwell’s equations.


It follows from Ehrenfest’s theorem that


Proof:  By locality the equal time commutator is zero. Using the Gupta-Bleuler gauge condition,
The Lorenz gauge condition fixes gauge up to the unobservable light-like polarisation. In classical electrodynamics one may choose a different gauge without affecting predictions, but here Lorenz gauge is fixed by the requirement of a first order covariant equation.

Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the interaction density
where j is the current observable,

Gauge Symmetry

The local phase transformation,
applied to the field operators, makes no difference to the current and so leaves the predictions of the theory unchanged (equivalently the transformation may be applied to the creation operators, remembering the sign change for antiparticles).


The term gauge symmetry is something of a misnomer. It was introduced by Herman Weyl, as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, Vladimir Fock and Fritz London adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown in the section Gauge Invariance.

Momentum in the Interacting Theory

In the absence of interactions, there is no issue with local gauge freedom. The phase of an electron wave function is fixed at the point of creation and becomes simply the global symmetry of the one particle theory, in which kets can be multiplied by constant phase without altering their meaning in quantum logic. When interactions are introduced the result is that the evolution of the wave function does not match the evolution of the field operator which created it, and which is defined on the non-interacting space. A difficulty arises because the momentum observable in the non-interacting theory,
extracts the frequency and wavelength of the the wave function. When the simplified notation, i∂^a, is used as an operator on ket space, the integral, the bra, and the ket are implicit. We would like to use Ehrenfest’s theorem to calculate the classical force due to the interaction, by differentiating the expectation of momentum,
but states evolve according to the full Hamiltonian, whereas the creation operators are defined on the Fock space of non-interacting particles, and create states obeying the Dirac equation. There is a real phase shift corresponding to change in momentum, which must be distinguished from the arbitrary phase in the definition of field operators.

To ensure that creation operators and states evolve identically, we define the field picture,
In the field picture states evolve as in the Schrödinger picture for non-interacting particles. The momentum operator in the field picture is
In the semi-classical correspondence, for small t, this may be treated as a perturbation to the evolution of a non-interacting particle, in which the interaction is replaced with an expectation. For a classical particle with position x and velocity , the classical current is . The expectation of the interaction Hamiltonian is
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation (courtesy of Hsin-Chia Cheng; for a general discussion, see Costella & McKellar^»). For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.

Replacing the interaction Hamiltonian with its expectation, the momentum operator in the field picture is
Thus the expectation, , of the operator which creates and annihilates photons, acts in the manner of a classical vector field, modifying energy and momentum. This is the standard formula for momentum in the presence of a field, but normally it would be assumed, on phenomenological grounds, i.e. because we know empirically that it gives correct predictions. A phenomenological treatment assumes the concept of a classical field defined on a spacetime background. The treatment here has assumed only particles, electrons and photons, and has arrived at the classical field in an approximation in which it is an expectation of photon creation/annihilation.

The Interacting Dirac Equation

With the replacement of the momentum operator for non-interacting particles with the corresponding operator taking interactions into account,
the Dirac equation,
becomes the interacting Dirac equation^».


The interacting Dirac equation describes the behaviour of an electron in a classical field, and predicts a value of 2 for the gyromagnetic moment, or g-factor, of the electron. Higher order corrections leading to a slightly larger value may be calculated from Feynman diagrams, and agree with the measured value to 14 significant figures.

The Lorentz Force Law

Working in the field picture, we have, from Ehrenfest’s theorem,
Using expectations for the interaction, as above, we have
Substituting for H, together with
and dropping the suffix F (the expectation is the same in any picture),
where the product rule of differentiation has been used to find the second term.

Classical force is defined as the rate of change of momentum, according to Newton’s second law.
Since laws of physics must obey general covariance, the parameter τ is the proper time of the matter on which the force acts, not coordinate time defined by a particular observer. The electromagnetic force on a charged particle will be evaluated in the rest frame of the particle, in which and the current is . In the rest frame,
So,
After Lorentz transformation, this is
where F^ab is the Faraday tensor. The classical E and B fields are defined as components of the Faraday tensor.


With this definition, 3-vector force obeys the Lorentz force law.

Gauge Invariance

The interacting Dirac equation,
can be written, in terms of creation operators acting on any ket ,
A local gauge transformation applied to the creation operators,
gives
So,
which is identical to the original form of the interacting Dirac equation apart from the replacement
But the Faraday tensor is also unchanged by this replacement,
So the local phase symmetry of the field operators is precisely equivalent to the well known symmetry of the classical electromagnetic field.

Maxwell’s Equations

Proof:
Differentiating the expectation of the photon field twice, using Ehrenfest’s theorem,
Using the Hamiltonian density for qed, I(x) = ej(x) · A(x),
The equal time commutor for photons is
Maxwell’s equations in Lorenz gauge follow immediately.


Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorenz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current density operator, and using properties of Dirac spinors.

I have given Maxwell’s equations in terms of the classical vector field, . The standard form of Maxwell’s equations follow.


(Proof). To convert to SI units, divide space derivatives (i.e. and B) by c, put , , and use ε₀μ₀ = c.

At no point has electromagnetism been assumed, but rather it has been found from an underlying structure consisting only of particles and interactions. The general requirements of a theory of measurement within such a structure have lead to relativity and to quantum theory. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded a conserved current, the Lorentz force law, and Maxwell’s equations. The new predictions of quantum electrodynamics come from the calculation of Feynman diagrams; it is sometimes suggested that qed is only valid as perturbation theory, but the derivations of the classical equations shows that it is, as Feynman described, a complete description of the interactions underlying electromagnetism.

Classical Electromagnetism ↑ Feynman Diagrams →