THE HAMILTONIAN APPROACH TO ELECTRODYNAMICS

We shall in what follows widely apply the so-called Hamiltonian method for the interpretation of a whole range of electrodynamical problems. When we use this method electrodynamics is formulated in a way which is strongly reminiscent of mechanics. The transition from classical to quantum mechanical electrodynamics is thus in the Hamiltonian framework completely analogous to the transition from classical, Newtonian mechanics to non-relativistic quantum mechanics. Nowadays much more sophisticated methods are predominant in quantum electrodynamics and in general in quantum field theory and there are strong arguments for using them. However, the use of the Hamiltonian method is still completely justified for the elucidation of a large number of physical aspects; this is, for instance, also done by Heitler (1947) in his book. Moreover, we shall in what follows apply the Hamiltonian method mainly to classical electrodynamics, both in vacuo and in a medium.

Before introducing the Hamiltonian method we shall give the main equations and relations and we shall do that in considerable detail for future convenience.

The usual form of the Maxwell equations in vacuo is:†

 H=4πcρv+∂E∂t,div E=4πρ,curl E=−1c∂H∂tdiv H=0. (1.1)

Here H is the magnetic field strength, E the electrical field strength, ρ the charge density, v the velocity of the charges, and c the velocity of light in vacuo. We assume for the sake of simplicity that there is a single point charge e, at position ri(t), in the electromagnetic field. In that casethe charge density is given by a δ-function

=eδ(r−ri(t)). (1.2)

It is well known that Eqs. (1.1) can be reduced to the equations for the electromagnetic potentials A and ϕ which are connected with the fields E and H sby the relations

=−1c∂A∂t−grad ϕ,H=curl A. (1.3)

The third and fourth of Eqs. (1.1) are automatically satisfied by virtue of (1.3), as can be verified by substitution.

From the first and second of Eqs. (1.1) we get, using (1.3) and the identity

  curl A=−∇2A+grad div A, (1.4)

equations for the potentials A and ϕ :

2A−1c2∂2A∂t2−grad(1c∂ϕ∂t+div A)=−4πcρv,∇2ϕ+1c∂∂tdiv A=−4πρ. (1.5)

The set of Eqs. (1.5) determines the potentials A and ϕ. The fields E and H can be found using Eqs. (1.3)

It is well known that the vector potential A and the scalar potential ϕ are not uniquely determined. Indeed, we can change to new potentials:

′=A+grad χ,ϕ′=ϕ−1c∂χ∂t, (1.6)

where χ is an arbitrary function of the coordinates and the time. This is called a gauge transformation. One can easily show that the fields E and H do not change under a gauge transformation. They can be expressed in terms of A′ and ϕ′ just as well as in terms of A and ϕ; one can verify this by substituting (1.6) into (1.3).

The fact that the definition of the potentials is not unique enables us to impose upon A and ϕ an additional condition. This condition can be chosen in such a way that the form of Eqs. (1.5)becomes as simple as possible.

For instance, we can impose as such a condition the relation:

 A+1c∂ϕ∂t=0. (1.7)

This is a relativistically invariant condition which is called the Lorentz condition, and the resulting gauge is called the Lorentz gauge. It can be written in the form

Ai∂xi=0, (1.7a)

where we have assumed summation over repeated indexes, as will be done everywhere in what follows. We use here and elsewhere in this book (apart from Chapter 12) the notation of Landau and Lifshitz (1975). We refer to that text for the definition of four-vectors, for the difference between covariant and components and for the summation convention.

One sees easily that if condition (1.7) is satisfied, the Maxwell equations take the following form:

A≡(∇2−1c2∂2∂t2)A=−4πcρv,□ϕ≡(∇2−1c2∂2∂t2)ϕ=−4πρ. (1.8)

One should not think that the condition (1.7) and the set of Eqs. (1.8) determine A and ϕ completely. We can still perform a gauge transformation of the form (1.6), where in the present case χ must satisfy the homogeneous equation χ = 0. The fields E and H remain invariant under the transformation.

Splitting the field into longitudinal and transverse components is important, especially in the Hamiltonian framework. We split the vectors E and H into components

=Eℓ+Etr,H=Htr, (1.9)

where div Etr = 0 and, by virtue of (1.1), div Htr = div H= 0.

We demand that the vector potential A describe only the transverse field; this means that we impose on it the condition

 A=0, (1.10)

instead of the additional condition (1.7). Sometimes the potential which satisfies the condition (1.10) is denoted by Atr.

If condition (1.10) is satisfied, the Eqs. (1.5) for A and ϕ take the form

2ϕ=−4πρ, (1.11)

2A−1c2∂2A∂t2=−4πcρv+1cgrad ∂ϕ∂t. (1.12)

We see that we have obtained the ‘static’ Poisson equation for the potential ϕ.

If ρ is the charge density (1.2) of a point charge, the solution is the well known one:

=e|r−ri(t)| (1.13)

where ri(t) is the position of the charge at time t. The vector potential A now describes only the transverse field. The gauge (1.10) is called the Coulomb gauge† The potentials A and ϕ are here determined apart from a gauge function χ(r,t) which satisfies the condition ∇2χ = 0.

We now evaluate the energy of the electromagnetic field,

=∫E2+H28πd3r (1.14)

We substitute here the expressions for the fields E and H in the form (1.9); it is clear that in the case of the Coulomb gauge (1.10) we have

tr=−1c∂A∂t,Eℓ=−grad ϕ. (1.15)

Substituting (1.9) and (1.15) into (1.14) we...