Of The Electromagnetic Field

We now use the techniques developed in Chapter 1 to quantize the electromagnetic (EM) field. This system is one of the most important in physics but is also one of the most complicated. The EM field appears to be two coupled three-vector fields, but through Maxwell's equations and gauge invariance, it can be reduced to a single four-vector field with only two independent components. The elimination of these redundant components, which are connected with the gauge invariance of the system, poses a new problem unlike any discussed in the previous chapter. The relativistic nature of the EM field is also a new feature which needs to be discussed.

This chapter begins with a description of the properties of Lorentz transformations and a discussion of gauge invariance, topics which must be addressed f ¿fore we can quantize the field. The particles which emerge from this quantization are photons, familiar from elementary studies. The vector nature of the EM field means that the photons have spin one as well as energy and momentum. The appearance of this spin, and its connection to the vector property of the field, will be the last topic covered in the chapter.

The goal of this chapter is to lay the foundation for the treatment of the interaction of the EM radiation field with matter, which will be discussed in Chapter 3.

2.1 LORENTZ TRANSFORMATIONS

We begin with a brief discussion of Lorentz four-vectors and transformations. The emphasis here will be on notation; the properties of the Lorentz group will be discussed in more detail in Chapter 5. In the natural system of units, the speed of light, c, is equal to unity, so that the space/time four-vector is denoted

= (t,r) = (t,x,y,z) = (t,rl) 2V = (t,-r) = (t,-x,-y,-z) = (t,-r') ,

where x** is the contravariant and x^ the covariant form. Note that Greek indices on four-vectors (such as fi) vary from 0 to 3, while Roman indices on three-vectors (such as i) vary from 1 to 3.* The invariant length of this four-vector is written x — 9¡¿I/— x^iT ~~' i f* — t x y

where gM„ = g^ is the metric tensor and a sum over repeated indices is always assumed. Note that (2.2) implies that the relation between the contravariant form of x (zM) and the covariant form (xM) is

A Lorentz transformation (LT) is any transformation which leaves the length of four-vectors, defined in (2.2), invariant. In general, a transformation A which operates on the space of four-vectors can be written*

In this notation the requirement that the four-vector length remain invariant becomes x'2 = x^g^x'» = A^axag^Al/0x& = gQ0xax0 , which leads to the following condition on A:

Any transformation which satisfies this relation is an LT. In Sec. 5.8 we will show that all of the transformations which satisfy (2.6) form a group in the mathematical sense.

*We will adopt the convention that the Roman indices on three-vectors will always be written as superscripts. Be careful to always include the minus sign when converting the spatial components of a covariant four-vector to a three-vector!

*Free indices on both sides of a relativistic equation must always be in the same position (either up or down), and indices on one side of an equation which are summed (or contracted) must always be paired, with one up and one down. This will insure that both sides of the equation transform in the same way. In three-vector equations, the position of the indices is arbitrary, and placement is by convention.

which is the same as the transformation law for a general covariant four-vector = Gx, as given in Eq. (2.9). If is a contravariant four-vector, the divergence is r\

Note that a plus sign appears in this equation, instead of the minus sign which might be naively expected.

The LT's are not necessarily orthogonal matrices. The rotations, which leave the time component of any four-vector unchanged (and also one direction in space, the rotation axis, unchanged), can be written

A R =

1

0

0

R

and are orthogonal. The boosts, which leave two directions in space invariant, are not orthogonal. A simple example is the boost in the ¿-direction

which leaves the x- and y-directions invariant. Note that A^ = Ab, not As1.

2.2 RELATIVISTIC FORM OF MAXWELL S THEORY

The Maxwell equations (with c = 1) are

dt dE

These are in rationalized Gaussian units where Coulomb's law for a point charge is V = e2/47rr and the fine structure constant is a = e2/47r. We replace two of these equations with potentials d

These solve the two homogeneous equations identically, leaving dE

Gauge Invariance

The electromagnetic Lagrangian has two special features not encountered before:

(i) The generalized momentum conjugate to the time component of the four-vector potential (which will be denoted by A0, instead of <j>, in this subsection) is zero. This follows from the fact that the Lagrangian density (2.14) does not depend on dA°/dt, and hence dC

Because of this, the Poisson bracket of A0 with 7r° (or, after we quantize the field, the commutator [A0,7r0]) must also be zero. If we attempt to quantize the field component A0 by turning it into an operator, it would therefore commute with all operators, and by Schur's Lemma would reduce to a c-number. The field component A0 is special.

(ii) If the current is conserved (and we have seen that consistency requires it), then the Lagrangian is invariant under the gauge transformation

where Ac is a scalar. Note that the Lagrangian density is not locally gauge invariant [i.e., is not invariant under the transformation (2.15) at every space-time point 1], because

not zero

However, the action f dtL (and hence the theory) is gauge invariant. To show this, use the fact that the fields are assumed to satisfy periodic boundary conditions, so that when integrating over all space any surface terms which might arise from any integrations by parts can be assumed to vanish or cancel. To justify dropping the surface terms from the time integration, assume that Ac = 0 at i = ±00. Therefore, integrating the non-zero term in (2.16) by parts gives

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