J xj Ac J d4x 0 Ac

In order to obtain a definite solution for the EM fields, the arbitrariness associated with the gauge freedom (2.15) must be removed so that the fields can be uniquely specified everywhere. This process is referred to as "gauge fixing" and involves imposing some constraints on the fields which will fix the gauge function Ac and remove the gauge freedom. Two popular choices for the constraint, or choice of gauge, are the Lorentz and Coulomb gauges, defined by the constraints

Lorentz gauge Coulomb gauge

There are advantages and disadvantages which accompany the use of each of these gauges, and the choice of gauge is closely related to how the time component of the four-vector potential, >1°, is to be treated. Since the time derivative of does not occur naturally in the Lagrangian, and since the gauge transformations give us some freedom to redefine the field in a convenient way, the solution of the electromagnetic problem may be approached in one of two ways:

• The quantity A0 may be eliminated from the Lagrangian by expressing it in terms of the remaining components of A11. This approach is simplified by using the Coulomb gauge.

• A new term may be added to the Lagrangian which contains the time derivative of ^4°. In this case, the Lorentz gauge is the preferred constraint.

Each of these approaches will now be discussed briefly.

To see what is involved in eliminating A0 from the Lagrangian, look at Eq. (2.13) when v = 0:

This equation is greatly simplified by imposing the Coulomb gauge, which reduces the equation to Poisson's equation

V2j4° = -p , and this equation has the unique solution

Coulomb's law

Coulomb's law

This solution is zero if p = 0. Had we chosen to use the Lorentz gauge, the equation for A0 which would result from (2.18) is

is the familiar wave operator. This equation is manifestly covariant, but the solutions of the wave equation may depend on time and are not zero even when p = 0. For these reasons the Coulomb gauge, which gives Coulomb's law, is used in the study of atomic and other low energy systems, and it will be used in Part I of this book. The disadvantage of this choice is that the Coulomb gauge condition is not manifestly covariant; to maintain this gauge condition in different frames requires that a new gauge function Ac be chosen for each frame, so all of the results obtained from this gauge will look non-covariant. The final results of any calculation will always turn out to be covariant, but often this is only apparent after the final answer is obtained.

Now consider the second approach to the study of the EM field, in which a term containing the time derivative of A0 is added to the Lagrangian. This must be done in such a way that the theory is not altered, and a convenient way to do this is to add the following gauge fixing term to the Lagrangian density:

This extra term can be regarded as a constraint, with the redundant field components related to Lagrange multipliers [see Itzykson and Zuber (1980)]. The parameter a is the gauge parameter and may assume any finite value. Two well-known choices are a = 1, the Feynman gauge, and a —* 0, the Landau gauge. Note that the overall theory is not affected by the addition of the gauge fixing term because it is zero after the gauge condition dMAM = 0 is imposed. These gauges are very convenient for the study of high energy scattering processes where it is desirable to maintain manifest Lorentz invariance, and using the method of Gupta [Gu 50] and Bleuler [B1 50] [see also Bogoliubov and Shirkov (1959)], it is possible to quantize all four components of Af as independent degrees of freedom. A modern approach, in which these gauges are used in conjunction with the method of path integrals, will be discussed in Chapter 15.

It is important to realize that the physics is unaffected by the choice of gauge. Any gauge may be used, as long as it is used consistently in all parts of the calculation. The intermediate steps may be very different, but the final result for any physical observable must be independent of the gauge used to calculate it.

For example, note that a scalar gauge function Ac can always be found so that either the Coulomb or Lorentz condition is satisfied. Suppose first that V • A ^ 0 and we wish to impose the Coulomb condition. Then change A to A' so that

This implies that

and we know that this equation (Poisson's equation again) can be solved. Similarly, suppose that dtlA^ / 0, and we wish to impose the Lorentz gauge. Then change A to A' so that d^A'» = (A" - d»Ac) = o .

This implies that

This is the inhomogeneous wave equation, which also can be solved. Since the physics does not depend on the scalar Ac, the physics also cannot depend on the gauge.

The Lagrangian in the Coulomb Gauge

The next task is to rewrite the Lagrangian density using Coulomb's law to define A0. The resulting Lagrangian will then depend only on the three components of the vector potential A' (and the charge and current densities, considered sources of the fields and not dependent on them). The three components of A' will be treated as independent fields, and the Lagrangian will be constructed so that the correct equations of motion for these fields will emerge naturally.

To see more clearly what this means, look at the equation for the vector potential. From the field equations (2.13), this equation is

where is referred to as the transverse current. Taking the divergence of both sides of this equation and assuming that Aq is given by Poisson's equation give dt

(by current conservation) . (2.23)

Hence, if we did not know that the Coulomb gauge condition V • A = 0 had been used to relate A0 to p, this equation would enable us to recover it in the following sense: if V • A =0 and d (V ■ A) /dt — 0 holds at one time, it will hold at all times. In this sense the Coulomb gauge condition can be regarded as a dynamical consequence of Eq. (2.22) for A. Our task is to construct a Lagrangian density which will give this equation.

To find the correct Lagrangian density, we will first separate out the A0 terms from the Lagrangian density (2.14). All three-vectors will be expressed in a "standard" form, which is taken to be A' —► A and Vj —» V. Hence we use AM — (A0, -A'), d" (ö°, -Vi), and obtain:

= - ±dßA° (<9M° - 9° A") + Id^ (3M4 + V.A") - PA° +j ■ A

= i Vj A0 Vj A0 + \doA^doA3 - \B2 + VjA°d0Aj - PA° +j-A .

The third term in the last line was obtained using the identity e^kCum. — bjtökm —

B2 = eijkV,AkeiemVeAm = S/jAkVjAk - V^V^ = VjAk [VjAfc - VfcAJ] .

A vector field V which has a zero three-divergence, V • V = 0, is said to be transverse. Physically, this condition means that the field is perpendicular to its momentum, as will be discussed below. After the equations of motion have been obtained, and the gauge condition applied, V • = 0, and only transverse radiation fields remain in £. The longitudinal component of E, sometimes denoted by J?«.

£|i = -Wl0 , is no longer a dynamical variable and is expressed in terms of p.

Finally, the Lagrangian derived from £ assumes a nice symmetrical form:

L = Jd3rC= Jd3r {^\E±(r,t)\2 — ^\B(r,t)\2 +j\{r,t) -A(r,t)}

Note the presence of the instantaneous Coulomb interaction, which makes this approach ideal for application to relativistic atoms. As advertised, L is no longer manifestly covariant.


To complete the picture, and to introduce interactions, add two spherically charged particles to the Lagrangian. The location of these particles will be described by generalized coordinates fla«

and their charge density will be denoted pa (\qa - r|), where \qa - r| is the length of the vector which connects the location qa of the ath particle to the field point r. To simplify the notation, the particle coordinates will usually be denoted simply by qa, and the charges by pa(r), although both depend on time. The four-current of each particle is j£{r)=Pa{r) (1, qa) , (2.27)

where qa = dqa(t)/dt and the total charge and current is the sum of the single particle charges and currents:

Note that the current of each particle is conserved:

dpajr) dt

where p'a(r) denotes the derivative of pa(r) with respect to its argument, \qa - r|. The Lagrangian of this field-particle system is composed of three terms:

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