and the argument in the exponential of the plane waves is the generalization of (1.7) to three space dimensions kn ■ x — wni knzx k-n^y kniz kn r
The plane wave solutions must satisfy the wave equation, which fixes the frequency kl = k2n = cWn = u2n ■ c = 1 . (2.35)
As in Chapter 1, the frequency will always be chosen to be positive, so that
and the negative frequency solutions are the complex conjugates of the positive frequency ones, with the phase ik • x.
The vectors e are referred to as polarization vectors. They carry the vector direction of a and are dependent on n. The Coulomb gauge condition requires that they must be orthogonal to kn:
Fig. 2.1 The relative orientation of the two polarization vectors of the photon and its momentum k.
and hence there can only be two independent vectors for each kn. To maintain the normalization for the a„iQ's introduced in Chapter 1, we require that these vectors be normalized to unity. Since they are in general complex, they will be defined so that _
There are many ways to choose independent c which satisfy (2.37). We will define a linearly polarized basis by choosing, for kn in the ¿-direction, e1 = x where the relative orientation of the two independent polarization vectors is shown in Fig. 2.1. There are only two independent e's, and they are both perpendicular to k. It is this property which leads to the description of the vector potential as transverse. There is no simple relation between e\ and er\,.
Before we turn to the quantization of the EM field, we will briefly discuss massive vector fields and the differences between massless and massive fields.
2.5 MASSIVE VECTOR FIELDS *
In order to highlight the unique properties of the EM theory, we consider the effect of adding a "mass" term to the Maxwell theory. Massive vector fields play a fundamental role in physics; the W± and Z bosons which mediate the electroweak interactions are examples of such fields, and these will be discussed in Sec. 9.10 and in Chapter 15 [see also Appendix D]. For now we are primarily interested in how the massive theory differs from the massless one.
"This section may be omitted on a first reading.
Start by adding a new term to the Lagrangian (2.14):
where, for now, M is simply regarded as a real parameter. Later we will see that it can be interpreted as the mass of the particles which emerge from the quantization of the field. As before, the four-current is the source of the field, and it is assumed to be conserved. The equations of motion obtained from this Lagrangian are known as the Proca equations:
Taking the four-divergence of both sides and remembering that F^u is antisymmetric give
Because the mass is not zero, the Lorentz condition emerges as a necessary constraint* We no longer have the freedom to choose another constraint (such as the Coulomb gauge condition) because the mass term is not gauge invariant. Under a gauge transformation
M2A^A" M2^A!» = M2 (ApA* - A«.A" - A^AC + ^Ac 3"AC) ±M2Av.Atl .
Using the Lorentz condition, the equations for the field simplify,
If the source is zero, this equation has plane wave solutions
Av ~ evke~lk x , provided the four-vector k satisfies the following equation:
This shows that the parameter M is indeed a mass. The Lorentz condition means that the polarization vectors accompanying these plane wave solutions must satisfy
which is satisfied by three independent polarization states (instead of only two as in the massless case). Two of these are the transverse states previously introduced for the EM field, and the third is a longitudinal state with a three-vector part
'Note that this constraint must hold for free fields, even if the current is not conserved.
in the direction of the particle momentum (for more detail see the discussion in Sec. 9.10).
The most general solution for a free massive vector field can therefore be written where En and satisfy the constraints (2.43) and (2.44), respectively. While this equation appears to be almost identical to the EM field expansion (2.33), it differs in two essential ways. First, the energy and four-momentum are those appropriate to a massive particle and, second, there are three independent polarization states instead of only two.
In conclusion, we restate some of the main points of the previous discussion. The Lagrangian for a massive vector theory, Eq. (2.39), still does not depend on dA°/dt, so that the time component of the field, j4°, must in some sense depend on the sources and other components, as was the case in the massless theory. However, because the massive theory is no longer gauge invariant, the Lorentz condition emerges automatically as the only appropriate constraint on the field, and the Lorentz condition is the constraint which fixes the component A0 in terms of the other components. Once this condition is taken into account, the free massive field can be expanded in plane waves with three independent polarization degrees of freedom. In the massless case, it is gauge invariance which allows (in fact, requires) us to remove two degrees of freedom from the field, which (in Coulomb gauge) amounts to removing the components A0 and A3 (if the momentum is in .he ¿-direction), leaving only two independent polarization states.
We now return to a discussion of the quantization of the EM field. Much of the following discussion will be extended to the massive case in Sec. 9.10.
We now quantize the theory, described by the Hamiltonian (2.31), for the interaction of the electromagnetic field with nonrelativistic particles, by turning all canonically conjugate variables into operators. The nonrelativistic particles are quantized by the replacements:
where the operator r is simply multiplication by r. Similarly, the EM field is quantized by turning A into an operator. The development used in Chapter 1 for the string will be followed again here. This involves two steps. The simple harmonic oscillators which describe the classical field must be found and described, and then they must be quantized.
The plane wave expansion (2.33) for the EM potential A is the solution to the first of these steps; it expresses the field A in terms of independent oscillators described by the quantities a„iQ. The second step, the quantization of the field, is done in precisely the same way it was done in Chapter 1; the quantities a„,Q are turned into operators by imposing the commutation relations
The only difference between Eq. (2.46) and the corresponding relations for the one-dimensional string is the fact that now there are three space dimensions and two polarizations. This means that Eq. (2.46) must describe many times the number of normal modes, and hence many times the number of independent operators, than were described before. However, this does not really change the result, since operators corresponding to independent normal modes still commute, and the commutation relation for a and a* for a single normal mode is the same. Thus Eqs. (2.33) and (2.46) give the complete description of the EM field and its quantization, and we will now use them to work out several details.
Because of the gauge condition, the forms of the canonical commutation relations for A and n differ from those found for the string. The CCR can be worked out 'com where x — (t,r) and x' = (t,r'). This can be further reduced using the fact that the polarization vectors, together with kn (the unit vector in the direction of kn), form a complete orthonormal set. Hence, for each n,
48 QUANTIZATION OF THE ELECTROMAGNETIC FIELD where <5T is the transverse ¿-function. This gives
Recalling that the sum over the plane wave states gives a delta function for each direction in space leads to the following expression for the CCR's for EM theory:
The extra Ôidj term is necessary in order that the CCR's be consistent with the gauge condition:
(Vr')j [^(r'.tU'^t)] = i [Vi - Vj] 63(r — r') = 0
Note also that
where the factor of 2 appears because there are two independent polarization states.
All of these commutation relations hold at equal times, and the commutators are zero if the two points are separated in space. Under Lorentz transformations, the interval (t-t')2-(r-r')2 = (x-x')2 is invariant, and thus one consequence of a relativistic generalization of the CCR's is that the field operators commute when their arguments are separated by a space-like interval [i.e., one for which the four-vector distance (a; - x')2 < 0]. This has a beautiful physical interpretation: it is impossible to exchange information between two points separated by a spacelike interval, and hence any physical observables (fields in this case) at two such points must be truly independent of each other. The mathematical expression of this independence is the statement that the operators corresponding to these quantities must commute. This is an important principle, referred to as local commutativity or microscopic causality, which can be used as a starting point for an axiomatic development of field theory [see Streater and Wightman (1964)].
Perhaps the appearance of the transverse ¿-function in the CCR's (2.48) could have been anticipated from the start, but in any case, it follows in a straightforward way from the commutation relations (2.46). These commutation relations between the creation and annihilation operators involve only the independent degrees of freedom, and hence are the same for all types of fields. It is for this reason that we have chosen to use them to begin the quantization of any field theory.
The form of the Hamiltonian and momentum operators can be inferred from the discussion of the string in Chapter 1. Here we will demonstrate that the Hamiltonian does indeed have the expected form. The proof that the momentum operator also has this form is deferred to Prob. 2.1.
For simplicity, assume that the polarization vectors are real. Then, recalling Eq. (2.31), the Hamiltonian for the free EM field is n ,r\' v a, a'
x | [WbUb.e" • 6°: + (kn x ) • (kn, xe£)'
x (<aan,,a, ci<fc--fc-')-I+4,iQ,an,Q e-W»-*»')^ -wnu>n. • - (kn x 6«) • (kn, x ei )
where the normal ordering prescription has required all cross terms to be written as a^a. Using the fact that fr L3 d3r f ±nkn-kn,)-x _ 6 ,
H = \ E ["» 6«<' + (*n><0-(*»><«#')] (<aan,Q, + afn Q,a„,a)
x (a„lQa-nio'e-2<w"t + aJliQat_niQ,ea<,--t)| .
n cn una
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