where Htrans is the energy stored in the transverse fields, (5.83) and

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is the Coulomb energy of the charged particles, (5.84)

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The first term in (5.84) is the sum of the Coulomb self-energies of each particle and the second term is the Coulomb interaction between pairs of charges. The former diverges. This is in part due to the fact that it is inconsistent to try to account, in a nonrelativistic theory, for the interaction of the particles with the high-frequency modes of the field. An immediate solution is to introduce a cutoff kor in all integrals over k in reciprocal space. For each particle 0: the cutoff must be on the order of

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merc/Ii so as to leave out interactions with modes whose energy is comparable to merc'2. Such a cutoff gives a finite value for the Coulomb self-energy on the order of

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(5.85)

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A proper solution to the problem of the divergency of the self-energy requires a relativistic theory. When a relativistic theory of quantum electrodynamics was first developed, however, this divergency associated with the Coulomb self-energy and others associated with the infinite number of degrees of freedom of the field remained unsolved. These divergencies were dealt with only by renormalization theory [202]. We do not discuss this problem here and refer the reader to [203, 313], for example. The Hamiltonian (5.82) is meaningless unless the associated canonical variables are defined. It turns out that the canonical variables do not involve the fields directly, but the electromagnetic potentials. This gives the potentials a physical significance in quantum mechanics that they do not possess in classical mechanics7 [9,318]. A complete discussion should provide a derivation of the Hamiltonian and Canonical variables from a Lagrangian. Such an approach is adopted by Cohen-Tannoudji et at. [108]. In this section I merely show that a given choice of canonical variables is possible. The potentials A and U are defined such that

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1 . E(r, t) = -VU(r, t) - -A(r, t),

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(5.86) (5.87)

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B(r, t) = V 1\ A(r, t). In reciprocal space, this becomes &(k, t)

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= -- A(k, t) c B(k, t) = ik 1\ A(k, t).

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A description of the electromagnetic field in terms of potentials introduces extra degrees of freedom that cannot possibly be independent. This is reflected in the fact that the fields remain unchanged under the following gauge transformations of the potentials:

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A(r, t)

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+ VF(r, t),

(5.90) (5.91)

U(r, t)

U(r, t) - -!'IF(r, t),

where F(r, t) is an arbitrary function of rand t. Indeed, when a particular gauge is chosen, the redundant degrees of freedom can be eliminated using the constraint

7Here we are referring to the transverse component of the vector potential. which is gauge invariant. This component can give rise to an observable physical effect in a region where there are no fields (BohmAharonov effect [9]).

AN ARBITRARY CHARGE DISTRIBUTION

relations introduced by the choice of gauge. Depending on the choice of gauge. however. the separation of the truly independent degrees of freedom of the fields from the rest might not be so straightforward as when we were dealing with the fields themselves. In non relativistic quantum electrodynamics. such a separation is possible by the adoption of the Coulomb gauge. The Coulomb gauge is the gauge where the vector potential A is transverse:

A(k, t) = A-L(k, t).

(5.92)

The transverse vector potential. however. has only two independent components in reciprocal space. These are components along two orthogonal directions on the plane perpendicular to k. We will denote a given choice of two such directions by the normalized vectors } (k) and 2(k). The components will be indicated by the subscripts } and 2. In the Coulomb gauge. the transverse fields depend on the vector potential only: 1 . (5.93) E-L(k,t) = -- A(k,t),

c 8(k. t) = ik 1\ A(k, t),

(5.94)

and the longitudinal component of the electric field depends only on the scalar potential: (5.95) 1I(k, t) = -ikU(k, t). So the vector potential is rid of all redundant degrees of freedom, with the scalar potential being determined completely by the dynamic variables of the charged particles. This is the great advantage of the Coulomb gauge. There are. however. some disadvantages. The Coulomb gauge is not so well suited to the relativistic theory because it is not manifestly covariant8 [108. 314J. Moreover. the instantaneous Coulomb scalar potential raises a causality question [108.314]. We do not address this problem here. I refer the reader to Brill and Goodman [75J or Cohen-Tannoudji et al. [108J for a detailed discussion. Now we show that the canonical variables of our system are ra. Pa. A(k, t), and II(k, t). where Pa and II(k, t), the conjugate momenta of ra and A(k, t), respectively. are given by