ABSTRACT
Let us start with a division of the transverse vector potential, AT (r, t), into its positiveand negative-frequency parts, i.e.,
AT (r, t) = A (+) T (r, t) +A
(−) T (r, t). (28.1)
As we know, the positive-frequency (analytic) part,
A (+) T (r, t) ≡ F(r, t), (28.2)
relates to the transverse photon (F) concept, and the negative-frequency part,
A (−) T (r, t) ≡ FA(r, t), (28.3)
to the transverse antiphoton (FA) concept. In free space the two parts satisfy the wave equations
F(r, t) = 0, (28.4)
and
FA(r, t) = 0, (28.5)
as a consequence of the fact that AT (r, t) = 0; cf. the analysis in Sec. 2.3. A transition four-current density concept between two positive-frequency transverse
vector-potential distributions F1(r, t) and F2(r, t), satisfying the free-space wave equations
F1(r, t) = F2(r, t) = 0, (28.6)
can be introduced in the following manner: Multiply F1 by the Hermitian conjugate F † 2
of F2 from the left, and subtract the Hermitian conjugate F † 2 of the second member of
Eq. (28.6) multiplied from the right by F1. Thus,
F†2F1 − ( F†2
) F1 = 0, (28.7)
or equivalently
∇ · [ F†2∇F1 −
( ∇F†2
) F1
] +
∂
c∂t
[( ∂
c∂t F†2
) F1 − F†2
∂
c∂t F1
] = 0. (28.8)
of
In Eq. (28.8), a notation has been used in which the ith (i = 1 − 3) component of a form α†∇β is defined by α†(∂β/∂xi), i.e., as the product of the row vector α∗ and the column vector ∂β/∂xi. From Eq. (28.7) and onward the dot (·) used when multiplying row and column vectors in Sec. 15.6 is omitted. Remembering the expressions for {∂µ} [Eq. (3.42)] and {∂µ} [Eq. (3.44)] it appears that Eq. (28.8) (multiplied by a certain constant K) can be written in the covariant form
∂µJ µ 1→2(x) = 0, (28.9)
where
Jµ1→2(x) = K [ F†2∂
µF1 − ( ∂µF†2
) F1
] . (28.10)
It makes sense to consider {Jµ1→2(x)} as a T-photon transition four-current density from the field distribution F1 to the distribution F2, and Eq. (28.9) as the related equation of continuity. In three-vector notation Eq. (28.9) is written as
∇ · J1→2(r, t) + ∂ ∂t
( c−1J01→2(r, t)
) = 0. (28.11)
By integrating Eq. (28.11) over the entire r-space, using Gauss theorem, and assuming (as usual) that the surface integral vanishes at infinity, it follows that the quantity
〈F2|F1〉 ≡ ∫ ∞ −∞
c−1J01→2(r, t)d 3r (28.12)
is time independent. As suggested by our notation,
〈F2|F1〉 = K c2
[( ∂
∂t F†2
) F1 − F†2
∂
∂t F1
] d3r (28.13)
is the relativistic definition of the inner product of F1 and F2 up to a constant, K. Below, this constant will be chosen in such a manner that the one-photon states are normalized, i.e., 〈F|F〉 = 〈Φ|Φ〉 = 1. I here remind the reader of the treatment given in Secs. 15.4-15.6.
