7.7 Negative energy waves 225while the Hermitian part remains the same. There is now a new term involvingki.With the
incorporation of this new term, Eq.(7.75) becomes
〈
E·
(
J+ε 0∂E
∂t)〉
=
2 ε 0 ωi
4̃E∗·
[
∂
∂ωω←→
K(ω)]
ω=ωr·E ̃+
ε 0
4E ̃∗·
[
2 ωki·∂
∂k←→
K(ω,k)]
ω=ωr,
k=kr· ̃E
e−^2 ki·x+2ωit(7.85)
where we have explicitly written the exponential space-dependent factorexp(− 2 ki·x).
What is the meaning of this new term involvingki? The answer to this question may be
found by examining the Poyntingflux for the situation wherekiis finite. Using the product
rule to allow for finitekishows that
〈∇·P〉 = 〈∇·(E×H)〉=1
4
∇·
[(
E ̃∗×H ̃+E ̃×H ̃∗
)
e−^2 ki·x+2ωit]
=
1
4
(
− 2 ki·(
E ̃∗×H ̃+ ̃E×H ̃∗
))
e−^2 ki·x+2ωit. (7.86)Comparison of Eqs.(7.85) and (7.86) show that the factor− 2 ki·acts as a divergence and
so the second term in Eq.(7.85) represents anenergyflux. Since the Poynting vectorPis
the energyflux associated with the electromagnetic field, this additional energyflux must
be identified as the energyflux associated with particle motion due to the wave. Defining
thisflux asTit is seen that
Tj=−ωε 0
4E ̃∗·
[
∂
∂kj←→
K(ω,k)]
·E ̃ (7.87)
in the limitki→ 0 .For small but finiteki,ωithe generalized Poynting theorem can be
written as
− 2 ki·(P+T)+2ωi(wE+wB+ ̄wpart)=0. (7.88)
We now define the generalized group velocityvgto be the velocity with which wave
energy is transported. This velocity is the total energyflux divided by the total energy
density, i.e.,
vg=P+T
wE+wB+ ̄wpart. (7.89)
The bar in the particle energy represents the fact that this is the difference between the
particle energy with the wave and the particle energy without the wave andit should be
recalled that this difference can be negative.
7.7 Negative energy waves
A curious consequence of this analysis is that a wave can have anegativeenergy density.
While the field energy densitieswEandwBare positive definite, the particle energy den-
sityw ̄partcan have either sign and in certain circumstances can be sufficientlynegative
to make the total wave energy density negative. This surprising possibility can occur be-
cause, as was shown in Eq.(7.78), the wave energy density is actually thechangein the