5.7 Exact Conservation Equations of the Collisional Equation
109
For the spatial components a = i we obtain trivially zero because microscopic
time-evolution conserves momentum so [
P, S] = 0. For the time component a = 0
the situation is a bit less trivial, because H asy. is not the full Hamiltonian H and
[H, H asy. ] = 0. Nevertheless, H asy. coincides with H on asymptotic states by definition, so the conservation of H implies
S H asy. |in = S H|in ≡ H S|in ≡ H |out = H asy. |out = H asy. S|in , (5.7.7)
for all |in, meaning that [S, H asy. ] = 0 on the states of interest. Thus, H asy. is not
necessarily constant during evolution, but the initial and final values at T → ∓∞
are the same, which concludes our proof
∇ b T
ab
= 0 .
(5.7.8)
For the electric current we proceed similarly and arrive at
∇ a J
a
(x) ∼ ∼S
†
[Q, S]] ρ(x) = 0 ,
(5.7.9)
where Q is the total charge operator (5.3.13), which is also exactly conserved during
microscopic evolution and therefore commutes with the S matrix. As for the total
entropy current (5.4.34), again the same procedure leads to
∇ a s
a
(x) =
s
d
3 p
(2π) 3 E p,s
C log
f ◦
f
ss
(x,
p) .
(5.7.10)
If we only consider the 2 → 2 scattering term, and work in the tilded basis where ˜
f ˜
s ˜
s
is diagonal, we recover the set-up of the standard Boltzmann equation, in which case
the H -theorem ∇ a s
a
≥ 0 is proved in the usual way. In the general case, however, it
is not clear to us how to proceed, so we will not consider this issue further.
5.8 Liouville-Transported Wave-Functions and Tensor
Distributions
The s indexation of the 1-particles states | |
p, s we have considered so far parametrizes
exactly the degrees of freedom of the QFT, so it is not in a one-to-one correspondence
with the quantum field components in general. The typical example is the (mesoscopic) quantum electromagnetic field A a (X ), which has four components, while
there are only two physical photon states. Another example is the Dirac field ψ(X ),
which has eight real components, but corresponds to only four physical states, two
for the particle and two for the anti-particle. The relation between the field indices
and the s indices is given by the so-called “wave-functions”. As we will see, these
wave-functions are also required in order to express the BUU equations in terms of
109
For the spatial components a = i we obtain trivially zero because microscopic
time-evolution conserves momentum so [
P, S] = 0. For the time component a = 0
the situation is a bit less trivial, because H asy. is not the full Hamiltonian H and
[H, H asy. ] = 0. Nevertheless, H asy. coincides with H on asymptotic states by definition, so the conservation of H implies
S H asy. |in = S H|in ≡ H S|in ≡ H |out = H asy. |out = H asy. S|in , (5.7.7)
for all |in, meaning that [S, H asy. ] = 0 on the states of interest. Thus, H asy. is not
necessarily constant during evolution, but the initial and final values at T → ∓∞
are the same, which concludes our proof
∇ b T
ab
= 0 .
(5.7.8)
For the electric current we proceed similarly and arrive at
∇ a J
a
(x) ∼ ∼S
†
[Q, S]] ρ(x) = 0 ,
(5.7.9)
where Q is the total charge operator (5.3.13), which is also exactly conserved during
microscopic evolution and therefore commutes with the S matrix. As for the total
entropy current (5.4.34), again the same procedure leads to
∇ a s
a
(x) =
s
d
3 p
(2π) 3 E p,s
C log
f ◦
f
ss
(x,
p) .
(5.7.10)
If we only consider the 2 → 2 scattering term, and work in the tilded basis where ˜
f ˜
s ˜
s
is diagonal, we recover the set-up of the standard Boltzmann equation, in which case
the H -theorem ∇ a s
a
≥ 0 is proved in the usual way. In the general case, however, it
is not clear to us how to proceed, so we will not consider this issue further.
5.8 Liouville-Transported Wave-Functions and Tensor
Distributions
The s indexation of the 1-particles states | |
p, s we have considered so far parametrizes
exactly the degrees of freedom of the QFT, so it is not in a one-to-one correspondence
with the quantum field components in general. The typical example is the (mesoscopic) quantum electromagnetic field A a (X ), which has four components, while
there are only two physical photon states. Another example is the Dirac field ψ(X ),
which has eight real components, but corresponds to only four physical states, two
for the particle and two for the anti-particle. The relation between the field indices
and the s indices is given by the so-called “wave-functions”. As we will see, these
wave-functions are also required in order to express the BUU equations in terms of