5.4 Expressing ρ and the Entropy Current in Terms of f ss
99
Defining
f ◦ (x,
p) := 1 + 1 ◦ f (x,
p) ,
(5.4.27)
we the density matrix (5.4.2) becomes
ρ(x) = Z
−1
(x) exp
d
3 p
(2π) 3
log
f (x,
p)
f ◦ (x,
p)
ss
N
p,ss
,
(5.4.28)
and
Z (x) = exp
V
d
3 p
(2π) 3 Tr
1 ◦ log f ◦ (x,
p)
,
(5.4.29)
while the entropy density (5.4.8) takes the form
s(x) =
d
3 p
(2π) 3 Tr
− f log f + 1 ◦ f ◦ log f ◦
(x,
p) .
(5.4.30)
We can also eliminate the singular V factor that appears in (5.4.28) through Z (x).
We first write
ρ(x) = exp
d
3 p
(2π) 3
log
f (x,
p)
f ◦ (x,
p)
ss
N
p,ss −
1 ◦ log f ◦ (x,
p)
ss V δ ss I
,
(5.4.31)
then use
V δ ss I ≡
a
p,s , a
†
p,s
|s||s |
≡ a
p,s a
†
p,s − (−1)
|s||s
| a
†
p,s a
p,s ,
(5.4.32)
and thus find
ρ(x) = exp
d
3 p
(2π) 3
log f (x,
p)
ss a
†
p,s a
p,s −
1 ◦ log f ◦ (x,
p)
ss a
p,s a
†
p,s
.
(5.4.33)
Note that we have used log A + log B = log( AB), which does not hold for generic
matrices, but here does because both can be simultaneously diagonalized.
Finally, from Eq. (5.4.30) we note that s(x) is not a Lorentz scalar, because the
measure d
3 p is not invariant. This fact could have also been inferred already from the
lonesome volume factor in Eq. (5.4.1). Rather, it is the time-component of a Lorentz
vector, the entropy current
s
a (x) :=
s
d 3 p
(2π) 3 E p,s
p
a
s
− f log f + 1 ◦ f ◦ log f ◦
ss
(x,
p) ,
s
0 (x) ≡ s(x) ,
(5.4.34)