5.6 Comparison with the Literature on the Collision Term
107
one rather relates ∂ λ
!
= ∂ T . Evaluating f ss (x,
p) on a specific geodesic we then get
∂ λ f ss (γ(λ),
k(λ))
!
= V
−1 Tr
∂ρ(γ)
∂T
N
p,s s
= −i V
−1 Tr
[H int. , ρ(γ)] N
p,s s
,
(5.6.1)
and, since the above equation must hold for all geodesic paths, one finally gets
L f ss (x,
p) = −i E p,s V
−1 Tr
[H int. , ρ] N
p,s s
≡ −i E p,s V
−1
N
p,s s , H int.
ρ .
(5.6.2)
We have to be careful, however, because ∂ λ is LLT-invariant, but ∂ T is not, so the
above equation apparently breaks that symmetry. In the finite derivative case (5.5.1)
this was not a problem, because the finite difference from T = −∞ to T = ∞ leads
to the S matrix, which is Lorentz invariant. To correct the situation in the present case,
we note that in the above construction ∂ λ leads to the Liouville operator L, which has
dimensions of mass squared, and that one can find an analogous Lorentz-invariant
generalization of ∂ T , namely
∂ T → p
a ∂
∂ X a .
(5.6.3)
With this, the combination E p,s H int. in (5.6.2) would generalize to − p
a
s P
int.
a , where
P
a
int. := (H int. ,
P). But, since this operator enters through a commutator with ρ, and
the later is homogeneous (5.3.17), the result would be again (5.6.2). This equation
is therefore LLT-invariant indeed, although not explicitly.
Thus, with the present prescription, instead of finding the full S matrix on the
right-hand side as in (5.5.2), i.e. a full scattering event from T = −∞ to T = ∞,
one gets the interaction Hamiltonian that generates an infinitesimal increment in
time. Although to lowest order in the interactions both approaches lead to the same
collision term, we believe that the one we chose (5.5.1) is more consistent with
the assumptions behind kinetic theory (dilute gas and molecular chaos) and behind
the applicability of perturbative QFT. Indeed, for the use of QFT amplitudes we
need the existence of asymptotic states, meaning that the particles in our gas must
be mostly free, up to sporadic interactions, i.e. the gas must be dilute. This is also
necessary for the applicability of the molecular chaos assumption, i.e. one needs
to have clearly separated “in” (T → −∞) and “out” (T → ∞) asymptotic regions
in order to justify the fact that ρ in := lim T →−∞ ρ(T ) is completely determined by
f
in
ss (
p), whereas ρ out := lim T →∞ ρ(T ) requires a full tower of correlation functions
f
out
s 1 ...s n
(
p 1 , . . . ,
p n−1 ). One can then follow the evolution of f ss := f
in
ss , express f
out
ss
as the fully scattered f ss information, and compute the difference, as we did in
Eq. (5.5.1). In contrast, in the usual approach where ∂ λ
!
= ∂ T , one identifies the
microscopic dynamics with the macroscopic ones, so that the particles are treated as
being in a continuous state of interaction. There are therefore no clear “in” and “out”
phases for the use of asymptotic QFT states to be justified and for the molecular chaos
hypothesis to be implemented unambiguously. Finally, from the purely mathematical
viewpoint, as already argued in Sect. 5.1, the time variable T of the microscopic QFT,
107
one rather relates ∂ λ
!
= ∂ T . Evaluating f ss (x,
p) on a specific geodesic we then get
∂ λ f ss (γ(λ),
k(λ))
!
= V
−1 Tr
∂ρ(γ)
∂T
N
p,s s
= −i V
−1 Tr
[H int. , ρ(γ)] N
p,s s
,
(5.6.1)
and, since the above equation must hold for all geodesic paths, one finally gets
L f ss (x,
p) = −i E p,s V
−1 Tr
[H int. , ρ] N
p,s s
≡ −i E p,s V
−1
N
p,s s , H int.
ρ .
(5.6.2)
We have to be careful, however, because ∂ λ is LLT-invariant, but ∂ T is not, so the
above equation apparently breaks that symmetry. In the finite derivative case (5.5.1)
this was not a problem, because the finite difference from T = −∞ to T = ∞ leads
to the S matrix, which is Lorentz invariant. To correct the situation in the present case,
we note that in the above construction ∂ λ leads to the Liouville operator L, which has
dimensions of mass squared, and that one can find an analogous Lorentz-invariant
generalization of ∂ T , namely
∂ T → p
a ∂
∂ X a .
(5.6.3)
With this, the combination E p,s H int. in (5.6.2) would generalize to − p
a
s P
int.
a , where
P
a
int. := (H int. ,
P). But, since this operator enters through a commutator with ρ, and
the later is homogeneous (5.3.17), the result would be again (5.6.2). This equation
is therefore LLT-invariant indeed, although not explicitly.
Thus, with the present prescription, instead of finding the full S matrix on the
right-hand side as in (5.5.2), i.e. a full scattering event from T = −∞ to T = ∞,
one gets the interaction Hamiltonian that generates an infinitesimal increment in
time. Although to lowest order in the interactions both approaches lead to the same
collision term, we believe that the one we chose (5.5.1) is more consistent with
the assumptions behind kinetic theory (dilute gas and molecular chaos) and behind
the applicability of perturbative QFT. Indeed, for the use of QFT amplitudes we
need the existence of asymptotic states, meaning that the particles in our gas must
be mostly free, up to sporadic interactions, i.e. the gas must be dilute. This is also
necessary for the applicability of the molecular chaos assumption, i.e. one needs
to have clearly separated “in” (T → −∞) and “out” (T → ∞) asymptotic regions
in order to justify the fact that ρ in := lim T →−∞ ρ(T ) is completely determined by
f
in
ss (
p), whereas ρ out := lim T →∞ ρ(T ) requires a full tower of correlation functions
f
out
s 1 ...s n
(
p 1 , . . . ,
p n−1 ). One can then follow the evolution of f ss := f
in
ss , express f
out
ss
as the fully scattered f ss information, and compute the difference, as we did in
Eq. (5.5.1). In contrast, in the usual approach where ∂ λ
!
= ∂ T , one identifies the
microscopic dynamics with the macroscopic ones, so that the particles are treated as
being in a continuous state of interaction. There are therefore no clear “in” and “out”
phases for the use of asymptotic QFT states to be justified and for the molecular chaos
hypothesis to be implemented unambiguously. Finally, from the purely mathematical
viewpoint, as already argued in Sect. 5.1, the time variable T of the microscopic QFT,