5.1 Collisions and the Microscopic Space-Time
85
The ± distinguishes between bosons (+) and fermions (−) and these O( f
3
) terms
implement the “Bose enhancement” and “Pauli blocking” effects. The fact that
C s (x,
p) is independent of ∂
n>0
μ
f s (x,
p) reflects the separation of scales, i.e. that
f s (x,
p) typically varies over macroscopic space-time scales, whereas the collision
term only captures the microscopic ones. Indeed, when taking the dilute limit from
non-equilibrium QFT [1–3], the collision term of the BUU equation appears as the
zeroth order result in a derivative expansion probing the inhomogeneities in x
μ .
Let us now discuss some important structural aspects of Eq. (5.1.2). First, from the
mathematical viewpoint, the fact that we are considering a full scattering process at
every x
μ means that the “space-time” in which the QFT is defined is not M. Rather,
by definition, it is the space-time that is Fourier dual to the p
a coordinates, i.e. the
one which is parametrized by the X
a coordinates defined at the end of Sect. 3.4.1
and appearing in Eq. (3.4.23) in particular. The fact that the p
a data coordinatize the
tangent spaces means that the X
a can be thought of as parametrizing an infinitesimal
space-time in the tetrad basis of the tangent space around each x
μ . In the usual
derivations of the Boltzmann equation, there is a single space-time coordinate x
μ
and the separation between the macroscopic and microscopic scales is performed by
simplifying/neglecting terms depending on their behaviour with respect to x
μ .
The present mathematical framework provides a radically different structure for
implementing this idea. Instead of separating scales with respect to a single spacetime coordinate x
μ , we literally have two such coordinates that already represent
“macroscopic” and “microscopic” spaces-times. The former is the x
μ coordinate on
M, the “macroscopic” space-time capturing the variation of the distributions. The
latter is the X
a coordinate of the “microscopic” space-time, the one on which we
compute the scattering matrix S through the Fourier dual parameters p
a . Note that this
is a Minkowski space-time, because the X
a only mix under Lorentz transformations
(3.4.24). The x
μ dependence of the Lorentz matrix
a
b (x) now reflects the fact
that one can choose different frames at each point x
μ , i.e. different observers for
each scattering event. From the viewpoint of X
a , however, these are global Lorentz
transformations, i.e. the usual symmetry of QFT.
Thus, an important property of this construction is that it allows one to match
the symmetries of GR to the ones of QFT on flat space-time, without compromising
the former. One can therefore directly plug the QFT amplitudes in the collision
term without performing any kind of approximation. The only approximation here is
the extreme separation between macro and micro scales and the classical treatment
of gravity, since the latter is by construction a “macro” entity. The disadvantage
of having this separation of scales “hardwired” into the mathematical structure is
that we do not have access to effects of intermediate scale, as one could recover
perturbatively in the usual approach. However, in the case of cosmology this is not
really a problem, as the separation between “macro” and “micro” scales is huge. On
the other hand, the advantages of this structure are important, especially for deriving
the desired generalization of (5.1.2), as we will soon discuss.
Finally, observe that C s depends on the momenta
p, not X
a , meaning that the
involved (statistical) states in the microscopic QFT are invariant under translations
X
a
→ X
a
+ c
a . Thus, every microscopic space must host itself a dilute, but very large
85
The ± distinguishes between bosons (+) and fermions (−) and these O( f
3
) terms
implement the “Bose enhancement” and “Pauli blocking” effects. The fact that
C s (x,
p) is independent of ∂
n>0
μ
f s (x,
p) reflects the separation of scales, i.e. that
f s (x,
p) typically varies over macroscopic space-time scales, whereas the collision
term only captures the microscopic ones. Indeed, when taking the dilute limit from
non-equilibrium QFT [1–3], the collision term of the BUU equation appears as the
zeroth order result in a derivative expansion probing the inhomogeneities in x
μ .
Let us now discuss some important structural aspects of Eq. (5.1.2). First, from the
mathematical viewpoint, the fact that we are considering a full scattering process at
every x
μ means that the “space-time” in which the QFT is defined is not M. Rather,
by definition, it is the space-time that is Fourier dual to the p
a coordinates, i.e. the
one which is parametrized by the X
a coordinates defined at the end of Sect. 3.4.1
and appearing in Eq. (3.4.23) in particular. The fact that the p
a data coordinatize the
tangent spaces means that the X
a can be thought of as parametrizing an infinitesimal
space-time in the tetrad basis of the tangent space around each x
μ . In the usual
derivations of the Boltzmann equation, there is a single space-time coordinate x
μ
and the separation between the macroscopic and microscopic scales is performed by
simplifying/neglecting terms depending on their behaviour with respect to x
μ .
The present mathematical framework provides a radically different structure for
implementing this idea. Instead of separating scales with respect to a single spacetime coordinate x
μ , we literally have two such coordinates that already represent
“macroscopic” and “microscopic” spaces-times. The former is the x
μ coordinate on
M, the “macroscopic” space-time capturing the variation of the distributions. The
latter is the X
a coordinate of the “microscopic” space-time, the one on which we
compute the scattering matrix S through the Fourier dual parameters p
a . Note that this
is a Minkowski space-time, because the X
a only mix under Lorentz transformations
(3.4.24). The x
μ dependence of the Lorentz matrix
a
b (x) now reflects the fact
that one can choose different frames at each point x
μ , i.e. different observers for
each scattering event. From the viewpoint of X
a , however, these are global Lorentz
transformations, i.e. the usual symmetry of QFT.
Thus, an important property of this construction is that it allows one to match
the symmetries of GR to the ones of QFT on flat space-time, without compromising
the former. One can therefore directly plug the QFT amplitudes in the collision
term without performing any kind of approximation. The only approximation here is
the extreme separation between macro and micro scales and the classical treatment
of gravity, since the latter is by construction a “macro” entity. The disadvantage
of having this separation of scales “hardwired” into the mathematical structure is
that we do not have access to effects of intermediate scale, as one could recover
perturbatively in the usual approach. However, in the case of cosmology this is not
really a problem, as the separation between “macro” and “micro” scales is huge. On
the other hand, the advantages of this structure are important, especially for deriving
the desired generalization of (5.1.2), as we will soon discuss.
Finally, observe that C s depends on the momenta
p, not X
a , meaning that the
involved (statistical) states in the microscopic QFT are invariant under translations
X
a
→ X
a
+ c
a . Thus, every microscopic space must host itself a dilute, but very large