Chapter 5
General-Relativistic Matrix Kinetic
Theory
Abstract In the case of the cosmic microwave background, the observable information is contained in the phase space distribution of photons. The latter are particles
with internal degrees of freedom, i.e. polarization, and the corresponding quantum
states can be superposed, which leads to the consideration of matrix-valued distributions. In this chapter we derive in detail the corresponding general-relativistic
Boltzmann equation, including also the case of fermions, since the latter interact with
photons. Our equations are derived within the tetrad formalism, which allows one to
embed exactly and straightforwardly the quantum field theory amplitudes describing microscopic collisions in the macroscopically curved geometry. As a concrete
example, we provide the lowest-order collision term for a fluid of photons, electrons
and protons that includes the polarization/spin information. Although our construction follows standard practice for the most part, our understanding of the involved
assumptions and symmetries leads us to a different definition of the collision term
than the one found in the literature, yielding in particular different results beyond the
lowest-order particle interactions.
5.1 Collisions and the Microscopic Space-Time
In Sect. 3.4.2 we have derived the evolution equation for the distribution functions
f s (x,
p) for a gas of “free” particles, i.e. particles whose trajectories are solely
altered by the non-trivial space-time geometry and electromagnetic field. Let us now
consider the presence of interparticle forces, i.e. “collisions”. In this case, one must
distinguish between two important space/time scales, namely, the typical separation
between two successive collisions L free and the typical space/time extent of the
collision event itself L coll .
In the case where the gas is “dilute” enough L free L coll , the particles spend most
of their time in free motion, so one can describe the effect of collisions as merely
changing a given free state to some other free state. In particular, this means that
the degrees of freedom of the gas are the ones of a collection of free particles. This
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_5
83
General-Relativistic Matrix Kinetic
Theory
Abstract In the case of the cosmic microwave background, the observable information is contained in the phase space distribution of photons. The latter are particles
with internal degrees of freedom, i.e. polarization, and the corresponding quantum
states can be superposed, which leads to the consideration of matrix-valued distributions. In this chapter we derive in detail the corresponding general-relativistic
Boltzmann equation, including also the case of fermions, since the latter interact with
photons. Our equations are derived within the tetrad formalism, which allows one to
embed exactly and straightforwardly the quantum field theory amplitudes describing microscopic collisions in the macroscopically curved geometry. As a concrete
example, we provide the lowest-order collision term for a fluid of photons, electrons
and protons that includes the polarization/spin information. Although our construction follows standard practice for the most part, our understanding of the involved
assumptions and symmetries leads us to a different definition of the collision term
than the one found in the literature, yielding in particular different results beyond the
lowest-order particle interactions.
5.1 Collisions and the Microscopic Space-Time
In Sect. 3.4.2 we have derived the evolution equation for the distribution functions
f s (x,
p) for a gas of “free” particles, i.e. particles whose trajectories are solely
altered by the non-trivial space-time geometry and electromagnetic field. Let us now
consider the presence of interparticle forces, i.e. “collisions”. In this case, one must
distinguish between two important space/time scales, namely, the typical separation
between two successive collisions L free and the typical space/time extent of the
collision event itself L coll .
In the case where the gas is “dilute” enough L free L coll , the particles spend most
of their time in free motion, so one can describe the effect of collisions as merely
changing a given free state to some other free state. In particular, this means that
the degrees of freedom of the gas are the ones of a collection of free particles. This
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_5
83