86
5 General-Relativistic Matrix Kinetic Theory
number of particles, so that we can reach statistical homogeneity. We are therefore
treating the cosmic fluids as a collection of infinitesimal homogeneous thermodynamic systems, one at each x
μ , in which scattering events take place. In this refined
description, the microscopic spaces at each x
μ will be renamed “mesoscopic”, since
each one of them hosts a full thermodynamic system, as opposed to the individual
particle interactions that occur at the truly “microscopic” scale L coll .
5.2 Quantum Superposition and Matrix Distributions
In the cases of physical interest, on top of 3-momenta
p, there are extra discrete labels
s k determining the 1-particle states | |
p, {s k }} in the QFT of the microscopic spacetime. These distinguish among different particle species, spin, polarization, flavor,
etc. Also, since this is the particle content of an interacting theory, it contains bound
states of the fundamental particles, as well as unstable particles whose life-time is
long enough ( L coll ) to be considered as part of the spectrum. Here we choose to
store all of these indices {s k } inside a single index s for notational simplicity, so the
1-particle states of the QFT read | |
p, s. For instance, if we only consider photons
and electrons, then we have that s takes four values. For example, “1” could denote
a photon state with positive (circular) polarization, “2” a photon state with negative
polarization, “3” an electron with spin “up” and “4” an electron with spin “down”.
Now one has to take into account the fact that quantum states can be superposed,
meaning that one cannot simply generalize f (x,
p) → f s (x,
p), as we did in Sect. 3.4
for the species indexation for instance. Indeed, this privileges some basis | |
p, s in the
underlying Hilbert space, thus neglecting all the possible state superpositions of the
form
s α s | |
p, s. If we chose to work with a different polarization basis | |
p, s →
U ss | |
p, s
, where U ss is a unitary matrix, then the corresponding distribution f s (x,
p)
would either not be real or would no longer be an array but a matrix.
As we will see in Sect. 5.3, the solution is to consider a hermitian matrix in the
discrete index f ss (x,
p) that arises naturally in the quantum context and serves as a
two-point correlation function, thus capturing the information of superposed states.
In particular, under a change of basis, one would now get (in matrix notation)
˜ f = U f U
†
,
(5.2.1)
which therefore remains consistently hermitian. Note that the s parametrization we
use here is non-redundant, i.e. each s value corresponds to a physical state, and is
obtained after decomposing the microscopic quantum fields φ ... (X ) in some basis of
wave-functions, to be discussed in Sect. 5.8. In particular, f ss is a set of scalars both
under MDs and LLTs.
One must also pay attention to the fact that some superpositions are forbidden by
(super-) selection rules [4]. For instance, one cannot have a superposition of bosonic
and fermionic 1-particle states, meaning
5 General-Relativistic Matrix Kinetic Theory
number of particles, so that we can reach statistical homogeneity. We are therefore
treating the cosmic fluids as a collection of infinitesimal homogeneous thermodynamic systems, one at each x
μ , in which scattering events take place. In this refined
description, the microscopic spaces at each x
μ will be renamed “mesoscopic”, since
each one of them hosts a full thermodynamic system, as opposed to the individual
particle interactions that occur at the truly “microscopic” scale L coll .
5.2 Quantum Superposition and Matrix Distributions
In the cases of physical interest, on top of 3-momenta
p, there are extra discrete labels
s k determining the 1-particle states | |
p, {s k }} in the QFT of the microscopic spacetime. These distinguish among different particle species, spin, polarization, flavor,
etc. Also, since this is the particle content of an interacting theory, it contains bound
states of the fundamental particles, as well as unstable particles whose life-time is
long enough ( L coll ) to be considered as part of the spectrum. Here we choose to
store all of these indices {s k } inside a single index s for notational simplicity, so the
1-particle states of the QFT read | |
p, s. For instance, if we only consider photons
and electrons, then we have that s takes four values. For example, “1” could denote
a photon state with positive (circular) polarization, “2” a photon state with negative
polarization, “3” an electron with spin “up” and “4” an electron with spin “down”.
Now one has to take into account the fact that quantum states can be superposed,
meaning that one cannot simply generalize f (x,
p) → f s (x,
p), as we did in Sect. 3.4
for the species indexation for instance. Indeed, this privileges some basis | |
p, s in the
underlying Hilbert space, thus neglecting all the possible state superpositions of the
form
s α s | |
p, s. If we chose to work with a different polarization basis | |
p, s →
U ss | |
p, s
, where U ss is a unitary matrix, then the corresponding distribution f s (x,
p)
would either not be real or would no longer be an array but a matrix.
As we will see in Sect. 5.3, the solution is to consider a hermitian matrix in the
discrete index f ss (x,
p) that arises naturally in the quantum context and serves as a
two-point correlation function, thus capturing the information of superposed states.
In particular, under a change of basis, one would now get (in matrix notation)
˜ f = U f U
†
,
(5.2.1)
which therefore remains consistently hermitian. Note that the s parametrization we
use here is non-redundant, i.e. each s value corresponds to a physical state, and is
obtained after decomposing the microscopic quantum fields φ ... (X ) in some basis of
wave-functions, to be discussed in Sect. 5.8. In particular, f ss is a set of scalars both
under MDs and LLTs.
One must also pay attention to the fact that some superpositions are forbidden by
(super-) selection rules [4]. For instance, one cannot have a superposition of bosonic
and fermionic 1-particle states, meaning