2.2 Advantages of the Tetrad Formulation of Differential Geometry
17
• A natural framework for kinetic theory with QFT amplitudes.
In kinetic theory, the QFT scattering amplitudes in momentum space A( p 1 , . . .
p n → q 1 , . . . q m ) carry the microscopic physics information of the collision terms for
phase space particle distributions f (x, p). These functions A are computed through
QFT on Minkowski space-time, so they are made of contractions of the momenta
with η. In the metric formalism of GR, where the Lorentz group no longer appears and
the momenta are diffeomorphism vectors p
μ , one can make the amplitude generallycovariant by simply replacing η → g(x). The (on-shell) momentum integration too
can be expressed in a covariant way by considering a 4-dimensional integral with
measure δ(g μν (x) p
μ p
ν
+ m
2
) d
4 p/
√ −g(x), which can then be reduced to a 3dimensional one at the price of breaking manifest covariance. In most of the literature,
which employs the metric formalism, one either works with the above prescription
(see for instance [1–6]), or one invokes a specific tetrad field as an intermediate step
to perform the computation of the collision term, but then turns back to the metric
description as soon as possible (see for instance [7–12]). In both cases the final result
is the same: the collision term depends on the gravitational fields.
Here we want to highlight that, in the tetrad formalism, and in particular in its
present use where we switch to Lorentz indices as soon as possible, the collision term
is much simpler. Indeed, by working with the Lorentz-indexed momenta p
a the QFT
amplitudes can be used as they are, since we are contracting the momenta with η
in the general-relativistic case too. In particular, the momenta all transform with the
same Lorentz matrix p
a
→
a
b (x) p
b , so the A functions are invariant, since this is a
symmetry of the S-matrix. The integration measure is simpler too, it is the Lorentzinvariant combination d
3 p/
m 2
s + p 2 which also appears in QFT. Therefore, the
collision term is much simpler, as it is completely independent of the gravitational
field e
a
μ (x), and its only dependence on x
μ comes from the distributions f s (x, p) and
their associated wave-functions in the matrix case. Thus, by working with Lorentzindexed quantities one not only considers directly the relevant parametrization for
observables, but one also obtains simpler collision integrals to compute in kinetic
theory.
2.3 A Word on the Other Common Interpretation of ˆ
e a
Finally, let us make contact with another, quite widespread interpretation of a tetrad,
when it is invoked only at a specific point, say ˆ
P. We note that (2.1.1), or its coframe
analogue
ˆ
g μν = η ab ˆ
e
a
μ ˆ
e
b
ν ,
(2.3.1)
take the form of a coordinate transformation from some arbitrary coordinate system
x
μ , with metric components g μν , to some other system x
a
ˆ
P
in which g ab is Minkowski
at ˆ
P, i.e. the system of an observer at ˆ
P. The tetrad matrix then appears as the Jacobian
of that coordinate transformation x
a
ˆ
P
(x) evaluated at ˆ
P
17
• A natural framework for kinetic theory with QFT amplitudes.
In kinetic theory, the QFT scattering amplitudes in momentum space A( p 1 , . . .
p n → q 1 , . . . q m ) carry the microscopic physics information of the collision terms for
phase space particle distributions f (x, p). These functions A are computed through
QFT on Minkowski space-time, so they are made of contractions of the momenta
with η. In the metric formalism of GR, where the Lorentz group no longer appears and
the momenta are diffeomorphism vectors p
μ , one can make the amplitude generallycovariant by simply replacing η → g(x). The (on-shell) momentum integration too
can be expressed in a covariant way by considering a 4-dimensional integral with
measure δ(g μν (x) p
μ p
ν
+ m
2
) d
4 p/
√ −g(x), which can then be reduced to a 3dimensional one at the price of breaking manifest covariance. In most of the literature,
which employs the metric formalism, one either works with the above prescription
(see for instance [1–6]), or one invokes a specific tetrad field as an intermediate step
to perform the computation of the collision term, but then turns back to the metric
description as soon as possible (see for instance [7–12]). In both cases the final result
is the same: the collision term depends on the gravitational fields.
Here we want to highlight that, in the tetrad formalism, and in particular in its
present use where we switch to Lorentz indices as soon as possible, the collision term
is much simpler. Indeed, by working with the Lorentz-indexed momenta p
a the QFT
amplitudes can be used as they are, since we are contracting the momenta with η
in the general-relativistic case too. In particular, the momenta all transform with the
same Lorentz matrix p
a
→
a
b (x) p
b , so the A functions are invariant, since this is a
symmetry of the S-matrix. The integration measure is simpler too, it is the Lorentzinvariant combination d
3 p/
m 2
s + p 2 which also appears in QFT. Therefore, the
collision term is much simpler, as it is completely independent of the gravitational
field e
a
μ (x), and its only dependence on x
μ comes from the distributions f s (x, p) and
their associated wave-functions in the matrix case. Thus, by working with Lorentzindexed quantities one not only considers directly the relevant parametrization for
observables, but one also obtains simpler collision integrals to compute in kinetic
theory.
2.3 A Word on the Other Common Interpretation of ˆ
e a
Finally, let us make contact with another, quite widespread interpretation of a tetrad,
when it is invoked only at a specific point, say ˆ
P. We note that (2.1.1), or its coframe
analogue
ˆ
g μν = η ab ˆ
e
a
μ ˆ
e
b
ν ,
(2.3.1)
take the form of a coordinate transformation from some arbitrary coordinate system
x
μ , with metric components g μν , to some other system x
a
ˆ
P
in which g ab is Minkowski
at ˆ
P, i.e. the system of an observer at ˆ
P. The tetrad matrix then appears as the Jacobian
of that coordinate transformation x
a
ˆ
P
(x) evaluated at ˆ
P