42
3 Mathematical Framework
3.4 Phase Space Fields
In this section we consider the dynamics of phase space distributions for free pointparticles. Starting from the cotangent bundle T
∗
M of the space-time manifold M,
we build the corresponding Lorentz bundle LM using the tetrad field and then the
on-shell phase space PM. We pay special attention to the symmetries associated
with each case. We then derive the Liouville equation for the distributions on PM,
generalize to the case of tensor/spinor valued ones and also define their moments
and corresponding conservation equations.
3.4.1 Geometrical Considerations
We start by considering the off-shell phase space of 1-particle dynamics (i.e. without
imposing any mass condition) that is the cotangent bundle manifold T
∗
M. This is an
8-dimensional manifold conventionally parametrized by the pair x
μ and P μ . Incidentally, this is the space in which the canonical line fields (γ
μ
(λ), K μ (λ)) introduced
in Sect. 3.3.2 take their values. As a manifold on its own right, one can consider the
corresponding group of diffeomorphisms T
∗
MD, whose passive version relates all
possible coordinate systems
˜
x
μ
= ˜
x
μ
(x, P) ,
˜
P μ = ˜
P μ (x, P) .
(3.4.1)
However, T
∗
M is not any manifold, but inherits its structure from M, so this constrains the set of admissible coordinate systems and thus the diffeomorphisms that
relate them. Indeed, the coordinate systems on T
∗
M (its “atlas”) are the ones related
only by the subgroup of the transformations (3.4.1) that corresponds to the following
representation of PMDs
˜
x
μ
= ˜
x
μ
(x) ,
˜
P μ =
∂x
ν
∂ ˜
x μ ( ˜
x(x)) P ν ,
(3.4.2)
and is therefore only 4-dimensional instead of the 8-dimensional T
∗
MD. For the
active version, i.e. acting on fields on T
∗
M, the corresponding 8-dimensional generating vector field ≡
ξ
μ
, π μ [ξ]
satisfies
ξ
μ
= ξ
μ
(x) ,
π μ = −∂ μ ξ
ν
(x) P ν ,
(3.4.3)
and we have again the corresponding Lie derivative that generates the transformation.
For instance, a scalar field f ∗ (x, P) on T
∗
M transforms as
δ ξ f ∗ = −L f ∗ + O(ξ
2
) = −ξ
μ
∂ μ f ∗ + ∂ μ ξ
ν P ν
∂ f ∗
∂ P μ
+ O(ξ
2
) .
(3.4.4)
3 Mathematical Framework
3.4 Phase Space Fields
In this section we consider the dynamics of phase space distributions for free pointparticles. Starting from the cotangent bundle T
∗
M of the space-time manifold M,
we build the corresponding Lorentz bundle LM using the tetrad field and then the
on-shell phase space PM. We pay special attention to the symmetries associated
with each case. We then derive the Liouville equation for the distributions on PM,
generalize to the case of tensor/spinor valued ones and also define their moments
and corresponding conservation equations.
3.4.1 Geometrical Considerations
We start by considering the off-shell phase space of 1-particle dynamics (i.e. without
imposing any mass condition) that is the cotangent bundle manifold T
∗
M. This is an
8-dimensional manifold conventionally parametrized by the pair x
μ and P μ . Incidentally, this is the space in which the canonical line fields (γ
μ
(λ), K μ (λ)) introduced
in Sect. 3.3.2 take their values. As a manifold on its own right, one can consider the
corresponding group of diffeomorphisms T
∗
MD, whose passive version relates all
possible coordinate systems
˜
x
μ
= ˜
x
μ
(x, P) ,
˜
P μ = ˜
P μ (x, P) .
(3.4.1)
However, T
∗
M is not any manifold, but inherits its structure from M, so this constrains the set of admissible coordinate systems and thus the diffeomorphisms that
relate them. Indeed, the coordinate systems on T
∗
M (its “atlas”) are the ones related
only by the subgroup of the transformations (3.4.1) that corresponds to the following
representation of PMDs
˜
x
μ
= ˜
x
μ
(x) ,
˜
P μ =
∂x
ν
∂ ˜
x μ ( ˜
x(x)) P ν ,
(3.4.2)
and is therefore only 4-dimensional instead of the 8-dimensional T
∗
MD. For the
active version, i.e. acting on fields on T
∗
M, the corresponding 8-dimensional generating vector field ≡
ξ
μ
, π μ [ξ]
satisfies
ξ
μ
= ξ
μ
(x) ,
π μ = −∂ μ ξ
ν
(x) P ν ,
(3.4.3)
and we have again the corresponding Lie derivative that generates the transformation.
For instance, a scalar field f ∗ (x, P) on T
∗
M transforms as
δ ξ f ∗ = −L f ∗ + O(ξ
2
) = −ξ
μ
∂ μ f ∗ + ∂ μ ξ
ν P ν
∂ f ∗
∂ P μ
+ O(ξ
2
) .
(3.4.4)