3.4 Phase Space Fields
43
Another special case of PT
∗
MDs are the ones which correspond to the U(1)GTs
and which we deduce from Eq. (3.3.13)
˜
x
μ
= x
μ
,
˜
P μ = P μ − q∂ μ θ(x) .
(3.4.5)
Now considering T
∗
M as the (off-shell) covariant phase space of a particle endows
it with more structure, because one has access to a larger subgroup of T
∗
MDs, the
“canonical transformations”. These can be defined as the T
∗
MDs that preserve the
canonical 1-form on T
∗
M
C := P μ dx
μ
,
(3.4.6)
up to a total derivative. One can then check that C is invariant under PMDs and
varies by a total derivative under AMDs and U(1)GTs, so these symmetries are
particular cases of canonical transformations. To make contact with the canonical
action-based geodesic formalism discussed in Sect. 3.3.2, if we consider a definite
path in phase space
(x
μ
, P μ ) = (γ
μ
(λ), K μ (λ)) ,
(3.4.7)
then evaluating the canonical 1-form on that path (i.e. its pullback to L) gives the
world-line 1-form
C(γ, K ) = K μ dγ
μ
≡ K μ ∂ λ γ
μ dλ .
(3.4.8)
This is the combination that enters the canonical action (3.3.12), by definition, and
thus allows one to identify the rest as (minus) the “Hamiltonian”. The canonical transformations leave this combination invariant, up to a boundary term in the canonical
action (3.3.12), so the latter remains in canonical form. Another property of the
canonical transformations is that they preserve the canonical volume form on T
∗
M
vol ∗ :=
1
(2π) 4 d
4 x ∧ d
4 P ,
(3.4.9)
where
d
4 x :=
1
4!
ε μνρσ dx
μ ∧ dx
ν ∧ dx
ρ ∧ dx
σ ,
d
4 P :=
1
4!
ε
μνρσ d P μ ∧ d P ν ∧ d P ρ ∧ d P σ .
(3.4.10)
The (2π)
−4 normalization comes from the fact that the elementary phase space
volume is the Planck constant h ≡ 2π, for each space-time dimension, and we use
the = 1 normalization. It is then conventional, in particle physics and cosmology,
to associate this normalization with the momentum coordinates and this is reflected
in the definition of the Fourier transform.
Let us now switch to the tetrad formalism, where the momentum components of
interest are the ones in the tetrad basis, and in particular the U(1)GT-invariant ones
p
a
:= e
aμ
(x)
P μ + q A μ (x)
,
(3.4.11)
43
Another special case of PT
∗
MDs are the ones which correspond to the U(1)GTs
and which we deduce from Eq. (3.3.13)
˜
x
μ
= x
μ
,
˜
P μ = P μ − q∂ μ θ(x) .
(3.4.5)
Now considering T
∗
M as the (off-shell) covariant phase space of a particle endows
it with more structure, because one has access to a larger subgroup of T
∗
MDs, the
“canonical transformations”. These can be defined as the T
∗
MDs that preserve the
canonical 1-form on T
∗
M
C := P μ dx
μ
,
(3.4.6)
up to a total derivative. One can then check that C is invariant under PMDs and
varies by a total derivative under AMDs and U(1)GTs, so these symmetries are
particular cases of canonical transformations. To make contact with the canonical
action-based geodesic formalism discussed in Sect. 3.3.2, if we consider a definite
path in phase space
(x
μ
, P μ ) = (γ
μ
(λ), K μ (λ)) ,
(3.4.7)
then evaluating the canonical 1-form on that path (i.e. its pullback to L) gives the
world-line 1-form
C(γ, K ) = K μ dγ
μ
≡ K μ ∂ λ γ
μ dλ .
(3.4.8)
This is the combination that enters the canonical action (3.3.12), by definition, and
thus allows one to identify the rest as (minus) the “Hamiltonian”. The canonical transformations leave this combination invariant, up to a boundary term in the canonical
action (3.3.12), so the latter remains in canonical form. Another property of the
canonical transformations is that they preserve the canonical volume form on T
∗
M
vol ∗ :=
1
(2π) 4 d
4 x ∧ d
4 P ,
(3.4.9)
where
d
4 x :=
1
4!
ε μνρσ dx
μ ∧ dx
ν ∧ dx
ρ ∧ dx
σ ,
d
4 P :=
1
4!
ε
μνρσ d P μ ∧ d P ν ∧ d P ρ ∧ d P σ .
(3.4.10)
The (2π)
−4 normalization comes from the fact that the elementary phase space
volume is the Planck constant h ≡ 2π, for each space-time dimension, and we use
the = 1 normalization. It is then conventional, in particle physics and cosmology,
to associate this normalization with the momentum coordinates and this is reflected
in the definition of the Fourier transform.
Let us now switch to the tetrad formalism, where the momentum components of
interest are the ones in the tetrad basis, and in particular the U(1)GT-invariant ones
p
a
:= e
aμ
(x)
P μ + q A μ (x)
,
(3.4.11)