44
3 Mathematical Framework
in analogy with Eq. (3.3.19). From the viewpoint of T
∗
M, the change of coordinates
x
μ
→ x
μ
,
P μ → p
a
(3.4.12)
corresponds to a passive T
∗
MD that is not a canonical transformation, so the volume
form transforms non-trivially to
vol ∗ → vol L =
1
(2π) 4 (e d
4 x) ∧
1
4!
ε abcd d p
a ∧ d p
b ∧ d p
c ∧ d p
d
,
e := det e
a
μ ≡
√ −g ,
(3.4.13)
where in the first round bracket we recognize the volume form on M.
10 By performing a non-canonical coordinate transformation, the corresponding space is technically
no longer T
∗
M, because we have changed the transformations that act on it and thus
its geometric structure. Indeed, now the PMDs are represented as
˜
x
μ
= ˜
x
μ
(x) ,
˜
p
a
= p
a
,
(3.4.14)
and we also have LLT coordinate transformations
˜
x
μ
= x
μ
,
˜
p
a
=
a
b (x) p
b
.
(3.4.15)
We have thus traded the cotangent bundle T
∗
M for a vector bundle based on M
with structure group SO(1, 3). We will refer to it as the “Lorentz” bundle and denote
it by LM. In order to get the action of the AMDs, we first transform our scalar to
the new coordinates
f L (x, p) := f ∗ (x, P( p)) ≡ f ∗ (x, e μa (x) p
a
− q A μ (x)) ,
(3.4.16)
and obtain
δ ξ f L = −ξ
μ
∂ μ f L + O(ξ
2
) .
(3.4.17)
Observe that, since now the LLTs are coordinate transformations on LM, i.e. a
passive transformation, they also have their active counterpart on fields over that
manifold, generated by the Lie derivative with respect to some vector field. We
proceed as in the MD case, i.e. we express (3.4.15) in terms of the generators
˜
x
μ
= x
μ
,
˜
p
a
=
a
b (x) p
b
= p
a
− θ
a
b (x) p
b
+ O(θ
2
) ,
(3.4.18)
and identify the generating vector field
=
0, θ
a
,
θ
a
:= −θ
a
b (x) p
b
.
(3.4.19)
Thus, writing down the coordinate transformation (3.4.15) for the scalar
10 This is obtained by noting that, whenever the exterior derivative d acts on either e a
μ (x) or A μ (x)
the corresponding terms vanish because they are proportional to the exterior product of five dx μ .
3 Mathematical Framework
in analogy with Eq. (3.3.19). From the viewpoint of T
∗
M, the change of coordinates
x
μ
→ x
μ
,
P μ → p
a
(3.4.12)
corresponds to a passive T
∗
MD that is not a canonical transformation, so the volume
form transforms non-trivially to
vol ∗ → vol L =
1
(2π) 4 (e d
4 x) ∧
1
4!
ε abcd d p
a ∧ d p
b ∧ d p
c ∧ d p
d
,
e := det e
a
μ ≡
√ −g ,
(3.4.13)
where in the first round bracket we recognize the volume form on M.
10 By performing a non-canonical coordinate transformation, the corresponding space is technically
no longer T
∗
M, because we have changed the transformations that act on it and thus
its geometric structure. Indeed, now the PMDs are represented as
˜
x
μ
= ˜
x
μ
(x) ,
˜
p
a
= p
a
,
(3.4.14)
and we also have LLT coordinate transformations
˜
x
μ
= x
μ
,
˜
p
a
=
a
b (x) p
b
.
(3.4.15)
We have thus traded the cotangent bundle T
∗
M for a vector bundle based on M
with structure group SO(1, 3). We will refer to it as the “Lorentz” bundle and denote
it by LM. In order to get the action of the AMDs, we first transform our scalar to
the new coordinates
f L (x, p) := f ∗ (x, P( p)) ≡ f ∗ (x, e μa (x) p
a
− q A μ (x)) ,
(3.4.16)
and obtain
δ ξ f L = −ξ
μ
∂ μ f L + O(ξ
2
) .
(3.4.17)
Observe that, since now the LLTs are coordinate transformations on LM, i.e. a
passive transformation, they also have their active counterpart on fields over that
manifold, generated by the Lie derivative with respect to some vector field. We
proceed as in the MD case, i.e. we express (3.4.15) in terms of the generators
˜
x
μ
= x
μ
,
˜
p
a
=
a
b (x) p
b
= p
a
− θ
a
b (x) p
b
+ O(θ
2
) ,
(3.4.18)
and identify the generating vector field
=
0, θ
a
,
θ
a
:= −θ
a
b (x) p
b
.
(3.4.19)
Thus, writing down the coordinate transformation (3.4.15) for the scalar
10 This is obtained by noting that, whenever the exterior derivative d acts on either e a
μ (x) or A μ (x)
the corresponding terms vanish because they are proportional to the exterior product of five dx μ .