38
3 Mathematical Framework
which therefore relates it to the space-time fields. This means that, if we pullback
the fields, then we must also move the geodesic in order to obtain the same physical
configuration and thus a symmetry of the geodesic equation. Therefore, by treating
γ as a “dynamical” object, i.e. one that has an action coupled to g μν , we force it to
transform as all other dynamical objects (the space-time fields) in order to maintain
a symmetry. The AMD transformation of a geodesic is therefore simply given by
the expression of the passive one in Eq. (3.3.4) in terms of the generating vector field
δ ξ γ
μ
= ξ
μ
(γ) + O(ξ
2
) .
(3.3.6)
If we do not transform γ
μ along with the fields, then we effectively obtain a configuration corresponding to a different path than the one we started with, so the geodesic
equation is not invariant. As a result, the PMDs and AMDs have basically no conceptual difference for a geodesic and thus also for the (scalar) space-time fields X
evaluated on it, i.e. we have respectively
˜
X ( ˜
γ) = X (γ) ,
δ ξ [X (γ)] = [δ ξ X ](γ) + δ ξ γ
μ
[∂ μ X ](γ) + O((δ ξ γ)
2
) = 0 .
(3.3.7)
For this reason, the “passive” and “active” prefixes will only refer to the way that
same transformation is usually expressed, i.e. in terms of a coordinate transformation,
or in terms of the generating vector field, respectively. This situation is ultimately
due to the fact that, for γ, the MD symmetries are internal symmetries, and there
is thus no distinction between passive and active versions, as it is for instance also
the case for LLTs. This is also why we only referred to PMDs in the transformation
rules of a cosmological observable in Sect. 2.2, because the AMDs are basically the
same transformation.
7 In contrast to the above remarks about MDs, the distinction
between PLDs and ALDs is important, because the solutions of observables will
involve integrals over L with non-trivial boundaries (see appendix 6.2).
3.3.2 Action for Point-Particles
We start by considering the following action
S =
dλ
1
2
−1 g μν (γ) ∂ λ γ
μ
∂ λ γ
ν
− q A μ (γ) ∂ λ γ
μ
−
1
2
m
2
,
(3.3.8)
where is the analogue of a tetrad covector on L, i.e. a “monad” (or “einbein”). Here
we will adopt the convention of λ having length dimensions, which means that
7 In fact, in the case of cosmological observables the δ ξ γ μ = 0 choice, i.e. not moving the geodesic
along, would lead to yet another problem. If the space-time fields move while the points are held
fixed, then it is not guaranteed that the resulting ˜
C(P, ˆ
P) will still be connecting two points that are
linked by a light-like geodesic. This means that only a subgroup of such active diffeomorphisms is
actually defined on a cosmological observable linking ˆ
P and P.
3 Mathematical Framework
which therefore relates it to the space-time fields. This means that, if we pullback
the fields, then we must also move the geodesic in order to obtain the same physical
configuration and thus a symmetry of the geodesic equation. Therefore, by treating
γ as a “dynamical” object, i.e. one that has an action coupled to g μν , we force it to
transform as all other dynamical objects (the space-time fields) in order to maintain
a symmetry. The AMD transformation of a geodesic is therefore simply given by
the expression of the passive one in Eq. (3.3.4) in terms of the generating vector field
δ ξ γ
μ
= ξ
μ
(γ) + O(ξ
2
) .
(3.3.6)
If we do not transform γ
μ along with the fields, then we effectively obtain a configuration corresponding to a different path than the one we started with, so the geodesic
equation is not invariant. As a result, the PMDs and AMDs have basically no conceptual difference for a geodesic and thus also for the (scalar) space-time fields X
evaluated on it, i.e. we have respectively
˜
X ( ˜
γ) = X (γ) ,
δ ξ [X (γ)] = [δ ξ X ](γ) + δ ξ γ
μ
[∂ μ X ](γ) + O((δ ξ γ)
2
) = 0 .
(3.3.7)
For this reason, the “passive” and “active” prefixes will only refer to the way that
same transformation is usually expressed, i.e. in terms of a coordinate transformation,
or in terms of the generating vector field, respectively. This situation is ultimately
due to the fact that, for γ, the MD symmetries are internal symmetries, and there
is thus no distinction between passive and active versions, as it is for instance also
the case for LLTs. This is also why we only referred to PMDs in the transformation
rules of a cosmological observable in Sect. 2.2, because the AMDs are basically the
same transformation.
7 In contrast to the above remarks about MDs, the distinction
between PLDs and ALDs is important, because the solutions of observables will
involve integrals over L with non-trivial boundaries (see appendix 6.2).
3.3.2 Action for Point-Particles
We start by considering the following action
S =
dλ
1
2
−1 g μν (γ) ∂ λ γ
μ
∂ λ γ
ν
− q A μ (γ) ∂ λ γ
μ
−
1
2
m
2
,
(3.3.8)
where is the analogue of a tetrad covector on L, i.e. a “monad” (or “einbein”). Here
we will adopt the convention of λ having length dimensions, which means that
7 In fact, in the case of cosmological observables the δ ξ γ μ = 0 choice, i.e. not moving the geodesic
along, would lead to yet another problem. If the space-time fields move while the points are held
fixed, then it is not guaranteed that the resulting ˜
C(P, ˆ
P) will still be connecting two points that are
linked by a light-like geodesic. This means that only a subgroup of such active diffeomorphisms is
actually defined on a cosmological observable linking ˆ
P and P.