3.3 World-Line Fields
41
is invariant under LDs and therefore qualifies as a “physical” derivative (it is the
analogue of ∂ a := e
μ
a ∂ μ on M). Moreover, it is convenient to define the U(1)GTinvariant 4-momentum in the tetrad basis
k a := e
μ
a (γ)
K μ + q A μ (γ)
.
(3.3.19)
Just as its diffeomorphism-indexed counterpart, the k
a are four LD scalars, but they
transform as a vector under LLTs
˜
k
a
=
a
b (γ) k
b
,
(3.3.20)
and are invariant under MDs.
9 We can now express the equations of motion of
the action (3.3.12) in a neat way. First, the variation with respect to k μ yields the
velocity/momentum relation (3.3.11), now reading
∂ γ
μ
= e
μ
a (γ) k
a
.
(3.3.22)
Next, the variation with respect to γ
μ leads to
∇ k
a
= q F
a
b (γ) k
b
,
(3.3.23)
where we have defined the covariant derivative ∇ with respect to LLTs on L
∇ X
a
:= ∂ X
a
+
a
bc (γ) k
c X
b
,
(3.3.24)
which therefore commutes with e
a
μ (γ). Equations (3.3.22) and (3.3.23) with q = 0
are nothing but the geodesic equation, in first-order form, and the ∼ q term is nothing
but the Lorentz force. Finally, varying the action (3.3.12) with respect to on finds
the mass-shell condition for a free point-particle
m
2
+ k a k
a
= 0 .
(3.3.25)
9 As a concrete example in the AMD case, we first note that
δ ξ e
a
μ = −L ξ e
a
μ + O(ξ
2 ) ,
δ ξ = 0 ,
(3.3.21)
so, dropping O(ξ 2 ) terms,
δ ξ k
a = δ ξ
e
a
μ (γ) ∂ γ
μ
= δ ξ
e
a
μ (γ)
∂ γ
μ + e
a
μ (γ) ∂ δ ξ γ
μ
=
δ ξ e
a
μ
(γ) +
∂ ν e
a
μ
(γ) δ ξ γ
ν
∂ γ
μ + e
a
μ (γ) ∂ δ ξ γ
μ = 0 .
.