Kiefer C. Quantum gravity

74 PARAMETRIZED AND RELATIONAL SYSTEMS
We introduce now a formal time parameter τ (‘label time’) and elevate t
(Newton’s ‘absolute time’) formally to the rank of a dynamical variable (this
is an example for an absolute structure in disguise, as mentioned above). We
therefore write q(τ) andt(τ). Derivatives with respect to τ will be denoted by a
dot, and restriction to ṫ>0 is made. The action (3.1) can then be rewritten as
∫ τ2
(
S[q(τ),t(τ)] = dτ ṫL q, ˙q ) ∫ τ2
≡ dτ
τ 1
ṫ
˜L(q, ˙q,ṫ) . (3.2)
τ 1
The Lagrangian ˜L possesses the important property that it is homogeneous (of
degree one) in the velocities, that is,
˜L(q, λ ˙q,λṫ) =λ˜L(q, ˙q, ṫ) , (3.3)
where λ ≠ 0 can be an arbitrary function of τ. Homogeneous Lagrangians lead
to actions that are invariant under time reparametrizations τ → ˜τ ≡ f(τ) in
the sense that they can be written as a ˜τ-integral over the same Lagrangian
depending now on dq
d˜τ . Assuming f>0gives ˙
∫ τ2
∫ ˜τ2
(
d˜τ
S = dτL(q, ˙q) =
τ 1 ˜τ 1 f˙
L q, dq ) ∫
f
d˜τ ˙
˜τ2
(
= d˜τ L q, dq )
. (3.4)
˜τ 1
d˜τ
The canonical momentum for q is found from (3.2) to read
˜p q = ∂ ˜L
∂ ˙q = ṫ ∂L
( ) 1
∂ ˙q ṫ = p q , (3.5)
ṫ
thus coinciding with the momentum corresponding to (3.1). But now there is
also a momentum canonically conjugate to t,
( )
p t = ∂ ˜L (q,
∂ṫ = L ˙q ) ∂L q, ˙q ṫ
+ ṫ
ṫ ∂ṫ
(
= L q, dq )
− dq ∂L(q, dq/dt)
= −H . (3.6)
dt dt ∂(dq/dt)
Therefore, t and −H (the negative of the Hamiltonian corresponding to the
original action (3.1)) are canonically conjugate pairs. The Hamiltonian belonging
to ˜L is found as
˜H =˜p q ˙q + p t ṫ − ˜L = ṫ(H + p t ) . (3.7)
But because of (3.6), this is constrained to vanish. It is appropriate at this stage
to introduce a new quantity called ‘Super-Hamiltonian’. It is defined as
H S ≡ H + p t , (3.8)
and one has the constraint
H S ≈ 0 . (3.9)
The ≈ in this and further equation(s) means ‘to vanish as a constraint’ or ‘weak
equality’ in the sense of Dirac (1964). It defines a subspace in phase space and