Kiefer C. Quantum gravity

3
PARAMETRIZED AND RELATIONAL SYSTEMS
In this chapter, we shall consider some models that exhibit certain features of GR
but which are much easier to discuss. In this sense they constitute an important
conceptual preparation for the canonical quantization of GR, which is the topic
of the next chapters. In addition, they are of interest in their own right.
The central aspect is reparametrization invariance and the ensuing existence
of constraints; see, for example, Sundermeyer (1982) and Henneaux and Teitelboim
(1992) for a general introduction into constrained systems. Kuchař (1973)
gives a detailed discussion of reparametrization-invariant systems, which we shall
partly follow in this chapter. Such invariance properties are often named as ‘general
covariance’ of the system because they refer to an invariance with respect to
a relabelling of the underlying space–time manifold. A more precise formulation
has been suggested by Anderson (1967), proposing that an invariance group is a
subgroup of the full covariance group which leaves the absolute, non-dynamical,
elements of a theory invariant; cf. also Ehlers (1995) and Giulini (2007). Such
an absolute element would, for example, be the conformal structure in the scalar
theory of gravity mentioned at the end of Section 2.1.1. In GR, the full metric
is dynamical and the invariance group coincides with the covariance group, the
group of all diffeomorphisms. According to Anderson (1967), general covariance
should be interpreted as absence of absolute structure, also called ‘background
independence’.
Whereas dynamical elements are subject to quantization, absolute elements
remain classical; see Section 1.3. Absolute elements can also appear in ‘disguised
form’ if a theory has been reparametrized artificially. This is the case in the
models of the non-relativistic particle and the parametrized field theory to be
discussed below, but not in GR or the other dynamical systems considered in
this chapter.
3.1 Particle systems
3.1.1 Parametrized non-relativistic particle
Consider the action for a point particle in classical mechanics
∫ t2
(
S[q(t)] = dtL q, dq )
dt
. (3.1)
t 1
It is only for simplicity that a restriction to one particle is being made. The
following discussion can be easily generalized to n particles. For simplicity, the
Lagrangian in (3.1) has been chosen t-independent.
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