By Claudio Simeone
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Additional info for Deparametrization and Path Integral Quan
In other theory with a quadratic Hamiltonian constraint, as it is gravitation, the interpretation may not be so simple, but this analogy will prove to be useful for understanding the results in the context of minisuperspace quantization. 2 The ideal clock Consider a mechanical system with canonical coordinates and momenta (qk,pk). Its action functional reads but as the dynamics remains unchanged if we add a total derivative of t to the integrand, we can write Examples 37 We can give the evolution in terms of an arbitrary parameter T by including the time t among the canonical coordinates, so that the conjugated momentum pt appears.
However, this is perhaps only a consequence of usually working with simple parametrized systems like, for example, the relativistic particle (see below); the formalism for these systems, when put in a manifestly covariant form, has the time included among the coordinates, and the evolution is given in terms of a physically meaningless time parameter. But while for these systems the time coordinate always refers to an external clock, this is clearly not the case in cosmology; for example, in the case of pure gravitational dynamics the coordinates are the elements of the metric gab over spatial slices, and in principle there is not necessarily a connection between gab and anything "external".
14) are not equivalent (see Chapter 6). In the next section we shall give a deparametrization procedure avoiding this, and we shall obtain a different expression for the propagator corresponding to this model. 3 A more general constraint Let us consider a Hamiltonian constraint of the form where G(R) > 0 and V($,fl)is the potential. We shall restrict our analysis to the cases in which the potential V($,0 ) has a definite sign. As the cases V > 0 and V < 0 are formally analogous, we shall consider only V > 0.