By S., Ed. Flugge

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Additional resources for Encyclopedia of physics, vol. 5-1. Principles of quantum theory I

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We have already remarked that the variation of the action of a parametrized system under a gauge transformation is equal to end point terms; the ac- 22 Depamrnetrization and path integral quantization tion of the gravitational field then does not have gauge invariance a t the boundaries and canonical gauges would not be admissible. But in the last section we pointed that if it was possible to define a canonical transformation such that the Hamiltonian constraint could be matched with a new momentum, the system could be turned into an ordinary gauge one; hence canonical gauge conditions could be imposed to select one path from each class of equivalent paths in phase space.

The key point has been that in terms of the variables of the gauge system we have a natural choice for a function whose Poisson bracket with the constraint is non vanishing everywhere. Moreover, the change to the new coordinates and momenta gives the constraint hypersurface a trivial topology which allows to fix the gauge in a way that clearly does not generate Gribov copies. As we shall see below in the context of minisuperspace deparametrization, the gauge fixation can be relaxed to allow for different definitions of time.

19) because and then We have then shown that by imposing a canonical gauge condition on the gauge system described by (Qi, Pi) we have identified a global phase time for the parametrized system given by (qi,pi). The key point has been that in terms of the variables of the gauge system we have a natural choice for a function whose Poisson bracket with the constraint is non vanishing everywhere. Moreover, the change to the new coordinates and momenta gives the constraint hypersurface a trivial topology which allows to fix the gauge in a way that clearly does not generate Gribov copies.

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Encyclopedia of physics, vol. 5-1. Principles of quantum theory I by S., Ed. Flugge


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