Reduced Hamiltonians

Physics

Scientific paper

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Quantum Gravity

Scientific paper

The relatively short history of the study of quantum gravity has found a plethora of problems, arguably the most perplexing of these is the one that has become known as the problem of time and it arises when we study theories of gravity in spatially closed manifolds, because for these the naive Hamiltonian is forced to vanish as a constraint. This treatise deals with the problem of finding a Hamiltonian that correctly describes the evolution of the physical variables of the theory and which can be used to canonically quantize whatever is found to be the correct theory of gravity. We will refer to the procedure described here as reduction and the resulting Hamiltonian will be called the reduced Hamiltonian. The reduction of a gauge theory consists of eliminating all spurious degrees of freedom by gauge fixing on the initial value surface, using the remaining physical variables to construct a set of canonical variables, and finally constructing a Hamiltonian that describes the appropriate time evolution of these, the reduced canonical variables. It must be pointed that this construction is not original to this work, in fact it dates to the end of the last century. What is original is its application to gravity. We begin by describing the procedure for a general model. Next we apply this method to a harmonic oscillator and to scalar QED in temporal gauge. After these simple examples we will be ready for the more exciting problem of finding the Hamiltonian for the theory of general relativity in the manifold T^3times R. We conclude by setting up the functional formalism where we discover that there is a simple relation connecting the matrix elements and expectation values of functionals of the reduced operators with those of the original unreduced theory.

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