By Claudio Simeone
During this e-book, homogeneous cosmological versions whose Hamilton-Jacobi equation is separable are deparametrized by means of turning their motion sensible into that of a regular gauge method. Canonical gauges imposed at the gauge process are used to outline a world part time when it comes to the canonical variables of the minisuperspaces. The process truly indicates how the geometry of the constraint floor restricts the alternative of time. the results that this has for direction imperative quantization are mentioned, and the transition amplitude is acquired for relativistic isotropic versions, relativistic anisotropic versions (Kantowski-Sachs and Taub) and isotropic string cosmologies. an entire bankruptcy concerning the program of the deparametrization application to the standard canonical quantization scheme is usually integrated.
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Extra info for Deparametrization and path integral quantization of cosmological models
1 Isotropic universes An isotropic and homogeneous model gives an acceptable description of the current state of the universe. For example, an essential feature of this model is its non-stationary character, which constitutes a good explanation for the observed redshift of far galaxies. The spatial line element of an isotropic model has the form dl2 = gabdxadxb where gab is the space metric, whose components are functions of time. The isotropy and homogeneity hypothesis lead to the fact that the curvature depends on only one parameter: for k = 0 we have a flat universe, for k — — 1 the universe is open, and for k = 1 the universe is closed.
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 Deparametrization and path integral quantization tion of the gravitational field then does not have gauge invariance at 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.
Note that, as / depends only on observables, h conmutes with the complete Hamiltonian K = NPQ + h, so that dh _ d2f dr dr2 Thus if / could be defined as a function linear in r we would be able to obtain a conserved Hamiltonian for the reduced system; when possible, we shall choose the reduction procedure leading to such a reduced Hamiltonian. The reduced Hamiltonian h could be both positive or negative-definite. The possibilility of a double sign is not necessarily a serious problem: for example, a double sign appears in the quantum theory for a relativistic free particle, and the interpretation is that of particles and antiparticles (see the next section); unless an interaction making the "effective mass" squared vanish -then allowing the two sheets of the constraint touch each other- is introduced, one can work with two disjoint theories [Barvinsky (1993)].
Deparametrization and path integral quantization of cosmological models by Claudio Simeone
Categories: Solid State Physics