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arXiv:gr-qc/0410131 · Classical and Quantum Gravity

Minisuperspace models in histories theory

Charis Anastopoulos, Ntina Savvidou

Published Apr 14, 2005• 24 citations
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Abstract

We study the Robertson–Walker minisuperspace model in histories theory, motivated by the results that emerged from the histories approach to general relativity. We examine, in particular, the issue of time reparametrization in such systems. The model is quantized using an adaptation of reduced state space quantization. We finally discuss the classical limit, the implementation of initial cosmological conditions and estimation of probabilities in the histories context.

DOI
10.1088/0264-9381/22/9/023

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We study the Robertson–Walker minisuperspace model in histories theory, motivated by the results that emerged from the histories approach to general relativity. We examine, in particular, the issue of time reparametrization in such systems. The model is quantized using an adaptation of reduced state space quantization. We finally discuss the classical limit, the implementation of initial cosmological conditions and estimation of probabilities in the histories context.
Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the SSS-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.
Inspired by the central equation of canonical quantum gravity, the timeless Wheeler-DeWitt equation, we investigate timeless properties for quantum cosmological models. In particular we are interested in the following question: What is the probability of the system entering certain regions of its configuration space without reference to time at all? For the classical case this is a property of the entire system trajectories and can be formulated using a phase space probability distribution 1. In the quantum case we propose and investigate a probability which we obtain by adapting the decoherent histories approach to quantum theory 2 to the timeless case and by using the induced inner product on the physical state space of quantum cosmological models 3. For the example of the Klein-Gordon equation we have investigated the case of space–like surfaces and our results agree with the standard operator methods 4. For a more general model we consider a semiclassical approximation as well as the effect of an environment 1. When the histories are decoherent, the probabilities approximately coincide with the classical case, with the phase space probability distribution replaced by the Wigner function of the quantum state. The influence functional of the environment respects the reparametrization invariance of the quantum cosmological Wheeler-DeWitt equation.
In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in nonrelativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the SSS matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit, and are closely related to an intersection number operator. The definitions of class operators given here handle the key case in which the underlying classical system has multiple crossings of the boundaries of the regions of interest. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.
The major purpose of this study is to consecrate specific approaches to some problems of quantum theory and quantum cosmology, in terms of decoherence histories formalism which is a very promising formalism for the canonical quantum gravity theories. The reason is to give a unified picture to these issues in order to be possible to compare them. Specifically, the first part contains two different approaches to the time of arrival in quantum mechanics, both of these use a histories formalism. The first is from Halliwell and Yearsly (2009) and the second from Anastopoulos and Saviddou (2012). By comparing them we deduce that both of them first gives a coarse-grain form of the Kijowski' s probability distribution. The second part concerns quantum cosmology. In this, we presented a decoherent histories approach to quantum cosmological probabilities, in which was used a complex potential, from Halliwell (2009). After that we present a histories approach to minisuperspace models by Anastopoulos and Savidou (2005). In this, minisuperspace models is written in terms of histories projector operator (HPO) formalism. The spectacular of this is that in that form general relativity satisfies both spacetime diffeomorfisms and Dirac algebra, which is very important because it is easier to be quantized.