The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path-integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The construction is complicated by the fact that the amplitudes (class-operators) calculated using a path integral typically do not satisfy the constraint equation everywhere, but we show how they may be systematically modified in such a way that they do satisfy the constraint. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the ``evolving constants'' method).
The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path-integral methods with initial states attached using the (positive definite) induced'' inner product between solutions to the constraint equation. The construction is complicated by the fact that the amplitudes (class-operators) calculated using a path integral typically do not satisfy the constraint equation everywhere, but we show how they may be systematically modified in such a way that they do satisfy the constraint. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cross such a surface many times, but we show that first and last crossings are in essence the only useful possibilities. Different possible results for the probabilities are obtained, depending on how the relativistic particle is quantized (using the Klein-Gordon equation, or its square root, with the associated Newton-Wigner states). In the Klein-Gordon quantization, the decoherence is only approximate, due to the fact that the paths in the path integral may go backwards and forwards in time. We compare with the results obtained using operators which commute with the constraint (the evolving constants'' method).
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.
This paper is concerned with the question of the existence of composition laws in the sum-over-histories approach to relativistic quantum mechanics and quantum cosmology, and its connection with the existence of a canonical formulation. In nonrelativistic quantum mechanics, the propagator is represented by a sum over histories in which the paths move forward in time. The composition law of the propagator then follows from the fact that the paths intersect an intermediate surface of constant time once and only once, and a partition of the paths according to their crossing position may be affected. In relativistic quantum mechanics, by contrast, the propagators (or Green functions) may be represented by sums over histories in which the paths move backward and forward in time. They therefore intersect surfaces of constant time more than once, and the relativistic composition law, involving a normal derivative term, is not readily recovered. The principal technical aim of this paper is to show that the relativistic composition law may, in fact, be derived directly from a sum over histories by partitioning the paths according to their first crossing position of an intermediate surface. We review the various Green functions of the Klein-Gordon equation, and derive their composition laws. We obtain path-integral representations for all Green functions except the causal one. We use the proper time representation, in which the path integral has the form of a nonrelativistic sum over histories but is integrated over time.The question of deriving the composition laws therefore reduces to the question of factoring the propagators of nonrelativistic quantum mechanics across an arbitrary surface in configuration space. This may be achieved using a known result called the path decomposition expansion (PDX). We give a proof of the PDX using a spacetime lattice definition of the Euclidean propagator. We use the PDX to derive the composition laws of relativistic quantum mechanics from the sum over histories. We also derive canonical representations of all of the Green functions of relativistic quantum mechanics, i.e., express them in the form 〈x''\ensuremath{\Vert}x'〉, where the {\ensuremath{\Vert}x〉} are a complete set of configuration-space eigenstates. These representations make it clear why the Hadamard Green function G(1) does not obey a standard composition law. They also give a hint as to why the causal Green function does not appear to possess a sum-over-histories representation. We discuss the broader implications of our methods and results for quantum cosmology, and parametrized theories generally. We show that there is a close parallel between the existence of a composition law and the existence of a canonical formulation, in that both are dependent on the presence of a timelike Killing vector. We also show why certain naive composition laws that have been proposed in the past for quantum cosmology are incorrect. Our results suggest that the propagation amplitude between three-metrics in quantum cosmology, as constructed from the sum over histories, does not obey a composition law.
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 S 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.
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 S-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 quantum cosmology, in which the wave function of the universe is annihilated by the total Hamiltonian, we consider the internal dynamics of a simple particle system in an energy eigenstate. Such a system does not possess a uniquely defined time parameter, and all physical questions about it must be posed without reference to time. We consider in particular the following question: what is the probability that the system's trajectory passes through a set of regions of configuration space without reference to time? We first consider the classical case, where the answer has a variety of forms in terms of a phase-space probability distribution function. We then consider the quantum case, and we analyze this question using the decoherent histories approach to quantum theory, adapted to questions which do not involve time. 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. For some initial states, decoherence requires an environment, and we compute the required influence functional and examine some of its properties. Special attention is given to the inner product used in the construction (the induced or Rieffel inner product), the construction of class operators describing the histories, and the extent to which reparametrization invariance is respected. Our results indicate that simple systems without an explicit time parameter may be quantized using the decoherent histories approach, with the expected classical limit extracted. The results support, for simple models, the usual heuristic proposals for the probability distribution function associated with a semiclassical wave function satisfying the Wheeler-DeWitt equation.
We study the Wheeler-DeWitt quantum cosmology of a spatially flat Friedmann cosmological model with a free massless scalar field. We compare the consistent histories approach with the de Broglie--Bohm theory when applied to this simple model under two different quantization schemes: the Schr\"odinger-like quantization, which essentially takes the square root of the resulting Klein-Gordon equation through the restriction to positive frequencies and their associated Newton-Wigner states, or the induced Klein-Gordon quantization, that allows both positive and negative frequencies together. We show that the consistent histories approach can give a precise answer to the question concerning the existence of a quantum bounce if and only if one takes the single frequency approach within a single family of histories; namely, a family containing histories concerning properties of the quantum system at only two specific moments of time: the infinity past and the infinity future. In that case, as shown by Craig and Singh [Phys. Rev. D 82, 123526 (2010)], there is no quantum bounce. In any other situation, the question concerning the existence of a quantum bounce has no meaning in the consistent histories approach. On the contrary, we show that if one considers the de Broglie--Bohm theory, there are always states where quantum bounces occur in both quantization schemes. Hence, the assertion that the Wheeler-DeWitt quantization does not solve the singularity problem in cosmology is not precise. To address this question, one must specify not only the quantum interpretation adopted but also the quantization scheme chosen.