Quantization: History and problems

Georges Lochak

Ntina Savvidou, Charis Anastopoulos

We develop a new algorithm for the quantization of systems with first-class constraints. Our approach lies within the (History Projection Operator) continuous-time histories quantization programme. In particular, the Hamiltonian treatment (either classical or quantum) of parameterized systems is characterized by the loss of the notion of time in the space of true degrees of freedom (i.e. the `problem of time'). The novel temporal structure of the HPO theory (two laws of time transformation that distinguish between the temporal logical structure and the dynamics) persists after the imposition of the constraints, hence the problem of time does not arise. We expound the algorithm for both the classical and quantum cases and apply it to simple models.

Nelson Pinto-Neto, F. T. Falciano, Roberto Pereira B +1 more

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.

J. J. Halliwell, Miguel Ortiz

This paper gives a brief description of the derivation of a composition law for the propagator of a relativistic particle, in a sum over histories quantization. The extension of this derivation to the problem of finding a composition law for quantum cosmology is also discussed. It has been argued that the sum over histories should be regarded as a more generally applicable method of quantization than the canonical framework, in the sense that the former method may be applied to a broader variety of theories. 1 A case in point is general relativity, which admits, at least formally, a sum over histories quantization, but may not admit a canonical one. The existence of a composition law is closely related to the derivation of a canonical formalismfrom the sum over histories. This observation provides the motivation for our discussion of the derivation of a composition law for the relativistic particle propagator. Although we know that in this particular case, composition laws, of the fo...

Maurice A. de Gosson