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arXiv:1403.1557

When does "Time" become "Observable" ?

Tadashi Fujimoto

Published Mar 5, 2014• 0 citations
arXivHTMLPDF

Abstract

We investigate some spectral properties of time operators which are obtained through Canonical Commutation Relation (CCR) and Positive Operator Valued Measure (POVM) of quantum physics. In addition, we re-interpret the spectral properties of time operators from the standpoint of the Copenhagen Interpretation, especially, of W.Heisenberg.
DOI
10.48550/arxiv.1403.1557

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The question of time in quantum mechanics has been a topic of intense discussion since its inception.Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations.A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible.Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a 'generalized' observable.Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures.Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures.This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
The question of time in quantum mechanics has been a topic of intense discussion since its inception. Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations. A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible. Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a generalized observable. Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures. Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures. This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.
Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.
We define a stationary Gaussian quantum stochastic process (GQSP) on the C*-algebra of the canonical commutation relations over a real symplectic Hilbert space. Physically GQSPs can describe diffusion with non-Markovian memory effects in quantum harmonic oscillators of arbitrary dimension. We find that in analogy with the commutative case a GQSP is completely specified by a operator-valued autocovariance function satisfying certain positive definiteness and reality conditions. The autocovariance function also determines the response of the system to a class of time-dependent generalized forces, and it has a spectral representation in terms of a positive operator-valued measure.