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Europhysics Letters (EPL)

Constrained event space and properties of the physical time observable

D. Arsenović, Nikola Burić, Dragomir M Davidović, Slobodan Prvanović

Published Jan 1, 2012• 6 citations
Publisher

Abstract

It is demonstrated that the important common property of operators representing various observable times in quantum mechanics, namely the fact that such an operator is always given by a nonorthogonal resolution of unity, can be obtained from a constraint on the possible physical events, i.e. on the extended space of the Hamiltonian formulation of the parametric dynamics. Operators which could generate meaningful probability distributions for various time measurements are suggested. Copyright (C) EPLA, 2012

DOI
10.1209/0295-5075/97/20013

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It is demonstrated that the important common property of operators representing various observable times in quantum mechanics, namely the fact that such an operator is always given by a nonorthogonal resolution of unity, can be obtained from a constraint on the possible physical events, i.e. on the extended space of the Hamiltonian formulation of the parametric dynamics. Operators which could generate meaningful probability distributions for various time measurements are suggested. Copyright (C) EPLA, 2012
Quantum mechanical time operator is introduced following the parametric formulation of classical mechanics in the extended phase space. Quantum constraint on the extended quantum system is defined in analogy to the constraint of the classical extended system, and is interpreted as the condition defining the space of physical events. It is seen that the peculiar properties of the time observable, otherwise obtained in the models of time measurement, are of the classical origin, (i.e.), due to the quantized classical constraint of the parametric Hamiltonian dynamics.
Although time is one of our most intuitive physical concepts, its understanding at the fundamental level is still an open question in physics. For instance, time in quantum mechanics and general relativity are two distinct and incompatible entities. While relativity deals with events (points in spacetime), with time being observer dependent and dynamical, quantum mechanics describes physical systems by treating time as an independent parameter. To resolve this conflict, in this work, we extend the classical concept of an event to the quantum domain by defining an event as a transfer of information between physical systems. Then, by describing the universe from the perspective of a certain observer, we introduce quantum states of events with space-time-symmetric wave functions that predict the joint probability distribution of a measurement (observation) at $(t,\stackrel{P\vec}{x})$. Under these circumstances, without assuming collapse, we propose that a well-defined instant of time, like any other observable, arises from a single event, thus being an observer-dependent property. In this manner, we obtain a stationary quantum state written as a sum of a sequence of normalized states of events with increasing memories. As a consequence, a counterfactual asymmetry along this sequence of events gives rise to the flow of time as being successive ``snapshots'' from the observer's perspective. In this proposal, which contrasts strikingly to the view in which time is an illusion, it is the many distinguishable states in which the observer stores information that makes the existence of time possible.
In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional Ψ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which Ψ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra C generated by constants of motion is of type I, Ψ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of C is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes.
In this paper we wish to explore the possibility of the existence of a quantity that can be regarded as the complement of the Hamiltonian for a quantum system with discrete energy levels, recognizing that if this quantity is represented by an operator the appropriate commutator must be a generalized form of Eq. 2. Although this quantity will have dimensions of time, it will not be appropriate to refer to such an operator as a time operator. As an operator it would represent an observable of the quantum system and not time in the abstract or coordinate sense or as a reading on an external clock. Its eigenstates would represent a state of the quantum system and some measurement on the system, at least in principle, should tell us about the quantity involved. Although this quantity is not time, we would hope that for a particular ideal system its expectation value may change linearly with time, for example, so that a measurement of the quantity would also give a measure of time. In this case, we would be using the system as a clock. Before seeking an operator for the complement of the Hamiltonian we first represent this quantity by a nonorthogonal probability-operator measure, which is a more general concept 6.
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics.