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Universe

Quantum Time and Quantum Evolution

ANDRZEJ GÓŹDŹ, Marek Góźdź, Aleksandra Pȩdrak

Published May 26, 2023• 9 citations
Publisher

Abstract

The problem of quantum time and evolution of quantum systems, where time is not a parameter, is considered. In our model, following some earlier works, time is represented by a quantum operator. In this paper, similarly to the position operators in the Schrödinger representation of quantum mechanics, this operator is a multiplication-type operator. It can be also represented by an appropriate positive operator-valued measure (POVM) which together with the 3D position operators/measures provide a quantum observable giving a position in the quantum spacetime. The quantum evolution itself is a stochastic process based on Lüder’s projection postulate. In fact, it is a generalization of the unitary evolution. This allows to treat time and generally the spacetime position as a quantum observable, in a consistent and observer-independent way.

DOI
10.3390/universe9060256

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The problem of quantum time and evolution of quantum systems, where time is not a parameter, is considered. In our model, following some earlier works, time is represented by a quantum operator. In this paper, similarly to the position operators in the Schrödinger representation of quantum mechanics, this operator is a multiplication-type operator. It can be also represented by an appropriate positive operator-valued measure (POVM) which together with the 3D position operators/measures provide a quantum observable giving a position in the quantum spacetime. The quantum evolution itself is a stochastic process based on Lüder’s projection postulate. In fact, it is a generalization of the unitary evolution. This allows to treat time and generally the spacetime position as a quantum observable, in a consistent and observer-independent way.
We discuss the problem of time in quantum mechanics. In the traditional formulation time enters the model as a parameter, not an observable, which follows from the famous Pauli theorem. It is now known, that Pauli's assumptions were too strong and that by removing some of them time can be represented as a quantum observable. In this case, instead of the unitary time evolution, other operators which map the space of initial states into the space of final states at each step of the evolution can be used. This allows to treat time as a quantum observable in a consistent way. We present the projection evolution model and show how the traditional Schroedinger evolution can be obtained from it. We propose the form of the time operator which satisfies the energy-time uncertainty relation. As an example, we discuss the temporal version of the double-slit experiment.
We discuss the problem of time in quantum mechanics. In the traditional formulation time enters the model as a parameter, not an observable, which follows from the famous Pauli theorem. It is now known, that Pauli's assumptions were too strong and that by removing some of them time can be represented as a quantum observable. In this case, instead of the unitary time evolution, other operators which map the space of initial states into the space of final states at each step of the evolution can be used. This allows to treat time as a quantum observable in a consistent way. We present the projection evolution model and show how the traditional Schroedinger evolution can be obtained from it. We propose the form of the time operator which satisfies the energy-time uncertainty relation. As an example, we discuss the temporal version of the double-slit experiment.
Quantum systems, described in terms of wave functions and Hermitian operators that are defined at any instant of time, should be capable of temporal evolution, and this is consistent with the first four Postulates. There is nothing in Postulate 1 to prevent us from considering state vectors Ψ( t ) which depend on time. A Hermitian operator  ( t ), representing a real observable A according to Postulate 2, depends on time either explicitly or through the position coordinate and conjugate momenta, following the correspondence rule introduced by Postulate 4.
Given a Hamiltonian HHH on a Hilbert space H\mathcal HH it is shown that, under the assumption that σ(H)=σac(H)=R+σ(H)=σ_{ac}(H)=R^+σ(H)=σac​(H)=R+, there exist unique positive operators TFT_FTF​ and TBT_BTB​ registering the Schrödinger time evolution generated by HHH in the forward (future) direction and backward (past) direction respectively. These operators may be considered as time observables for the quantum evolution. Moreover, it is shown that the same operators may serve as time observables in the construction of quantum stochastic differential equations and quantum stochastic processes in the framework of the Hudson-Parthasarathy quantum stochastic calculus. The basic mechanism enabling for the definition of the time observables originates from the recently developed semigroup decomposition formalism used in the description of the time evolution of resonances in quantum mechanical scattering problems.
The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of Hilbert space quantum mechanics (QM) into a broader noncontextual, hence local, framework, reinterpreting quantum probabilities as conditional (in a nonstandard sense). We have proven in a previous paper that each generalized observable introduced by the ESR model is represented by a family of positive operator valued measures (POVM) parametrized by the pure states of the physical system Ω that is considered. We show here that each mixture is represented in the ESR model by a family of density operators parametrized by the physical properties characterizing Ω. This representation implies predictions that may differ from those of QM and avoids some deep problems that arise in the interpretation of mixtures provided by QM. We also show that the state transformations induced by idealized nondestructive measurements can be obtained by means of a nontrivial generalization of the Lüders postulate, and discuss our results in the special case of discrete generalized observables.
Abstract Discussions of quantum mechanics often loosely claim that time evolution logically must be unitary, in order for the probabilistic interpretation of the amplitudes of the state vector to make sense at all times. We discuss from first principles whether this claim is true: if we assume only that the time-evolution operator is linear, then does the stronger requirement that it be unitary follow from the other axioms of quantum mechanics? The answer is subtle. We discuss two mathematically distinct but physically equivalent formulations of the axioms of quantum mechanics, and consider generalizing each to postulate only that time evolution is linear. Within one formulation, the unitarity of time evolution follows logically from the other axioms – but within the other formulation, it does not. Allowing the time-evolution operator to be (a priori) arbitrarily linear does not change the physical observables in one formulation of quantum mechanics, but changes the other formulation to a distinct (internally consistent) physical theory that allows new phenomenology like (e.g.) faster-than-light communication. Therefore, the unitarity of time evolution is arguably better thought of as a logically independent and experimentally falsifiable axiom of quantum mechanics, not as a tautological consequence of the other axioms.