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arXiv:1910.11198

Projection evolution of quantum states. Part I

ANDRZEJ GÓŹDŹ, Marek Góźdź, Aleksandra Pȩdrak

Published Oct 24, 2019• 1 citation
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Abstract

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.

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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.
In quantum mechanics, time is introduced as a non-measurable quantity, as there is no possibility to build a hermitian operator canonically conjugated to the Hamiltonian. We cannot have, therefore, the time operator, which means that the temporal structure of the evolution of quantum systems is ill-defined. We present an extension to the model, in which the time evolution is based on the projection postulate rather than the unitary operator. This approach is in agreement with all other aspects of quantum mechanics and allows to discuss time as an observable. Using this framework we present a description of the temporal double-slit experiment in which a~single particle interferes with itself from a different instant of time. Such behaviour has already been observed experimentally but lacked a consistent theoretical explanation.
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.
In the Schrödinger evolution of a quantum state time enters as a real parameter representing the coordinate. In a more consistent approach time should be defined as a quantum observable, with the evolution taking place in a four-dimensional spacetime. This is possible in the projection evolution model in which the wave function is defined in both space and time. This allows to construct the time operator and discuss the temporal structure of quantum processes. In this paper we discuss a photon travelling through a Mach-Zehnder interferometer, focusing the description on the temporal profile of the wave function. We show that in this approach the delayed-choice experiments can be explained by the temporal overlap of the photon and the devices in the interferometer.
In the Schrödinger evolution of a quantum state, time enters as a real parameter representing the coordinate. In a more consistent approach, time should be defined as a quantum observable, with the evolution taking place in a four-dimensional spacetime. This is possible in the projection evolution model in which the wave function is defined in both space and time. This allows to construct the time operator and discuss the temporal structure of the quantum processes. In this paper, we discuss a photon travelling through a Mach-Zehnder interferometer, focusing the description on the temporal profile of the wave function. We show that in this approach the delayed-choice experiments can be explained by the temporal overlap of the photon and the devices in the interferometer.
W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥ μ (with Ĥ 0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂ μ ψ = Ĥ μ ψ and Heisenberg equation d /d x μ = ∂ μ + i[Ĥ μ , ] are obtained, from which we have: (1) t is to Ĥ 0 as x j is to Ĥ j ( j = 1, 2, 3); likewise, t is to i∂ 0 as x j is to i∂ j ; (2) the proposed time operator is canonically conjugate to i∂ 0 rather than to Ĥ 0 , therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δ x μ Δ p μ ≥ 1/2 and the Mandelstam–Tamm treatment Δ x μ Δ H μ ≥ 1/2, have been formulated.
In this paper, we consider time as a dynamical variable. In particular, we present the explicit realization of the time operator within four-dimensional nonrelativistic spacetime. The approach assumes including events as a part of the evolution. The evolution is not driven by the physical time, but it is based on the causally related physical events. The usual Schrödinger unitary evolution can be easily derived as a special case of the three-dimensional projection onto the space of simultaneous events. Also the time—energy uncertainty relation makes clear and mathematically rigorous interpretation.