Time operator and quantum projection evolution
- PDF / 603,697 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 37 Downloads / 187 Views
SECOND INTERNATIONAL WORKSHOP ON SUPERINTEGRABLE SYSTEMS IN CLASSICAL AND QUANTUM MECHANICS Theory
Time Operator and Quantum Projection Evolution* *** ´ z´ ** and M. Debicki A. Go´ zd ¸
Institute of Physics, University of Marie Curie-Skłodowska, Lublin, Poland Received May 16, 2006
Abstract—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 Schrodinger 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. PACS numbers: 03.65.-w; 03.65.Ta; 42.50.Xa DOI: 10.1134/S106377880703012X
1. INTRODUCTION ¨ The Schrodinger evolution law in quantum mechanics treats the time as a parameter which enumerates the subsequent events. The time, as a parameter, is independent from the system dynamics. But in many cases, it is required that the physical time be a dynamical variable. It is especially important when we want to measure the time of some event occurrences, e.g., when a particle reaches a detector [1, 2], or explain the effect of the so-called time interference [3]. In addition, the time–energy uncertainty relation could be absorbed into the quantum mechanics in a natural way for the case of time being a quantum mechanical observable. Some attempts in order to apply the concept of time as an observable were suspended after it had been proven that the time operator cannot be defined properly within standard quantum mechanics. The Pauli theorem [4–6] states that the self-adjoint time operator implies an unbound continuous energy spectrum. It means that it is impossible to build a self-adjoint time operator canonically conjugate to a Hamiltonian bounded from below. However, Galapon [7] showed that, in the Pauli theorem, no mathematical rigor was used because no attention was paid to the domain operators involved. He proved that it is possible to construct the bounded selfadjoint time operator conjugate to the Hamiltonian. ∗
The text was submitted by the authors in English. E-mail: [email protected] *** E-mail: [email protected] **
Still, there is a problem of a physical meaning of such an operator. It is also possible, for example, to build a proper time operator canonically conjugate, not to a Hamiltonian, but to its extension, i.e., in the simplest case ˆ ≡ sgn(Pˆ )H ˆ 0 , where the energy operator is comH pleted with the particle’s momentum sign [2]. We can ask, however, is it still a time operator and does it predict properly the measurement results? Pauli’s arguments can also be easily overcome within the POVM (Positive Operator Valued Measures) formalism where the self-adjointness is not required [8]. Owing to Gleason
Data Loading...
