Time operators and time crystals: self-adjointness by topology change

In the standard formulation of quantum mechanics and quantum field theory, time is not an observable but merely a parameter. There are two open problems which can promote time to an observable, namely 1) how to define self-adjoint time operators and 2) how to obtain systems called quantum time crystals.

In this presentation, we investigate time operators in the context of quantum time crystals in ring systems [1,2]. A generalized commutation relation called the generalized weak Weyl relation [3] is used to derive a class of self-adjoint time operators for ring systems. The conventional Aharonov-Bohm time operator, which describes the time of arrival of a particle on a one-dimensional line, is obtained by taking the infinite-radius limit. We also reveal the relationship between our time operators and a PT-symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.

We surmise that time operators and time crystals are closely related and also that topology change is important when time promotes from a parameter to an observable.

[1] K. Nakatsugawa, T. Fujii, A. B Saxena, and S. Tanda, J. Phys. A in press (2019)
[2] K. Nakatsugawa, T. Fujii, and S. Tanda, Phys. Rev. B 96, 094308 (2017)
[3] A. Arai, Rev. Math. Phys. 17, 1017 (2005)