TY - JOUR

T1 - Self-adjoint Lyapunov variables, temporal ordering, and irreversible representations of Schrödinger evolution

AU - Strauss, Y.

N1 - Funding Information:
This research was supported by the Israel Science Foundation (Grant Nos. 1169/06 and 1282/05). The author wishes to thank Professor I. E. Antoniou for discussions motivating the present work.

PY - 2010/2/1

Y1 - 2010/2/1

N2 - In nonrelativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable-i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admits a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proven that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.

AB - In nonrelativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable-i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admits a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proven that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.

UR - http://www.scopus.com/inward/record.url?scp=77952295439&partnerID=8YFLogxK

U2 - 10.1063/1.3276419

DO - 10.1063/1.3276419

M3 - Article

AN - SCOPUS:77952295439

SN - 0022-2488

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 2

M1 - 001002JMP

ER -