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arXiv:quant-ph/0302116

Time operator for a quantum many-body system with SU(1,1) dynamical symmetry

Ivan Andrić, M. Martinis

Published Feb 14, 2003• 0 citations
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Abstract

The time operator canonically conjugated to the Hamiltonian of $N$ interacting particles on the line is constructed using SU(1,1) as a dynamical symmetry. This hidden conformal symmetry enables us to make a group theoretic analysis of the time operator in terms of SU(1,1) generators. At distances very far from the interacting region the time operator is represented as a generalization of the quantum "time-of-arrival" operator.

DOI
10.48550/arxiv.quant-ph/0302116

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The time operator canonically conjugated to the Hamiltonian of NNN interacting particles on the line is constructed using SU(1,1) as a dynamical symmetry. This hidden conformal symmetry enables us to make a group theoretic analysis of the time operator in terms of SU(1,1) generators. At distances very far from the interacting region the time operator is represented as a generalization of the quantum "time-of-arrival" operator.
The problem of existence of a self-adjoint time operator conjugate to a Hamiltonian with SU(1,1) dynamical symmetry is investigated. In the space spanned by the eigenstates of the generator K3K_3K3​ of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form ω2x2+g/x2ω^2x2 + g/x2ω2x2+g/x2 is constructed explicitly, and shown that it is related to the time-of-arrival operator of Aharonov and Bohm. Our construction is fully algebraic, involving only the generators of the SU(1,1) group.
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.
For a quantum (many-body) system with dynamical symmetry described by a given Lie group, we study the trace of exponential operators with complex coefficients in one of the irreducible subspaces in terms of the boson realization. By using this approach, for compact groups, we obtain the result of the trace that is consistent with the well-known Weyl character formula. For non-compact groups (with SU(1,1) as an application), convergent condition of the trace is also obtained. This approach may be a powerful tool to study the thermodynamics of quantum (many-body) systems in equilibrium state or nonequilibrium processes.
All covariant time operators with normalized probability distribution are derived for Hamiltonians with continuous eigenvalues. Symmetry criteria are invoked to arrive at a unique expression for a given Hamiltonian. As an application, a well-known result for the arrival time distribution of a free particle is generalized and extended. Interestingly, the resulting arrival time distribution operator is connected to a particular, positive, quantization of the classical current. For particles in a potential, we also introduce and study the notion of conditional arrival-time distribution.
All covariant time operators with normalized probability distribution are derived for Hamiltonians with continuous eigenvalues. Symmetry criteria are invoked to arrive at a unique expression for a given Hamiltonian. As an application, a well-known result for the arrival time distribution of a free particle is generalized and extended. Interestingly, the resulting arrival time distribution operator is connected to a particular, positive, quantization of the classical current. For particles in a potential, we also introduce and study the notion of conditional arrival-time distribution.