The time operator, an operator which satisfies the canonical commutation relation with the Hamiltonian, is investigated, on the basis of a certain algebraic relation for a pair of operators T and H, where T is symmetric and H self-adjoint. This relation is equivalent to the Weyl relation, in the case of self-adjoint T, and is satisfied by the Aharonov-Bohm time operator T_0 and the free Hamiltonian H_0 for the one-dimensional free-particle system. In order to see the qualitative properties of T_0, the operators T and H satisfying this algebraic relation are examined. In particular, it is shown that the standard deviation of T is directly connected to the survival probability, and H has no point spectrum. Hence, it is concluded that the existence of the operator T implies the existence of scattering states. It is also shown that the minimum uncertainty states do not exist. Other examples of these operators T and H, than the one-dimensional free-particle system, are demonstrated.The time operator, an operator which satisfies the canonical commutation relation with the Hamiltonian, is investigated, on the basis of a certain algebraic relation for a pair of operators T and H, where T is symmetric and H self-adjoint. This relation is equivalent to the Weyl relation, in the case of self-adjoint T, and is satisfied by the Aharonov-Bohm time operator T_0 and the free Hamiltonian H_0 for the one-dimensional free-particle system. In order to see the qualitative properties of T_0, the operators T and H satisfying this algebraic relation are examined. In particular, it is shown that the standard deviation of T is directly connected to the survival probability, and H has no point spectrum. Hence, it is concluded that the existence of the operator T implies the existence of scattering states. It is also shown that the minimum uncertainty states do not exist. Other examples of these operators T and H, than the one-dimensional free-particle system, are demonstrated.

Jun 19, 200029 citations

GENERALIZED WEAK WEYL RELATION AND DECAY OF QUANTUM DYNAMICS

Asao Arai

Let H be a self-adjoint operator on a Hilbert space [Formula: see text], T be a symmetric operator on [Formula: see text] and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on [Formula: see text]. We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te-itHψ = e-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of [Formula: see text], an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e-itHϕ〉|2 as |t| → ∞ for all vectors ψ and ϕ in a subspace of [Formula: see text], where 〈·,·〉 denotes the inner product of [Formula: see text]. We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.Let H be a self-adjoint operator on a Hilbert space [Formula: see text], T be a symmetric operator on [Formula: see text] and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on [Formula: see text]. We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te-itHψ = e-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of [Formula: see text], an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e-itHϕ〉|2 as |t| → ∞ for all vectors ψ and ϕ in a subspace of [Formula: see text], where 〈·,·〉 denotes the inner product of [Formula: see text]. We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.

Reviews in Mathematical PhysicsOct 1, 200536 citations

Time Operators and Time Crystals

Keiji Nakatsugawa, Toshiyuki Fujii, Avadh Saxena +1 more

We investigate time operators in the context of quantum time crystals in ring systems. A generalized commutation relation called the generalized weak Weyl relation is used to derive a class of self-adjoint time operators for ring systems with a periodic time evolution: The conventional Aharonov-Bohm time operator is obtained by taking the infinite-radius limit. Then, we discuss the connection between time operators, time crystals and real-space topology. We also reveal the relationship between our time operators and a $\mathcal{PT}$ -symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.We investigate time operators in the context of quantum time crystals in ring systems. A generalized commutation relation called the generalized weak Weyl relation is used to derive a class of self-adjoint time operators for ring systems with a periodic time evolution: The conventional Aharonov-Bohm time operator is obtained by taking the infinite-radius limit. Then, we discuss the connection between time operators, time crystals and real-space topology. We also reveal the relationship between our time operators and a $\mathcal{PT}$-symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.

arXiv:1711.10179Nov 28, 20170 citations

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

Keiji Nakatsugawa, Toshiyuki Fujii, Avadh Saxena +1 more

Abstract We investigate time operators in the context of quantum time crystals in ring systems. A generalized commutation relation called the generalized weak Weyl relation is used to derive a class of self-adjoint time operators for ring systems with a periodic time evolution: The conventional Aharonov–Bohm time operator is obtained by taking the infinite-radius limit. Then, we discuss the connection between time operators, time crystals and real-space topology. We also reveal the relationship between our time operators and a -symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.Abstract We investigate time operators in the context of quantum time crystals in ring systems. A generalized commutation relation called the generalized weak Weyl relation is used to derive a class of self-adjoint time operators for ring systems with a periodic time evolution: The conventional Aharonov–Bohm time operator is obtained by taking the infinite-radius limit. Then, we discuss the connection between time operators, time crystals and real-space topology. We also reveal the relationship between our time operators and a -symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.

Journal of Physics A Mathematical and TheoreticalAug 30, 20197 citations

Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics

Y. Strauss, Jonathan Silman, Shai Machnes +1 more

In [Y. Strauss, Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution, J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced that has the property that it indicates the direction of time within the framework of standard quantum mechanics, in the sense that as a function of time its expectation value decreases monotonically for any initial state. In this paper we study some of this operatorʼs properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle.In [Y. Strauss, Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution, J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced that has the property that it indicates the direction of time within the framework of standard quantum mechanics, in the sense that as a function of time its expectation value decreases monotonically for any initial state. In this paper we study some of this operatorʼs properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle.

arXiv:1101.3969Comptes Rendus MathématiqueOct 1, 201110 citations

q-Equivalent Hamiltonian Operators

Mohsen Razavy

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

Canadian Journal of PhysicsSep 1, 197219 citations

\mathcal{PT}

-symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.