Weak Weyl relations, time operators and their applications

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

Generalised Weyl Operators

RL Hudson, Kalyanapuram Rangachari Parthasarathy

Lecture notes in mathematicsJan 1, 19844 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

\mathcal{PT}

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