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Asao Arai - One of the best experts on this subject based on the ideXlab platform.

  • on the uniqueness of weak weyl representations of the Canonical Commutation Relation
    Letters in Mathematical Physics, 2008
    Co-Authors: Asao Arai
    Abstract:

    Let (T, H) be a weak Weyl representation of the Canonical Commutation Relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space $${\mathcal{H}}$$ satisfying the weak Weyl Relation: for all $${t \in \mathbb{R}}$$ (the set of real numbers), e−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and $${T{\rm e}^{-itH}\psi = {\rm e}^{-itH}(T+t)\psi, \forall t \in \mathbb{R}, \forall\psi \in D(T)}$$ . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let $${\mathcal{H}}$$ be separable. Assume that H is bounded below with $${\varepsilon_0 := \inf \sigma(H)}$$ and $${\sigma(T)=\{z \in \mathbb{C}|{\rm Im} z \ge 0\}}$$ , where $${\mathbb{C}}$$ is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then $${(\overline{T}, H)}$$ ( $${\overline{T}}$$ is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation $${(-\overline{p}_{\varepsilon_0,+}, q_{\varepsilon_0,+})}$$ on the Hilbert space $${L^2((\varepsilon_0,\infty))}$$ , where $${q_{\varepsilon_0,+}}$$ is the multiplication operator by the variable $${\lambda \in (\varepsilon_0,\infty)}$$ and $${p_{\varepsilon_0,+} :=-i{\rm d}/{\rm d}\lambda}$$ with $${D({\rm d}/{\rm d}\lambda)=C_0^{\infty}((\varepsilon_0,\infty))}$$ . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation $${(\overline{T}, H)}$$ .

  • construction of a weyl representation from a weak weyl representation of the Canonical Commutation Relation
    Letters in Mathematical Physics, 2008
    Co-Authors: Asao Arai, Yasumichi Matsuzawa
    Abstract:

    Weak Weyl representations of the Canonical Commutation Relation (CCR) with one degree of freedom are considered in Relation to the theory of time operator in quantum mechanics. It is proven that there exists a general structure through which a weak Weyl representation can be constructed from a given weak Weyl representation. As a corollary, it is shown that a Weyl representation of the CCR can be constructed from a weak Weyl representation which satisfies some additional property. Some examples are given.

Yasumichi Matsuzawa - One of the best experts on this subject based on the ideXlab platform.

Eric A Galapon - One of the best experts on this subject based on the ideXlab platform.

  • solutions to the time energy Canonical Commutation Relation for harmonic oscillator potential
    Proceedings of the Samahang Pisika ng Pilipinas, 2018
    Co-Authors: John Jaykel Ponte Magadan, Eric A Galapon
    Abstract:

    Different solutions to the time-energy Canonical Commutation Relation (TE-CCR) for the harmonic oscillator potential were obtained and were differentiated by comparing their symmetries. Their dynamics, in particular the dynamics of their eigenfunctions, were investigated. The eigenfunction of the TOA operator that satisfies the time reversal symmetry has the sharpest arrival with the arrival occurring at tits corresponding eigenvalue. Meanwhile the eigenfunctions of the TOA operators that do not satisfy the time reversal symmetry do not have good unitary arrival and can not be considered as legitimate TOA operators.

  • characterizing multiple solutions to the time energy Canonical Commutation Relation via internal symmetries
    Physical Review A, 2010
    Co-Authors: Roland Cristopher F Caballar, Leonard R Ocampo, Eric A Galapon
    Abstract:

    Internal symmetries can be used to classify multiple solutions to the time-energy Canonical Commutation Relation (TE-CCR). The dynamical behavior of solutions to the TE-CCR possessing particular internal symmetries involving time reversal differ significantly from solutions to the TE-CCR without those particular symmetries, implying a connection between the internal symmetries of a quantum system, its internal unitary dynamics, and the TE-CCR.

  • characterizing multiple solutions to the time energy Canonical Commutation Relation via quantum dynamics
    Physics Letters A, 2009
    Co-Authors: Roland Cristopher F Caballar, Eric A Galapon
    Abstract:

    Abstract We address the multiplicity of solutions to the time–energy Canonical Commutation Relation for a given Hamiltonian. Specifically, we consider a particle spatially confined in a potential free interval, where it is known that two distinct self-adjoint and compact time operators conjugate to the system Hamiltonian exist. The dynamics of the eigenvectors of these operators indicate that different time operators posses distinguishing properties that can unambiguously associate them to specific aspects of the quantum time problem.

Roland Cristopher F Caballar - One of the best experts on this subject based on the ideXlab platform.

Hideyasu Yamashita - One of the best experts on this subject based on the ideXlab platform.

  • hyperfinite dimensional representations of Canonical Commutation Relation
    Journal of Mathematical Physics, 1998
    Co-Authors: Hideyasu Yamashita
    Abstract:

    This paper presents some methods of representing Canonical Commutation Relations in terms of hyperfinite-dimensional matrices, which are constructed by nonstandard analysis. The first method uses representations of a nonstandard extension of the finite Heisenberg group, called hyperfinite Heisenberg group. The second is based on hyperfinite-dimensional representations of so(3). Then, the cases of infinite degree of freedom are argued in terms of the algebra of hyperfinite para-Fermi oscillators, which is mathematically equivalent to a hyperfinite-dimensional representation of so(n).

  • hyperfinite dimensional representations of Canonical Commutation Relation
    arXiv: Quantum Physics, 1997
    Co-Authors: Hideyasu Yamashita
    Abstract:

    This paper presents some methods of representing Canonical Commutation Relations in terms of hyperfinite-dimensional matrices, which are constructed by nonstandard analysis. The first method uses representations of a nonstandard extension of finite Heisenberg group, called hyperfinite Heisenberg group. The second is based on hyperfinite-dimensional representations of so(3). Then, the cases of infinite degree of freedom are argued in terms of the algebra of hyperfinite parafermi oscillators, which is mathematically equivalent to a hyperfinite-dimensional representation of so(n).