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Asao Arai - One of the best experts on this subject based on the ideXlab platform.
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on the uniqueness of weak weyl representations of the canonical Commutation Relation
Letters in Mathematical Physics, 2008Co-Authors: Asao AraiAbstract: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 $${(\overline{T}, H)}$$ .
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construction of a weyl representation from a weak weyl representation of the canonical Commutation Relation
Letters in Mathematical Physics, 2008Co-Authors: Asao Arai, Yasumichi MatsuzawaAbstract: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.
Tatsu Takeuchi - One of the best experts on this subject based on the ideXlab platform.
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effect of the minimal length uncertainty Relation on the density of states and the cosmological constant problem
Physical Review D, 2002Co-Authors: Lay Nam Chang, Djordje Minic, Naotoshi Okamura, Tatsu TakeuchiAbstract:We investigate the effect of the minimal length uncertainty Relation, motivated by perturbative string theory, on the density of states in momentum space. The Relation is implemented through the modified Commutation Relation [x_i,p_j]=i hbar[(1 + beta p^2) delta_{ij} + beta' p_i p_j]. We point out that this Relation, which is an example of an UV/IR Relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the Relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.
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effect of the minimal length uncertainty Relation on the density of states and the cosmological constant problem
Physical Review D, 2002Co-Authors: Lay Nam Chang, Djordje Minic, Naotoshi Okamura, Tatsu TakeuchiAbstract:We investigate the effect of the minimal length uncertainty Relation, motivated by perturbative string theory, on the density of states in momentum space. The Relation is implemented through the modified Commutation Relation $[{x}_{i},{p}_{j}]=i\ensuremath{\Elzxh}[(1+\ensuremath{\beta}{p}^{2}){\ensuremath{\delta}}_{\mathrm{ij}}+{\ensuremath{\beta}}^{\ensuremath{'}}{p}_{i}{p}_{j}].$ We point out that this Relation, which is an example of a UV/IR Relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the Relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.
Eric A Galapon - One of the best experts on this subject based on the ideXlab platform.
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solutions to the time energy canonical Commutation Relation for harmonic oscillator potential
Proceedings of the Samahang Pisika ng Pilipinas, 2018Co-Authors: John Jaykel Ponte Magadan, Eric A GalaponAbstract: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.
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characterizing multiple solutions to the time energy canonical Commutation Relation via internal symmetries
Physical Review A, 2010Co-Authors: Roland Cristopher F Caballar, Leonard R Ocampo, Eric A GalaponAbstract: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.
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characterizing multiple solutions to the time energy canonical Commutation Relation via quantum dynamics
Physics Letters A, 2009Co-Authors: Roland Cristopher F Caballar, Eric A GalaponAbstract: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.
Yasumichi Matsuzawa - One of the best experts on this subject based on the ideXlab platform.
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construction of a weyl representation from a weak weyl representation of the canonical Commutation Relation
Letters in Mathematical Physics, 2008Co-Authors: Asao Arai, Yasumichi MatsuzawaAbstract: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.
Lay Nam Chang - One of the best experts on this subject based on the ideXlab platform.
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effect of the minimal length uncertainty Relation on the density of states and the cosmological constant problem
Physical Review D, 2002Co-Authors: Lay Nam Chang, Djordje Minic, Naotoshi Okamura, Tatsu TakeuchiAbstract:We investigate the effect of the minimal length uncertainty Relation, motivated by perturbative string theory, on the density of states in momentum space. The Relation is implemented through the modified Commutation Relation [x_i,p_j]=i hbar[(1 + beta p^2) delta_{ij} + beta' p_i p_j]. We point out that this Relation, which is an example of an UV/IR Relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the Relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.
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effect of the minimal length uncertainty Relation on the density of states and the cosmological constant problem
Physical Review D, 2002Co-Authors: Lay Nam Chang, Djordje Minic, Naotoshi Okamura, Tatsu TakeuchiAbstract:We investigate the effect of the minimal length uncertainty Relation, motivated by perturbative string theory, on the density of states in momentum space. The Relation is implemented through the modified Commutation Relation $[{x}_{i},{p}_{j}]=i\ensuremath{\Elzxh}[(1+\ensuremath{\beta}{p}^{2}){\ensuremath{\delta}}_{\mathrm{ij}}+{\ensuremath{\beta}}^{\ensuremath{'}}{p}_{i}{p}_{j}].$ We point out that this Relation, which is an example of a UV/IR Relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the Relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.