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数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu

Strong Time Operators in Algebraic Quantum Mechanics and Quantum Field Theory (Spectral and Scattering Theory and Related Topics)

Asao Arai

Published Apr 1, 2010• 0 citations
OpenAlex

Abstract

Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},

Related papers

Strong Time Operators in Algebraic Quantum Mechanics and Quantum Field Theory

Asao Arai

Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described. As for the facts presented in QFT, they have essentially been established in a previous paper (A. Arai, Rev. Math. Phys. 17(2005), 1071–1109).

•Apr 1, 2010•1 citation

Intermediate Spectral Theory and Quantum Dynamics

César R. de Oliveira

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models. The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.

Birkhäuser Basel eBooks•Jan 1, 2009•134 citations

Comment on “Arrival time in quantum mechanics” and “Time of arrival in quantum mechanics”

Jerzy Kijowski

Contrary to claims contained in papers by Grot, Rovelli, and Tate [Phys. Rev. A 54, 4676 1996)] and Delgado and Muga [Phys. Rev. A 56, 3425 (1997)], the ``time operator,'' which I have constructed [Rep. Math. Phys. 6, 361 (1974)] in an axiomatic way, is a self-adjoint operator existing in a usual Hilbert space of (nonrelativistic or relativistic) quantum mechanics.

Physical Review A•Jan 1, 1999•45 citations

Introduction

Jan Nesemann

In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.

Vieweg+Teubner eBooks•Jan 1, 2011•0 citations

Lieb-Thirring bound and generalized weak time operators associated with Schrodinger operators (Mathematical Aspects of Quantum Fields and Related Topics)

文生 廣島

This is a short version of [Hir15].A weak time operator $T$ associated with a given self-adjoint operator $H$ is a symmetric operator such that $(H\phi, T\psi)-$ $(T\phi, H\psi)=-i(\phi, \psi)$ for $\phi,$ $\psi\in D$ with some domain $D$ .In this paper we generalize weak time operators as a densely defined symmetric quadratic form, and a generalized weak time operator $T_{H}$ associated with a Schr\"odinger operator of the form $H=-\Delta/2+V$ on $\mathscr{H}=L^{2}(\mathbb{R}^{d})$ is constructed.It is assumed that the quadratic moment of the negative eigenvalues $\{E_{j}\}_{j=1}^{\infty}$ of $H$ is finite, i.e., $\sum_{j=1}^{\infty}E_{j}^{2}<\infty$ .This is ensured by the Lieb-Thirring inequality.Then we can construct $T_{H}$ ) : $\mathscr{H}\cross \mathscr{H}arrow \mathbb{C}$ such that $T_{H}(H\phi, \psi)-T_{H}(\phi, H\psi)=-i(\phi, \psi)$ for all $\phi,$ $\psi\in \mathscr{D}$ with some domain $\mathscr{D}.$

数理解析研究所講究録•Aug 1, 2015•0 citations

Mathematical Foundations of Quantum Mechanics

Fumitomo Maeda

Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.

Hiroshima Mathematical Journal•Jan 1, 1937•48 citations
Some aspects of strong time operators in an abstract algebraic quantum mechanics (including quantum statistical mechanics) and in quantum field theory (QFT) are described.As for the facts presented in QFT, they have essentially been established in a previous paper (A.Arai, Rev. Math.Phys.17(2005), 10711109). §1. Introduction: Strong Time Operators and Their FundamentalProperties This paper is concerned with time operators in algebraic quantum mechanics (including quantum statistical mechanics) and quantum field theory.The concept of time operators itself is a general one and there are some types on them [1, 2, 5, 9].In the present paper we concentrate our attention on strong time operators as defined below and their realizations in algebraic quantum mechanics and quantum field theory.Let \ ma t h f r a k { H} be a complex Hilbert space.We denote the inner product and the norm of \ ma t h f r a k { H} by \l a ng l e \c dot , \ r a n g l e _ { \ m a t h f r a k { H } } (anti-linear in the first variable) and \ V e r t \ c d o t \ V e r t _ { \ ma t h f r a k { H} } respectively.If there is no danger of confusion, then the subscript \ ma t h f r a k { H} in \l a ng l e \c do t , \ r a n g l e _ { \ m a t h f r a k { H } } and \ operator A on a Hilbert space, \mat hrm{D}(A) (resp. Ran (A)) denotes the domain (resp. range) of A . IfA is closable, then we denote its closure by \ov e r l i ne { A} .Let H be a self-adjoint operator on \ ma t h f r a k { H} and T be a symmetric operator on \ ma t h f r a k { H} .If there exists a bounded self-adjoint operator C\neq 0 on \ ma t h f r a k { H} such that \mathrm{D}(C)=\mathf rak{H}, (1.1) e^{-i tH}\mathrm{D}(T)\subset \mathrm{D}(T) , \foral l t\i n \mathbb{R},
The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models. The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
This is a short version of [Hir15].A weak time operator TTT associated with a given self-adjoint operator HHH is a symmetric operator such that (Hϕ,Tψ)−(H\phi, T\psi)-(Hϕ,Tψ)− (Tϕ,Hψ)=−i(ϕ,ψ)(T\phi, H\psi)=-i(\phi, \psi)(Tϕ,Hψ)=−i(ϕ,ψ) for ϕ,\phi,ϕ, ψ∈D\psi\in Dψ∈D with some domain DDD .In this paper we generalize weak time operators as a densely defined symmetric quadratic form, and a generalized weak time operator THT_{H}TH​ associated with a Schr\"odinger operator of the form H=−Δ/2+VH=-\Delta/2+VH=−Δ/2+V on H=L2(Rd)\mathscr{H}=L^{2}(\mathbb{R}^{d})H=L2(Rd) is constructed.It is assumed that the quadratic moment of the negative eigenvalues {Ej}j=1∞\{E_{j}\}_{j=1}^{\infty}{Ej​}j=1∞​ of HHH is finite, i.e., ∑j=1∞Ej2<∞\sum_{j=1}^{\infty}E_{j}^{2}<\infty∑j=1∞​Ej2​<∞ .This is ensured by the Lieb-Thirring inequality.Then we can construct THT_{H}TH​ ) : $\mathscr{H}\cross \mathscr{H}arrow \mathbb{C}$ such that TH(Hϕ,ψ)−TH(ϕ,Hψ)=−i(ϕ,ψ)T_{H}(H\phi, \psi)-T_{H}(\phi, H\psi)=-i(\phi, \psi)TH​(Hϕ,ψ)−TH​(ϕ,Hψ)=−i(ϕ,ψ) for all ϕ,\phi,ϕ, ψ∈D\psi\in \mathscr{D}ψ∈D with some domain D.\mathscr{D}.D.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.