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On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation
| Title: | On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation |
| Authors: | Arai, Asao Browse this author →KAKEN DB |
| Keywords: | canonical commutation relation | | Hamiltonian | | weakWeyl representation | | Weyl representation | | spectrum | | time operator | | MSC=81Q10 | | MSC=47N50 |
| Issue Date: | Jul-2008 |
| Publisher: | Springer |
| Journal Title: | Letters in Mathematical Physics |
| Volume: | 85 |
| Issue: | 1 |
| Start Page: | 15 |
| End Page: | 25 |
| Publisher DOI: | 10.1007/s11005-008-0252-9 |
| 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 H satisfying the weak Weyl relation: For all t ∈ 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 Te^[-itH]ψ = e^[-itH](T + t)ψ, ∀t ∈ R, ∀ψ ∈ 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 H be separable. Assume that H is bounded below with ε_0 := inf σ(H) and σ(T) = {z ∈ C|Im z ≧ 0}, where C is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Suppose that {T‾, T*, H} (T‾ is the closure of T) is irreducible. Then (T‾,H) is unitarily equivalent to the weak Weyl representation (-p‾_[ε_[0,]+], q_[ε_[0,]+]) on the Hilbert space L^2((ε_0,∞)), where q_[ε_[0,]+] is the multiplication operator by the variable λ ∈ (ε_0,∞) and p_[ε_[0,]+] := -id/dλ with D(d/dλ) = C_0^∞((ε_0,∞)). Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation (T‾,H). |
| Rights: | The original publication is available at www.springerlink.com |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/38135 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 新井 朝雄
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