Weyl Commutation Relations Research Articles

The parabolic algebra revisited

The parabolic algebra Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document} is the weakly closed operator algebra on L2(ℝ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L^2}(\\mathbb{R})$$\\end{document} generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions eiλx,λ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${e^{i\\lambda x}},\\lambda\\ge 0$$\\end{document}. It is reflexive, with an invariant subspace lattice LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson’s notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} is not synthetic relative to the H∞(ℝ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${H^\\infty}(\\mathbb{R})$$\\end{document} subalgebra of Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document}. Also, various new operator algebras, derived from isometric representations and from compact perturbations of Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document}, are defined and identified.

On Geometry of p-Adic Coherent States and Mutually Unbiased Bases.

This paper considers coherent states for the representation of Weyl commutation relations over a field of p-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proven that the bases of coherent states corresponding to different lattices are mutually unbiased, and that the operators defining the quantization of symplectic dynamics are Hadamard operators.

Universality of Weyl unitaries

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p×p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v, satisfy up=vp=1p (the p×p identity matrix) and the commutation relation uv=ζvu, where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d×d unitary matrices such that up=vp=1d and uv=ζvu, then there exists a unital completely positive linear map ϕ:Mp(C)→Md(C) such that ϕ(u)=u and ϕ(v)=v. We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails.There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p≥3, producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

Constructions of locally distinguishable sets of maximally entangled states which require two-way LOCC

Most of known locally distinguishable sets of maximally entangled states (MESs) can be perfectly distinguished with one-way local operations and classical communication (1-LOCC), and there are not many known locally distinguishable sets which require two-way LOCC (2-LOCC). Recently some such sets of MESs have been built, that is, they can be locally distinguished only with 2-LOCC. In this work, we show that there exist a lot of locally distinguishable MESs which require 2-LOCC (denote such sets by 2-LOCC sets). Firstly generalized Bell states are employed to construct lots of 1-LOCC indistinguishable sets of MESs in the quantum systems and . Then we build a lot of 2-LOCC sets by using Weyl commutation relation and two constructive methods. Especially, it is shown that there are at least 8 and 105 2-LOCC sets in quantum systems and respectively.

Non-commutative birational maps satisfying Zamolodchikov equation, and Desargues lattices

We present new solutions of the functional Zamolodchikov tetrahedron equation in terms of birational maps in totally non-commutative variables. All the maps originate from Desargues lattices, which provide geometric realization of solutions to the non-Abelian Hirota–Miwa system. The first map is derived using the original Hirota’s gauge for the corresponding linear problem, and the second one is derived from its affine (non-homogeneous) description. We also provide an interpretation of the maps within the local Yang–Baxter equation approach. We exploit the decomposition of the second map into two simpler maps, which, as we show, satisfy the pentagonal condition. We also provide geometric meaning of the matching ten-term condition between the pentagonal maps. The generic description of Desargues lattices in homogeneous coordinates allows us to define another solution of the Zamolodchikov equation, but with a functional parameter that should be adjusted in a particular way. Its ultra-local reduction produces a birational quantum map (with two central parameters) with the Zamolodchikov property, which preserves Weyl commutation relations. In the classical limit, our construction gives the corresponding Poisson map, satisfying the Zamolodchikov condition.

The Covariant Stone–von Neumann Theorem for Actions of Abelian Groups on $$ C^{*} $$-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),\alpha) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert $ C^{\ast} $-modules to significantly shorten the length of Iain Raeburn's well-known proof of Takai-Takesaki Duality.

The Regular Ground States of the Linear Boson Field in Terms of Soft Modes

We supplement the characterization of the regular ground states of the linear boson field, as detailed in Segal (Illinois J. Math. 6 (1962), 500–523), Weinless (J. Funct. Anal. 4 (1969), 350–379) and Baez, Segal & Zhou (Introduction to algebraic and constructive quantum field theory, Princeton University Press, 1992, Princeton), by using more developed operator algebraic methods for general Weyl systems and by continuing ideas of Honegger & Rieckers (Photons in Fock space and beyond I, World Scientific, 2015, Singapore). For a linear boson field in the sense of Segal (Illinois J. Math. 6 (1962), 500–523) the originally real symplectic test function dynamics is given in terms of a continuous unitary group. The quantized field dynamics is realized via a continuous unitary group in a representation space of the Weyl commutation relations and specifies a quasifree automorphism group, which does not act strongly continuous on the C*-Weyl algebra. If the generator of the test function dynamics is strictly positive, we show that the quasifree dynamical automorphism group is R -central. Under this condition, we determine the form of all “partially regular” ground states, which are regular only over a subspace of test functions, introduced in Segal (Illinois J. Math. 6 (1962), 500–523). All regular ground states are then functional integrals , in some weakened sense, over the bare vacua dressed by singular classical modes. If the ground states are nuclear continuous these functional integrals are proper ones. If there are also time-invariant test functions then the related invariant quantum modes intervene into these superpositions of the pure regular ground states and complicate particularly the fully regular case. Global features of the sets of (partially) regular ground states are detailed and compared with the ground state sets of C*-dynamical systems. By way of example, the vacuum dressing is interpreted as a soft-boson cloud, forming collective classical soft modes.

Infinite degrees of freedom Weyl representation: Characterization and application

By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.

Phase–number uncertainty from Weyl commutation relations

We derive suitable uncertainty relations for characteristics functions of phase and number of a single-mode field obtained from the Weyl form of commutation relations. Some contradictions between the product and sums of characteristic functions are noted.

Dissipative and nonunitary solutions of operator commutation relations

We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine group $\cG$, the group of affine transformations of the real axis preserving the orientation. If $A$ is a symmetric operator, the group $G$ induces an action/flow on the operator unit ball of contractive transformations from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow give rise to solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of non-unitarily equivalent representations. In the case of deficiency indices $(1,1)$, our general results can be strengthened to the level of an alternative.

The pentagon relation and incidence geometry

We define a map \documentclass[12pt]{minimal}\begin{document}$S:{\mathbb {D}}^2\times {\mathbb {D}}^2 \dashrightarrow {\mathbb {D}}^2\times {\mathbb {D}}^2$\end{document}S:D2×D2⤏D2×D2, where \documentclass[12pt]{minimal}\begin{document}${\mathbb {D}}$\end{document}D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev–Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure—the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

Non-abelian Weyl commutation relations and the series product of quantum stochastic evolutions

We show that the series product, which serves as an algebraic rule for connecting state-based input-output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.

Toeplitz and Translation Operators on the q-Fock Spaces

In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z); and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq ; and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .

Self-adjoint time operators and invariant subspaces

The question of existence of self-adjoint time operators for unitary evolutions in classical and quantum mechanics is revisited on the basis of Halmos-Helson theory of invariant subspaces, Sz.-Nagy-Foias dilation theory and Misra-Prigogine-Courbage theory of irreversibility. It is shown that the existence of self-adjoint time operators is equivalent to the intertwining property of the evolution plus the existence of simply invariant subspaces or rigid operator-valued functions for its Sz.-Nagy-Foias functional model. Similar equivalent conditions are given in terms of intrinsic randomness in the context of statistical mechanics. The rest of the contents are mainly a unifying review of the subject scattered throughout an unconnected literature. A well-known extensive set of equivalent conditions is derived from the above results; such conditions are written in terms of Schrrdinger couples, the Weyl commutation relation, incoming and outgoing subspaces, innovation processes, Lax-Phillips scattering, translation and spectral representations, and spectral properties. Also the natural procedure dealing with symmetric time operators in standard quantum mechanics involving their self-adjoint extensions is illustrated by considering the quantum Aharonov-Bohm time-of-arrival operator.

Extremal quantum states in coupled systems

Let H 1 , H 2 be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose ρ i is a state in H i , i = 1 , 2 . Let C ( ρ 1 , ρ 2 ) be the convex set of all states ρ in H = H 1 ⊗ H 2 whose marginal states in H 1 and H 2 are ρ 1 and ρ 2 respectively. Here we present a necessary and sufficient criterion for a ρ in C ( ρ 1 , ρ 2 ) to be an extreme point. Such a condition implies, in particular, that for a state ρ to be an extreme point of C ( ρ 1 , ρ 2 ) it is necessary that the rank of ρ does not exceed ( d 1 2 + d 2 2 − 1 ) 1 / 2 , where d i = dim H i , i = 1 , 2 . When H 1 and H 2 coincide with the 1-qubit Hilbert space C 2 with its standard orthonormal basis { | 0 〉 , | 1 〉 } and ρ 1 = ρ 2 = 1 2 I it turns out that a state ρ ∈ C ( 1 2 I , 1 2 I ) is extremal if and only if ρ is of the form | Ω 〉 〈 Ω | where | Ω 〉 = 1 2 ( | 0 〉 | ψ 0 〉 + | 1 〉 | ψ 1 〉 ) , { | ψ 0 〉 , | ψ 1 〉 } being an arbitrary orthonormal basis of C 2 . In particular, the extremal states are the maximally entangled states. Using the Weyl commutation relations in the space L 2 ( A ) of a finite Abelian group we exhibit a mixed extremal state in C ( 1 n I n , 1 n 2 I n 2 ) .

Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

In this paper we study a class of generalized Fock spaces associated with the Dunkl operators. Next we introduce the commutator relations between the Dunkl operators and multiplication operators which lead to a generalized class of Weyl commutation relations for the Dunkl kernel.

On the torus automorphisms: analytic solution, computability and quantization

The exact solution of the classical torus automorphism, which partial case is Arnold Cat map is obtained and compared with the numerical solution. The torus, considered as the classical phase space admits the quantization in terms of the Weyl pair. The remarkable fact is that quantum map, as the evolution with respect to the discrete time, preserves the Weyl commutation relation. We have obtained also the operator solution of this quantum torus automorphism.

On the Local Operator Commutation Relations and Extensions of Locally Defined Positive Definite Functions on the Heisenberg Group

It is proved that if a group of unitary operators and a local semigroup of isometries satisfy the Weyl commutation relations then they can be extended to groups of unitary operators which also satisfy the commutation relations. As an application a result about the extension of a class of locally defined positive definite functions on the Heisenberg group is obtained.

On the realization of the Weyl commutation relation HR = qRH

A generic solution of the Weyl relation HR = qRH is found, where H and R are discrete tridiagonal Schrödinger operators with real coefficients. This solution describes the two-parameter non-linear unitary automorphism of the quantum SL q (2) algebra. This leads to a new class of exactly solvable discrete Schrödinger equations having SL q (2) as dynamical symmetry.

Characterizations of intrinsically random dynamical systems

We show that intrinsically random dynamical systems with the Prigogine operator Λ of the form of a random Laplace transform, can be characterized as Kolmogorov flows (K-flows). We also obtain a spectral characterization in the language of the Weyl commutation relation. As a consequence we conclude that the dynamical system is intrinsically random if and only if its Liouvillian and time operators form a Schrödinger couple.