共轭线性对称性及其对PT-对称量子理论的应用

传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件.     

Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean–Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.     

Concurrence and Foliations Induced by Some 1-Qubit Channels

We start with a short introduction to the roof concept. An elementary discussion of phase-damping channels shows the role of antilinear operators in representing their concurrences. A general expression for some concurrences is derived. We apply it to 1-qubit channels of length 2, getting the induced foliations of the state space, the optimal decompositions, and the entropy of a state with respect to these channels. For amplitude-damping channels one obtains an expression for the Holevo capacity allowing for easy numerical calculations.     

An equation for the quantum dissipative system in statistical mechanics

A formalism for describing quantum dissipative systems in statistical mechanics is developed. A new equation of the Lindblad type with a quadratic superoperator consisting of Hermitian dissipative operators is derived from the Bloch equation for temperature density matrix using the Feynman integral over the trajectories with a modified Menskii weight functional. By way of example, this equation is solved for a one-dimensional quantum harmonic oscillator with linear dissipation. Applying the projection operator technique, an integral-differential equation for a reduced temperature statistical operator is obtained, which is analogous to the Zwanzig equation in statistical mechanics, and its formal solution is found as a convergent series. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 30–34, December, 2006.     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

Hermitian Operators Conjugate to Two-Mode Number-Difference Operator Studied in Entangled State Representation

In similar to the derivation of phase angle operator conjugate to the number operator by Arroyo Carrasco-Moya Cessay we deduce the Hermitian phase operators that are conjugate to the two-mode number-difference operator and the three-mode number combination operator. It is shown that these operators are on the same footing in the entangled state representation as the one of Turski in the coherent state representation.     

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.     

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.     

Non-Lie and Discrete Symmetries of the Dirac Equation

Abstract

New algebras of symmetries of the Dirac equation are presented, which are formed by linear and antilinear first–order differential operators. These symmetries are applied to decouple the Dirac equation for a charged particle interacting with an external field.     

Pseudo-Hermiticity and Theory of Singular Perturbations

Pseudo-Hermitian operators are studied within the theory of singular perturbations. Necessary and sufficient conditions for the spectra of such operators to be real are presented. A criterion for the similarity of a pseudo-Hermitian operator to an Hermitian one is established.     

Quantum Brachistochrone problem and the geometry of the state space in pseudo-Hermitian quantum mechanics

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.     

关联函数的长时间渐近行为——波对弛豫过程的影响

本文从非平衡态统计的一般理论出发,系统地讨论了宏观体系中波对弛豫过程的影响,特别是关联函数的非指数型渐近行为。由于表象选择对处理耗散过程的重要性,我们必须从元激发表象出发(而不是自由粒子表象)计算关联函数或输运系数(久保公式)。为此,应用了C代数的基本概念,发展了一套算子展开的方法,用之将流算子在元激发二次量子化表象中一般地表示出来。我们从刘维算子的预解式出发,对元激发间的剩余相互作用作微扰,应用了投影算子方法后,可以比较简单地得到关联函数非指数衰减部分的一般表达式。根据对元激发处理过程的分析,我们将元激 关键词：     

Tempered distributions in infinitely many dimensions

The space of testing functions for tempered distributions is characterized in an abstract way as the maximal space in a certain class of locally convex topological vector-spaces. The main characteristic of this class is stability under the differentiation and multiplication operators.The ensuing characterization of tempered distributions may readily be generalized to the case of infinitely many dimensions, and a certain class of such generalizations is studied. The spaces of testing elements are required to be stable under the action of the canonical field operators of the quantum theory of free fields, and it is shown that extreme spaces of testing elements exist and have simple properties. In fact, the maximal space is a Montel space, and the minimal complete space is a direct sum of such spaces.The formalism is applied to the problem of extending the canonical field operators, and a number of extension theorems are derived. In a forthcoming paper the theory of tempered distributions in infinitely many variables will be applied to a structurally simple linear operator equation.     

Quantum Encoding and Entanglement in Terms of Phase Operators Associated with Harmonic Oscillator

Realization of qudit quantum computation has been presented in terms of number operator and phase operators associated with one-dimensional harmonic oscillator and it has been demonstrated that the representations of generalized Pauli group, viewed in harmonic oscillator operators, allow the qudits to be explicitly encoded in such systems. The non-Hermitian quantum phase operators contained in decomposition of the annihilation and creation operators associated with harmonic oscillator have been analysed in terms of semi unitary transformations (SUT) and it has been shown that the non-vanishing analytic index for harmonic oscillator leads to an alternative class of quantum anomalies. Choosing unitary transformation and the Hermitian phase operator free from quantum anomalies, the truncated annihilation and creation operators have been obtained for harmonic oscillator and it has been demonstrated that any attempt of removal of quantum anomalies leads to absence of minimum uncertainty.     

Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments fQ _f from compactly supported C functions f to bounded linear operators Q _f on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.     

Invariant*-products of holomorphic functions on the hyperbolic hermitian spaces

In this paper, we give an explicit basis of the invariant bidifferential operators on the Hermitian symmetric spaces of rank 1. As an application we prove that on noncompact Hermitian symmetric spaces of rank 1, an invariant^*-product coincides with the usual product for holomorphic functions.Aspirant du Fonds National Belge de la Recherche Scientifique.