Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

有限维算符整函数幂级数展开的简单实现

在量子力学算符可表示为有限维矩阵形式的情况下,利用线性代数中Hamilton—Cayley定理,针对算符整函数的无穷幂级数展开,给出化简方法．     

构造阶梯算符的定量及其应用

给出一个基本定理，提供了构造二阶线性常微分算符相应的阶梯算符的普遍适用的方法，利用这个定理得到量子力学中常见的本征问题的解。     

构造阶梯算符的定理及其应用

给出一个基本定理，提供了构造二阶线性常微分算符相应的阶梯算符的普遍适用的方法.利用这个定理得到量子力学中常见的本征问题的解.     

Algebraic aspects of the wigner distribution in quantum mechanics

The algebraic structure underlying the method of the Wigner distribution in quantum mechanics and the Weyl correspondence between classical and quantum dynamical variables is analysed. The basic idea is to treat the operators acting on a Hilbert space as forming a second Hilbert space, and to make use of certain linear operators on them. The Wigner distribution is also related to the diagonal coherent state representation of quantum optics by this method.     

量子力学混合态表象

在以往的文献中量子力学的表象都是纯态表象,在本文中我们从算符的合理排序和概率统计的正态分布思想出发,首次提出了量子力学混合态表象的概念,并证明了其完备性和正交性.量子力学混合态表象的优点是可以反映算符的多种表示以及其相应的排序规则.     

Quantum mechanics and geometric analysis on manifolds

A central idea of modern geometric analysis is the assignment of a geometric structure, usually called thesymbol, to a differential operator. It is known that this operation is closely related to quantum mechanics. For a class of linear operators, including the Dirac operator, a geometric structure, called aco-Riemannian metric, is assigned to such symbols. Certain other topics related to the geometric structure of quantum mechanics, e.g., the symplectic structure of the projective space of Hilbert space, are briefly treated.     

On dynamical symmetry algebra for aq-extension of the harmonic oscillator

Dynamical symmetry algebra for aq-analogue of the linear harmonic oscillator in quantum mechanics is explicitly constructed in terms ofq-difference raising and lowering operators, which factorize governing Hamiltonian for this model.     

A Non-Associative Quantum Mechanics

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.     

Probability Representation of Quantum States

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born’s rule and recently suggested method of dequantizer–quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical–like equations for the probability distributions determining the quantum system states. Relations to phase–space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.     

On Realization of Partially Ordered Abelian Groups

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating ?-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of ?. A similar characterization is derived for an order determining set of ?-maps and symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.     

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.     

Rigid Motions in the Presence of a Magnetic Field

It is shown that, in the standard framework of non-relativistic quantum mechanics, the presence of a magnetic field implies that there are no operators representing those translations or rotations that do not leave invariant the magnetic field, and the corresponding components of the linear or angular momentum are undefined. Pacs: 03.65.-w. 02.20.-a     

P-adic Schrödinger-type equation

A Schrödinger-type equation is considered in relation to p-adic quantum mechanics. We discuss the appropriate notion of differential operators. A solution of the Schrödinger-type equation is given and a new set of vacuum states for the p-adic quantum harmonic oscillator is presented. The correspondence principle with the standard quantum mechanics is also discussed.     

A Generalized Quantum Theory

In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus (which is a quantum logical model of the Lüders-von Neumann measurement process). It is shown how positive groups, automorphism groups, Lie algebras and statistical operators emerge from one major postulate—the non-existence of third-order interference [third-order interference and its impossibility in quantum mechanics were discovered by Sorkin (Mod Phys Lett A 9:3119–3127, 1994)]. This again underlines the power of the combination of the conditional probability calculus with the postulate that there is no third-order interference. In two earlier papers, its impact on contextuality and nonlocality had already been revealed.     

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.     

Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom

This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schrödinger equation. Using complex canonical variables, a formal proof of the quantization axiom p → pˆ = −i, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov–Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.