Necessary and sufficient postulates of quantum mechanics

We describe a system of axioms that, on one hand, is sufficient for constructing the standard mathematical formalism of quantum mechanics and, on the other hand, is necessary from the phenomenological standpoint. In the proposed scheme, the Hilbert space and linear operators are only secondary structures of the theory, while the primary structures are the elements of a noncommutative algebra (observables) and the functionals on this algebra, associated with the results of a single observation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 510–529, March, 2005     

Locality problem in quantum theory

We discuss the locality problem in relativistic and nonrelativistic quantum theory. We show that there exists a formulation of quantum theory that, on one hand, preserves the mathematical apparatus of the standard quantum mechanics and, on the other hand, ensures the satisfaction of the locality condition for each individual event including the measurement procedure. As an example, we consider the scattering from two slits. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 327–343, May, 2008.     

Normal form of a quantum Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point

According to classical result of Moser [1] a real-analytic Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point can be reduced to the normal form by a real-analytic symplectic change of variables. In this paper the result is extended to the case of the non-commutative algebra of quantum observables.We use an algebraic approach in quantum mechanics presented in [2] and develop it to the non-autonomous case. We introduce the notion of quantum non-autonomous canonical transformations and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a non-commutative normal form and prove that a time-periodic quantum observable with one degree of freedom near a hyperbolic fixed point can be reduced to a normal form by a canonical transformation. Unlike traditional results, where only formal theory of normal forms is constructed, we prove a convergence of the normalizing procedure.      

Possibility of reconciling quantum mechanics with general relativity theory

We show that the mathematical formalisms of general relativity and of quantum mechanics can be reconciled based on an algebraic approach. In this case, gravity does not need to be quantized.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.     

Center-of-mass tomography and probability representation of quantum states for tunneling

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 371–387, February, 2005.     

A geometric approach to the canonical reformulation of quantum mechanics

The measure of distinguishability between two neighboring preparations of a physical system by a measurement device naturally defines a line element on the preparation space of the system. We show that quantum mechanics can be derived from the invariance of this line element in the canonical formulation. We also discuss the canonical formulation of quantum statistical mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 479–486, June, 2006.     

Evolution of probability measures associated with quantum systems

We derive the evolution equation for probability distributions and characteristic functions of the quantum tomograms associated with the linear and nonlinear evolutions of quantum states.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 365–370, February, 2005.     

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self‐delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program‐size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so‐called universal probability m is introduced first, and then H (s) is defined as –log₂ m (s) without reference to the concept of program‐size. Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator‐valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour‐El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi‐POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi‐POVM. The validity of this definition is discussed. In what follows, we introduce an operator version (s) of H (s) in a Hilbert space of infinite dimension using a universal semi‐POVM, and study its properties. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Real reduction theory

On a Borel bar space (E,B,-), the following concepts are introduced in a suitable way: measurable fields of real Hilbert spaces, real measurable fields of vectors, operators and Von Neumann (VN) algebras: ξ (⋅),a (⋅),M (⋅). Then a satisfactory real reduction theory is obtained: a real VN algebraM can be represented as a direct integral where each VN algebraM(t) in this field will be simpler.     

Topological quantum field theory with corners based on the Kauffman bracket

We describe the construction of a topological quantum field theory with corners based on the Kauffman bracket, that underlies the smooth theory of Lickorish, Blanchet, Habegger, Masbaum and Vogel. We also exhibit some properties of invariants of 3-manifolds with boundary. Received: March 4, 1996     

Toward a vertex operator construction of quantum affine algebras

We describe a construction of the quantum-deformed affine algebras using vertex operators in the free field theory. We prove the Serre relations for the Borel subalgebras of quantum affine algebras; in particular, we consider the case in detail. We also construct the generators corresponding to the positive roots of . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 240–248, February, 2008.     

从投掷问题到概率论的创立

点数问题的解决是概率论创立的标志.该问题最终由帕斯卡和费马圆满解决,正是这些新思想奠定了概率论基础.惠更斯的<论赌博中的计算>第一次把概率论建立在公理、命题和问题上而构成较完整的理论体系.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

We show that in contrast to a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. We consider an approximation based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the second-order term. To escape technical difficulties related to the infinite dimensionality of the phase space for quantum mechanics, we consider finite-dimensional quantum mechanics. On one hand, this is a simple example with high pedagogical value. On the other hand, quantum information operates in a finite-dimensional state space. Therefore, our investigation can be considered a construction of a classical statistical model for quantum information. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 278–291, August, 2007.     

The quantum double of a factorizable weak Hopf algebra

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k. We first construct a weak Hopf algebra [Δ(1)(H?H)Δ(1)]^R, which is based on the subalgebra of the tensor product algebra H?H. Next we verify that if H is factorizable, then the Drinfeld’s quantum double of H is isomorphic to [Δ(1)(H?H)Δ(1)]^R.