On Fuzzy Probability Theory

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach [Demonestratio Mathematica 31(3), 1998, 235–254; Foundations of Physics, 30, 1663–1678] to fuzzy probability theory, the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and presenting an example that indicates that the converse is not true, we prove the strong law of large numbers according to this definition.     

On probability theory and probabilistic physics—Axiomatics and methodology

A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin of the Schrödinger equation is elucidated. Attention is given to the true meaning of the wave-corpuscle duality, and the incompleteness of nonrelativistic quantum mechanics is explained.     

Classical stark mixing at ultralow collision energies

Exact solutions of the time-dependent classical equations are obtained for the full array of angular momentum mixing transitions nl-->nl(') in atomic hydrogen induced by collisions with charged particles at ultralow energies. A novel classical expression for the transition probability P(l(')l) is presented. The exact classical results for P(l(')l)(alpha) as a function of l,l(') and the Stark parameter alpha agree exceptionally well with (exact) quantal results. They complement the quantal results by revealing essential characteristics which remain obscured in the quantal treatment.     

Localization for Yang-Mills Theory on the Fuzzy Sphere

We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

On Probability Domains II

We continue our studies of the foundation of probability theory using elementary category theory. We propose a classification scheme of probability domains in terms of cogenerators and their algebraic and topological properties and use the scheme to describe the transition from classical to fuzzy probability. We show that ?ukasiewicz tribes form a category of natural probability domains in which ??-fields of sets are ??minimal?? and measurable [0,1]-valued functions are ??maximal?? objects. The maximal objects form an epireflective subcategory in which both the classical and fuzzy probability can be modelled. This leads to a better understanding of the transition.     

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary * operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.     

奇异拉氏量系统的整体量子正则对称性质

从奇异拉氏量系统相空间路径积分的量子化形式出发，导出了系统在增广相空间整体变换下的广义正则Ｗａｒｄ恒等式和量子水平的守恒荷，一般这些守恒荷有别于经典Ｎｏｅｔｈｅｒ荷．给出了在杨-Ｍｉｌｓ场论中的应用，找到了新守恒荷     

De Broglian probabilities in the double-slit experiment

A new probability interpretation of interference phenomena in the double-slit experiment is proposed. It differs from the standard interpretation (based on elementary events happening in complementary, mutually exclusive setups—arrivals of waves to the screen when one of the slits is closed) which encounters the paradox that the law of total probability is violated. This new interpretation is free of such difficulties and paradoxes since it is based on compatible elementary events (events happening in the same setup in which happenall events considered—arrivals of quantons to the screen when both slits are open). Quantum objects—quantons—possess simultaneously particle and wave properties. Compatible statistical interpretation synthesizes in a consistent way the superposition principle for waves and the law of total probability applied to compatible events. Such synthesis is a theoretical expression of de Broglie's observation, now fully confirmed by experiments, that the interference fringes obtained on a photographic plate result from an infinite number of tiny local spots which display arrival of quantons, while the set of fringes is a statistical effect of the wave aspect.     

States as Morphisms

Using elementary categorical methods, we survey recent results concerning D-posets (equivalently effect algebras) of fuzzy sets and the corresponding category ID in which states are morphisms. First, we analyze the canonical structures carried by the unit interval I = [0,1] as the range of states and the impact of “states as morphisms” on the probability domains. Second, we analyze categories of various quantum and fuzzy structures and their relationships. Third, we describe some basic properties of ID and show that traditional probability domains such as fields of sets and bold algebras can be viewed as full subcategories of ID and probability measures on fields of sets and states on bold algebras become morphisms. Fourth, we discuss the categorical aspects of the transition from classical to fuzzy probability theory. We conclude with some remarks about generalized probability theory based on ID.     

Delinearization of quantum logic

The algebraic structure of the set of elementary observables of a delinearized quantal theory is described. As the delinearization procedure provides a kind of classical representation for any quantal theory, its relation to the traditional hypothesis of hidden variables is discussed.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

The global minimum of energy is not always a sum of local minima—A note on frustration

A classical lattice gas model with translation-invariant, finite-range competing interactions, for which there does not exist an equivalent translation-invariant, finite-range nonfrustrated potential, is constructed. The construction uses the structure of nonperiodic ground-state configurations of the model. In fact, the model does not have any periodic ground-state configurations. However, its ground-state—a translation-invariant probability measure supported by ground-state configurations—is unique.     

Electron tunneling through a barrier that depends linearly on time

We discuss the creation of a negative molecular ion via the decay of a self-captured electron state — the fluctuon — in a nonideal dipole-molecule gas. The creation probability of the negative ion is evaluated in the classical approximation using Green's function methods. We assume that the potential barrier through which the electron tunnels from the potential well of the fluctuon to that of the molecule depends linearly on time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 84–88, September, 1987.     

Finite Euler hierarchies and integrable universal equations

Recent work on Euler hierarchies of field theory Lagrangians iteratively constructed from their successive equations of motion is briefly reviewed. On the one hand, a certain triality structure is described, relating arbitrary field theories,classical topological field theories — whose classical solutions span topological classes of manifolds — and reparametrisation invariant theories — generalising ordinary string and membrane theories. On the other hand,finite Euler hierarchies are constructed for all three classes of theories. These hierarchies terminate withuniversal equations of motion, probably defining new integrable systems as they admit an infinity of Lagrangians. Speculations as to the possible relevance of these theories to quantum gravity are also suggested.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.The author would like to thank A. Morozov and especially D. B. Fairlie for a very enjoyable and stimulating collaboration, and the organisers of the Colloquium for their efficient organisation of a most pleasant and informative meeting. This work is supported through a Senior Research Assitant position funded by the S.E.R.C.     

Accurate integration of scalar equations in multiscalar-tensor theory

The integration of scalar equations in theories generalizing Brans—Dicke—Jordan—Fierz scalar—tensor theory is considered. Conditions under which these equations may be integrated by complete variable separation are established. Under these conditions, the scalar equations take the form of classical equations of motion for a single particle moving in scalar space in an external force field.Institute of High-Power Electronics, Siberian Branch, Russian Academy of Sciences. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 79–84, February, 1995.     

Statistical interpretation of the p-adic quantum field theory

A p-adic generalization of the frequency theory of probability is developed. Within the framework of this theory frequency meaning is imparted to probabilities belonging to the field of p-adic numbers. The Bargmann-Fock representation is constructed for the p-adic field theory. A frequency interpretation of quantum states in the Bargmann-Fock representation is proposed. The p-adic generalization is essentially an introduction of new quantum states which are meaningless from the point of view of the standard theory of probability based on Kolmogorov's axiomatics.Moscow Institute of Electronic Engineering. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 51–55, November, 1992.     

Variational principles of magnetostatics of superconductors. II

Simple examples are provided, where demonstrating the implementation of algorithms formulated by the variational principles of the first part of this paper is quite difficult, because even for the simplest body of practical interest —a loop — solving the problem by classical methods of potential theory is rather difficult. For this reason, presenting the corresponding results requires substantially more space, and obtaining these results can be considered as an independent study.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 103–109, September, 1991.