Wigner distributions for finite dimensional quantum systems: An algebraic approach

We discuss questions pertaining to the definition of ‘momentum’, ‘momentum space’, ‘phase space’ and ‘Wigner distributions’; for finite dimensional quantum systems. For such systems, where traditional concepts of ‘momenta’ established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail     

Infinity and Newton’s Three Laws of Motion

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible.     

Reply to “quantum tunneling times: A crucial test for the causal program?”

Because Bohm’s Interpretation models particles with continuous trajectories, a natural property to attribute to a Bohmian particle is atunneling time, the time it takes for a particle to pass through a barrier. We also attribute a property-a different property-named ‘tunneling time’ to Copenhagen systems, systems that do not have particles with continuous trajectories. Cushing presents a discussion of the possibility of measuring Bohmian particle tunneling time; however, as becomes clear when considering the differences between properties named ‘tunneling time,’ he incorrectly argues that if such a measurement were possible, the measurement might constitute an empirical test between the Copenhagen interpretation and Bohm’s interpretation.     

Modification of Lytle’s theory: Interpretation of the extended fine structure associated with LIII absorption discontinuity of some ytterbium systems

In Lytle’s theory for the extended fine structure in x-ray absorption spectra, the potential at the boundary of the ‘equivalent sphere’ around the absorbing atom, having volume equal to that of the Wigner-Seitz cell is considered to be infinite. It has been observed that Lytle’s theory is applicable only in the case of metals and metallic systems. In the present paper the extended fine structure associated with the L_III absorption spectra of some systems of ytterbium is interpreted on the basis of Lytle’s model, modified by using a finite potential instead of an infinite one at the boundary of the equivalent sphere. The values of this potential are estimated for eight systems of ytterbium. It has been shown that there exists a correlation between the potentials and covalency of the compounds.     

Generalized Fock spaces,new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ‘infinite’, Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are: new algebras for infinite statistics,q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ‘doubly-infinite’ statistics, many representations of orthostatistics, Hubbard statistics and its variations.     

Turbulence. 2. Pfaff’s systems in turbulence models

Based on the classification of dynamic coordinates presented in Part 1 of this work and analogy with the classical systems constrained by nonholonomic coupling, Pfaff’s systems of turbulent dynamics are constructed. A method of constructing trajectory bundles for particles forming a vortex sheet is described. Thermodynamic interpretation of Pfaff’s coefficients is suggested.     

Proof of absence of spooky action at a distance in quantum correlations

I prove that there is no spooky action at a distance and nonlocal state-reduction during measurements on quantum entangled systems. The prediction of quantum theory as well as experimental results are in conflict with the concept of nonlocal state-reduction, as conclusively shown here under very general assumptions. This has far-reaching implications in the interpretation of quantum mechanics in general, and demands a radical change in its present interpretation of measurements on entangled multiparticle systems. Motivated by these results we re-examine Bell’s theorem for correlations of entangled systems and find that the correlation function used by Bell fails to incorporate phase correlations at source. It is the use of such an unphysical correlation function, and not failure of locality, that leads to the Bell’s inequalities.     

Systems of Hess-Appel’rot Type

We construct higher-dimensional generalizations of the classical Hess- Appel’rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the “class of systems of Hess-Appel’rot type”.     

Algebraic structure of Nambu mechanics

Abstracting from Nambu’s work [1] on the generalization of Hamiltonian mechanics, we obtain the concept of a classical Nambu algebra of type I (CNA-I). Consistency requirement of time evolution of the trilinear Nambu bracket leads to a new five point identity (FPI). Incorporating the FPI into CNA-I, we obtain a classical Nambu algebra of type II (CNA-II). Nambu’s algorithm for generalized classical mechanics turns out to be compatible with CNA-II. Tensor product composition of two CNA-I’s results in another CNA-I whereas that of two CNA-II’s does not. This implies that interacting systems cannot be consistently treated in Nambu’s framework. It is shown that the recent generalization of Nambu mechanics based on an arbitrary Lie group (instead of the particular case of the rotation group as in the case of Nambu’s original algorithm) suggested by Biyalinicki-Birula and Morrison [2], is compatible with CNA-I but not with CNA-II. Relaxation of the commutative and associative observable product by making it nonassociative so as to arrive at the quantum counterpart meets with serious difficulties from the view point of tensor product composition property. Thus neither CNA-I nor CNA-II have quantum counterparts. Implications of our results are discussed with special reference to existing work on Nambu mechanics in the literature.     

On the Gaussian Approximation for Master Equations

We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen’s expansion approach (the fact that the probability distribution is Gaussian at first order). We analyze the scaling of the error with a large parameter of the system and compare it with van Kampen’s method. Our theoretical analysis and the study of several examples shows that the Gaussian approximation turns out to be more accurate than van Kampen’s expansion at first order. This could be specially important for problems involving stochastic processes in systems with a small number of particles.     

Analysis of protein folds using protein contact networks

Proteins are important biomolecules, which perform diverse structural and functional roles in living systems. Starting from a linear chain of amino acids, proteins fold to different secondary structures, which then fold through short- and long-range interactions to give rise to the final three-dimensional shapes useful to carry out the biophysical and biochemical functions. Proteins are defined as having a common ‘fold’ if they have major secondary structural elements with same topological connections. It is known that folding mechanisms are largely determined by a protein’s topology rather than its interatomic interactions. The native state protein structures can, thus, be modelled, using a graph-theoretical approach, as coarse-grained networks of amino acid residues as ‘nodes’ and the inter-residue interactions/contacts as ‘links’. Using the network representation of protein structures and their 2D contact maps, we have identified the conserved contact patterns (groups of contacts) representing two typical folds — the EF-hand and the ubiquitin-like folds. Our results suggest that this direct and computationally simple methodology can be used to infer about the presence of specific folds from the protein’s contact map alone.      

Reichenbachian Common Cause Systems Revisited

According to Reichenbach’s principle of common cause, positive statistical correlations for which no straightforward causal explanation is available should be explained by invoking the action of a hidden conjunctive common cause. Hofer-Szabó and Rédei’s notion of a Reichenbachian common cause system is meant to generalize Reichenbach’s conjunctive fork model to fit those cases in which two or more common causes cooperate in order to produce a positive statistical correlation. Such a generalization is proved to be unsatisfactory in the light of a probabilistic conception of causation. Accordingly, an alternative model for systems of multiple common causes is offered, which is capable of emulating the explanatory efficacy of Reichenbachian common cause systems, while overcoming their major conceptual shortcomings at the same time.     

Quasiparticles in the theory of fermion condensation

It is shown that Landau’s quasiparticle formalism continues to work in systems with a fermion condensate. In the case of a finite system this formalism is suitable for describing the restructuring of states at the Fermi surface. It also works in an infinite system, and the idea of quasiparticles at low temperature as well-defined excitations at the Fermi surface remains valid. The quasiparticle lifetime is directly proportional to the temperature, and the density of states is inversely proportional to the temperature. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 9, 719–723 (10 May 1996)     

Factorised steady states and condensation transitions in nonequilibrium systems

Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium —for example phase transitions in one-dimensional systems. In this talk I will review a simple model of a nonequilibrium system known as the ‘zero-range process’ and its recent developments. The nonequilibrium stationary state of this model factorises and this property allows a detailed analysis of several ‘condensation’ transitions wherein a finite fraction of the constituent particles condenses onto a single lattice site. I will then consider a more general class of mass transport models, encompassing continuous mass variables and discrete time updating, and present a necessary and sufficient condition for the steady state to factorise. The property of factorisation again allows an analysis of the condensation transitions which may occur.     

All basic condensed matter physics phenomena and notions mirror in biology — A hypothesis,two examples and a novel prediction

A few billion years of evolutionary time and the complex process of ‘selection’ has given biology an opportunity to explore a variety of condensed matter phenomena and situations, some of which have been discovered by humans in the laboratory, that too only in extreme non-biological conditions such as low temperatures, high purity, high pressure etc., in the last centuries. Biology, at some level, is a complex and self-regulated condensed matter system compared to the ‘inanimate’ condensed matter systems such as liquid ⁴He, liquid water or a piece of graphite. In this article I propose a hypothesis that ‘all basic condensed matter physics phenomena and notions (already known and ones yet to be discovered) mirror in biology’. I explain this hypothesis by considering the idea of ‘Bose condensation’ or ‘momentum space order’ and discuss two known example of quantum magnetism encountered in biology. I also provide some new and rather speculative possibility, from light harvesting in biological photosynthesis, of mesoscopic excition condensation related phenomena at room temperature.     

Application of green functions to finite Fermi systems with fixed numbers of particles and strong pairing

The single-particle Green function formalism (Gor’kov’s method) is applied to calculate the energies of finite superconducting Fermi systems with a finite number of particles. It is shown that strong pairing leads to equations similar to those from the Bardeen-Cooper-Schrieffer theory.     

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.     

The effects of time delay on the decline and propagation processes of population in the Malthus-Verhulst model with cross-correlated noises

This contribution presents a derivation of the steady-state distribution of velocities and distances of driven particles on a onedimensional periodic ring, using a Fokker-Planck formalism. We will compare two different situations: (i) symmetrical interaction forces fulfilling Newton’s law of “actio = reactio” and (ii) asymmetric, forwardly directed interactions as, for example in vehicular traffic. Surprisingly, the steady-state velocity and distance distributions for asymmetric interactions and driving terms agree with the equilibrium distributions of classical many-particle systems with symmetrical interactions, if the system is large enough. This analytical result is confirmed by computer simulations and establishes the possibility of approximating the steady state statistics in driven many-particle systems by Hamiltonian systems. Our finding is also useful to understand the various departure time distributions of queueing systems as a possible effect of interactions among the elements in the respective queue [Physica A 363, 62 (2006)].