Entropy andP-particle observables. II. The two-particle entropyS 2

This paper defines, and then evaluates perturbatively, an information-theoretic notion of entropyS ₂ for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving pairs of particles, but is completely ignorant of the form of any higher-order correlations involving three or more particles. By construction, thisS ₂(t) involves only the reduced two-particle distribution functions, or density matrices,f ₂(i,j) at timet, and, though the implementation of a subdynamics,dS ₂ (t)/dt can be realized in terms of thef ₂(i, j)'s at retarded timest–. A similar line of reasoning demonstrates that the most probable three-particlef ₃(i,j, k) consistent with a knowledge of thef ₂'s is precisely thatf ₃ suggested by the Kirkwood, or cluster, decomposition.     

DYNGA: a general purpose QM-MM-MD program. I. Application to water

We present a general purpose QM-MM-MD engine (DYNGA) designed to test alternative hybrid Hamiltonians geared towards the treatment of problems of interest in structural biology including the use of experimental data constraints. In this first presentation we use DYNGA to explore the behaviour of a traditional QM-MM approach in the treatment of the water—water interaction. We find the potential energy hypersurface for the water dimer computed with the HF 4–31G*/TIP3P hybrid Hamiltonian tends to be too flat. We also explore the effect of using traditional QM-MM techniques on proton wires and conclude there is a need for improvement, possibly addressed by using polarizable force fields.     

Local observables and particle statistics I

We consider the family of those states which become asymptotically indistinguishable from the vacuum for observations in far away regions of space. The pure states of this family may be subdivided into superselection sectors labelled by generalized charge quantum numbers. The principle of locality implies that within this family one may define a natural product composition (leading for instance from single particle states ton-particle states). Intrinsically associated with then-fold product of states of one sector there is a unitary representation ofP ⁽ⁿ⁾, the permutation group ofn elements, analogous in its role to that arising in wave mechanics from the permutations of the arguments of ann-particle wave function. We show that each sector possesses a statistics parameter which determines the nature of the representation ofP ⁽ⁿ⁾ for alln and whose possible values are 0, ±d ^–1 (d a positive integer). A sector with 0 has a unique charge conjugate (antiparticle states); if =d ^–1 the states of the sector obey para-Bose statistics of orderd, if =–d ^–1 they obey para-Fermi statistics of orderd. Some conditions which restrict to ± 1 (ordinary Bose or Fermi statistics) are given.     

Fields,observables and gauge transformations I

Starting from an algebra of fields and a compact gauge group of the first kind , the observable algebra is defined as the gauge invariant part of . A gauge group of the first kind is shown to be automatically compact if the scattering states are complete and the mass and spin multiplets have finite multiplicity. Under reasonable assumptions about the structure of it is shown that the inequivalent irreducible representations of (sectors) which occur are in one-to-one correspondence with the inequivalent irreducible representations of and that all of them are strongly locally equivalent. An irreducible representation of satisfies the duality property only if the sector corresponds to a 1-dimensional representation of . If is Abelian the sectors are connected to each other by localized automorphisms.On leave of absence from Instituto di Fisica G. Marconi, Università di Roma.     

Doubly metric formalism and observables in general relativity theory

The gravitational field is described as a standard physical field in two-dimensional space (a radical variant of doubly metric formalism). By a systematic execution of this approach, one is able to clarify certain difficulties inherent in doubly metric theories. The analysis of properties of a gravitational field in small spatial regions shows a close analogy to electrodynamics, which enables one to reduce a number of problems of system mechanics in the gravitational field to similar problems in electrodynamics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 69–74, March, 1977.     

Entropy concept and ordering of states. I

In an earlier paper it was proposed to regard the Second Law of Thermodynamics, in the form of the Principle ofCarathéodory, primarily as an essential link in an argument leading directly to the existence of a certain order amongst the states of any adiabatically enclosed system, and to the consequent definition of a continuous empirical entropy function. The previous line of reasoning was unfortunately afflicted with certain errors which require rectification. In the present paper, the first of two, this is achieved in the first place by falling back upon the existence of a continuous and differentiable energy function. This means that the full content of the First Law of Thermodynamics has been invoked. However, in order to go some way towards a clear separation between the contents of the First and Second Laws, it is then shown that the results already obtained may also be deduced by essentially similar arguments if one replaces the First Law by rather weaker assumptions.     

The noncommutative values of quantum observables

We discuss the notion of representing the values of physical quantities by the real numbers, and its limits to describe the nature to be understood in the relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values for a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutative algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex numbers, the idea makes reasonable sense theoretically as well as practically.     

Natural waves in azimuthally magnetized bigyrotropic medium I. The general case

A study is made of natural electromagnetic waves in media whose permittivity and magnetic permeability are described by complex tensors of the second rank. In the general case, the analytic solution obtained for the Maxwell equations constitutes a linear combination of four solutions of one type in the form of generalized series expansion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 18–22, April, 1979.     

The entropy of observables on quantum logic

The entropy of an abstract observable on quantum logic is defined as an informational property of the corresponding sublogic of a quantum logic associated with the physical system. The main properties of such quantity are stated. It is proved that the entropy is completely characterized by the entropies of the corresponding finite resolutions of the unit (experiments). The connection with the entropy of a state is also mentioned.     

Nonequilibrium statistical mechanics in the general theory of relativity I. A general formalism

This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.     

Metric-affine variational principles in general relativity. I. Riemannian space-time

Within the confines of conventional general relativity, variational principles are analyzed in which the metric tensor and the asymmetric linear connection are varied independently. The constraint that space-time remain Riemannian is introduced by means of the Lagrange multiplier technique. The Lagrange multiplier which effects this constraint, the hypermomentum current, is closely related to the constraint force which keeps space-time Riemannian and should be a measure for the violation of the Riemannian constraint at the microscopic level.Alexander von Humboldt Fellow.     

The CPT-theorem in two-dimensional theories of local observables

Let be a von Neumann algebra with cyclic and separating vector , and letU(a) be a continuous unitary representation ofR with positive generator and as fixed point. If these unitaries induce for positive arguments endomorphisms of then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translation only, can be imbedded into a theory of local observables covariant under the whole Poincaré group. This theory is also covariant under the CPT-transformation.     

Spin and torsion in general relativity: I. Foundations

From physical arguments space-time is assumed to possess a connection \(\Gamma _{ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} + S_{ij}^{ k} - S_{j i}^{ k} + S_{ ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} - K_{ij}^{ k} \) . \(\left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\}\) is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor \(S_{ij} ^k = \tfrac{1}{2}(\Gamma _{ij}^k - \Gamma _{ji}^k )\) and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above in this series of articles, we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing, in a unified way, gravitation and a very weak spin-spin contact interaction. In section 1 we start with the well-known dynamical definition of the energy-momentum tensor σ ^ij ～ δ?/δg _ij , where ? represents the Lagrangian density of matter (section1.1). In sections1.2,3 we will show that due to geometrical reasons, the connection assumed above leads to a dynamical definition of the spin-angular momentum tensor according to τ^k _ji ～ δ?/δK _ij ^k . In section1.4, by an ideal experiment, it will become clear that spin prohibits the introduction of an instantaneous rest system and thereby of a geodesic coordinate system. Among other things in section1.5 there are some remarks about the rôle torsion played in former physical theories. In section 2 we sketch the content of the theory. As in GR, the action function is the sum of the material and the field action function (sections2.1,2). The extension of GR consists in the introduction of torsion S _ij ^k as a new field. By variation of the action function with respect to metric and torsion we obtain the field equations in a general form (section2.3). They are also valid for matter described by spinors; in this case, however, one has to introduce tetrads as anholonomic coordinates and slightly to generalize the dynamical definition of energy-momentum (sections2.4,5).