Entropy andp-particle observables. I. The general program

An information-theoretic notion of entropy is proposed 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 some numberp of the particles but is completely ignorant of the form of any higher-order correlations involving more thanp particles. The idea is to define a generic measure of entropyS[ ] = –Tr log for an arbitrary density matrix or distribution function , and then, given the trueN-particle, to define a reduced _R ^P which reflects the observer's partial knowledge. The result, at any timet, is a chain of inequalitiesS[ _R ¹ ]S[ _R ² ]...S[ _R ^N ]S[], with true equalityS[ _R ^p ]=S[ _R ^p+1 ] if and only if the true factorizes exactly into a product of contributions involving all possiblep-particle groupings. It follows further than (1) if, at some initial timet ₀, the true factorizes in this way, thenS[ _R ^p (]S[ _R ^p (t ₀)] for all finite timest>t ₀, with equality if and only if the factorization is restored, and (2) the initial response of the system must be to increase itsp-particle entropy.     

A non-relativistic model of two-particle decay II. Reduced resolvent

We continue discussion of the Lee-type decay model described in the first part of this paper. After separation of the centre-of-mass motion, we deduce meromorphic structure of the reduced resolvent for small values of the coupling constant.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Time Evolution in Macroscopic Systems. II. The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.     

Local observables and particle statistics II

Starting from the principles of local relativistic Quantum Theory without long range forces, we study the structure of the set of superselection sectors (charge quantum numbers) and its implications for the particle aspects of the theory. Without assuming the commutation properties (or even the existence) of unobservable fields connecting different sectors (charge-carrying fields), one has a particle-antiparticle symmetry, an intrinsic notion of statistics for identical particles, and a spin-statistics theorem. Particles in pseudoreal sectors cannot be their own antiparticles (a variant of Carruthers' theorem). We also show how scattering states and transition probabilities are obtained in this frame.Partly supported by CNR.     

Fields,observables and gauge transformations II

We wish to study the construction of charge-carrying fields given the representation of the observable algebra in the sector of states of zero charge. It is shown that the set of those covariant sectors which can be obtained from the vacuum sector by acting with localized automorphisms has the structure of a discrete Abelian group. An algebra of fields can be defined on the Hilbert space of a representation of the observable algebra which contains each of the above sectors exactly once. The dual group of acts as a gauge group on in such a way that is the gauge invariant part of is made up of Bose and Fermi fields and is determined uniquely by the commutation relations between spacelike separated fields.     

Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–36, August, 2007.     

Casimir effects: An optical approach II. Local observables and thermal corrections

We recently proposed a new approach to the Casimir effect based on classical ray optics (the “optical approximation”). In this paper we show how to use it to calculate the local observables of the field theory. In particular, we study the energy–momentum tensor and the Casimir pressure. We work three examples in detail: parallel plates, the Casimir pendulum and a sphere opposite a plate. We also show how to calculate thermal corrections, proving that the high temperature ‘classical limit’ is indeed valid for any smooth geometry.     

Entropy concept and ordering of states. II

In this paper, the second of two, the Second Law of Thermodynamics is again regarded primarily as an essential link in an argument leading directly to the existence of a certain order amongst the states of an adiabatically enclosed system, and to the consequent definition of a continuous empirical entropy function. The mode of reasoning is now topological in character. To achieve the desired result the following physically acceptable statement is assumed to be true: “If a state\(\mathfrak{S}\) of an adiabatically enclosed system is inaccessible from a state\(\mathfrak{S}'\) then every state in a certain neighbourhood of\(\mathfrak{S}'\) is inaccessible from every state in a certain neighbourhood of\(\mathfrak{S}\).” A clear-cut separation is thus effected between the contents of the Second Law on the one hand and that of the other principal Laws of Thermodynamics on the other, the First Law in particular no longer being invoked at all.     

A non-relativistic model of two-particle decay III. The pole approximation

In the third part of the paper, we are concerned mostly with the problem of justifying the approximation in which the reduced resolvent is replaced by the pole term alone. Imposing additional regularity assumptions on the function, which specifies the interaction, we are able to estimate the difference of the corresponding reduced propagators. This result is used further to derive an estimate of the deviations from the exponential decay law which results from the pole approximation. With exception of very small and very large times, the obtained bound is proportional to fourth power of the coupling constant. We prove also Fermi golden rule for the model under consideration, and compare the present method to the one previously used by Demuth.     

Quantum mechanics as relativistic statistics. I: The two-particle case

It is shown that non-relativistic quantum mechanics can be treated as a kind of relativistic statistical theory, which describes the indeterministic motion of classical particles. The theory is relativistic in the sense that the relativistic notion of the state and two-time equations of motion are used. The principles and relations of quantum mechanics are obtained from the principles of statistics and those of classical mechanics.     

Entropy Production in Thermostats II

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies for every 2-plane σ, where K _w is the sectional curvature of the associated Weyl connection and is the orthogonal projection of E onto σ. Then the entropy production of any SRB measure is positive unless E has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.     

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.     

Calculable observables in quantum chromodynamics: (II). Heavy quark production

It is argued that the color-averaged inclusive cross section dσ/d³p for the production of a heavy quark of given momentum p and given flavor is infrared-singularity free and therefore calculable in perturbative QCD if it is evaluated away from heavy quark thresholds and if the typical reaction energy Q is such that α_s(Q)ln[Q/m(Q)] is small, α_s and m being the strong coupling constant and the heavy quark mass respectively. An interpretation of this cross section is proposed in terms of heavy flavor hadronic production.     

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.     

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.