Statistical mechanics of quantum mechanical particles with hard cores

The states of a quantum mechanical system of hard core particles are characterized as a convex weak *compact subset of the states over aC* algebra associated with the canonical (anti-) commutation relations. It is shown that the mean conditional entropy, i.e. entropy minus energy, can be defined as an affine upper semi-continuous function over theG-invariant hard core states whereG is an invariance group containing space translations. An abstract definition of the pressure and equilibrium states is given in terms of the maximum of the conditional entropy and it is shown that the pressureP _S obtained in this way satisfiesPP _S P whereP andP are the thermodynamic pressures obtained from the usual Gibbs formalism with elastic wall, and repulsive wall, boundary conditions respectively. A number of additional results concerning the equilibrium states are also given.     

The maximum entropy principle as a consequence of the principle of Laplace

The maximum entropy principle states that the probability distribution which best represents our information is the one which maximizes the entropy with the given evidence as constraints. We prove that this principle is implied from the Laplace principle of equiprobabilities applied to the setS of allN-term sequences of results which are compatible with the given evidence. We generalize to the information gain method of Kullback.     

Differential entropy and tiling

This paper relates the differential entropy of a sufficiently nice probability density functionp on Euclideann-space to the problem of tilingn-space by the translates of a given compact symmetric convex setS with nonempty interior. The relationship occurs via the concept of the epsilon entropy ofn-space under the norm induced byS, with probability induced byp. An expression is obtained for this entropy as approaches 0, which equals the differential entropy ofp, plusn times the logarithm of 2/, plus the logarithm of the reciprocal of the volume ofS, plus a constantC(S) depending only onS, plus a term approaching zero with. The constantC(S) is called the entropic packing constant ofS; the main results of the paper concern this constant. It is shown thatC(S) is between 0 and 1; furthermore,C(S) is zero if and only if translates ofS tile all ofn-space.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Measures on the Splitting Subspaces of an Inner Product Space

Let S be an inner product space and let E(S) (resp. F(S)) be the orthocomplemented poset of all splitting (resp. orthogonally closed) subspaces of S. In this article we study the possible states/charges that E(S) can admit. We first prove that when S is an incomplete inner product space such that dim S/S < , then E(S) admits at least one state with a finite range. This is very much in contrast to states on F(S). We then go on showing that two-valued states can exist on E(S) not only in the case when E(S) consists of the complete/cocomplete subspaces of S. Finally we show that the well known result which states that every regular state on L(H) is necessarily -additive cannot be directly generalized for charges and we conclude by giving a sufficient condition for a regular charge on L(H) to be -additive.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

An Index for 4 Dimensional Super Conformal Theories

We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S ³ × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS ₅ × S ⁵ and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS ₅, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the theory. We note that entropy for some small supersymmetric AdS ₅ black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the Yang Mills theory.     

Optimal Concentration for SU(1, 1) Coherent State Transforms and An Analogue of the Lieb-Wehrl Conjecture for SU(1, 1)

We derive a lower bound for the Wehrl entropy in the setting of SU(1, 1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1, 1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1, 1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on . To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on . Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. Work partially supported by U.S. National Science Foundation grant DMS 06-00037.     

Hyperon decays and quark-quark correlations

The nonleptonic weak Hamiltonian strongly favours the annihilation and creation of correlated quark pairs in spin zero and colour antitriplet states. It is demonstrated that bothS- andP-wave nonleptonic hyperon decays are well described by this mechanism if supplemented by factorization contributions. Apart from the use of mass values and the -nucleon coupling constant, the results are obtained in a parameter free way. Since the diquark properties applied are identical to the ones used in the calculation ofK2 decays and the long range contribution to theK _L –K _S mass difference, a coherent physical picture of dominant nonperturbative effects in the low energy domain is reached.Supported in part by the Bundersministerium für Forschung und Technologie, Bonn, FRG     

An analysis of the mass formulae forS- andP-wave mesons

Mass regularities forS- andP-wave mesons and relations between their masses are discussed. A detailed analysis is given forS-wave mesons which extends the investigations onP-wave mesons reported earlier. Masses for theS- andP-states of all interesting -systems (including toponium states) are predicted. Partial understanding of the mass formulae is obtained within a general potential model approach. Scaling arguments are presented which support the empirical scaling behaviour found for the expectation values determining the spin-splittings in the potential picture.     

Essay: The Holography of Gravity Encoded in a Relation Between Entropy, Horizon Area, and Action for Gravity

I provide a general proof of the conjecture that one can attribute an entropy to the area of any horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature T = ^–1 and evaluating the exact partition function Z(). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z exp[S – E], where S = (1/4)4 a ² and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic.     

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS ² classical spins asL] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has diagonal form).Supported by USNSF Grant MCS-78-01885     

Effect of Phase-Sensitive Reservoir on the Decoherence of Pair-Cat Coherent States

We study the decoherence of a superposition of four coherent states under the action of a phase sensitive reservoir. We verify that the decoherence times _k,l, k,l=1,2,3,4, between any two coherent states of the superposition can be controlled through the reservoir parameters. The decoherence time between two components of any pair, for instance _1,2 or _3,4, can be significantly increased, compared with the decoherence time when the state is acted by a thermal reservoir. However, this occurs at the expense of decreasing the decoherence time between the ``cat states" (1,2) and (3,4). This can be useful in quantum computation.     

A Kinetic Model of Quantum Jumps

A new class of models describing the dissipative dynamics of an open quantum system S by means of random time evolutions of pure states in its Hilbert space is considered. The random evolutions are linear and defined by Poisson processes. At the random Poissonian times, the wavefunction experiences discontinuous changes (quantum jumps). These changes are implemented by some non-unitary linear operators satisfying a locality condition. If the Hilbert space of S is infinite dimensional, the models involve an infinite number of independent Poisson processes and the total frequency of jumps may be infinite. We show that the random evolutions in are then given by some almost-surely defined unbounded random evolution operators obtained by a limit procedure. The average evolution of the observables of S is given by a quantum dynamical semigroup, its generator having the Lindblad form.⁽¹⁾ The relevance of the models in the field of electronic transport in Anderson insulators is emphasised.     

Entropy balance in “pure” interactions of open quantum systems

Processes are considered in which a statistical ensemblew of quantum systems is split into ensembles, or channels (w _i ), conditional to the occurrence, with respective probabilities (p _i ^w ), of associated macroscopic effects. These processes are described here by a family of operationsT _i :w p _i ^w w _i ^J , which remarkably generalize the usual state reductions of the nondestructive measurements. In a previous work it was proved that the microscopic entropy of the given open system decreases or at most remains constant if all theT _i are pure operations, i.e., they transform pure states into pure states; it is proved here that the increase in entropy of the external world, computed asS ^Jm (w)=– _i p _i ^w lg _i ^w , is sufficient to compensate for such an entropy decrease whenever theT _i are all pure operations of the first kind, whereas whenever someT _i is pure of the second kind (or nonpure, too), the total entropy, computed as above, may decrease.     

On the 2S meson nonets

A scheme is proposed to identify the first radial excitations of the well established pseudoscalar and vector meson nonets. The states nominated all lie in the mass rangem=1.2–1.5 GeV; and each finds some occurrence in experimental literature, although much confirmation is needed. Identification is aided by a simple model that depends heavily onm ² differences among the states. This model gives a good fit to the ground (1S) states; with one exception, no surprises are found on extension to 2S systems. The exception is the comparative absence till now of observed widths for the 2S pseudoscalars in the predicted range of 2 5 keV. Further investigation is desirable but faces severe difficulties with experimental statistics.     

Quantum qualitative dynamics

We describe our work on qualitative methods for visualizing the quantum eigenstates of systems with nonlinear classical dynamics. For two-degree-of-freedom systems, our approach is based on the use of generalized coherent states, and allows systems with nonoscillator kinematics to be investigated. The general approach is illustrated with two examples involving vibration-rotation interaction in polyatomic molecules. We apply the coherent states of the Lie groupH ₄SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, andSU(2)SU(2) coherent states to study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction. In both systems, quantum states are visualized on the rotational surface of section and compared with the corresponding classical phase space structure. Striking classical-quantum correspondence is observed. We then describe recent results on the quantum states of (N 3)-dimensional systems of coupled nonlinear oscillators, which reveal a quantum delocalization that is reminiscent of classical Arnold diffusion.     

On quantum systems in thermal contact

It is shown that under rather general conditions two K.M.S. states ₁ and ₂ of systemsS ₁ andS ₂, respectively, can be simultaneously extended to a K.M.S. state of a system composed ofS ₁ andS ₂, provided both systems have equal temperatures. This result gives further support to the conjecture that K.M.S. states are equilibrium states. In the second part, a model of thermal coupling is constructed which satisfies the assumptions of the first part, thereby showing that the result is also valid in the interesting case of systemsS ₁ andS ₂ in thermal contact.     

A Class of k-Quantum Nonlinear Coherent States and Some of Their Properties

A class of k-quantum nonlinear coherent states, i.e., the k eigenstates of the powers B ^k (k 3) of the annihilation operator of f-oscillators, is obtained and its completeness is investigated. An alternative method to construct them is proposed. We introduce a new kind of higher-order squeezing and sub-Poissonian distribution. The quantum statistical properties of the k states are studied. The result shows that all of the eigenstates can be generated by a linear superposition of k Roy-type nonlinear coherent states. These states may form a complete Hilbert space, and the M-th order [M = (n + 1/2)k; n = 0,1, ...] squeezing effects exist in all of the k states when k is even. There is the sub-Poissonian distribution in all of the states.     

Entanglement and Density Matrix of a Block of Spins in AKLT Model

We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension . This subspace is described by non-zero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to . NSF Grant DMS-0503712.