Semimodularity and the logic of quantum mechanics

If (, ,P, ) is an event-state-operation structure, then the events form an orthomodular ortholattice (, , ) and the operations, mappings from the set of states into , form a Baer *-semigroup (S, , *, ). Additional axioms are adopted which yield the existence of a homomorphism from (S, , *, ) into the Baer *-semigroup (S(), , *, ) of residuated mappings of (, , ) such thatx S maps states while _x S () maps supports of states. If (, , ) is atomic and there exists a correspondence between atoms and pure states, then the existence of provides the result: (, , ) is semimodular if and only if every operationx S is a pure operation (maps pure states into pure states).Supported in part by the United States Atomic Energy Commission and in part by the Fonds National Suisse.     

On log-Sobolev inequalities for infinite lattice systems

For a system on an infinite lattice, we show that a Gibbs measure for a smooth local specification ={E } satisfying the Dobrushin uniqueness theorem also satisfies log-Sobolev inequality, provided it is satisfied for one-dimensional measures E _l .     

Generalized solutions of the radiative transfer equations in a singular case

This paper is devoted to the study of the radiative transfer equations: First, we prove a global existence theorem, which allows a blow-up of the opacity _v() when 0. Thus, it extends Mercier's previous result [13]. This proof relies mainly on a nonlinear version of Hille-Yosida theorem: see Crandall-Ligett [9].Then, we prove the uniqueness of the semigroup solving (TR), and some regularity results (in the class of functions with bounded variation).Finally, we prove the convergence of some splitting algorithms associated to (TR).     

Zeta function continuation and the Casimir energy on odd and even dimensional spheres

The zeta function continuation method is applied to compute the Casimir energy on spheresS ^N. Both odd and even dimensional spheres are studied. For the appropriate conformally modified Laplacian the Casimir energy is shown to be finite for all dimensions even though, generically, it should diverge in odd dimensions due to the presence of a pole in the associated zeta function (s). The residue of this pole is computed and shown to vanish in our case. An explicit analytic continuation of (s) is constructed for all values ofN. This enables us to calculate exactly and we find that the Casimir energy vanishes in all even dimensions. For odd dimensions is never zero but alternates in sign asN increases through odd values. Some results are also derived for the Casimir energy of other operators of Laplacian type.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

Classical foundations of quantum logic

We construct a languageL for a classical first-order predicate calculus with monadic predicates only, extended by means of a family of statistical quantifiers. Then, a formal semantic model is put forward forL which is compatible with a physical interpretation and embodies a truth theory which provides the statistical quantifiers with properties that fit their interpretation; in this framework, the truth mode of physical laws is suitably characterized and a probability-frequency correlation principle is established. By making use ofL and , a set of basic physical laws is stated that hold both in classical physics (CP) and in quantum physics (QP), which allow the selection of suitable subsets of primitive predicates ofL (the set _P of pure states; the sets _o and _E of operational and exact effects, respectively) and the introduction on these subsets of binary relations (a preclusion relation # on _P , an order relation < on _E . By assuming further physical laws, ( _E , <) turns out to be a complete orthocomplemented lattice [mixtures and atomicity of ( _E , <) also can be introduced by means of suitable physical assumptions]. Two languagesL _E ^x andL _E ^S are constructed that can be mapped intoL; the mapping induces on them mathematical structures, some kind of truth function, an interpretation. The formulas ofL _E ^x can be interpreted as statements about properties of a physical object, and the truth function onL _E ^x is two valued. The formulas ofL _E ^S can be endowed with two different interpretations as statements about the frequency of some physical property in some class (state) of physical objects; consequently, a two-valued truth function and a multivalued fuzzy-truth function are defined onL _E ^S . In all cases the algebras of propositions of these logics are complete orthocomplemented lattices isomorphic to ( _E , <). These results hold both in CP and in QP; further physical assumptions endow the lattice ( _E , <), henceL _E ^x andL _E/S , with further properties, such as distributivity in CP and weak modularity and covering law in QP. In the latter case,L _E ^S andL _E ^S , together with their interpretations, can be considered different models of the same basic mathematical structure, and can be identified with standard (elementary) quantum logics. These are therefore founded on the classical extended languageL with semantic model .     

Letter: Cosmic Rays and Large Extra Dimensions

We have proposed that the cosmic ray spectrum knee, the steepening of the cosmic ray spectrum at energy E 10^15.5 eV, is due to new physics, namely new interactions at TeV cm energies which produce particles undetected by the experimental apparatus. In this letter we examine specifically the possibility that this interaction is low scale gravity. We consider that the graviton propagates, besides the usual four dimensions, into an additional , compactified, large dimensions and we estimate the graviton production in pp collisions in the high energy approximation where graviton emission is factorized. We find that the cross section for graviton production rises as fast as ( /M _f)²⁺, where M _f is the fundamental scale of gravity in 4 + dimensions, and that the distribution of radiating a fraction y of the initial particle's energy into gravitational energy (which goes undetected) behaves as y ^{– 1}. The missing energy leads to an underestimate of the true energy and generates a break in the inferred cosmic ray spectrum (the knee). By fitting the cosmic ray spectrum data we deduce that the favorite values for the parameters of the theory are M _f 8 TeV and = 4.     

Quasi-Periodic Solutions for Two-Level Systems

We consider the Schrödinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small, and we study the problem of finding quasi-periodic solutions of a related generalized Riccati equation. We prove the existence of quasi-periodic solutions of the latter equation for a Cantor set of values of around the origin which is of positive Lebesgue measure: such solutions can be obtained from the formal power series by a suitable resummation procedure. The set can be characterized by requesting infinitely many Diophantine conditions of Melnikov type.     

A Generalized Hypergeometric Function II. Asymptotics and <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">4</Emphasis></Subscript> Symmetry

In previous work we introduced and studied a function that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function , with parameters ⁴ related to the couplings c⁴ by a shift depending on a ₊ , a _– . We show that the -function is invariant under all maps w(), with w in the Weyl group of type D ₄ . Choosing a ₊ , a _– positive and real, we obtain detailed information on the |Re v| asymptotics of the -function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R.     

Embedding of posets into lattices in quantum logic

It has been proposed that some posets of quantum logic could be embedded into lattices in order to recover the lattice structure avoiding the introduction of ad hoc axioms. We consider here the embedding _s of any posetS into the complete lattice _s of its closed ideals (normal embedding ofS) and show that _s can be characterized (up to a lattice isomorphism) either by means of a density property or by means of a minimality property. Both of these suggest that the normal embedding satisfies some intuitive conditions which make it preferable with respect to other possible embeddings ofS. We consider the poset of all the effects associated to yes-no experiments and briefly comment on the application of the normal embedding in this case. The possibility of giving a physical interpretation to the elements of is also discussed.Research sponsored by CNR and INFN (Italy).     

On diffusion equations for dynamical systems driven by noise

A unified formalism is presented to study Hamiltonian linear systems driven by noise. With this formalism, the phase averaging approximation, valid at weak noise, is easily performed. Already known results are straightforwardly recovered and new ones are obtained. After introducing this formalism on the exactly solvable one-degree-of-freedom problem with uncorrelated noise, one studies the corresponding exponentially correlated case. The validity of the approximate results thus obtained is considered by investigating the systematic weak-disorder expansion beyond the quasilinear approximation. In particular, it is argued that this expansion behaves uniformly for weak and large correlation time. The two-degrees-of-freedom problem is completely solved at the low-disorder approximation and this result is applied to the two-channel Anderson localization problem. The invariant measure and the two positive Lyapunov exponents are computed at all coupling between the channels. For systems withn degree of freedom the phase averaging leads to a Fokker-Planck equation for the measure in action space describing the system. However, it is argued that it is not solvable except in a special case which is explicitly displayed and solved. Nevertheless, in the large-n limit, it is possible to compute the largest Lyapunov exponent. Moreover, generalized Lyapunov exponents are calculated in this limit, and they do not exhibit a dispersion: in particular, log/log1, where is the energy of the system and where the brackets denote averaging over the noise. On the other hand, it is possible to compute at weak noise the sum of all the positive Lyapunov exponents. Taking into account all these results allows more insight on the whole spectrum of Lyapunov exponents.     

The Polyakov path integral over bordered surfaces (the open string amplitudes)

We use the Feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic BDHP string in a position representation. The consistent functional integral representation of the open string propagator is derived and evaluated. This result is used as a starting point for two kinds of constructions of the off-shell multiloop open string amplitudes. The general idea of the presented approach is to consider the off-shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space /₊.     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

Partition function zeros for the one-dimensional ordered plasma in Dirichlet boundary conditions

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacity in an applied electric fieldE with Dirichlet boundary conditions. The system has a phase transition from a low-coupling phase with equally spaced particles to a high-coupling phase with particles clustered into dipolar pairs. An exact expression for the partition function is developed. In zero applied field the zeros in the plane occupy the imaginary axis from –i to –i_c and i_c to i for some _c. They also occupy the diamond shape of four straight lines from ±i_c to _c and from ±i_c to –_c. The fugacity acts like a temperature or coupling variable. The symmetry-breaking field is the applied electric fieldE. A finite-size scaling representation for the partition in scaled coupling and scaled electric field is developed. It has standard mean field form. When the scaled coupling is real, the zeros in the scaled field lie on the imaginary axis and pinch the real scaled field axis as the scaled coupling increases. The scaled partition function considered as a function of two complex variables, scaled coupling and scaled field, has zeros on a two-dimensional surface in a domain of four real variables. A numerical discussion of some of the properties of this surface is presented.     

Muon catalyzed fusion: old and new aspects of energy production

Muon catalyzed fusion of deuterium and tritium (CF) yields the same energy gain per reaction as fusion with magnetic or inertial confinement (17.6 MeV). The crucial points of Cf are, however, very different, namely (a) the energy costW ₍₎ for production of one ^– and (b) the numbern of reactions a single muon can catalyze on the average. (b) is ultimately limited by theeffective sticking probability _f :n1/ _f. With standard methods one hasW ₍₎5 GeV, _f=0.5%. Hence a standard CF reactor can never reach a net energy gain. To solve this problem, ways discussed since about a decade are to increase the efficiency by both (i) energy multiplication using a fissionable blanket and (ii) breeding. A new way to increase the safety of fission devices mostly due to Yu. Petrov is outlined. On the other hand there is a hope to lowerW ₍₎ slightly and _f drastically, the latter by artificial reactivation. New theoretical results for beam cooling in an omegatron type driven integrated CF reactor, important forW ₍₎ and, in particular, _f, is presented.     

Geometric localization of the threshold in two-dimensional Ising ±J spin glasses forT=0

Following an approach of Toulouse, ground states in random 2D Ising ±J spin glasses (without external magnetic field), on square lattices, and with concentrations 0p0.5 of antiferromagnetic bonds are studied by means of minimal matchings of frustrated plaquettes. Lete(p) be the ground-state energy per spin in the thermodynamic limit. Then the well-known equatione(p)=–2+(p)f(p) holds, wheref(p) is the concentration of frustrated plaquettes and(p) is the average connection length between paired frustrated plaquettes in minimal matchings. Introducing (p) as the probability that a frustrated plaquette is matched to another frustrated plaquette by a connection of length (in a minimal matching), the average length(p) can be rewritten asgl(p)=(p). The study of(p) and its components (p) leads to an intervalp ^*pp ₂ (p ^*0.121±0.008,p ₂0.161±0.008) where the threshold between ferromagnet and paramagnet forT=0 lies. Analyzing a similar so-called adjoined average lengthl(p) admits further insight.     

Elastic constants,the ΔE effect,and retarded polarization processes in a ferroelectric niobate ceramic

A resonant method was used to determine the ultrasonic velocity in circular plates of a ferroelectric niobate ceramic having the composition (Pb_0.57Ba_0.43)Nb₂O₆, over the frequency range 3–9 MHz at 19 °C. The average values of the elastic constants are calculated for unpolarized samples. A study is made of the Young's modulus (the E effect) and of the polarization vector P as functions of the polarizing field and the time for which this field is applied. The E effect and P, which increase monotonically but nonlinearly with increasing , approach limiting values as functions of the polarization time in different manners. A comparative analysis of the E and polarization curves shows that the domain structure can be thought of as slightly different from that found from a scheme proposed for its behavior during the polarization process.Translated from Izvestiya VUZ. Fizika, No. 7, pp. 61–65, July, 1970.     

Gentle perturbations

By introducing a specific type of perturbation,A, in the Hamiltonian, we define a class of gently perturbed states, ,A, of a canonical ensemble, . The perturbations are chosen so as to preserve a relationship of the form ,A constant ×. Applications in ergodic theory and phase transitions are described.     

On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.     

Implementation of comparative probability by normal states. Infinite dimensional case

Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on .