A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t _n =[(P ^n/p )₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{ ₀(z, ₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz ^p andA _c :=z/z–z+c. This requirement is satisfied, for arbitraryp, if ₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW as a double matrix Laplace transform in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour ^-+ ^- defined by ^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.     

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.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

On Relativistic Collisional Invariants

Two simple proofs of the result that a relativistic summational invariant is a linear combination of the momentum four-vector p are given by assuming that is a continuous and differentiable function of class C ². The results can be extended to the case when is just assumed to be a generalized function.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

ИССЛЕДОВАНИЕ НЕКОТО РЫХ СЛОЖНЫХ ПОЛУПРОВОДНИКОВЫХ Т ВДРДЫХ РАСТВОРОВ И СОЕДИНЕНИЙ НА ОСНОВ Е ИНДИЯ

A(III)B(V)-A₂(III)B₃(VI) - , , , In₂S₃, In₂Se₃ In₂Te₃, . , , : InP-In₂Se₃ — 50% , InAs-In₂Se₃ — 60% , InAs-In₂Te₃ — — . . , A(III)B(V). - - In₄SbTe₃, 6,12₈ Å. , .

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Investigation of some complex semi-conducting solid solutions and compounds on the basis of indium

In order to determine the possibility of preparing solid solutions on the principle of heterovalent substitution between semi-conducting compounds with undefective and defective structures A(III) B(V)-A₂(III) B₃(VI) systems on the basis of indium-indium phosphide, indium arsenide and indium antimonide on the one hand and In₂S₃, In₂Se₃ and In₂Te₃ on the other, are investigated.It was found by means of X-ray structure analysis and thermal and metallographical examinations that immediately after the synthesis, solid solutions with the structure of zinc blende are created in the following systems: InP-In₂Se₃ — in the range of 50% concentration of the initial compounds, InAs-In₂Se₃ — in the range of 60% concentration of the initial compounds, InAs-In₂Te₃ — unlimited solid solutions.Certain electric properties of the alloys within the range of existence of the solid solutions were measured.Regarding the rest of the systems, observable solubility is either completely missing or is limited within a narrow region near the compounds of the A(III) B(V)-type.In the indium-antimony-tellurium system a new compound, with empirical formula In₄SbTe₃ was discovered which showed an NaCl-type structure and lattice parameter equal to 6·12₈ Å. Preliminary study of the electrical characteristics concerning this compound leads to the conclusion that it possesses semi-conducting properties.

     

Cantor spectrum and singular continuity for a hierarchical Hamiltonian

We study the spectrum of the HamiltonianH onl ₂() given by (H)(n)=(n+1)+(n–1)+V(n)(n) with the hierarchical (ultrametric) potentialV(2 ^m (2l+1))=(1–R ^m )/(1–R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0<R<1,R=1 andR>1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese _n (1)<e _n (2)<...<e _n (2 ⁿ –1) of the Dirichlet problemH=E, (0)=(2 ⁿ )=0,n1. In the gap opening ate _n (k) the integrated density of states takes on the valuek/2 ⁿ . The spectrum is purely singular continuous forR1 when the potential is unbounded, and the Lyapunov exponent vanishes in the spectrum. The spectrum is purely continuous forR<1 in (H)[–2, 2] and =0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.Work supported by the Fonds National Suisse de la Recherche Scientifique, Grant No. 2.042-0.86 (H.K. and R.L.) and 2.483-0.87 (A.S.)On leave from the Dipartimento di Fisica, Università degli Studi di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Clifford algebras and Hestenes spinors

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl _1,3 M₂ (H). Hestenes invented first in 1966 hisideal spinors and later 1967/75 he recognized the importance of hisoperator spinors Cl _1,3 ⁺ M₂ (C).This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC Cl _1,3 andnot inC M₄ (R)=M₄ (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ₀=0, ₀₀₁₂₃=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.Projection operators P_±, _± are resettled among their new relatives in End(Cl _1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl _1,3 to the (graded) opposite algebra Cl _3,1 without resorting to complex numbers, that is, not using a replacement i.     

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).     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Contribution of the new vector mesons to the total cross section of hadron photoabsorption

The new vector mesons with mass 1.1 < M < 2 GeV as well as the ⁰,, ,, and mesons are included in the sum rules that relate the total cross section of hadron photoabsorption on the proton to the total cross sections for interaction of vector mesons with the proton. Satisfactory agreement with experiment is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 108–111, September, 1977.     

Fuzzy quantum logic II. The logics of unsharp quantum mechanics

A survey of the main results of the Italian group about the logics of unsharp quantum mechanics is presented. In particular partial ordered structures playing with respect to effect operators (linear bounded operatorsF on a Hilbert space such that, 0¦F²) the role played by orthomodular posets with respect to orthogonal projections (corresponding to sharp effects) are analyzed. These structures are generally characterized by the splitting of standard orthocomplementation on projectors into two nonusual orthocomplementations (afuzzy-like and anintuitionistic-like) giving rise to different kinds of Brouwer-Zadeh (BZ) posets: de Morgan BZ posets, BZ^* posets, and BZ³ posets. Physically relevant generalizations of ortho-pair semantics (paraconsistent, regular paraconsistent, and minimal quantum logics) are introduced and their relevance with respect to the logic of unsharp quantum mechanics are discussed.     

On the geometric structure underlying the eikonal equation

Given the eikonal equation _i=1 ³ (/x ⁱ ) ² =n ², we investigate the geometric structure that underlies the law of propagation of the wavefronts (x ¹,x ²,x ³) —ct=0. It turns out that Huygens' principle for the propagation of wavefronts is given in terms of a contact structure. Wavefronts are carried into wavefronts by contact transformations. As regards the wave-particle duality principle that arises in quantum mechanics, there is a natural geometric structure, a symplectic manifold (M ²ⁿ , ), which unifies Fermat's principle and the eikonal equation (Huygens' principle).On leave of absence from Institut für Angewandte Mathematik, Fachbereich Mathematik der Universität Mainz, Mainz, German Federal Republic.     

Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.     

Short distance scale invariance and high-energy processes in field theory

The dynamics of short distance interaction is considered from viewpoint of the quark field theory. If the effective coupling of quarks (the invariant charge(p ²)) decreases with distance (with increasingp ²) the field picture unifies different phenomenological models: reggeology in the region of smallp and partonology in the region of highp but(p ² ) lnp ² 1 and predict deviations with a subsequent increase ofp . A character of the deviations depends on the behaviour of whenp ². The variants of the asymptotic freedom (0) and scale invariance (g ₂ ⁰ ) are discussed. The strong suppression of—3 decay is considered as a support for the latter.Presented at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.     

The revival of Schrödinger's interpretation of quantum mechanics

A distinction is made between two wave functions(x) and(x), The former describing a continuous distribution of electronic matter for a single system, the latter describing the regularities in repeated experiments. The classical field(x) necessarily includes the self energy and accounts for all the radiative processes without the probability interpretation.     

The velocity autocorrelation function of a finite model system

We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particlesN and the length of the boxL approach infinity andN/L , the velocity autocorrelation function(t) is given simply by c² exp(–2ct@#@). For a finite system, the function _N(t) is periodic with period 2L/c. We also show that for more general velocity distribution functions (particles can have velocities ±c_i,i = 1,...), _N(t) is an almost periodic function oft. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keept fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our _N (t) will be very close to(t) as long ast is small compared to the effective size of the system, which is 2L/c for the first model.Supported in part by the AFOSR under Contract No. F44620-71-C-0013.     

Measurement of the refractive index and optical absorption spectra of epitaxial bismuth substituted yttrium iron garnet films at uv to near-ir wavelengths

Refractive-index and optical-absorption spectra of Bi-substituted yttrium iron garnet films, epitaxially grown by liquid-phase epitaxy, have been measured in the spectral regime 0.26 m1.9 m by thin-film interference for 0.52 m and by ellipsometry for0.52 m. The Y_3–x–y Bi _x Pb _y Fe_5–z Pt _z O₁₂ films contain bismuth in the range Ox 1.42, lead in the range 0.01 y0.08 and platinum in the range 0.005<=z0.03. There is satisfactory coincidence between the results from ellipsometry and thin-film interference in the overlapping wavelength region. The materials investigated are the same as reported earlier from this laboratory in ter mof their magnetic and magnetooptic properties.     

Interacting spinor and scalar fields: Exact self-consistent solutions in Bianci I space

Exact self-consistent solutions of the equations that describe a system of interacting spinor and massless scalar fields with the interaction Lagrangian L_int=_,^,(S), where (S) is an arbitrary function of the invariant S=, are obtained in Bianci I space. The possibility of excluding the initial singularity is studied for the case of a power-law function (S), and isotropic expansion of the space as t is established.Russian University of International Amity. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 53–58, July, 1995.