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.     

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.     

On the extreme relativistic limit of arbitrary spin wave equations

The extreme relativistic limit (E-representation) of the wave equation in the Schrödinger formi/t =H describing particles and anti-particles of spin s and non-zero rest mass m is presented here. As the wave function has just the minimum number of 2(2s+1) components, the necessity of avoiding redundant components by auxiliary conditions does not arise. Relevant expressions are given for the infinitesimal generators of the Poincaré group and for the operators representing the observables in this representation.     

Generalized Bargmann potentials

Conditions are found for the coefficient functions of a linear ordinary differential equation of the kth order ^(k)+u₁^(k-2)+...+u_k-1=^k, when its solution has the following analytical dependence on the parameter: =exp(z _i=1 ⁿ (+a_i)(a_ja_j(Z)). The problem is closely related to the finding of n-soliton solutions of the simplest form for the periodic Toda system, corresponding to A _nand C _nseries.     

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.     

Primes in Short Intervals

Contrary to what would be predicted on the basis of Cramérs model concerning the distribution of prime numbers, we develop evidence that the distribution of (x+H)–(x), for 0xN, is approximately normal with mean H and variance HlogN/H, when NHN^1–.Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthdayResearch supported by NSF grants DMS–0070720 and DMS–0244660.Research supported by the American Institute of Mathematics and NSF grants.     

Maximum uniform deformation in tension

An equation is derived for the hyperbola which touches the true stress curve S =f(), where is the contraction of the specimen at the point _p (uniform contraction), S_B (true ultimate strength). With a flat maximum of the tensile force, this hyperbola coincides with the true stress curve at a part corresponding to extension by the maximum force. The use of the tangent hyperbola for determining _p and S_B is demonstrated.It is found that for those metals and alloys which are at present known to have a convex true stress curve in the uniform plasticity range, the uniform contraction _p cannot exceed 0.5, corresponding to a uniform elongation _p 1, while the true (logarithmic) uniform elongation _p 0.693. The limiting values of the hardening modulus and of the ratio S_B/_B are also found.     

The ground state energy of a Bose gas with Coulomb interaction II

LetH _N be the quantum mechanical Hamiltonian for a neutral system of 2_N charged particles, each of unit charge. The HamiltonianH _N is assumed to act on wave functions inL ²(^6N ) which satisfy Bose statistics. It is shown that if the kinetic energy of is sufficiently small, then |H _N |–CN ^7/5 for some universal constantC.Research supported by U.S. National Science Foundation Grant DMS 8600748     

Geometrical interpretation of the bilinear equations for the KP hierarchy

The Grassmann manifold approach to the KP hierarchy, in the spirit of Segal and Wilson, is used to define the wavefunction _W and its adjoint _W. From the fact that _W and _W are the orthogonal, we derive the bilinear equations. The modified equations are treated at the same time.     

Duality-invariant orthogonality relation between the magnetic monopole and associated quark charges

Duality invariance of the Dirac-Schwinger charge-symmetric theory for electromagnetism leads one to consider the complex-valued amplitudes ₁ and ₂ for the separation between the magnetic monopole and quarks in the logarithmic charge plane. It is observed that the orthogonality relation on the latter amplitudes, Re( ₁ ^* ₂)=0, is equivalent to the equation (ln 9 ^–1)(ln 2)=(1/2) ², which is indeed satisfied by the experimental value fora to within 0.027%. In addition to fixing the unit of electric charge at a primary physical value, the orientation of ₁, ₂ may also prescribe the Cabibbo angle to have the theoretical value 12.4438.     

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.     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

A\overline {Poincar\acute{e}} gauge theory of gravity

We develop a gauge theory of gravity on the basis of the principal fiber bundle over the four-dimensional space-timeM with the covering groupP¯ ₀ of the proper orthochronous Poincaré group. The field components are constructed with the connection coefficients , and with a Higgs-type field. A Lorentz metricg is introduced with , which are then identified with the components of duals of the Vierbein fields. Associated with there is a spinor structure onM. For Lagrangian densityL, which is a function of , ,, matter field , and oftheir first derivatives, we give the conditions imposed by the requirement of the gauge invariance. The Lagrangian densityL is restricted to be of the formL =L ^tot (, T ^klm ,R ^klmn , _k , ), in whichT ^klm ,R ^klmn are the field strengths of , , respectively. Identities and conservation laws following from the gauge invariance are given. Particularly noteworthy is the fact that the energy momentum conservation law follows from theinternal translational invariance. The field equation of is automatically satisfied, if those of and of are both satisfied. The possible existence of matter fields with intrinsic energy momentum is pointed out. When is a field with vanishing intrinsic energy momentum, the present theory practically agrees with the conventional Poincaré gauge theory of gravity, except for the seemingly trivial terms in the expression of the spin-angular momentum density. A condition leading to a Riemann-Cartan space-time is given. The field holds a key position in the formulation.     

Effective potential for the four-fermion interaction in topologically non-trivial spaces

We consider the influence of the space-time topology on the formation of a fermion condensate in a theory with a four-fermion interaction. We study the dependence of the behavior of the effective potential on the magnitude of the fermion condensate ¯ and find the dependence of the magnitude ¯ on the periodicity length of the wave functions in space-time with the topology R₁ × R₁ × S₁.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 5–8, March, 1988.     

Surface channeling of protons and electrons on reflection by a single crystal

Computer modeling is used to investigate the reflection of 0.67-MeV protons and 1-MeV electrons by the (110) face of an Au single crystal for angles of divergence < _cr of the incident beam with the direction 110 lying in the surface. The angular distributions and reflection coefficients of protons and electrons are compared. Several maxima are found in the angular distributions. In additon to the specular-reflection maximum at = there is a maximum at > for protons and a maximum at < for electrons. The maxima in the angular distributions are attributed to multiple scattering at atomic chains of the crystal surface. An analytic approximation of the reflection coefficient for protons K_cor() is given. The great contribution to the reflected flux from particles channeled in the surface layer allows it to be used for structural analysis of crystal surfaces.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 85–90, March, 1979.     

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.     

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.     

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     

Percolation of level sets for two-dimensional random fields with lattice symmetry

Let (x),x², be a random field, which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of . Percolation methods are used to analyze the sizes of the connected components of level sets {x:(x)=h} and sets {x:(x)h} in several classes of random fields with lattice symmetry. In typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component. In some examples, however, there is a nondegenerate interval of values ofh where components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every region of the lattice.     

Determination of random variations in ionosphere height from phase fluctuations of VLF signals

Conclusions A feature of our problem of restoration of fluctuations in ionosphere height is that the experimental data(t, _k) obtained at a fixed reception point are functions of time, whereas the function ₁(u, y₂) to be restored is a function of the coordinates. If we use the assumption that the irregularities migrate transversely, coordinate y₂ can be exchanged for time t. Restoration with respect to the second coordinate u=x_i–x₀/2 is in effect obtained by using data(t, _k) on some set of carrier frequencies.The resultant solution of the restoration problem is in the form of an expansion of ₁(u, y₂) in known functions determined from the observed data(t, _k). We have evaluated the solution accuracy, which depends on the overall power signal-to-noise ratio at all frequencies used. We have demonstrated that the restoration algorithm contains an optimum number (with respect to accuracy) of coefficients to be evaluated.Khar'kov Aviation Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 20, No. 8, pp. 1138–1145, August, 1977.