Volume 15, Issue 11, November 1974
Index of content:

Zeros of the partition function for higher‐order spin systems
View Description Hide DescriptionA theorem is proved which is useful in determining information regarding the location of zeros of the partition function for lattice models with arbitrary spin. This theorem is a generalization to higher‐order spin systems of a theorem for spin‐1/2 systems proved by Ruelle. The total partition function for the system can be constructed by contracting (a generalization of the Asano contraction procedure) a set of lower‐order ``partition function‐like'' polynomials. The theorem presented relates information regarding the location of zeros of the lower‐order polynomials to the location of zeros of the partition function. This theorem is then used to establish the Lee‐Yang unit circle theorem for several higher‐order spin models.

Double spectral representations of single loop amplitudes with k vertices: k ≥ 4
View Description Hide DescriptionA method developed in several previous papers is combined with the method of induction to derive double dispersion relations, with Mandelstam boundary, for the class of single loop amplitudes with four or more vertices. The spectral functions are expressed as integral representations and restrictions on the masses and kinematic invariants for which dispersion relations are valid are found. It is also discussed how representations for the low order single loop amplitudes can be obtained for wider ranges of these variables.

Principle of compensation of dangerous diagrams for boson systems. III. Finite temperature
View Description Hide DescriptionThe principle of compensation of dangerous diagrams (PCDD) is derived at finite temperature for boson systems by minimizing the average number of Bogoliubov quasiparticles in the system. The conditions obtained state that (a) the amplitude for the creation (or annihilation) of a single quasiparticle is zero and (b) the amplitude for the creation (or annihilation) of a pair of quasiparticles is zero. These conditions are expanded in finite‐temperature perturbation theory, using both the density matrix and Green's function methods. In first order the resulting equations are the Hartree‐Fock‐Bogoliubov equations for a homogeneous boson system at finite temperature which can also be obtained from a free energyvariational principle.

Exact next nearest neighbor degeneracy
View Description Hide DescriptionIt is shown that A [n _{11}, n _{101}, n _{111}, q, N], the arrangement degeneracy arising when q indistinguishable particles are placed on a one‐dimensional lattice of N equivalent compartments so that n _{11} occupied nearest neighbors, n _{101} next nearest neighbors of the 101‐type, and n _{111} next nearest neighbors of the 111‐type are created, is given by . The normalization and first moment of the next nearest neighbor density are determined. Similar results for the vacant next nearest neighbor degeneracy are also presented.

Schrödinger equation and quantum state codons in discrete transform space
View Description Hide DescriptionQuantum states of simple systems are shown to have composite root‐pole structure in the complex transform plane. The Schrödinger condition becomes the inverse of a continuity equation expressing invariance of the Ψ‐transform codon under discrete displacement. Four distinct quotient polynomial solutions model the Legendre, Hermite, Laguerre, and Jacobi polynomial families. Schrödinger coefficients are identified with pole strengths of quotient polynomials and are geometrically interpreted in terms of universal root and pole interactions.

Coherent pulse propagation, a dispersive, irreversible phenomenon
View Description Hide DescriptionThe initial value problem for the propagation of a pulse through a resonant two‐level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.

Factorizability of resonance poles in multiparticle amplitudes
View Description Hide DescriptionUsing the energy‐analytic representation of Green's functions and relying on certain explicitly stated properties of the off‐shell scattering elements, it is shown that resonance poles in the S matrix contribute poles to the off‐shell scattering amplitude, and the residues there have the same factorizable form as that associated with bound state particles.

The renormalization group and the large n limit
View Description Hide DescriptionThe basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self‐contained and of pedagogical value to a wide audience.

Quantization on hyperboloids and full space‐time field expansion
View Description Hide DescriptionBeginning with free field quantization on hyperboloids in the forward lightcone, we extend field expansions to the full space‐time. The role of boundary conditions on the propagator and of the particle‐antiparticle distinction in establishing a unique field expansion is discussed. Some difficulties with the possibility of a surface to surface development of the S matrix are encountered in the massless case. Even in the massive case hyperboloids are unsuitable quantization surfaces when x ^{2}<0, and we develop in some detail an alternative set of surfaces for that region in two dimensions.

Propagation through an anisotropic random medium
View Description Hide DescriptionAn expression is derived, Eq. (59), for the mutual coherence of an initial plane wave signal that has propagated a distance, z, into an anisotropic random medium. This expression is valid for cases in which the characteristic radiation wavelength divided by 2π is roughly speaking of the same order as, or greater than, all characteristic correlation lengths in the direction perpendicular to the mean propagation direction, which is taken to lie in a horizontal plane. The exact condition is given in Eq. (48). We require to be much smaller than all characteristic correlation lengths in the horizontal plane. Two derivation procedures are used. One follows that introduced by Beran (J. Opt. Soc. Am. 56, 1475 (1966)] for cases in which is much smaller than all characteristic correlation lengths and one is based on introducing simplifications in a Bethe‐Salpeter equation. We discuss the problem of the loss of spatial coherence of an acoustic signal due to scattering by the ocean temperature microstructure in the light of the theory presented.

The plane‐wave expansion method
View Description Hide DescriptionA proof of the uniform convergence of the plane‐wave expansion method is given. Also, an alternative method to obtain the expansion coefficients of the plane‐wave expansion is derived through the use of the Rayleigh‐Ritz variational method.

Multiplicative stochastic processes, Fokker‐Planck equations, and a possible dynamical mechanism for critical behavior
View Description Hide DescriptionA derivation of the Fokker‐Planck equations for additive stochastic processes is given which involves treating the continuity equation in the configuration space representation of the additive stochastic process as a multiplicative stochastic process. The average of the continuity equation becomes the Fokker‐Planck equation. A presentation of the ``multiplicative stochastic, Markov approximation'' follows. This approximation is applied to the analysis of the dynamics of a heavy particle in a molecular fluid as described by Hamilton's equations. The nonperturbative approximation technique leads to the Fokker‐Planck equation for simple Brownian motion. As part of the analysis, ``intrinsic diffusion'' is discovered and used to show ergodicity for the autocorrelation formula which appears during the Brownian motion calculation. An account of how these methods might be used to study the dynamical origins of critical behavior is given.

Weyl tensor decomposition in stationary vacuum space‐times
View Description Hide DescriptionThe electric and magnetic parts of the Weyl tensor are represented as symmetrized derivatives of gradient vector fields in stationary vacuum space‐times. It is shown that a necessary and sufficient condition for a stationary vacuum space‐time to be static is that the Weyl tensor be electric type. It is further shown that the only stationary vacuum space‐time with vanishing electric type Weyl tensor is flat space.

The maximal solvable subgroups of SO(p,q) groups
View Description Hide DescriptionA recursive procedure is developed that makes it possible to determine all conjugacy classes under both SO(p,q) and O(p,q) of the maximal solvable subalgebras of the Lie algebrasLO(p,q) [and the continuous maximal solvable subgroups of SO(p,q)]. The cases of greatest physical interest with p ≥ q ≥ 0 and p + q ≤ 6 are considered in detail (they include the Lorentz group, de Sitter groups, and the conformal group of space‐time). Formulas (in terms of Fibonacci numbers) are given for the number of O(p,q) [and SO(p,q)] equivalence classes of maximal solvable subalgebras of LO(p,q).

Invariant imbedding and Fredholm integral equations with degenerate kernels
View Description Hide DescriptionIn a manner similar to that given in preceding papers by Bellman and Ueno, with the aid of the Bellman‐Krein formula for the resolvent, we show how to solve Fredholm integral equations of the second kind with degenerate kernels. The standard procedure for solution is to convert it into an equivalent matrix equation, but in this paper it is transformed into a Cauchy problem which can be solved effectively by high speed digital computers.

Kinetic theory and the Lorentz gas
View Description Hide DescriptionThe high density properties of the velocity autocorrelation function and the diffusion coefficient are discussed in a one‐ and three‐dimensional Lorentz gas on the basis of kinetic theory.

Diffraction characteristics of a slit formed by two staggered parallel planes
View Description Hide DescriptionThe diffraction of a plane electromagnetic wave by a slit formed by two staggered parallel planes is investigated using an asymptotic Wiener‐Hopf technique. By following a standard procedure the problem is formulated in terms of two coupled Wiener‐Hopf equations. For large edge‐edge separation, the decoupling of the equations is accomplished by evaluating certain integrals by the saddle point method of integration. The results thus obtained can be conveniently identified as rays emanating from the two edges. It is shown that various changes in transmission coefficient and diffraction pattern of a slit can be obtained by changing the angle of stagger of the planes. Plots of transmission coefficients and diffraction patterns are presented for various slit widths and angles of stagger to show these characteristics.

Asymptotic behavior of Markoffian kinetics
View Description Hide DescriptionThe ergodic state with the least mean recurrence time in an irreducible Markoffian process having countable states, between any pair of which transition probability rates are definable, is the state with the least irreversible decay rate if neither the microscopic reversibility nor the doubly stochastic property holds.

Quantumlike formulation of stochastic problems
View Description Hide DescriptionIt is shown that a stochastic process can be viewed as a set of states (normalized positive linear functionals) over an Abelian C*‐algebra. Alternatively, the stochastic process can be associated with a set of representations of the algebra as a subalgebra of the (noncommutative) C*‐algebra of bounded operators in a Hilbert space. Then, an operator equation can be associated with every stochastic equation in some general conditions. The formalism is applied to Brownian motion. Then, we study the nonrelativistic motion of a single particle in stochastic electrodynamics, a theory which has been proposed as a possible alternative to quantum mechanics. The equations of motion, which are derived, coincide with the basic ones of quantum mechanics. The differences between this theory and quantum mechanics are summarized.

The analytic noncharacteristic Cauchy problem for nonlightlike isometries in vacuum space‐times
View Description Hide DescriptionThe analytic noncharacteristic problem for the existence of a spacelike isometry in vacuum space‐time, given its existence on the hypersurfaces, on the Cauchy data, is posed and solved using the ADM equations. The timelike case is also solved. In both cases the isometry will locally exist.