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Dec 1973

Volume 14, Issue 12, pp. 1725-2018

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Gauge invariant canonical mechanics for charged particles

R. J. Torrence and W. M. Tulczyjew

J. Math. Phys. 14, 1725 (1973); http://dx.doi.org/10.1063/1.1666246 (8 pages) | Cited 4 times

Online Publication Date: 3 November 2003

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The mechanics of charged particle motion is presented in a five‐dimensional form compatible with the five‐dimensional Kaluza theory of the electromagnetic field. Gauge dependence is given an intrinsic geometrical interpretation and the role of generalized momenta is clarified. The theory is a new example of a mechanical system with constraints and offers an interesting exercise in canonical quantization with a nonstandard Poisson bracket.

Power statistics for wave propagation in one dimension and comparison with radiative transport theory

W. Kohler and G. C. Papanicolaou

J. Math. Phys. 14, 1733 (1973); http://dx.doi.org/10.1063/1.1666247 (13 pages) | Cited 48 times

Online Publication Date: 3 November 2003

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We consider a one‐dimensional medium with random index of refraction or a transmission line with random capacitance per unit length, allowing for impedance mismatch at the load and generator. We compute the expected value of the incident and reflected powers at any point between the generator and load in the limit of weak fluctuations and a long line. The results are compared with those of radiative transport theory and discrepancies show the limitations of that theory. Finally, we consider the spreading of pulses due to random fluctuations.

Time evolution of a two‐dimensional model system. I. Invariant states and time correlation functions

J. Hardy, Y. Pomeau, and O. de Pazzis

J. Math. Phys. 14, 1746 (1973); http://dx.doi.org/10.1063/1.1666248 (14 pages) | Cited 67 times

Online Publication Date: 3 November 2003

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This paper is the first one of a series devoted to the study of a particularly simple two‐dimensional system of classical particles. The model is presented and some general feautres of it are established. We prove that, among states without correlations between particles with different velocities, there is a unique time invariant state with given density and hydrodynamic velocity. This ``equilibrium state'' is studied in detail. In particular its ergodic and mixing properties are investigated. We propose an approximation in order to estimate the asymptotic part of the time correlation functions and show that the long time tail is ruled by the ``hydrodynamic'' behavior of the model, namely by the evolution of the long wavelength perturbations.

Duffin‐Kemmer‐Petiau subalgebras: Representations and applications

Ephraim Fischbach, Michael Martin Nieto, and C. K. Scott

J. Math. Phys. 14, 1760 (1973); http://dx.doi.org/10.1063/1.1666249 (15 pages) | Cited 18 times

Online Publication Date: 3 November 2003

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A reduction of the Duffin‐Kemmer‐Petiau algebra to a direct sum of irreducible subalgebras for spin‐0 and spin‐1 bosons is presented. The subalgebras are defined by multiplication rules for the linearly independent basis elements. In the representations discussed the spin projection operators are independent basis elements of the subalgebras. The formal utility of these representations is demonstrated by obtaining the reduction of arbitrary operator products and trace theorems. The practical utility is demonstrated by application to the analysis of free and interacting boson field currents. Most importantly, one can understand the differences between DKP nonconserved currents and those obtained from second‐order wave equations.

Diffusion, Einstein formula and mechanics

Gérard G. Emch

J. Math. Phys. 14, 1775 (1973); http://dx.doi.org/10.1063/1.1666250 (9 pages) | Cited 32 times

Online Publication Date: 3 November 2003

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A mathematically rigorous discussion of the diffusion equation and of its connection with the Einstein relation linking the diffusion coefficient and the velocity autocorrelation function is presented. Diffusion is then propounded as a typical case for which the logical consistency of a purely mechanistic theory of dissipative phenomena can be established.

Parastatistics and the quark model

A. J. Bracken and H. S. Green

J. Math. Phys. 14, 1784 (1973); http://dx.doi.org/10.1063/1.1666251 (10 pages) | Cited 18 times

Online Publication Date: 3 November 2003

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An extension of ordinary parastatistics is considered which makes use of all the representations of the parastatistics algebra obtained from the usual ansatz. Govorkov's demonstration that such an extension, for parastatistics of order 2, implies a U(2) symmetry, is generalized for parastatistics of order p. The parastatistics algebra, restricted to N dynamical states, is characterized by the irreducible representations of U(N), S O(2N), and S O(2N + 1) which it contains. It is shown that these representations have multiplicities equal to the dimensions of associated representations of U(p), O(p) and C(p), respectively, where C(p) is a subalgebra of the enveloping algebra of O(p), but is not a Lie algebra. The symmetric group S(p) also appears, as a subalgebra of the enveloping algebra of C(p). It is shown how a nondegenerate vacuum state may be defined for the generalized parastatistics algebra of order p, and how to construct state vectors corresponding to arbitrary numbers of quarklike particles and antiparticles. Such states belong to irreducible representations of U(N), and can be obtained by the application of one kind of creation and annihilation operators to certain basic states, here called reservoir states, which correspond to the different irreducible representations of S O(2N + 1). The specialization to parastatistics of order 3 is discussed in detail with the application to a quark model of the hadrons in view. It is shown how to define isospin and hypercharge in a significant way in this model, which, however, differs in some respects from Gell‐Mann's well‐known 3‐fermion model, and also from Greenberg's 3‐parafermion model. Some of the physical implications are examined.

Generalizing Riemannian geometry

Richard H. Hudgin

J. Math. Phys. 14, 1794 (1973); http://dx.doi.org/10.1063/1.1666252 (6 pages) | Cited 4 times

Online Publication Date: 3 November 2003

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The properties of Riemannian geometry necessary to relativity have been used as a basis to derive a more general geometry. Emphasis is placed on a natural development with the result of considerable generalization. Several examples are discussed including the Brans‐Dicke field equation which are but one special case of the new manifolds. The scalar field is not introduced ad hoc but is a natural geometrical quantity.

Scattering theory in a model of quantum fields. I

Sergio Albeverio

J. Math. Phys. 14, 1800 (1973); http://dx.doi.org/10.1063/1.1666253 (17 pages) | Cited 2 times

Online Publication Date: 3 November 2003

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We study a model of quantum field theory with ``Yukawa‐like'' interaction λ ∫ Φb+(x)Φb−(x)Φa(x)dx between nucleons (b) and mesons (a). It is a version of Nelson's model with relativistic kinematics and has been renormalized by J. P. Eckmann. The infinite mass renormalization is a power series in λ2, chosen in such a way as to confer on the renormalized Hamiltonian Ĥ the correct relativistic single particle spectrum. Physical one nucleon states are given by a modified Friedrichs one‐particle expansion constructed by Eckmann. The Heisenberg picture's creation‐annihilation operator for dressed nucleons and mesons are studied in detail, as a preparation for the construction of the correspondent asymptotic fields, carried through, in this paper, for the mesons fields in general and for the nucleon fields on particular states (the general case is treated in the second paper of this series). Analytic properties of the interacting fields in λ are proved and commutation relations of the asymptotic fields are established. Moreover, strong asymptotic states are constructed as well as isometric wave operators. Finally some reduction formulas for the meson‐nucleon scattering are derived.

A restricted Bäcklund transformation

David W. McLaughlin and Alwyn C. Scott

J. Math. Phys. 14, 1817 (1973); http://dx.doi.org/10.1063/1.1666254 (12 pages) | Cited 32 times

Online Publication Date: 3 November 2003

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The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term ``restricted Bäcklund transformation'' (hereafter abbreviated R‐B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein‐Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R‐B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation ϕx y = F(ϕ), a theorem is established which states that R‐B transformations exist if and only if the nonlinearity F(⋅) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R‐B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.

Kinetic equations for the quantized motion of a particle in a randomly perturbed potential field

Ioannis M. Besieris and Frederick D. Tappert

J. Math. Phys. 14, 1829 (1973); http://dx.doi.org/10.1063/1.1666255 (8 pages) | Cited 16 times

Online Publication Date: 3 November 2003

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Within the framework of the first‐order smoothing approximation and the long‐time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase‐space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ‐correlated (in time), homogeneous, wide‐sense stationary, Gaussian process.

Two‐magnon bound states in Heisenberg ferromagnets

A. W. Sáenz and W. W. Zachary

J. Math. Phys. 14, 1837 (1973); http://dx.doi.org/10.1063/1.1666256 (16 pages) | Cited 5 times

Online Publication Date: 3 November 2003

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See Also: Erratum

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A generalized theory of bound two‐magnon states in three‐dimensional isotropic Heisenberg ferromagnets is given and the passage to the limit in which the total number of spins tends to infinity is handled rigorously. Powerful methods, mostly of the trace‐inequality type, are developed for determining upper and lower bounds to the number of such bound states in the latter limit. These methods constitute the central contribution of this paper. In the latter we apply them to investigate the existence of bound two‐magnon states in body‐centered Heisenberg ferromagnets whose nonvanishing exchange interactions are those of the nearest‐neighbor type. In work reported elsewhere, we have employed these methods to study spin‐wave impurity states in Heisenberg ferromagnets. They should be useful for determining bounds on the number of localized states in solids in many cases when interactions extending over several orders of neighbors are operative.

Deformations of inhomogeneous classical Lie algebras to the algebras of the linear groups

Charles P. Boyer and Kurt Bernardo Wolf

J. Math. Phys. 14, 1853 (1973); http://dx.doi.org/10.1063/1.1666257 (7 pages) | Cited 6 times

Online Publication Date: 3 November 2003

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We study a new class of deformations of algebra representations, namely, i2so(n)⇒sl(n,math), i2u(n)⇒sl(n,math)⊕u(1) and i2sp(n)⊕sp(1)⇒sl(n,Q)⊕sp(1). The new generators are built as commutators between the Casimir invariant of the maximal compact subalgebra and a second‐rank mixed tensor. These algebra deformations are related to multiplier representations and manifold mappings of the corresponding Lie groups. Behavior of the representations under Inönü‐Wigner contractions is exhibited. Through the use of these methods we can construct a principal degenerate series of representations of the linear groups and their algebras.

Asymptotic behavior of atomic bound state wave functions

Reinhart Ahlrichs

J. Math. Phys. 14, 1860 (1973); http://dx.doi.org/10.1063/1.1666258 (4 pages) | Cited 28 times

Online Publication Date: 3 November 2003

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In the present paper we investigate the asymptotic properties of an exact bound state wave function ϕ of an n‐electron atomic system within the infinite nuclear mass approximation and neglecting relativistic effects. An explicit upper bound is derived for ‖riμφ‖, where ri denotes the distance of the ith electron from the nucleus and μ = 1,2,3,⋯. We are then able to derive upper bounds for expressions like ‖h(ri)ϕ‖, where h (x) is an exponentially increasing function. We finally indicate an exponentially decreasing pointwise bound for ϕ.

Energy eigenvalues of a bounded centrally located harmonic oscillator

Richard Vawter

J. Math. Phys. 14, 1864 (1973); http://dx.doi.org/10.1063/1.1666259 (7 pages) | Cited 14 times

Online Publication Date: 3 November 2003

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In the study of the effects of finite boundaries on the magnetic properties of a solid, one encounters the problem of finding the energy eigenvalues of a one‐dimensional harmonic oscillator located in a potential enclosure. Series expansion techniques are applied to solve this problem for a harmonic oscillator located at the center of an infinitely high potential well. An analytical expression for the energy eigenvalues is found as a function of the size potential enclosure L, the quantum state n, oscillator frequency ω, and the mass of the particle m. The first order approximation of this expression is given by E = Eosccoth(Eosc∕Ebox) where Eosc = ℏ ω(n + 1∕2) is the energy eigenvalues of an unbound harmonic oscillator and Ebox = (2m∕ℏ2) (n + 1)2π2∕L2 is the energy eigenvalues of a free particle in an infinitely high well. For the ground state this approximation is better than 1% for all values of L, ω, and m.

Griffiths' inequalities for Ashkin‐Teller model

Ching Tsung Lee

J. Math. Phys. 14, 1871 (1973); http://dx.doi.org/10.1063/1.1666260 (5 pages)

Online Publication Date: 3 November 2003

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The two Griffiths' inequalities for the correlation functions of Ising ferromagnets and two others added by Kelly and Sherman and by Sherman are extended to what we call generalized Ashkin‐Teller model. In this model we consider a system of N particles; each can exist in r possible states. Let a collection of pairs of particles be represented by a graph with particles as vertices and pairs of particles as edges. The ``many‐body interaction'' among a cluster of particles represented by such a graph G(A) is ‐ JA(JA ≥ 0) when the particles in each connected component of G(A) all exist in the same state; it is 0 otherwise. For the special case with r = 2 and two‐body interactions only, the Ashkin‐Teller model is equivalent to Ising model. Therefore, what we present in this paper can be considered as yet another way of proving the original correlation inequalities for Ising ferromagnets with two‐body interactions. We have also discovered another new inequality, namely 〈δA〉 〈δABRS〉 + 〈δAB〉 〈δARS〉 ‐ 〈δAR〉 〈δABS〉 ‐ 〈δAS〉 〈δABR〉 ≥ 0.

Irreversible behavior of a thermally insulated system

Julio F. Fernández

J. Math. Phys. 14, 1876 (1973); http://dx.doi.org/10.1063/1.1666261 (7 pages)

Online Publication Date: 3 November 2003

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The time evolution of a system coupled to a reservoir, in such a way that no energy exchange at all takes place, is examined, considering the system plus reservoir as dynamically closed. An H‐theorem is proved, more precisely, it is shown quite generally that if the system and reservoir are initially uncorrelated, then the Gibbs‐Jaynes entropy at time t, S(t), obeys the relation S(t) ≥ S(0). We have previously exhibited a simple model where S(t) > S(0). Some relaxation times of S(t), for an x‐y model (system) interacting (no energy exchange) with an Ising model (reservoir), are obtained approximately. The values obtained seem reasonable. It is also shown, for this particular model, that if the motion of the system, but not the reservoir, is inverted at time T, then its Gibbs‐Jaynes entropy τ, later SMI (τ; T), is given by SMI(τ; T) ≃ S(T + τ), i.e., the entropy keeps evolving in time as if the motion inversion had not taken place.

Integral equations for scattering in potentials of infinite range

D. F. Freeman and J. Nuttall

J. Math. Phys. 14, 1883 (1973); http://dx.doi.org/10.1063/1.1666262 (3 pages) | Cited 1 time

Online Publication Date: 3 November 2003

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We show how, by finding an approximation to the exact Green's function, to obtain integral equations with compact kernels for two scattering problems involving potentials of infinite range. They are one particle scattering in a Coulomb potential for a given partial wave and one particle scattering in three dimensions in a potential having r−2 behavior at infinity. In the second case, we solve only an inhomogeneous version of Schrödinger's equation.

Symmetry mappings of constrained dynamical systems and an associated related integral theorem

Jack Levine and Gerald H. Katzin

J. Math. Phys. 14, 1886 (1973); http://dx.doi.org/10.1063/1.1666264 (6 pages) | Cited 4 times

Online Publication Date: 3 November 2003

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By use of Lie derivatives symmetry mappings of constrained conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetries are called ``natural trajectory collineations'' in that the total energy has the same fixed value along each trajectory of the natural family, this value being preserved by the symmetry. It is found that these natural trajectory collineations must be conformal motions subject to an additional restriction dependent upon the potential. The corresponding groups of natural trajectory collineations are obtained for a flat configuration space with potential energy functions with rotational invariance about a point. A specialization of the theory to an indefinite Riemannian space‐time shows that homothetic transformations are necessary and sufficient to map a natural family of time (space)‐like geodesics into itself. A related integral theorem for constrained dynamical systems admitting linear or quadratic constants of the motion is obtained and illustrated. This theorem shows that in general a new constant of the motion will be obtained by deformation of an existing constant of the motion under a natural trajectory collineation.

Gravitation and the new improved energy‐momentum tensor

Hans C. Ohanian

J. Math. Phys. 14, 1892 (1973); http://dx.doi.org/10.1063/1.1666265 (6 pages) | Cited 5 times

Online Publication Date: 3 November 2003

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We give precise definitions of the weak and strong principles of equivalence and show that the new gravitational theory based on the improved energy‐momentum tensor of Callan, Coleman, and Jackiw [Ann. Phys. (N.Y.) 59, 42 (1970)] satisfies both of these principles. As a consequence of the equality between the 4‐momentum given by the canonical energy‐momentum tensor and the ``momentum'' given by the pseudotensor that is the source of gravitation, the weak principle is also shown to hold in more general Einstein theories. Investigation of the interactions of a scalar field in the new gravitational theory shows that, besides the familiar long range interaction, there exists a new short range gravitational interaction between any scalar field and other matter.

Scintillations of randomized electromagnetic fields. II

Leonard S. Taylor and Danai Lekhyananda

J. Math. Phys. 14, 1898 (1973); http://dx.doi.org/10.1063/1.1666266 (6 pages) | Cited 1 time

Online Publication Date: 3 November 2003

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We study the propagation of an electromagnetic wave which has suffered changes of phase and amplitude in its passage through a randomizing medium such as a turbulent dielectric. General formulas are derived for the n th‐order autocorrelation and spectrum of intensity fluctuations in terms of the 2n th‐order mutual coherence function on the initial plane. Simple expressions are obtained when the wave is initially weakly phase distributed. The probability distribution of intensity fluctuations is shown to be Gaussian in the limit of vanishing phase perturbation.

Wigner and Racah coefficients for SU3

J. P. Draayer and Yoshimi Akiyama

J. Math. Phys. 14, 1904 (1973); http://dx.doi.org/10.1063/1.1666267 (9 pages) | Cited 93 times

Online Publication Date: 3 November 2003

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A general yet simple and hence practical algorithm for calculating SU3⊃SU2×U1 Wigner coefficients is formulated. The resolution of the outer multiplicity follows the prescription given by Biedenharn and Louck. It is shown that SU3 Racah coefficients can be obtained as a solution to a set of simultaneous equations with unknown coefficients given as a by‐product of the initial steps in the SU3⊃SU2×U1 Wigner coefficient construction algorithm. A general expression for evaluating SU3⊃R3 Wigner coefficients as a sum over a simple subset of the corresponding SU3⊃SU2×U1 Wigner coefficients is also presented. State conjugation properties are discussed and symmetry relations for both the SU3⊃SU2×U1 and SU3⊃R3 Wigner coefficients are given. Machine codes based on the results are available.

Linear integral transformations generated by the three‐dimensional neutron transport kernel

V. C. Boffi and G. Spiga

J. Math. Phys. 14, 1913 (1973); http://dx.doi.org/10.1063/1.1666268 (5 pages) | Cited 8 times

Online Publication Date: 3 November 2003

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Two theorems are established by which the theory of linear integral transformations in a Lebesgue space Lp(D), p ≥ 1, can be appropriately extended to solve linear integral equations with kernel of nonfinite double norm with respect to the considered Lp(D). An application of these theorems to a physical problem of three‐dimensional neutron transport theory is illustrated.

Infinite spin dimensionality limit for nontranslationally invariant interactions

Hubert J. F. Knops

J. Math. Phys. 14, 1918 (1973); http://dx.doi.org/10.1063/1.1666269 (3 pages) | Cited 40 times

Online Publication Date: 3 November 2003

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A proof is presented that a nontranslationally invariant system of classical n ‐dimensional spins approaches in the limit n → ∞ a suitable generalized spherical model.

Probability density function and moments of the field in a slab of one‐dimensional random medium

R. H. Lang

J. Math. Phys. 14, 1921 (1973); http://dx.doi.org/10.1063/1.1666270 (6 pages) | Cited 13 times

Online Publication Date: 3 November 2003

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The problem of a plane wave normally incident on a slab of one‐dimensional random medium is studied. The refractive index variations of the random medium are taken to be a stationary Gaussian‐Markov process. By employing an invariant imbedding technique and by using the Markov property of the refractive index variations, two cascaded diffusion equations are obtained for the probability density function of the reflection coefficient and the field in the slab. These equations are then solved approximately for small refractive index fluctuations and an expression is obtained for mean intensity in the slab interior.

Some rigorous results for the vertex model in statistical mechanics

H. J. Brascamp, H. Kunz, and F. Y. Wu

J. Math. Phys. 14, 1927 (1973); http://dx.doi.org/10.1063/1.1666271 (6 pages) | Cited 5 times

Online Publication Date: 3 November 2003

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It is shown that the free and periodic boundary conditions are completely equivalent for the ice‐rule (six‐vertex) models in zero field. With an external direct or staggered field, we establish that in an ice‐rule model the free and periodic boundary conditions are equivalent, and also equal to some special boundary conditions, either at sufficiently low temperatures or with sufficiently high fields in the appropriate direction. Regions of constant direct polarization are found. We also establish the existence of the spontaneous staggered polarization in an antiferroelectric using the Peierls argument.
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