bluuebunny/crossref_metadata_2025 huggingface dataset:
Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators
Authors: Lewis H. R.
Published: February 2004

A class of exact invariants for oscillator systems whose Hamiltonians areH=(1/2ε)[p2+Ω2(t)q2]is given in closed form in terms of a function ρ(t) which satisfies ε2d2ρ/dt2+Ω2(t)ρ−ρ−3=0. Each particular solution of the equation for ρ determines an invariant. The invariants are derived by applying an asymptotic theory due to Kruskal to the oscillator system in closed form. As a consequence, the results are more general than the asymptotic treatment, and are even applicable with complex Ω(t) and quantum systems. A generating function is given for a classical canonical transformation to a class of new canonical variables which are so chosen that the new momentum is any particular member of the class of invariants. The new coordinate is, of course, a cyclic variable. The meaning of the invariants is discussed, and the general solution for ρ(t) is given in terms of linearly independent solutions of the equations of motion for the classical oscillator. The general solution for ρ(t) is evaluated for some special cases. Finally, some aspects of the application of the invariants to quantum systems are discussed.
Representations of the Three-Dimensional Rotation Group in Terms of Direction and Angle of Rotation
Authors: Carmeli Moshe
Published: February 2004

We find the irreducible representations of the three-dimensional pure rotation group by Weyl's method, which makes use of the homomorphism of the special unitary group of order two onto the whole three-dimensional rotation group. The representations are realized, however, in terms of the angle of rotation in a specified direction and the spherical angles of the direction of the rotation rather than in terms of the familiar Euler angles. The results are then compared with those obtained by different methods and the advantages of the present technique are pointed out. We also derive the differential operators corresponding to infinitesimal rotations about the coordinate axis in terms of the new variables.
Generating Functions for the Exact Solution of the Transport Equation. II. Time-Dependent with Anisotropic Scattering
Authors: Abu-Shumays I. K. , Bareiss E. H.
Published: February 2004

Part I of this series [J. Math. Phys. 9, 1722 (1968)] introduced in detail the general method of transforming integrodifferential transport equations to partial differential equations. The treatment there is restricted to isotropic transport in slab geometry. This part extends the method to time-dependent anisotropic transport for slab geometry. Generating functions are used as an analytic tool to define appropriate transformations whose inverses are known. The general solution of the transport equation considered are expressed in terms of expansion modes. The expansion coefficients are determined by a combination of Fourier transforms and orthogonality relations. Fourier transforms in the time variable are used instead of the usual Laplace transforms. The solutions of the initial-value problem and its analog with the role of space and time interchanged are given.
Relativistic Dynamics of a Point Charge in a Magnetic-Monopole Field
Authors: Mitchell Thomas P. , Burns J. A.
Published: February 2004

This paper derives and interprets the constants of the charge's motion. The physical meaning of these constants and their use in discussing the over-all motion of the charge are presented.
On the Clebsch-Gordan Series of a Semisimple Lie Algebra
Authors: Radhakrishnan D.
Published: February 2004

The problem of determining the multiplicity of an irreducible representation of a semisimple Lie algebra, in the decomposition of the product of two such representations, is reduced to one of solving a system of linear equations. This is achieved by using some properties of partition functions which occur in the formulas for the multiplicity of a weight in a representation. It is shown that one need not know the partition function explicitly.
Theorems on the Ising Model with General Spin and Phase Transition
Authors: Suzuki Masuo
Published: February 2004

The theorem of Lee and Yang has been extended to the ferromagnetic Ising model with arbitrarily mixed spin values of Sj = ½, 1, and 32, including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic-field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling-law approach.
Eigenvalue Problem for Lagrangian Systems. III
Authors: Barston E. M.
Published: February 2004

The quadratic Lagrangian eigenvalue problem [λ2P + λQ - (L + B)]ζ = 0 and the associated time-dependent problem Pζ¨+iQζ̇+(L+B)ζ(t)=0 are investigated for the case where P, Q, and B are bounded linear Hermitian operators in Hilbert space, P is positive and invertible, L possesses a positive completely continuous Hermitian inverse, and L + B > 0. Existence and completeness theorems for the eigenvectors as well as variational characterizations of the eigenvalues are given, and the general solution of the time-dependent problem is obtained in terms of an eigenvector expansion. Finally, these results are applied to the problem of small oscillations of a rotating elastic string.
Master Analytic Representation: Reduction of O(2, 1) in an O(1, 1) Basis
Authors: Kuriyan Jacob G. , Mukunda N. , Sudarshan E. C. G.
Published: February 2004

We display the reduction of the pseudo-orthogonal group O(2, 1) with respect to a noncompact O(1, 1) basis. After the explicit solution is obtained, we rederive the results using the method of master analytic representations.
Auxiliary Variables in Statistical Mechanics: Variational Principle
Authors: Speisman Gerald
Published: February 2004

The variational principle of Bohm and Pines for the ground-state energy of the electron gas with uniform neutralizing positive-charge background, employing auxiliary variables, is reviewed as an illustration of the past use of auxiliary variables and as an example of the type of physical system to which the variational principle of this paper can be applied. We then develop this variational principle for the logarithm of the partition function of a physical system at nonzero temperatures, employing auxiliary variables. The variational principle contains a trial Hamiltonian H′ in an extended Hilbert space. For H′ equal to its optimal value H, the variational expression ln Q′ is equal to the logarithm of the partition function ln Q. For H′ ≠ H, it is shown that ln Q ≥ ln Q′ for H′ – H sufficiently small or temperatures sufficiently high, or for sufficiently low temperatures when an additional assumption is made, which reduces to one made by Bohm and Pines when applied to the electron gas. The variational expression ln Q′ contains more complicated trace formulas than are usually encountered in quantum statistical mechanics; one possible method of evaluation is sketched leading to a simpler approximate formula for ln Q′. Corrections to the variational approximation for ln Q are provided by the second- and higher-order terms in a certain perturbation expansion of ln Q. The variational principle developed here implies the variational principle of Bohm and Pines in the zero-temperature limit; in the ``no auxiliary variable'' limit it reduces to a modified form of Peierls' variational theorem. It is shown how the variational principle can be applied to any physical system containing charged particles in which the long-range collective effects of the Coulomb interaction are important.
Differential Equations for Half-Off-Shell Matrix Elements
Authors: Sobel M. I.
Published: February 2004

A first-order differential equation is found for the quasiphase parameter which describes the half-off-shell scattering matrix. The quasiphase is given as Δ(∞), where Δ(r) obeys an equation involving the potential but not the wavefunction and Δ(0) = 0. The equation is used to develop low-momentum expansions for Δ, and to derive upper and lower bounds for the expansion parameters.
Analyticity of the Envelope Diagrams
Authors: Risk Clifford
Published: February 2004

Analyticity of the envelope and open envelope diagrams is investigated, and solutions to the Landau equations are obtained which do not satisfy the usual symmetry property. The Landau curves which these solutions generate are found to have the following characteristics when the anomalous thresholds are on the unphysical sheet: (1) they are also on the unphysical sheet; (2) they have asymptotes that depend on the external masses of the diagrams; (3) the Landau curve for the envelope diagram has as an asymptote a line u = const, although the diagram has no u channel; (4) a new type of effective intersection is found.
On the Ising Model with Long-Range Interaction
Authors: Siegert A. J. F. , Vezzetti David J.
Published: February 2004

We have obtained an expansion of the free energy per spin of an Ising model with long-range interaction in the absence of an external field, for temperatures above the Curie temperature of the Weiss-Bragg-Williams approximation (BWCP). We use as expansion parameter the reciprocal (γ) of an effective number of neighbors. Terms through order γ2 are obtained by extracting factors from a representation of the partition function as an average over random fields. For terms of higher order, we give a diagrammatic series in which all terms through order γn are contained in the diagrams with not more than 2(n − 1) bonds. The terms of order γ3 are given explicitly. For temperatures below the BWCP we have calculated terms through order γ. Since after a few finite terms the coefficients in this γ expansion become infinite at the BWCP, we exhibit a modification of the random field representation which avoids this difficulty. We have compared our results with those of previous authors wherever available—that is, through order γ for the general case and through order γ2 for the one-dimensional model with exponential interactions above the BWCP. The results of the modified random field representation are in formal agreement with those of previous authors. In this previous work, an equation is given, whose solution is identified as the approximate Curie point. We give arguments to show that this interpretation is not justified.
Class of Solutions of the Physical 3-Particle Unitarity Equations in Static S-Matrix Theory
Authors: Lebrun Jean-Pierre
Published: February 2004

A class of formal exact solutions of the 3-particle unitarity equations are obtained under the restriction that the S matrix be invariant under a contraction of the Poincaré group known as the static limit. The main tool is the study of a homogeneous, multidimensional, Riemann-Hilbert boundary-value problem in the complex plane of a subenergy variable. It allows one to construct an ansatz for the S matrix that satisfies the four sets of 3-particle unitarity equations recurrently; i.e., the sets of Eqs. (II), (III), and (IV) are automatically satisfied, provided that (I) is satisfied. The set (I) is then reduced to a nonlinear boundary-value problem on the inelastic cut in the complex energy plane of the scattering amplitudes. The most general solution is obtained in terms of a particular solution; an algorithm is then described that produces such a particular solution. Finally, the relation of crossing to our exact 3-particle unitary S matrix is discussed.
Intrinsic Spinor Techniques with Applications to the Lorentz Group and the Dirac Equation
Authors: Luehr C. P. , Rosenbaum M.
Published: February 2004

An intrinsic spinor formalism is developed by extending the axiomatic approach of a previous paper [J. Math. Phys. 9, 284 (1968)] in which abstract tensor techniques in Minkowski space were discussed. Comparable notational and calculational advantages are achieved, and, in particular, the cumbersome spinor indices are not required. The advantages and practicality of the method are evidenced in a discussion of the Dirac equation and a novel derivation of the two- and four-dimensional spinor representations of the homogeneous Lorentz group.
Cayley Transform of the S Matrix
Authors: Landshoff P. V.
Published: February 2004

The requirement that the S matrix be unitary is expressed in simple form by use of its Cayley transform (K matrix), which is also of value for practical computations. Here the physical-region structure of the K matrix is analyzed, and it is used to derive the nature of the ``simple'' physical-region singularities of the S matrix.
Dirac Algebra and the Six-Dimensional Lorentz Group
Authors: ten Kate A.
Published: February 2004

We establish the relationship between the Dirac algebra and the six-dimensional Lorentz group. By considering in an appropriate way the fifteen excentrical basis elements of the Dirac algebra as the components of an antisymmetrical tensor in six dimensions, the commutation as well as the anti-commutation rules of the algebra can be written in a six-covariant way. The group of automorphisms of the real Dirac algebra turns out to be isomorphic to the proper six-dimensional Lorentz group. However, this result is very sensitive to the specific choice of Lorentz metric.
Structure of the Poincaré Generators
Authors: Fleming Gordon N.
Published: February 2004

The Poincaré generators for an open system augmented by the interaction parts of the full Poincaré generators are shown to satisfy a closed set of coupled differential equations having a form which is independent of the nature of the interaction parts. The differential equations are formulated in the hyperplane formalism, the differentiation being with respect to the hyperplane parameters. The general solutions of the equations are studied, yielding relations among the augmented generators that must be preserved in the limit of zero interaction, i.e., for a closed system. Introducing a hyperplane-dependent Hamiltonian density in a manner not implying local field theory, the obtained relations are shown to yield expressions for all the generators of a closed system in terms of the Hamiltonian density and its derivatives alone.
Unitary Representations of the Affine Group
Authors: Aslaksen Erik W. , Klauder John R.
Published: February 2004

The unitary representations of the affine group, or the group of linear transformations without reflections on the real line, have been found previously by Gel'fand and Naimark. The present paper gives an alternate proof, and presents several properties of the representations which will be used in a later application of this group to continuous representations of Hilbert space. The development follows closely that used by von Neumann to prove the uniqueness of the Schrödinger operators.
Clebsch-Gordan Coefficients for Space Groups
Authors: Litvin D. B. , Zak J.
Published: February 2004

It is shown that to find Clebsch-Gordan coefficients of space groups (both single and double), the representations of the groups of k alone are required. This is another example demonstrating the well-accepted fact that in applications of space groups it is sufficient to know the representations of the groups of k. Final formulas are derived that enable the calculation of the Clebsch-Gordan coefficients from the representations of the groups of k. As an example the spin-orbit coupling in solids is considered.
Remarks on the Nature of Relativistic Particle Orbits
Authors: Kerner Edward H.
Published: February 2004

Instantaneous action-at-a-distance relativistic particle dynamics, embraced in Newtonian-like equations of motion χ̈i = Fi(χi, χ̇i) with suitable F's, is examined in once-integrated or ``kinematical'' form χ̇i = fi(χi, Vi) with Vi a set of first integrals transforming as velocities. The Lorentz covariance requirements on fi are worked out and illustrative examples are given, including a family of many-valued ones. A general meaning for integrals of χ̈i = Fi being in involution is adduced, and general counterparts to some well-known theorems in Hamiltonian dynamics are obtained accordingly. A novel elementary proof of the zero-interaction theorem is appended.
Asymptotic Theory of Cerenkov Radiation in Inhomogeneous Media
Authors: Handelsman R. A. , Lewis R. M.
Published: February 2004

The problem of Cerenkov radiation in infinite inhomogeneous media is considered. The mathematical description of this phenomenon is given by the integro-differential system of equations for the electromagnetic field in a dispersive medium. The leading term of the asymptotic expansion of the electromagnetic field is obtained by applying an expansion procedure called the ``ray method.'' In this method all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. The source which gives rise to the radiation is taken to be quite general. In fact, it is shown that any multipole moving along an arbitrary trajectory is a special case of the general source considered. From the expansion of the fields an expression for the total energy of the radiation is determined. Then, as an example, the case of plane-stratified media is treated in detail.
Intrinsic Vector and Tensor Techniques in Minkowski Space with Applications to Special Relativity
Authors: Luehr C. P. , Rosenbaum M.
Published: February 2004

This paper describes an abstract formalism for tensor analysis in Minkowski space which entails considerable notational simplicity and calculational advantages, as evidenced when compared with the usual component techniques. The need to express tensor equations in component form is eliminated, and manipulations become formally the same as those in Euclidean spece. The method is based on an extension to Minkowski space of the intrinsic concepts of vectors, differential operators, and polyadics in three-dimensional Euclidean spece. Several examples from special relativity have been selected to illustrate the advantages of the formalism. In the first example, expressions in dyadic form for the Euler-Lagrange equations and canonical energy-momentum tensor are obtained and specialized to the electromagnetic field. From the electromagnetic-field dyadic, invariants and other useful relations are derived easily and economically. The dyadic form of the field equations is also shown to be particularly amenable for a derivation of the Dirac-like form of Maxwell's equations with the base elements of the Pauli algebra emerging in a most natural way. Further illustration of the practical utility of the method is given by considering several properties of the restricted homogeneous Lorentz transformations. Various dyadic expressions for these are obtained, and a detailed derivation of their eigenvalues and eigenvectors is given. By combining some of the results from the discussions of Lorentz transformations and the Dirac-like form of Maxwell's equations, it is shown how an isomorphism between the three-dimensional complex orthogonal group and the Lorentz group can be established in a simpler and different manner from other approaches appearing in the literature.
Semiclassical and Quantum Descriptions
Authors: Misra S. P. , Shankara T. S.
Published: February 2004

In the semiclassical descriptions, it is usual to describe a quantum-mechanical system in a classical language with (i) a correspondence between classical functions and operators of quantum mechanics and (ii) with a real, but not necessarily positive, probability density function in phase space corresponding to a particular quantum-mechanical state. The general forms of such semiclassical descriptions is discussed. The conditions for the two descriptions to be equivalent are also examined.
Onset of ODLRO and the Phase Transition in the Ideal Boson Gas
Authors: Luban Marshall , Revzen Michael
Published: February 2004

The divergence of the constant-pressure specific heat CP and the isothermal compressibility KT as one lowers the temperature of the ideal boson gas to the transition temperature Tc is discussed in terms of the onset of off-diagonal long-range order (ODLRO) of the one-particle density matrix ρ1.
Electron Correlations, Magnetic Ordering, and Mott Insulation in Solids
Authors: Richmond Peter , Sewell Geoffrey L.
Published: February 2004

A model is constructed for the purpose of investigating electron correlations pertinent to magnetic ordering and Mott insulation in solids. The model consists of an assembly of interacting itinerant electrons in a periodic atomic lattice, such that the intra-atomic coupling between electrons is extremely strong. The correlations due to this latter coupling serve to prevent electrons of opposite spin from occupying the same atomic state, except in virtual transitions. Thus their net effect is to renormalize certain interactions and, also, to confine the state vectors of the entire system to a subspace, H, of the Hilbert space, H0, that is kinematically available to them. The observables are thus represented by operators on H, whose algebraic properties are different from thsoe of the corresponding operators on H0. Thus, the correlations due to intra-atomic forces are imbedded in the theory in the form of the new algebra. In cases of one electron per atom, these correlations lead simply to both magnetic ordering and Mott insulation. In cases of nonintegral number of electrons per atom, they can lead to magnetic ordering, subject to specified conditions.
Foundations for Quantum Statistics
Authors: Giles Robin
Published: February 2004

A derivation of Fermi and Bose statistics is given, based on the general structure of quantum mechanics, together with a simple axiom of direct physical significance. The axiom concerns an operation, denoted by ∘, forming the union of two states; ψ ∘φ denotes the state of a compound system whose parts are in the states ψ and φ. Let ψ and φ denote 1-particle states and assume: (a) ψ ∘φ exists whenever ψ ⊥ φ; (b) ψ ∘φ=φ ∘ψ; (c) the transition probability between ψ ∘φ and ψ′ ∘φ′ is zero if ψ ⊥ ψ′ and ψ ⊥ φ′, and (d) the product of the transition probabilities from ψ to ψ′ and from φ to φ′ if ψ ⊥ φ′ and φ ⊥ ψ′. It is then shown that, at least in so far as 2-particle states are concerned, the particles obey either Fermi or Bose statistics.
State Labeling of the Irreducible Representations of SUn
Authors: Devi V. Syamala , Prasad L. S. R. K.
Published: February 2004

State labeling of the irreducible representations of SU3 is done by using Littlewood's rules for the analysis of products of representations of unitary groups. The method is generalized to any SUn.
Embeddings of the Plane-Fronted Waves and Other Space-Times
Authors: Collinson C. D.
Published: February 2004

The plane-fronted waves of general relativity are embedded in a six-dimensional pseudo-Euclidean space of signature −2. Two distinct families of embeddings are found. Embeddings of several well-known space-times are obtained. Certain results of J. Rosen are improved.
Asymptotic Properties of Perturbation Theory
Authors: Krieger Joseph B.
Published: February 2004

The perturbation expansions are derived by a technique which does not assume that convergent expansions exist. The theory is shown to be asymptotic, and criteria are developed to determine if a finite number of terms underestimates or overestimates the exact result for sufficiently small values of the coupling constant.
Asymptotic Gravitational Field of the ``Electron''
Authors: Martin A. W. , Pritchett P. L.
Published: February 2004

The Einstein-Maxwell equations appropriate to the exterior metric and fields of a source characterized by mass, electric charge, and magnetic dipole moment are formulated. Because of the presence of a nonvanishing Poynting vector, the metric tensor must include off-diagonal elements. A rigorous reduction of the metric tensor to three unknown functions is accomplished and the field equations are solved to second order in the gravitational constant. Using the parameters of the electron and proton, we find that the magnetic dipole terms dominate the metric at small distances and that general relativistic effects become important at distances of the order of 10−22 cm. Possible applications of the asymptotic metric are discussed.
Extension Theorem and Representations
Authors: Sankaranarayanan A.
Published: February 2004

An extension theorem prescribing a method for constructing the (n + 1)-dimensional Lorentz group O(n, 1) from the n-dimensional inhomogeneous rotation group, proved earlier, has been recapitulated and some comments on the theorem are made. The prescription is used to construct the unitary irreducible representations of the Lorentz groups O(2, 1), O(3, 1), and O(4, 1), and it is found that only a limited number of representations are allowed.
Green's Theorem and Invariant Transformations
Authors: Goldberg Joshua N.
Published: February 2004

Conservation laws are derived with the use of Green's theorem. These are studied specifically for the wave equation. A trivial class of invariant transformations exists which maps solutions into solutions. The mapping consists in the addition of a particular solution to all solutions. Because of the linearity of the wave equation, this sum will again be a solution. It is shown that this class of transformations has the Newman-Penrose constants among its generators. Calculations are carried out explicitly for the scalar wave equation and for Maxwell's equations.
Frequency Spectra of Harmonic Lattices with Weak Long-Range Interactions
Authors: Zwanzig Robert
Published: February 2004

Vibrational-frequency spectra are calculated analytically for certain harmonic lattice models with weak long-range interactions, in one and two dimensions. The force constants are chosen to decay, for large separations n, approximately as exp (−γn), where γ is an inverse range parameter. In the limit of infinite γ, standard nearest-neighbor results are recovered. In the limit of vanishing γ, or infinite interaction range, the frequency spectra have entirely different singularities. In the examples studied here, these can be either poles or branch points at the maximum frequency. When γ is small but not zero, then the singularities are still qualitatively the same as in the corresponding nearest-neighbor models; but the general shapes of the spectra are dominated by ``false'' singularities lying outside the allowed frequency domain. Except for a small region near the maximum frequency, spectra obtained in the limit of vanishing γ are good approximations to the correct spectra for small γ.
Electrostatic Interactions of the Configuration d
n−1sp
Authors: Roth Charles
Published: February 2004

The matrix elements of the Coulomb interactions of the configuration dn−1sp were obtained for L-S coupling in the form of linear combinations of certain radial integrals. The method used can be extended to the configuration ln−1l′l″.
Some Inequalities Involving Traces of Operators
Authors: Mehta C. L.
Published: February 2004

We prove that for arbitrary completely continuous operators A1, A2, ⋯, An and for positive numbers p1, p2, ⋯, pn with ∑k=1npk−1=1, the inequalities,|Tr (A1A2⋯An)|≤ ∏ k=1n [Tr (Ak†Ak)12pk]pk−1|Tr exp (A1+A2+⋯+An)|≤ ∏ k=1n [Tr exp {12pk(Ak†+Ak)}]pk−1hold. Further if a1, a2, ⋯, an are the annihilation operators of an N-dimensional harmonic oscillator, m and n are any positive integers, and ρ is a nonnegative definite operator, we prove the inequality |Tr (ρai1†⋯aim†aj1⋯ajn)|≤ ∏ k=1m [Tr ρ(aik†aik)m]1/2m ∏ l=1n [Tr ρ(ajl†ajl)n]1/2n. Some consequences of these inequalities, related results, and some applications to correlation functions of the quantized electromagnetic field are discussed.
Orear Behavior in Potential Scattering
Authors: Johnson Porter
Published: February 2004

The object of this paper is to find and study a class of potentials for which the corresponding scattering amplitude f decreases rapidly in energy at fixed (nonforward) angles. Specifically, we ask that f(k,θ)=O[exp {−b(θ)k}] as k → ∞ for θ fixed. It is shown here that this relation is valid for certain potentials V(r) which are even functions of r analytic in a strip about the real r axis. With further restrictions on the potentials we show that the scattering amplitude converges to its first Born approximation at high energies for fixed nonforward angles.
Integral Representation for Systems of Interacting Particles
Authors: Alexanian Moorad
Published: February 2004

This is a sequel to a previous paper, designated as I, which dealt primarily with nonequilibrium systems. The ideas in I are extended and include the study of systems of interacting particles in equilibrium. An integral representation is obtained for the distribution function of interacting particles as a superposition of distribution functions for noninteracting particles. In particular, it is shown that an interacting Bose gas need not show Bose condensation and that the behavior near the point E = 0 cannot be more singular than that of a simple pole.
New Field-Theory Approach to Singular Potentials
Authors: Aly H. H. , Taylor J. G.
Published: February 2004

An approach similar to the methods of renormalization of the Green's functions equations of quantum field theory is adopted to the singular potential in the Lippman-Schwinger equation. The close relation between our approach and the one used in field theory gives a method to be applied to nonrenormalizable field theories. The physical implication of this approach is discussed.
Realization of Poincaré-Group Generators on a Light Cone
Authors: Peres Asher
Published: February 2004

The ten generators of the proper inhomogeneous Lorentz group are explicitly constructed for spin-0 and spin-½ particles, in the case when wavefunctions are given on a light cone (rather than on an equal time hypersurface, as usual). The advantage of this formulation is that the generators J (spatial rotations) and K (Lorentz boosts) involve only elementary local operators. The Hamiltonian H and momentum operators P also contain the inverse radial momentum (∂/∂r)−1, but do not involve any square roots. Moreover, only two-component spinors are required for spin-½ particles.
Global Existence of Spin Structures for Gravitational Fields
Authors: Bichteler Klaus
Published: February 2004

The notion of a spin structure for a gravitational field is defined, and it is shown that a spin structure exists if and only if the second Stiefel-Whitney class of the space-time M vanishes. The number of different spin structures is then equal to the number of elements in H1(M, Z2).
Lie-Group and Lie-Algebra Inhomogenizations
Authors: Berzi Vittorio , Gorini Vittorio
Published: February 2004

A systematic formulation of the concept of inhomogenization is given both for Lie groups and for Lie algebras, and the connection between the two structures is clarified in terms of the notion of semidirect product. Special emphasis is devoted to the classification of the inhomogenizations of semisimple Lie algebras. As an application, a lemma due to O'Raifeartaigh is generalized to a wide class of inhomogeneous structures.
Ising-Model Spin Correlations
Authors: Hartwig Robert E. , Stephenson John
Published: February 2004

In this paper we study a class of planar Ising lattices which have two or more directions along which the pair-correlation function admits a Toeplitz determinant representation. On the basis of the recently developed theory of Toeplitz determinants, we discuss the asymptotic behavior of these correlations at the critical point.
Exact Quantization Conditions
Authors: Rosenzweig C. , Krieger J. B.
Published: February 2004

The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.
Convergence of the Born Series
Authors: Corbett John V.
Published: February 2004

Three types of Born series which can be associated with a transition amplitude are discussed, and the criteria for the convergence of the different series are compared. With regard to the divergence of the Born series, the ordering, matrix-element series diverges implies vector series diverges implies operator series diverges, is obtained for the natural vector and operator Born series that can be abstracted from the expression for the transition amplitude. The conclusion that the divergence of the operator Born series does not ensure that the Born series of physical matrix-elements divergences is applied to an example of three-body rearrangement scattering.
Are Bloch Bands at Finite Field Adiabatically Connected to Those at Zero Field?
Authors: Wannier G. H. , Van Dyke J. P.
Published: February 2004

It has been shown previously that if a potential consists of a superposition of a periodic part and a uniform electric field, then for a particle moving in this field there are Bloch bands closed in time. The present paper addresses itself to the question whether these bands may be identified with the field-free bands. The most natural thing is to expect that the bands are slightly field dependent, but converge toward the field-free bands as E goes to zero. Bands for which this is true are said to be adiabatically connected to corresponding bands at zero field. In Sec. 2 of the paper, two model cases are given for which this adiabatic connection pertains. Section 3 is the central part of the paper and provides the conclusion that the answer to the question in the title is almost always negative. In this proof the positive cases serve an essential function. It is shown that the parameters of the periodic potential must obey at least one supplementary condition to allow adiabatic connection, and that the collected cases precisely obey this condition. Adiabatic connection is thus generally not possible. Section 4 provides an explicitly soluble case which does not allow adiabatic connection. An infinite number of field values E converging toward zero are found at each of which the two bands under consideration switch identity (hyperbolic rather than linear connection at energy crossings). The connection postulated in the effective-mass approximation must therefore be of a nonadiabatic nature. It probably involves the ``sudden'' approximation of quantum theory.
Electromagnetic Two-Body Problem for Particles with Spin
Authors: Schild A. , Schlosser J. A.
Published: February 2004

A two-body system is considered for classical particles with charge, spin, and magnetic moment. The particles move in circular orbits, with spins orthogonal to the orbital plane, and they interact through time-symmetric electromagnetic fields. Rigorous relativistic equations of motion and rigorous expressions for the total energy and angular momentum of the system, including contributions from the field, are obtained.
Simple Generalization of Schrödinger's Equation
Authors: de la Peña-Auerbach L. , García-Colín Leopoldo S.
Published: February 2004

A possible generalization of quantum mechanics is examined by showing that the motion in phase space of a classical Brownian particle may be described by a complex probability amplitude depending on the phase-space coordinates and the time, and obeying a Schrödinger-like equation. However formal this result may seem, the usual dynamical operators may be defined whose physical meaning stems directly from the theory. An outstanding feature of the formalism is that ordinary quantum mechanics in configuration space may be recovered in a limiting process whereby the velocity variable, defined now through a statistical distribution, is eliminated. Therefore, it plays the role of a hidden variable. This result supports recent reinterpretations of von Neumann's theorem on the nonexistence of such variables in quantum mechanics and serves as a counterexample of the usual interpretation of his theorem.
Local Field Theory and Isospin Invariance. I. Free-Field Theory of Spinless Bosons
Authors: Carruthers Peter
Published: February 2004

The properties of free-field theories of spinless bosons are investigated. The self-conjugate boson field theories of isospin ½, 32, 52, ⋯ are shown to be nonlocal: the energy, isospin, and number densities fail to commute for spacelike separations. The difficulty is traced to the necessary occurrence of non-independent field variables. For the anomalous case the dependent variables are nonlocally related to the independent ones; elimination of the former makes superficially local quantities depend nonlocally on the independent fields. The connection with canonical field theory is investigated. The p and q coordinates have anomalous commutation relations in the nonlocal case. Although the Hamiltonian appears normal in every case, the anomalous theories are characterized by zero (integrated) Lagrangian. Antiparticle conjugation is investigated in detail with attention to phase questions. For anomalous theories two types of conjugation are found, one of which nonlocally relates the field to a superposition of its adjoint. A unitary transformation is constructed which converts one type to the other. Finally, CPT transformation properties of normal and abnormal theories are derived and compared.
Matrix Elements of Operators of Class I Representations of SL(n, C)
Authors: Fronsdal C. , Ar-Rashid Harun
Published: February 2004

Let D be an irreducible representation of the group SL(n, C) by operators Tg in a Hilbert space H. We introduce a canonical basis of vectors h ∈ H, and show how to evaluate the ``D-functions'' (h1, Tgh2). It has been suggested that one such function has the physical interpretation of the electric-charge form factor of the pion. This function is evaluated in detail.
Path-Integral Calculation of the Quantum-Statistical Density Matrix for Attractive Coulomb Forces
Authors: Storer R. G.
Published: February 2004

A method for the calculation of the two-particle statistical density matrix for attractive Coulomb forces is described. The path-integral expression for the density matrix is reduced to a modified path integral which involves summation over only one-dimensional paths. This expression is then approximated by an iteration procedure using direct numerical quadratures. The results obtained are related directly to the quantum-mechanical radial-distribution function for a plasma at small ion-electron separations.
Combinatorial Theorem for Graphs on a Lattice
Authors: Nagle J. F. , Temperley H. N. V.
Published: February 2004

Several problems in lattice statistical mechanics, such as the spin-½ Ising problem and the monomer-dimer problem, can be formulated in terms of the p-generating function for the weak subgraphs of a regular lattice. This paper presents an algebraic transformation theorem which allows the p-generating function for the weak subgraphs of a lattice to be determined from the p-generating function for the far less numerous subset consisting of the closed weak subgraphs. This result will be especially useful in reducing the labor required to obtain exact finite series for various problems. The theorem also enables one to give a straightforward proof of the Ising susceptibility graph theorem due to Sykes.
Rotation and Lorentz Groups in a Finite Geometry
Authors: Beltrametti E. G. , Blasi A. A.
Published: February 2004

The introduction in physics of a finite geometry approximating the ordinary Euclidean one poses the problem of studying the relativity groups over such a geometry. We present a detailed analysis of the structure and irreducible representations of the rotation, Lorentz, and Poincaré groups. It is found that, besides the usual quantum numbers, a new two-valued label is necessary to specify the representations.
Plethysm and the Theory of Complex Spectra
Authors: Smith P. R. , Wybourne B. G.
Published: February 2004

Notable progress in developing the theory of complex spectra has come from the exploitation of the symmetry properties of atomic wavefunctions using the theory of continuous groups. Many seemingly simple results have previously been obtained by what would appear to be unreasonably complex methods. Many of these complexities may be removed, and new results derived, using the algebra of plethysm first developed by Littlewood. Applications to three central problems in the theory of complex spectra are discussed: (1) the classification of the atomic states of n-electron configurations; (2) the analysis and classification of the N-particle operators that arise in the application of perturbation theory to atomic problems; (3) the derivation of selection rules for the matrix elements of operators.
Relativistic Effects in Atomic Fine Structure. II
Authors: Armstrong Lloyd
Published: February 2004

The methods of second quantization are used to investigate matrix elements of the Breit equation in mixed configurations. In the special case of a single configuration, the results are shown to be identical to those obtained in the previous paper of this series using equivalent operators. Application of second-quantization methods to calculations involving relativistic atomic states is shown to be extremely straightforward and to provide a simplicity and clarity unattainable using the equivalent-operator technique.
Construction of an Invariance from a Conservation Law, and Vice Versa, in Classical Mechanics
Authors: Pearle Philip
Published: February 2004

Special classes of problems (each class is called a mechanics) are discussed for which it is either always possible to construct a conservation law when one is given an invariance of the equations of motion, or else it is always possible to construct an invariance when one is given a conservation law. Of chief interest is the introduction of a set of mechanics obeying the latter alternative in the form: (A) Given any constant of the motion, one can always construct a linear combination of its first derivatives (with fixed constant coefficients) which will be an invariance generator. Hamiltonian mechanics is, roughly speaking, ``derived'' as a consequence of (A) and the additional postulate: (B) One can always find a function from which, when one constructs the linear combination of its first derivatives, one obtains the motion generator. Another interesting mechanics satisfying (A) [but not (B)] is discussed. This is a class of harmonic-oscillator problems, called oscillator mechanics. It is shown that there is a class of problems belonging to both Hamiltonian mechanics and oscillator mechanics. This class of problems has an algebra that is the same as the algebra of quantum mechanics.
New Methods for Reduction of Group Representations. II
Authors: Gabriel John
Published: February 2004

A better algorithm for reduction of a group representation than that given in an earlier paper [J. R. Gabriel, J. Math. Phys. 5, 494 (1964)] is derived.
Tamm-Dancoff Method in the 2V Sector of the Lee Model
Authors: Scarfone Leonard M.
Published: February 2004

The 2V sector of the Lee model with boson sources at zero separation is considered from the Tamm- Dancoff point of view. This involves three singular integral equations which are solved by appealing to results previously gained in the Lehmann-Symanzik-Zimmermann method of solution. Two scattering amplitudes and a production amplitude are rederived via the state vectors and an equation for the determination of the two-meson exchange potential is discussed.
Derivation of Newton's Law of Gravitation from General Relativity
Authors: Katz Amnon
Published: February 2004

A static situation of two objects held apart by a strut is considered within the framework of general relativity, and the gravitational attraction between the objects is inferred from the stress in the strut. In the Newtonian limit—the objects are well separated and the field weak everywhere except in their immediate vicinity—Newton's law of gravitation is reproduced. This check goes well beyond verifications of the Newtonian limit based on considerations of ``test particles,'' since arbitrarily strong self-fields are not excluded for either object. It is also an explicit verification of the equality of active and passive gravitational masses.
Particlelike Solutions to Nonlinear Complex Scalar Field Theories with Positive-Definite Energy Densities
Authors: Rosen Gerald
Published: February 2004

It is shown that a self-interacting complex scalar field theory with a positive-definite energy density can admit spatially localized singularity-free particlelike solutions. A condition on the self-interaction energy density, sufficient to guarantee the existence of such solutions, is that its derivative should be nonincreasing and not identically constant as the squared absolute value of the field increases from zero.
Charged Particlelike Solutions to Nonlinear Complex Scalar Field Theories
Authors: Rosen Gerald
Published: February 2004

It is shown that spatially localized singularity-free particlelike solutions exist for Lorentz-covariant complex scalar field theories with minimal gauge-invariant electromagnetic coupling, a positive-definite energy density, and suitably prescribed nonlinear self-interaction. Such a theory provides a perfectly consistent structural model on the classical level for a charged elementary particle of finite positive energy.
Time-Ordered Products in Two-Dimensional Field Theories
Authors: Challifour John L.
Published: February 2004

It is shown that the vacuum expectation values (VEV) of ⋮eλσ⋮(t,f)=∫dxf(x) ⋮eλσ⋮(t,x) are continuous functions of the time for test functions which are C∞ and of rapid decrease, with λ in some neighborhood of the origin in the complex plane. The field σ(x) is the pseudopotential derived from the pseudovector current of a free two-component massive field in two-dimensional space-time. A consequence of this result is the existence of Green's functions in the Federbush model. An essential technique in the proof is a theorem by Jaffe on the boundary values of limits of sequences of analytic functions.
Canonical Realizations of the Galilei Group
Authors: Pauri M. , Prosperi G. M.
Published: February 2004

The general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions Q(y),P(y),J(y) are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants dρσ appearing in the fundamental Poisson bracket relations among the infinitesimal generators ({yρ,yσ}=cρστyτ+dρσ) cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type which describes the free mass-point, and another of a regular type which describes a classical particle spin. A number of physical significant examples corresponding to nonirreducible realizations are thereafter discussed and the related typical forms are constructed: specifically, the cases dealt with are the rigid rod (linear rotator), the rigid body, and a system of two interacting mass points. It is shown that the problem of the construction of the variables of the typical form is equivalent to the determination of an appropriate solution of the time-independent Hamilton-Jacobi equation.