bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarks Concerning Reciprocity in Quantum Mechanics
Authors: Bilhorn D. E. , Foldy L. L. , Thaler R. M. , Tobocman W. , Madsen V. A.
Published: January 2005

A new derivation of the reciprocity theorem is given. The general invariance property of the Hamiltonian which leads to symmetry of the Green's function for a quantum mechanical system is exhibited. It is found that reciprocity does not necessarily imply Hermiticity of the Hamiltonian, so that the ``complex optical model potential,'' for example, satisfies the reciprocity relations. The concept of reciprocity is then generalized to include a somewhat wider class of symmetry properties. Some properties of antiunitary transformations are discussed.
Exact Conditions for the Preservation of a Canonical Distribution in Markovian Relaxation Processes
Authors: Andersen H. C. , Oppenheim I. , Shuler Kurt E. , Weiss George H.
Published: January 2005

Necessary and sufficient conditions have been determined for the exact preservation of a canonical distribution characterized by a time-dependent temperature (canonical invariance) in Markovian relaxation processes governed by a master equation. These conditions, while physically realizable, are quite restrictive so that canonical invariance is the exception rather than the rule. For processes with a continuous energy variable, canonical invariance requires that the integral master equation is exactly equivalent to a Fokker-Planck equation with linear transition moments of a special form. For processes with a discrete energy variable, canonical invariance requires, in addition to a special form of the level degeneracy, equal spacing of the energy levels and transitions between nearest-neighbor levels only. Physically, these conditions imply that canonical invariance is maintained only for weak interactions of a special type between the relaxing subsystem and the reservoir. It is also shown that canonical invariance is a sufficient condition for the exponential relaxation of the mean energy. A number of systems (hard-sphere Rayleigh gas, Brownian motion, harmonic oscillators, nuclear spins) are discussed in the framework of the above theory. Conditions for approximate canonical invariance valid up to a certain order in the energy are also developed and then applied to nuclear spins in a magnetic field.
Generalized Shmushkevich Method: Proof of Basic Results
Authors: Macfarlane A. J. , Mukunda N. , Sudarshan E. C. G.
Published: January 2005

We here derive certain orthogonality properties of the Clebsch-Gordan (CG) coefficients of an arbitrary compact group G. Our discussion recognizes the fact that the irreducible representations (IR's) of G need not be equivalent to their complex conjugates and that the same IR can appear more than once in the reduction of the direct product of two IR's of G. The properties obtained allow the development of a generalized Shmushkevich method for directly writing down consequences of the invariance of particle interactions under G. The discussion given is sufficiently general to apply to the currently interesting cases of SU3 and G2.
Intensive Observables in Quantum Theory
Authors: Streater R. F.
Published: January 2005

The notion of strictly intensive observables is introduced in a theory of local observables (such as that of R. Haag). If O1, O2 are two disjoint regions, then the value of a strictly intensive observable in O1 ∪ O2 is the sum of its values in O1 and O2. It is shown that energy-momentum can never be strictly intensive. This result is used to prove that the algebras of observables is not of Type I for some regions. By analogy with the energy-momentum tensor density for the free field, the definition is weakened, so that an intensive observable in a region O is only ``approximately'' in a(O). This leads to the introduction of germs of intensive observables. It is proved that the unitary intensive operators form a sheaf 𝔉 of groups, and the Hermitian intensive operators form a sheaf 𝔊 of Abelian groups with operators in 𝔉, and on which the inner derivative Y → i(XY − YX) is defined.
Coulomb Green's Functions and the Furry Approximation
Authors: Hostler Levere
Published: January 2005

The Coulomb Green's function for the nonrelativistic Schrödinger equation is obtained in closed form starting from the partial-wave expansion and using an integral representation for a product of two Whittaker functions with different arguments. The Neumann's series for Jv(kz) is required in evaluating the sum on states. Using the same methods, the Coulomb Green's functions for the Klein-Gordon and iterated Dirac equations are obtained in closed form in the ``Furry approximation,'' a2/(J + ½)2 ≪ 1, a = Ze2/4πh/c. The Klein-Gordon Green's function in this approximation is shown to be at the same time the exact Green's function for the Klein-Gordon equation without the potential squared term. An alternate and very simple derivation of the approximate Green's function for the iterated Dirac equation is given using perturbation theory. From this Green's function, an approximate Coulomb Green's function in closed form for the Dirac equation itself is constructed. Certain known results for Coulomb wavefunctions with modified plane-wave behavior at large distances are rederived using the foregoing methods and results.
Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations
Authors: Lax Peter D.
Published: January 2005

In a recent paper Zabusky has given an accurate estimate of the time interval in which solutions of the nonlinear string equation ytt = c2(1 + εyx)yxx exist. A previous numerical study of solutions of this equation disclosed an anomaly in the partition of energy among the various modes; Zabusky's estimate shows that at the time when the anomaly was observed the solution does not exist. The proof of Zabusky uses the hodograph method; in this note we give a much simpler derivation of the same result based on an estimate given some years ago by the author.
Existence of a New Conservation Law in Electromagnetic Theory
Authors: Lipkin Daniel M.
Published: January 2005

Ten new extensive quantities that appear to be independent of stress-energy, but that analogously characterize the physical state of an electromagnetic field, are exhibited and are shown to be conserved in vacuum because of Maxwell's equations. These new quantities are shown to be capable of retrograde flow in a circularly polarized plane-wave field.
Segal's Quantization Procedure
Authors: Prosser Reese T.
Published: January 2005

The procedure proposed by I. E. Segal for the quantization of nonlinear problems of field theory is here applied to the one-dimensional oscillator. It is shown that for the linear oscillator Segal's procedure is equivalent to the canonical procedure, but that for the nonlinear oscillator the two procedures lead to quite different results. The differences are reflected in the equations of motion, the energy spectra, and the scattering cross sections.
Convergent Perturbation Expansions for Certain Wave Operators
Authors: Prosser Reese T.
Published: January 2005

This paper establishes rigorously the validity of Dyson's perturbation expansion for the Mo/ller wave operators under suitable restrictive assumptions on the interaction potential.
Class of Ensembles in the Statistical Theory of Energy-Level Spectra
Authors: Leff Harvey S.
Published: January 2005

The investigation of a large class of ensembles in the statistical theory of energy-level spectra is initiated. Each member of this class is characterized by a joint probability density for N consecutive eigenvalues of the form PNβ(λ1,…λN)=ΩNβ−1{ ∏ i=1N f(λi)}  ∏ k<l |λk−λl|β,where a ≤ λi ≤ b, and β may be 1, 2, or 4. Formal calculations of the nearest-neighbor spacing distribution and the level density are made for β = 2. Results are in terms of asymptotic properties of orthogonal polynomials. It is conjectured that spacing distributions are relatively insensitive to the function f(λ) and the interval [a, b]. When f(λ) = 1 and b = −a = 1, the resulting (Legendre) ensemble has the same spacing distribution as the Gaussian and Dyson ensembles. The level density is concave upward and rapidly increasing for λ ≥ 0, qualitatively resembling actual nuclear and atomic densities. This feature is not present in previously investigated ensembles. Certain invariant matrix ensembles introduced by Dyson, which are of the above type, have the same level density and nearest-neighbor spacing distribution as the Legendre ensemble.
Theory of Scattering in Solids
Authors: Callaway Joseph
Published: January 2005

The general theory of the scattering of excitations in solids by localized imperfections is discussed. The solid-state analog of the usual partial-wave expansion of the scattering amplitude is derived. In an appendix, the applicability of the general theory to phonons and spin waves as well as electrons is demonstrated.
Continuous-Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics
Authors: McKenna James , Klauder John R.
Published: January 2005

A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase-space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.
Momentum Distribution in the Ground State of the One-Dimensional System of Impenetrable Bosons
Authors: Lenard A.
Published: January 2005

Girardeau has shown that an exact analytical formula may be given for the ground-state wave-function of a system of one-dimensional impenetrable bosons. Starting with this formula, we give a mathematically rigorous analysis leading to the determination of major features of the momentum distribution in the limit of an infinitely large system.
On the Stability of Flow of a Thermally Stratified Fluid under the Action of Gravity
Authors: Koppel Donald
Published: January 2005

The equation for the small disturbances from the plane-parallel flow of a thermally stratified fluid under the influence of gravity acting perpendicular to the plane of stratification is derived. It was found necessary to include not only viscosity but also heat conductivity to preclude the resulting differential equation from having a singularity. Asymptotic solutions of the sixth-order differential equation thus derived are obtained. They show the presence of a Stokes point. The limiting form of the differential equation near the Stokes point is next obtained and an exact solution of this equation is derived by means of a Laplace transformation. In the general case the integrand of the Laplace transformation involves Whittaker's confluent hypergeometric functions. In the special case of a Prandtl number of 1, the integrand is considerably simpler and for this case asymptotic representations of the solutions on both sides of the Stokes point have been derived from the Laplace transformation solution by the method of steepest descent. The connection formulas between the solutions are the same as that previously derived by Tollmien and Lin for the case when stratification, gravity, and heat conduction are neglected.
Inversion of Cyclic Matrices
Authors: Calais Jean-Louis , Appel Klaus
Published: January 2005

The inversion of the overlap matrix in LCAO calculations of crystals is discussed for the general case, when an arbitrary number of neighbors is taken into account. Of the several alternative methods that are possible in one dimension, one is singled out—which can be used as well in three dimensions—to calculate the inverse to any desired accuracy. Particular attention is paid to the numerical aspects of the problem.
Nested Hilbert Spaces in Quantum Mechanics. I
Authors: Grossmann Alexander
Published: January 2005

A nested Hilbert space is a pair of Hilbert spaces H0, H1, each of which is in a certain sense identified with a dense subset of the other. These structures are used here to study analytic continuation into ``unphysical sheets'' and to discuss nonnormalizable states of quantum-mechanical systems.
Upper and Lower Bounds on Generalized Fourier Coefficients
Authors: Delves L. M.
Published: January 2005

Methods are given for obtaining variational upper and lower bounds on the nth Fourier coefficient Cn2=|∫g*σUndτ/2 of a function g relative to a sequence of eigenfunctions Un. The methods differ in their ease of application and in the amount of information required concerning the eigenvalues associated with the Un. Some illustrative examples are given.
Upper and Lower Bounds on the Matrix Elements of an Arbitrary Bounded Operator
Authors: Delves L. M.
Published: January 2005

Methods are given for constructing variational upper and lower bounds on the matrix element Wmn=∫Um*WUndτ of an arbitrary bounded operator W between two eigenstates Um, Un of a Hamiltonian H. Numerical examples of the method are given.
Many-Channel Bargmann Potentials
Authors: Cox Joseph R.
Published: January 2005

A generalization of the Bargmann potentials to include the case of nonrelativistics-wave scattering of a particle by a target having a finite number of discrete excited states is presented. This generalization allows the explicit construction of a large class of many-channel S-matrices meromorphic on their energy Riemann surfaces as well as the explicit construction of the corresponding potential matrices. A two-channel example is treated in detail.
On the Theory of Randomly Dilute Ising and Heisenberg Ferromagnetics
Authors: Rushbrooke G. S.
Published: January 2005

The Brout expansion for the free energy of the Ising or Heisenberg model is formally summed over all interaction graphs with not more than m vertices: the result is expressed in terms of the partition functions of isolated physical clusters, again having not more than m vertices. These partition functions are multiplied by occurrence factors closely related to the occurrence factors for the corresponding isolated physical clusters in a randomly dilute ferromagnet at low concentrations of the magnetic elements. Comparison is made with earlier work on the randomly dilute Ising and Heisenberg models.
Kinetic Equation for an Inhomogeneous Plasma far from Equilibrium
Authors: Balescu R. , Kuszell A.
Published: January 2005

A kinetic equation for a nonuniform plasma is derived from the Liouville equation by a general diagram technique. It describes the evolution of a small spatial inhomogeneity in a plasma whose velocity distribution is far from equilibrium (and hence time-dependent). The equation is valid for short and long times, within the ring approximation. Its explicit form is obtained by the exact closed solution of a singular integral equation. The kinetic equation is non-Markoffian and, contrary to the corresponding homogeneous equation, keeps a trace of this character even in the limit of long times. Only when the velocity distribution and the two-body correlation function reach thermal equilibrium does the equation reduce to a Markoffian limit. The latter is identical with the kinetic equation derived earlier by Guernsey. The treatment of unstable inhomogeneous plasmas is briefly indicated.
Equation of State for a Gas with a Weak, Long-Range Positive Potential
Authors: Newman David S.
Published: January 2005

A one-dimensional fluid model in which the pair interaction potential is exponential and repulsive is considered, and the equation of state in the ``long-range'' limit is determined exactly. This model is complementary to one studied by Kac, in which the pair potential consists of a hard core and an exponential, attractive tail. In that model a phase transition occurs, but in the current model there is no phase transition. This fact lends support to a conjecture of Ruelle that no phase transition occurs if the potential is bounded. The possibility of applying the method of Kac to a wider class of potentials is suggested, and some of the mathematical difficulties yet to be overcome are outlined.
Microwave Surface Impedance of a Periodic Medium
Authors: Fischer Gaston
Published: January 2005

The surface impedance of a medium is calculated, the complex but scalar conductivity of which is a periodic function in a direction parallel to the surface. Radiation is assumed to impinge vertically onto the surface. It is shown that there are two independent field configurations according to whether the induced microwave current flows in a direction parallel to the conductivity pattern or perpendicular to it. These two configurations lead to different surface impedances, the one obtaining in the perpendicular configuration being generally larger. The calculation is carried through when the conductivity is of the form σ(y) = σ0 + 2σ1 cos (ky), and a local relation between current and fields is assumed. Equations are also derived for the more general problem when the periodic conductivity is an arbitrary Fourier series and a nonlocal relation between current and fields is assumed. These problems are of interest in measurements of the microwave surface impedance of hard super-conductors.
On the Integrodifferential Equations of the Nonlocal Theory of Superconductivity
Authors: Odeh Farouk M.
Published: January 2005

The nonlocal model of superconductivity has, in recent years, received strong support on both microscopic theoretical and experimental grounds. In the present report two of the mathematical aspects of the theory are discussed. One concerns the question of well-posedness of the equations and boundary conditions. The other concerns some methods of obtaining approximate solutions. Existence and uniqueness theorems are given for the solutions of the integrodifferential equations governing the vector potential, under suitable restrictions on the kernels involved. Both diffuse and specular reflection types of boundary conditions are considered in both scalar (one-dimensional) and vector (three-dimensional) problems. An existence theorem is also given for the time-dependent equations. Approximate solutions for the problem of a film in a parallel magnetic field are given in two limiting cases. In one the small-coherence-length limit in the Pippard model under diffuse scattering boundary conditions is considered, and a boundary-layer-type solution is given. In the other a thin-film approximation to the diffuse problem is found. An exact series representation is also given for the solution of the problem of a circular cylinder in a parallel magnetic field with specular reflection boundary conditions.
Asymptotic Expansion of Certain Integrals Containing the Bessel Function J0(x)
Authors: Schmidt Paul W.
Published: January 2005

An asymptotic expansion is calculated for the integral 2π ∫ 01dxf(x)J0(kx)(1−x2)12where J0(x) is the zero-order Bessel function of the first kind, and the function f(x) and all its derivatives are continuous for 0 ≤ x ≤ 1. To find the asymptotic expansion, the integral is converted to a Fourier integral, to which asymptotic expansion techniques are applied. The method can be generalized to apply to related integrals.
Interaction of Nonrelativistic Particles with a Quantized Scalar Field
Authors: Nelson Edward
Published: January 2005

We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on the interaction, we obtain a well-defined self-adjoint operator. The solution of the Schrödinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a constant infinite phase shift due to the self-energy of the particles. In the Heisenberg picture, we obtain a solution in the limit of no cutoff. We use a canonical transformation due to Gross to separate the divergent self-energy term. It is shown that the canonical transformation is implemented by a unitary operator, and that the transformed Hamiltonian, with an infinite constant subtracted, can be interpreted as a self-adjoint operator.
Quantization of Electrodynamics in the Axial Gauge
Authors: Yao York-Peng
Published: January 2005

The so-called axial gauge condition A3(x) = 0 is shown to be inconsistent with the condition of Lorentz invariance. This inconsistency is resolved herein.
Triality Type and its Generalization in Unitary Symmetry Theories
Authors: Hagen C. R. , Macfarlane A. J.
Published: January 2005

Within the context of an extension of the SU3-symmetry theory recently suggested by Gell-Mann and further developed by the authors, certain aspects of the theory of the special unitary groups are examined. The plurality type of a given representation is introduced as the generalization of the triality concept to SUn+1 and is shown to be associated with a multiplicative conservation law. Theorems for the reduction of representations of SUn+1 with respect to SUn⊗ U1(n) are derived which are subsequently used to relate plurality type to the existence of fractional eigenvalues for the generator Y1(n) of U1(n).
Formal Solution of a Nonhomogeneous Differential Equation with a Double Transition Point
Authors: Jordan P. F. , Shelley P. E.
Published: January 2005

A formal solution of a fairly general nonhomogeneous, linear, second-order differential equation with a large parameter and a double transition point is presented. This equation arises, for instance, in the quasilinear theory of pressurized membrane shells of revolution. The fact that the solution which is of foremost importance in practical applications converges at infinity makes it convenient to use a direct approach, avoiding any transcendental transformation. The solution is described by means of influence functions which arise from a formal inductive process. The more important influence functions are tabulated. The results of two approximate asymptotic procedures are compared with the exact solution.
Relation between the Onsager and Pfaffian Methods for Solving the Ising Problem. I. The Rectangular Lattice
Authors: Hurst C. A.
Published: January 2005

An algebraic proof is given showing the equivalence of the Pfaffian and Onsager methods of solution of the Ising problem for the two-dimensional rectangular lattice with free edge conditions. With cyclic conditions on the rows and columns and with helical edge conditions it is shown how the two solutions differ. The relation between the appearance of crossed long-range bonds and the appearance of unwanted negative signs in the Pfaffian method is shown explicitly for this particular lattice.
Continuous-Representation Theory. V. Construction of a Class of Scalar Boson Field Continuous Representations
Authors: Klauder John R. , McKenna James
Published: January 2005

A large class of continuous representations of separable Hilbert spaces is constructed with the aid of representations of the canonical commutation relations (CCR) for a scalar boson field φ(f) and its canonical conjugate π(g). A representation of the CCR for a scalar boson field consists of two operator-valued functions V[g] and W[f], defined for all f, g in Schwartz's space 𝒮 of real test functions, where V[g] and W[f] are unitary operators defined on some separable Hilbert space ℌ, and which satisfy the commutation relations V[g]W[f] = e−i(f,g) W[f]V[g]. These unitary operators are related to the field and its momenta by V[g] = e−iπ(g), W[f] = eiφ(f). We explicitly construct a family of such representations with the help of von Neumann's theory of infinite direct products of Hilbert spaces, the pertinent parts of which are reviewed. A continuous representation ℌ of the Hilbert space ℭ is composed of a linear vector space of complex, bounded, continuous functionals defined on 𝒮 × 𝒮. These functionals are defined for all Ψ∈H by ψ(f, g)=(V[g]W[f]Φ0, Ψ). In this definition, Φ0 is a fixed unit vector in ℌ. The properties of the functions in ℭ depend on the choice of the representation of the CCR and on the choice of Φ0. When ℭ is constructed with the aid of an irreducible representation of the CCR, an inner product can be defined for all pairs of functionals in ℭ by an intuitively meaningful, rigorously defined integral in the sense of Friedrichs and Shapiro. With this inner product, ℭ is a complete Hilbert space congruent with ℌ. As in all continuous representations, a reproducing kernel exists and determines the functions in the continuous representation. One such space is closely related to a space of analytic functionals introduced by Segal and Bargmann. The representation of various operators as kernels and as functional derivatives is discussed. Finally, the construction of a vast number of unitary invariants for a representation of the CCR is used to establish the unitary inequivalence of uncountably many of the representations that we construct.
Cluster Properties of Multiparticle Systems
Authors: Hunziker W.
Published: January 2005

The object of investigation is a system of N particles in nonrelativistic quantum mechanics. The particles interact via two- or many-body potentials, for which a sufficient condition is that they be square integrable in the relative coordinates of the interacting particles. Cluster properties are derived for the time translation operator, for the wave operators, for the transition probabilities, and for the S operator.
Invariant Approach to the Geometry of Spaces in General Relativity
Authors: Brans Carl H.
Published: January 2005

A procedure is described for obtaining a complete, invariant classification of the local, analytic geometries and matter fields in general relativity by a finite number of algebraic steps. The approach is based on an extension of the classification scheme to include differential invariants of all orders and to provide maximally determined standard frames of vectors at each point. It is further shown that the resultant invariant functions can be replaced, in a finite number of algebraic steps, by special invariant functions which, while still uniquely representative of the geometry, can be assigned arbitrarily to produce all possible local, analytic solutions to the Einstein equations, in this representation. It is suggested that this type and special function scheme, obtainable from ideal geometric measurements in a finite number of steps, could be useful in general relativity. Unfortunately, due to the extensive algebra involved, this scheme has not yet been explicitly calculated, even for empty spaces.
Analytic Continuation ofa(ℓ,s)for Re ℓ<N
Authors: Zimmerman Robert L.
Published: January 2005

The positions of the singularities of the partial wave amplitude in the complex l plane are investigated for that case of a relativistic, two-particle elastic scattering amplitude satisfying the Mandelstam representation. For a finite number of intermediate states it is shown that the convex hull of singularities in the l plane is related to a growth indicator function. For an infinite number of intermediate states it is demonstrated that the imaginary parts of the singularities in the l plane are unbounded from either above or below.
Effects of Finite Sample Size on the Measurement of Transport Quantities
Authors: Nossal Ralph
Published: January 2005

A determination is made of the temporal evolution of the momentum autocorrelation function for a noninteracting gas constrained by walls. A frequency-dependent dielectric constant is calculated from the latter by applying an appropriate Kubo relation. Deviations from the Drude conductivity are obtained which, for high frequency and/ or large systems, are found to be ∼ o(102/βmω2L2), where L is the spatial dimension of the assembly and ω the frequency of the applied field.

A similar calculation is performed to determine the effects of boundaries upon measurements of scattering cross sections for slow neutrons. It is found that, for realistic experimental conditions, wall perturbations are unlikely to be of importance.
Use of Analytic Signals in Third- and Fourth-Order Coherence Functions
Authors: Beran M. , Corson P.
Published: January 2005

In this paper complex forms of the third- and fourth-order coherence functions are defined using analytic signals. The relations between the complex forms and the real forms are given in detail. It is shown how to use the complex forms in the solution of radiation problems. The quasimonochromatic approximation greatly simplifies the calculations and this simplification is discussed for radiation problems in terms of the complex forms. The paper concludes with a discussion of quasimonochromatic radiation from a modulated incoherent source representing, perhaps, a self-luminous turbulent region.
Exact Solution of a Many-Fermion System and Its Associated Boson Field
Authors: Mattis Daniel C. , Lieb Elliott H.
Published: January 2005

Luttinger's exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi-Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter n̄k, and find: ∂n̄k/∂k|kF = ∞ (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.
Lepton Scattering Amplitudes in Two Model Field Theories
Authors: Erickson G. W. , Fried H. M.
Published: January 2005

Two lepton-lepton scattering amplitudes are considered within the context of a no-recoil Bloch-Nordsieck model, with emphasis on the singularities in that configuration-space variable conjugate to momentum transfer. For the interaction L′=−gψ̄Aψ, renormalizable in four dimensions, and in the approximation of including only the exchange of all possible bosons between a pair of leptons, a light cone singularity no worse than that of the one-boson-exchange graph is found. Similar statements may be made for the same interaction, nonrenormalizable in six dimensions, provided certain continuations in the center-of-mass energy variable are employed; otherwise, an essential singularity appears. A remark illustrating the formation of bound states is made for the renormalizable interaction. No argument is given to establish the relevance of these models to the actual field-theoretic situations.
Interacting Fermions in One Dimension. I. Repulsive Potential
Authors: McGuire J. B.
Published: January 2005

The exact energies and wavefunctions for the ground state and low-lying excited states of a system of N − 1 one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large-volume, finite-density limit when the particles interact via a repulsive delta function potential. A number of properties of the system such as pair correlation functions and the effective mass of a certain class of excitations are also discussed.
Self-Consistent Approximations in Many-Body Systems
Authors: Revzen Michael
Published: January 2005

Baym's definition of a self-consistent approximation is rephrased in a more diagrammatic way and compared with formulations of the exact many-body problem of Balian, Bloch, and DeDominicis. It is shown that many of the results of these authors are valid for any self-consistent approximation as defined by Baym.
Poles in Feynman Diagrams with Several Loops
Authors: Halpern Francis R. , Wilson William
Published: January 2005

A formula is given that extracts the pole contribution to a Feynman diagram having an internal line. The result is more complicated than earlier results for diagrams with only a single loop since the result is not directly expressible as the pole multiplied by a numerator that is the sum of the reduced diagrams. The apparent discrepancy with Cutkosky's formula for the residue of the pole is reconciled. The algebraic techniques employed are in principle applicable to the problem of determining the Landau surface when the diagram has several loops. A derivation of Stokes' formula for Feynman parametric integrals is given.
Transformation Having Applications in Quantum Mechanics
Authors: Heffner H. , Louisell W. H.
Published: January 2005

By properly ordering functions of noncommuting operators, a one-to-one transformation between operator functions and corresponding functions of commuting algebraic variables can be made. With this transformation, boson operator equations such as the Schrödinger equation can be converted to differential equations for the transformed functions, the resulting equations containing solely commuting variables. Once the solution to the transformed equation is obtained, the inverse transformation may be applied to yield the solution to the original operator equation. The method is extended to include angular momentum operators.
High-Order Limit of Para-Bose and Para-Fermi Fields
Authors: Greenberg O. W. , Messiah A. M. L.
Published: January 2005

Para-Bose and para-Fermi quantization are schemes of second quantization which generalize the usual Bose and Fermi schemes. The different cases of para-Bose (para-Fermi) quantization are labeled by a positive integer p, which is called the ``order.'' For p = 1, the usual Bose (Fermi) quantization is recovered. The high-p limit of para-Bose (para-Fermi) quantization is studied, and it is shown that, to a large extent, the high-p limit of a para-Bose (para-Fermi) field is a Fermi (Bose) field. The nature of the limit is studied in detail, and a paradox relating to the connection of spin and statistics is resolved.
Proof that Scattering Implies Production in Quantum Field Theory
Authors: Aks Stanley O/.
Published: January 2005

A proof of the necessity of production processes in quantum field theory is carried out in the axiomatic framework. It is shown that a field theory that is assumed to have a nontrivial scattering amplitude violates crossing symmetry, if production processes are absent. The proof is based on the rigorous analytic properties of the scattering amplitude, particularly the analyticity in the invariant-scattering variables s, t, and u. In the case of a scalar theory with pairing symmetry, the scattering amplitude is known to be analytic within the domain |stu| < 7168m6, except for the usual cuts. Under the working assumption that production processes are null, it is shown that this domain can be enlarged by applying the elastic unitarity conditions beyond the (usual) elastic region. The domain is enlarged sufficiently to include the first Landau singularity of the absorptive part of the scattering amplitude. This singularity is not symmetric in s and t within the extended domain, and this is incompatible with the crossing symmetry of the scattering amplitude. In order to avoid a contradiction, the discontinuity across this Landau singularity must be null. It follows that the scattering amplitude must itself be null.

In the course of the proof it is shown that the conclusion is valid for a scattering amplitude satisfying the requirements of an S-matrix theory embodied in the Mandelstam representation.
On a Special Class of Type-I Gravitational Fields
Authors: Trümper Manfred
Published: January 2005

This paper contains an investigation of the algebraic structure and the analytic properties of a class of normal hyperbolic Riemannian 4-spaces restricted by the following condition: There exists a timelike unit vector ua such that the Riemann tensor satisfies *Rabcdubud = 0. This condition is shown to be equivalent to the statement that the conform tensor is Petrov type I with real eigenvalues, ua being a principal vector and an eigenvector of the Ricci tensor. This means that there is no flux of nongravitational energy relative to an observer travelling with 4-velocity ua.

The eigen null directions (Debever vectors) of the conform tensor lie in a timelike hyperplane spanned by ua and the two eigenvectors of εac ≡ −Cabcdubud belonging to the eigenvalues with largest absolute value. The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer ua is an eigendirection of εab.

The complete set of Bianchi identities is examined. It yields an expression for the covariant eigentime derivative of εab and an algebraic relation linking the rotation and shear of ua to the curvature tensor of the Riemannian space.

The general results are applied to special Einstein spaces (Rab = 0) admitting a congruence of timelike curves without shear and rotation. We get a new simplified proof of the theorem that in the case of nondegeneracy such spaces are static, the curves of the congruence being paths of an isometric motion.
New Homogeneous Solutions of Einstein's Field Equations with Incoherent Matter Obtained by a Spinor Technique
Authors: Ozsváth István
Published: January 2005

Einstein's field equations with incoherent matter are solved for the case of homogeneous spacetime, i.e., for metrics allowing a four parametric simply transitive group of motions. Two families of new solutions are obtained by use of a spinor technique. As a special result a proof emerges for Gödel's theorem, which states that there exist only two homogeneous solutions of Einstein's field equations with incoherent matter and rigid rotation, namely the Gödel cosmos and the Einstein static universe.
Excitation Operators and Intrinsic Hamiltonians
Authors: Nesbet R. K.
Published: January 2005

An operator A† that satisfies [H, A†] = h/ωA† converts a stationary-state eigenfunction of the Hamiltonian H into another eigenfunction with energy eigenvalue increased by h/ω. Such operators describe collective excitations of many-particle systems, and their properties can be used to construct an intrinsic Hamiltonian that is dynamically independent of the collective degrees of freedom, without introducing subsidiary conditions. The procedure developed by Lipkin, valid when h/ω is real and positive, is extended to make possible the construction of an intrinsic Hamiltonian when h/ω vanishes and A† is Hermitian, and also when h/ω is complex. The nuclear cranking model is shown to be a special case of the proposed general method for vanishing h/ω in which effective moments of inertia occur as eigenvalues of linear equations. Several examples are worked out in detail, all dealing with an interacting phonon-electron system in the random-phase approximation. Results derived are the explicit screened Coulomb interaction resulting from electronic plasma excitations, a verification of the renormalized phonon frequency spectrum and phonon-electron interaction derived in the adiabatic approximation, and the resulting screened Coulomb and phonon-induced electronic interactions obtained when plasma and phonon excitations are treated simultaneously.
Upper Bound for the Double Spectral Function in Potential Scattering
Authors: Bessis D.
Published: January 2005

For potentials V(z) holomorphic in Re z > 0 and bounded by |V(z)|<K/|z|ρ  ρ<2  for  |z|≤1,|V(z)|7/4,we show that the double spectral ``function'' ρ(s, t) is a continuous function of s and t in s > 0, t > 0, and we obtain an upper bound for it. This upper bound shows clearly that the double integral of the Mandelstam representation in fact exists and defines an analytic function of s and t in two cut planes. We indicate how to generalize these results to the case when ρ(s, t) is no longer a function but a distribution.
High-Energy Behavior at Fixed Angle for the Five-Point Function in Perturbation Theory
Authors: Greenman J. V.
Published: January 2005

The validity of the d-line method used by Halliday in studying the high-energy behavior at fixed angle of the four-point function is examined in more detail. This is carried out within the context of determining the high-energy behavior at fixed angle for the five-point function. The leading asymptotic behavior of the sum of all planar five-point graphs in a φ3 theory at fixed angle is s−2.
A Set of Harmonic Functions for the Group SU(3)
Authors: Bég Mirza A. B. , Ruegg Henri
Published: January 2005

We construct a set of harmonic functions carrying all the irreducible representations of the group SU(3). Some features of these functions are discussed in detail.
Operators that Lower or Raise the Irreducible Vector Spaces of U
n−1 Contained in an Irreducible Vector Space of Un
Authors: Nagel John G. , Moshinsky Marcos
Published: January 2005

We define operators that lower or raise the irreducible vector spaces of a semisimple subgroup of a semisimple Lie group contained in an irreducible vector space of the group. We determine the lowering and raising operators for the canonical subgroup Un−1 of the unitary group Un. With the help of these operators, which are polynomial functions of the generators of Un, and the corresponding operators for the subgroups in the canonical chain Un⊃ Un−1⊃ …⊃ U2⊃ U1 we can obtain, in this chain, the full set of normalized basis vectors of an irreducible vector space of Un from any given normalized basis vector of the vector space. In particular we can obtain, using only the lowering operators, the set of basis vectors from the basis vector of highest weight of the vector space. This result is of importance in applications to many-body problems and in the determination of the Wigner coefficients of Un. In future papers we plan to determine the lowering and raising operators for the orthogonal and symplectic groups.
Tensor Methods and a Unified Representation Theory of SU3
Authors: Mukunda N. , Pandit L. K.
Published: January 2005

Starting with irreducible tensors, we develop an explicit construction of orthonormal basic states for an arbitrary unitary irreducible representation (λ, μ) of the group SU3. A knowledge of the simple properties of the irreducible tensors can then be exploited to obtain a variety of results, which ordinarily require more abstract algebraic methods for their derivation. As illustrative applications, we (i) derive Biedenharn's expressions for the matrix elements of the generators of SU3, (ii) compute the matrix elements of octet-type operators for the case (λ, μ) → (λ, μ), and (iii) develop an explicit unitary transformation connecting the isospin and the U-spin states in any arbitrary irreducible representation.
Fluctuations in Equilibrium and Nonequilibrium Ensembles
Authors: Jones Gerald L. , Davies Michael
Published: January 2005

Using the results of a previous paper, the probability of a fluctuation occurring in a given time interval, of certain dynamical variables, is considered for a rather general nonequilibrium situation and also a more specialized equilibrium case. In the first case we show that this probability vanishes at least as fast as 1/N. In the second case we obtain a much stronger decrease of the probability with increasing N.
Unified Dirac-Von Neumann Formulation of Quantum Mechanics. I. Mathematical Theory
Authors: Marlow A. R.
Published: January 2005

In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.
Weight Factors for the Two-Dimensional Ising Model
Authors: Baxter Glen
Published: January 2005

A group 𝔊 of local weights is constructed for the square, honeycomb, and triangular lattices which counts for any closed path in the lattice 1/2π times the change in the argument of the tangent vector (mod 2) and the number of enclosed units of area (mod 2). These weights are used to evaluate the partition function of the two-dimensional Ising model with nearest-neighbor interaction and with a particular, imaginary external magnetic field. For the square lattice, the method gives a result announced by Lee and Yang.
Tensorial Description of Neutrinos
Authors: Penney R.
Published: January 2005

It is shown that the equation Tνω(Tρλ+Tdρλ)|ρ+Tμω(Tμν−Tdμν)|λ=0, where Tμν is a null antisymmetric tensor and Td its dual, is necessary and sufficient that there be a two-component Majorana neutrino field. That is, the neutrino field may be described solely in terms of tensorial quantities and operations, without the need for spinors.
Geometrization of a Massive Scalar Field
Authors: Penney R.
Published: January 2005

A set of necessary and sufficient conditions involving only the metric tensor and the Einstein tensor Gμν are given which thereby guarantee that Gμν may represent the stress-energy tensor of a massive real scalar field. One of the conditions is nonlocal, and is a demand that the energy tensor vanish at spatial infinity.
Quantum Electrodynamics Without Indefinite Metric
Authors: Peres Asher
Published: January 2005

The electromagnetic potentials Aα(x) are quantized in such a way that their space components are Hermitian and their time component anti-Hermitian. On the other hand, the metric in Hilbert space and the Hamiltonian are positive definite. The above formalism, which leads to the usual commutator [Aα(x), Aβ(y)] = igαβD0(x − y), is shown to be Lorentz-covariant over the manifold of physical states. The latter contain neither longitudinal nor scalar photons (rather than an equal number of them, as in the usual theory). This definition is also Lorentz covariant. The S-matrix is not unitary, but satisfies PS*PSP = P, where P is the projection operator over physical states. In other words, the S-matrix is unitary over the subspace of physical states, this being sufficient for the interpretation of the theory.