bluuebunny/crossref_metadata_2025 huggingface dataset:
Off-Diagonal Long-Range Order and Generalized Bose Condensation
Authors: Girardeau M. D.
Published: January 2005

It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
Sufficient Conditions for an Attractive Potential to Possess Bound States. II
Authors: Calogero F.
Published: January 2005

A condition sufficient to secure the existence of a least one bound state for each angular momentum l ≤ L is given by the inequality − ∫ 0∞ d(qr)V(r)/[(qr)2Lq2−(qr)−2LV(r)]>1,where q is an arbitrary constant and V(r) an everywhere-attractive potential.
Variational Principle for Diffraction of Elastic Waves
Authors: Korringa J.
Published: January 2005

The theory of diffraction of elastic waves is developed for the case of two homogeneous elastic media, welded together at an interface of arbitrary shape. The Green's function of the problem, defined to be the displacement at a point P1, caused by a periodic force at a point P0, is expressed in terms of the displacement and tension on the interface. A set of Fredholm integral equations for these interface functions is obtained. From them a variational principle is derived which gives the Green's function, minus the ``free'' Green's function, as the stationary value of a functional. It is analogous to the variational theorem of Levine and Schwinger for optical diffraction at an aperture in a screen. The variational equations of this functional are the above-mentioned integral equations. Explicit expressions are obtained for the case of isotropic elastic solids and for liquids. A generalization to the case of pulsed waves is indicated.
Evaluation of Feynman's Functional Integrals
Authors: Zimmerman Robert L.
Published: January 2005

Several relationships for Feynman's functional integrals are derived. From these relationships, we construct two different schemes for approximating Feynman's functional integral. The methods of approximation are expected to converge sufficiently rapidly in many cases so that only the lowest orders of the approximation are required to give reliable answers.
Asymptotic Expansions of Solutions of Differential Equations
Authors: Fikioris J. G.
Published: January 2005

A generalization of Ford's method, concerning the asymptotic expansions of solutions of differential equations with polynomial coefficients and with three or more regular singular points and one irregular at infinity, is presented. The analysis is subsequently extended to the special case of integral values for the difference of exponents of the differential equation, thus providing the complete asymptotic expansion of the second, logarithmic solution of the equation. Explicit formulas for the evaluation of the constant coefficients of the expansions are given; each coefficient is expressed in terms of a single solution of the adjoint difference equation associated with the original differential equation. Differential equations possessing singularities in excess of the hypergeometric equation (3 regular) or its variations, appear as separated solutions of the wave equation in certain stratified media whose index of refraction is a continuous function of position.
On Polynomial Systems in a Banach Ring
Authors: Taylor J. G.
Published: January 2005

We define and discuss equations on Banach rings (algebras) which are of polynomial form. We prove a local uniqueness theorem for the homogeneous case, and an existence and local uniqueness theorem for the nonhomogeneous case. In order to apply these results to the equations of Lagrangian quantum field theory we find it necessary to extend the concept of a ring to that of an n-ring. The resulting theory is applied to a simple model equation arising in quantum field theory.
Proof and Refinements of an Inequality of Feynman
Authors: Symanzik Kurt
Published: January 2005

A lower bound given by Feynman for the quantum mechanical free energy of an oscillator is proved, refined, and generalized. The method, application of a classical inequality to path integrals, also gives upper bounds for one-temperature Green's functions.
Singularities of Analytic Functions Having Integral Representations, with a Remark about the Elastic Unitarity Integral
Authors: Gilbert R. P. , Howard H. C. , Aks S.
Published: January 2005

In this paper a survey is given of some results which have been obtained recently concerning the singularities of holomorphic functions having integral representations. These results are all essentially extensions or modifications of those developed by Hadamard (for the proof of his multiplication of singularities theorem) to the case of several complex variables. As a concluding remark we consider the connection between the original Hadamard idea and the elastic unitarity integral of the quantum theory of fields.
Solution of the Integral Equation for Vθ Scattering in the Lee Model
Authors: Sommerfield Charles M.
Published: January 2005

A deductive procedure is presented for obtaining the solution of the integral equation for Vθ scattering in the Lee model.
On the Approximation of Quantum Field Theories
Authors: Jaffe Arthur
Published: January 2005

We assume that the Wightman distributions are tempered and that they are boundary values of functions analytic in the forward tube. Under these conditions a sequence of such distributions may converge in Borchers' topology. The necessary and sufficient conditions for such convergence are spelled out in terms of the corresponding analytic functions. The two cases with and without the assumption of the spectral condition are separately treated. A discussion of other topologies and some examples of the use of this technique are given.
Mixing of Regge Poles and Cuts in a Field-Theory Model
Authors: Swift Arthur R.
Published: January 2005

The Regge poles generated by ladder diagrams in a λ φ3 theory are mixed with the moving cuts generated by a class of nonplanar graphs containing an internal ladder. Since the contribution from the cut-generating diagram is of order t−2ln mt, where t is the asymptotic variable, the t−2 behavior of the pure ladder graphs is examined and the trajectory of the Regge pole near ℓ = −2 is calculated. The Mellin transform method is used throughout. The transformed amplitude corresponding to a single cut insertion is given by a product of the form pole-cut-pole, where it is only the second Regge trajectory that mixes with the cut. The cut itself depends on the leading trajectory. This result substantiates the predictions as to form of other work based on unitarity, but differs in that the cut and pole depend on different trajectory functions. Finally, multiple insertions of the cut diagrams are shown to generate an amplitude with two moving poles on each sheet of the cut.
Variational Principle for the Time-Dependent Schrödinger Equation
Authors: Gurtin Morton E.
Published: January 2005

In this paper a variational principle is established which fully characterizes the initial-value problem associated with the time-dependent Schrödinger equation.
N-Particle Kinematics and Group-Theoretical Treatment of Phase Space I. Nonrelativistic
Authors: Lévy-Leblond Jean-Marc , Lurçat François
Published: January 2005

This paper is a group-theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase-space manifold and preserves the phase-space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three-dimensional rotation group.

Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.
Three-Particle Nonrelativistic Kinematics and Phase Space
Authors: Lévy-Leblond Jean-Marc , Lévy-Nahas Monique
Published: January 2005

The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.
Invariance and Conservation Laws in Classical Mechanics
Authors: Denman Harry H.
Published: January 2005

It is commonly supposed that, in classical mechanics, invariance of a physical system to coordinate translation implies conservation of linear momentum. ``Invariance'' may be defined in a number of ways. If it is defined to mean invariance of the equation of motion, it is shown that invariance of this equation with respect to coordinate translation does not imply conservation of linear momentum. The effects of scale transformation and coordinate inversion invariance are also investigated. Both the Lagrangian approach and Newton's (second) law approach are considered. It is shown that each of the above invariances implies a condition on the equation of motion, while a combination of these and time-inversion invariance is needed to obtain ordinary momentum conservation.
Relativistic Three-Particle SU3 States
Authors: Dragt A. J.
Published: January 2005

The scheme for classifying three-particle states according to SU3 is extended to the relativistic domain. A discussion is also given of the basic group theory lying behind the scheme.
Matrix Representation of the Angular Momentum Projection Operator
Authors: Shapiro Joseph
Published: January 2005

Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.
Solution of an Atomic Integral Containing Three Odd Powers of Interelectronic Separation Coordinates
Authors: Burke E. A.
Published: January 2005

A wavefunction expressed as a linear combination of terms each involving only one interelectronic separation coordinate requires the solution of matrix elements of the Hamiltonian which contain three odd powers of the interelectronic separation coordinates. This paper discusses in detail the integration of [triple integral operator]ρ1kρ2mρ3nρ23ρ13−1ρ12exp −(aρ1+bρ2+cρ3)dτ1dτ2dτ3.
On the Bound-State Wavefunctions of a Nonlocal Solvable Potential
Authors: Catara F. , Di Toro M.
Published: January 2005

The Schrödinger equation is solved in the momentum representation with a nonlocal factorable potential, with at most two point eigenvalues. When the potential is a bilinear form in the Bessel transforms of Yukawa functions, we prove that it is impossible to have two bound states whose asymptotic behavior is determined by the binding energy.
On a Set of Coupled Second-Order Differential Equations
Authors: Chen Joseph C. Y.
Published: January 2005

An alternative method for solving a set of coupled second-order differential equations, which often appears in theoretical treatments of many-body problems, is proposed. This method makes use of both mathematical relations derived in matrix theory and physical properties of the potentials provided by the set of equations to be solved.
Remarks on the Geometric Nature of Quantum Phase Space
Authors: Hermann Robert
Published: January 2005

Several geometric remarks on the foundations of quantum mechanics are presented. For example, quantum phase space is the tangent bundle to quantum configuration space.
Note on the Dynamics of Gravitational Sources
Authors: Newman Ezra T. , Unti T. W. J.
Published: January 2005

This paper is concerned with multipole structure for the sources of the gravitational field in the full (axially symmetric) nonlinear gravitational theory. Definitions proposed recently by Janis and Newman for the mass of the source, its linear momentum, dipole moment, and its spin or angular momentum are examined. These definitions are contrasted with those given by other authors. The advantages of the new definitions are discussed and the manner in which they enter into the dynamics of the system is studied. An appendix considers some transformation properties of these multiple moments.
Oblique Incidence on Plane Boundary between Two General Gyrotropic Plasma Media
Authors: Unz H.
Published: January 2005

The problem of a characteristic electromagnetic wave incident obliquely on a plane boundary between two different gyrotropic plasma media is solved. Two characteristic transmitted waves and two characteristic reflected waves will result. The corresponding reflection and transmission coefficients have been evaluated. The particular degenerate cases of free space-gyrotropic medium, gyrotropic medium-free space, and gyrotropic medium-perfect conductor are solved in the appendices.
On the Unitary Representations of the Galilei Group. II. Two-Particle Systems
Authors: Voisin J.
Published: January 2005

Mackey's theory is used to study the kinematics of a nonrelativistic two-particle system. We consider the helicity coupling and the l-S coupling and derive a Jacob-Wick's formula relating these two couplings (Glebsch-Gordan coefficients).
On the Representations of the Semisimple Lie Groups. V. Some Explicit Wigner Operators for SU3
Authors: Baird G. E. , Biedenharn L. C.
Published: January 2005

A general method for the calculation of the [21 … 10] Wigner operators is presented and used to obtain specific results for the SU3 group. General expressions for the SU3 reduced Wigner operators are given for tensor operators transforming like the representations [100], [110], and [210].
Momentum-Space Analyticity Properties of the Bergman-Weil Integral for the Three Point Function
Authors: Källén G. , Toll J. S.
Published: January 2005

The momentum-space analyticity domain of the Bergman-Weil integral representation of the vertex function or three-point function is investigated using the assumption that thresholds can be introduced simply as lower limits of integration for the mass variables in the representation. It is shown that this assumption leads to a regularity domain which is larger than the domain following from general physical assumptions of Lorentz invariance, local commutativity and reasonable mass spectrum. To simplify the discussion we also assume that the vertex function is regular when all three complex variables lie in the same half-plane. Standard techniques for the evaluation of Feynman diagrams have proved to be inconvenient for this investigation and we have developed new methods taking explicit advantage of the fact that we only have two external vectors and, hence, can work in a two-dimensional Lorentz space. Further, the vanishing of the masses for certain internal lines has also been exploited. The techniques we have used here might be of interest also in other connections.
Green's Functions and Superfluid Hydrodynamics
Authors: Kane Joseph W. , Kadanoff Leo P.
Published: January 2005

The two-fluid hydrodynamics of Landau and Tisza is related to the thermodynamic Green's function formulation of the many-body problem. A specific approximation for the self-energies yields directly the superfluid hydrodynamics in the limit of slow space and time variation. The approximation chosen is the simplest possible form for the self energies which includes the effects of collisions and which also satisfies the differential conservation laws for the mass, momentum, and energy. Of course these self energies are not adequate for a realistic description of 4He. Nonetheless the structure of this model may provide some insight into the exact theory. For example, in the model employed here the self-energies and Green's functions obey several integral identities which are utilized in the derivation of the two-fluid hydrodynamics, and which yield useful expressions for some thermodynamic quantities. It is speculated that the identities and the consequent expressions for the various thermodynamic quantities may have a wider range of validity than their derivation.
Note on Orthogonal Polynomials which are ``Invariant in Form'' to Rotations of Axes
Authors: Maldonado C. D.
Published: January 2005

The precedure of Bhatia and Wolf for constructing orthogonal sets whose elements are ``invariant in form'' with respect to rotations of axes, is extended to include sets which are defined over the entire two-dimensional plane. Use was made of the Gram-Schmidt process to derive general expressions for generating elements of many unique and complete sets corresponding to different circularly symmetric weight functions for two cases, where in the first case the elements are only functions of the real variables x and y while in the other case they are also functions of the real variable r = (x2 + y2)½. These general expressions were used to obtain two new, unique and complete sets corresponding to Gaussian and exponential weight functions, respectively. The radial polynomials for these two sets were found to be closely related to the Laguerre polynomials. The generating functions for these radial polynomials are also given.
Reduction of a One-Loop Feynman Diagram with n Vertices in m-Dimensional Lorentz Space
Authors: Petersson Bengt
Published: January 2005

An explicit formula is given for the evaluation of a diagram with one loop and n vertices in m-dimensional Lorentz space (n > m.). The result is given as a sum of terms each corresponding to a one-loop diagram with m vertices and with a coefficient which can be obtained from rules analogous to the rules of residue calculus.
The Froissart-Gribov Continuation and Reggeon Unitarity Conditions. I
Authors: Polkinghorne J. C.
Published: January 2005

An identity is derived which is useful in the discussion of unitary integrals. It is used to discuss single Regge pole insertions in the Froissart-Gribov continuation and with the help of perturbation-theory models a derivation is given of reggeon unitarity conditions. The general form suggested by Gribov, Pomeranchuk, and Ter-Martirosyan is confirmed. Finally cancellation mechanisms are discussed and their relation to the mechanism generating Regge poles is emphasized.
A Model of Interacting Radiation and Matter. II
Authors: Willis Charles R.
Published: January 2005

We extend the investigation of the long time behavior of a model consisting of N two-level atoms interacting with radiation to include dissipation, pumping, and center-of-mass motion. We show that when the effective expansion parameter γ̃Nγ̃ is small, the self-consistent field approximation remains the solution to lowest order in γ̃Nγ̃. The inclusion of the center-of-mass motion introduces another dimensionless parameter β into the theory. We show that when this parameter is small, the electromagnetic field amplitude varies slowly on the time scale of the center-of-mass motion. We solve the equations of motion in this slowly varying limit by an extension of the Bogoliuboff-Kryloff theory of quasilinearity to the problem of time-dependent integral kernals. We find the unique stable stationary state and show that in the slowly varying limit the stationary state is approached independent of initial conditions. We calculate the frequency shift to second order in β. The first-order frequency shift is the same as that calculated by Lamb. We compare our steady-state solution with recent experiments with lasers. We include the effect of collisions in the steady state.
Cluster-Star Inversion and the Möbius Formula
Authors: Sherman S.
Published: January 2005

The cluster-star equations are inverted by using the general Möbius inversion formula. Since Uhlenbeck and Ford have many instances of systems of equations to which general Möbius inversion can be applied, this paper first presents a general survey of Möbius inversion, then some graph terminology, and finally the cluster-star inversion.
Atomic Terms for Equivalent Electrons
Authors: Karayianis N.
Published: January 2005

The problem of determining the number of times, b(l N L S), a given term occurs in an atomic configuration of the form (nl)N is considered by the methods of group theory. The b are shown to be related to the classical problem of partitioning numbers. Several expressions relating b to other, simpler partitions are obtained, which result in a recursion relationship for the b. A table for b(4 N L S) is included to complement existing tables for the range 1 ≤ l ≤ 3, which are found in several places in the literature.
Quantum Electrodynamics with Vanishing Bare Fermion Mass
Authors: Bose S. K. , Biswas S. N.
Published: January 2005

Quantum electrodynamics is studied in the approximation suggested by Johnson, Baker, and Willey. The (approximate) Dyson equation of this model is transformed into a second-order, nonlinear differential equation by the application of a method due, originally, to Green. The solutions of this equation are then studied under various simplifying assumptions.
On Irreducible Representations of Space Groups
Authors: Rudra P.
Published: January 2005

A logical extension has been made of Seitz-Koster's method of space-group representations to the points on the surface of the Brillouin zone, using projective representations. This method has been applied in the case of the space group D34.
Fokker Action Principle for Particles with Charge, Spin, and Magnetic Moment
Authors: Schild A. , Schlosser J. A.
Published: January 2005

A Fokker action principle is obtained for a system of particles with charge, spin, and magnetic moment, interacting through time-symmetric electromagnetic fields. Conservation laws are derived for the energy, and the linear and angular momentum of the system. A limiting process is performed, which involves a renormalization of mass, in order that the magnitude of the spin of each particle remain constant.
Geometric Theory of Neutrinos
Authors: Penney R.
Published: January 2005

It is shown that the conditions Rμμ=0, R00≥0, RαβRβλ=14RρτRρτδαλ, Rαβ|σ=(RρτRρτ)|σRαβ, in the limit that the scalar RαβRαβ vanishes, reproduce the physics of neutrinos. That is, these conditions ensure the existence of a two-component spinor which obeys the Weyl equation and represents a field of completely polarized neutrinos.
Improved Method for Quantum Mechanical Three-Body Problems. I. Attractive Potentials
Authors: Eyges Leonard
Published: January 2005

The quantum-mechanical ground-state problem for three identical particles bound by attractive interparticle potentials is discussed. For this problem it has previously been shown that it is advantageous to write the wavefunction in a special functional form, for which an integral equation which is equivalent to the Schrödinger equation was derived. In this paper a new method for solving this equation is presented. The method involves an expansion in a set of ``effective'' two-body functions; these are the eigenfunctions of a two-body problem with a potential of the same shape as the interparticle potential in the three-body problem, but of enhanced strength. The integral equation then becomes a set of coupled equations for certain functions fl(k), the ``coefficients'' in this expansion. By choosing the effective two-body strength properly, one can optimize the convergence of this expansion so that a good approximation is got by retaining only the lowest (l̂ = 0) term. The resultant single integral equation for f0(k) can then be solved by an approximate method. As a test and check, the one-dimensional three-body problem with δ-function interactions is treated and the results are compared with the known exact one. It is found that the first approximation in the method, which is almost trivial to apply, yields an eigenvalue and eigenfunction accurate to a few percent. The method has also been applied to the three-dimensional problem with exponential interparticle potentials, and a comparison of the results with the best numerical calculations is given.
Concerning Space-Time, Symmetry Groups, and Charge Conservation
Authors: Sudarshan E. C. G.
Published: January 2005

Under the restriction to a symmetry group whose generators can be linearly expressed in terms of the generators of the Lorentz group and of an internal symmetry group, we show that charge conservation implies that the symmetry group is a direct product of the Lorentz group and the internal symmetry group. Only local structure of the groups and their unitary representations are considered.
Reduction of Representations of SUm+n with respect to the Subgroup SUm ⊗ SUn
Authors: Hagen C. R. , Macfarlane A. J.
Published: January 2005

A direct and rapid method is described for the reduction of representations of the special unitary group SUm+n with respect to its SUm ⊗ SUn subgroup. The technique is illustrated by specific reference to the case of SU6 and its SU2 ⊗ SU4 subgroup in the context of the recent extension of Wigner's (SU4) supermultiplet theory into the domain of particle physics. A tabulation of results is given for a slight generalization of this case where reduction is carried out with respect to U1 ⊗ SU2 ⊗ SU4.
Symmetrical Coupling of Three Angular Momenta
Authors: Lévy-Leblond Jean-Marc , Lévy-Nahas Monique
Published: January 2005

We introduce here a new coupling scheme for three angular momenta. It relies on the properties of an operator which depends ``democratically'' upon the three individual angular momenta; this is in fact their mixed product. This operator, the total angular momentum and one of its components form a complete set of commuting observables. In the case where the three individual angular momenta are equal, the eigenstates of this set possess remarkable symmetry properties with respect to the permutation group S3.
The Gravitational Compass
Authors: Szekeres P.
Published: January 2005

A purely covariant approach to general relativity, using the equation of geodesic deviation, is adopted. The physical interpretation is essentially that due to Pirani, but instead of using clouds of particles to analyze the gravitational field, a ``gravitational compass'' is proposed which fulfills the same purpose. Particular attention is focussed on the different roles played by the matter and the free gravitational field. The latter splits up conveniently into a super-position of a transverse wave component, a longitudinal component, and a ``Coulomb'' field, all of which introduce ``shearing'' forces on the gravitational compass, while the matter contributes a general contraction. Applications to the Friedmann cosmological models and the problem of interacting gravitational waves are discussed.
Structure of Fermion Density Matrices. II. Antisymmetrized Geminal Powers
Authors: Coleman A. J.
Published: January 2005

Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.
Local Approximation in Renormalizable and Nonrenormalizable Theories. I
Authors: Bardakci K. , Schroer B.
Published: January 2005

We present a framework of quantum field theory which is wide enough to incorporate renormalizable, as well as nonrenormalizable theories. A universal high-energy bound for matrix elements of fields is derived. The ``string'' approximation of the two-point function for various couplings is studied.
Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. I. Simple-Cubic and Face-Centered-Cubic Lattices
Authors: Miasek Maria
Published: January 2005

In certain problems in solid-state physics, the radial functions gj(r) in the expansion χ(r) = Σj=0∞ gj(r)Kj(θ, φ), where χ(r) is a known function and the Kj's are Kubic Harmonics, are of interest. This paper deals with the functions χ(r) ≡ N−1 Σk eik·r, where the sum runs over the first Brillouin Zone of a crystal. In particular, the functions χ(r) for simple cubic and face-centered cubic lattices are expanded into series of Kubic Harmonics and the radial functions gj(r) for several values of j are found using Houston's method, in which the expansion into series of Kubic Harmonics contains only a finite number of terms with lowest j's. g0(r) is calculated using 3, 6, and 9-term expansion, g2(r) and g3(r) using only 3 and 6-term expansion. Comparing gj(r) obtained from the formulas with different numbers of terms it is established that for r in the region 〈0, 2a〉, where a is the lattice constant, the 6-term approximation is very good. In practice, the functions gj(r) usually occur in integrands, together with atomic orbitals, and the tabulated results are expected to be particularly useful in the study of Wannier functions in the OPW scheme.
Translations of the Discrete Bose-Einstein Operators
Authors: Shelupsky David
Published: January 2005

The effects on a certain family of representations of the commutation relations of the operator-translations ak → ak + ck, ak* → ak* + ck*, are investigated. Here ak and ak* are the annihilation and creation operators of the representations, the ck are complex numbers (scalar operators), and the representations considered are the discrete representations described by Gårding and Wightman, and Schweber and Wightman, generalizations of the usual Fock representation. Necessary and sufficient conditions on the translations are obtained for the resultant representations to remain in the family of discrete representations, and the resultant translation groups are investigated for their relevant structure.
Local Approximations in Renormalizable and Nonrenormalizable Theories. II
Authors: Bardakci K. , Schroer B.
Published: January 2005

The investigations of a previous paper are generalized to two-point matrix elements. A principle is formulated, which yields unique finite Feynman rules in the renormalizable case, i.e., permits a unique separation of counterterms. For nonrenormalizable theories this principle yields uniqueness up to a ``scaling'' parameter. The results are generalized to a large class of Feynman graphs. For this subset of graphs, field-theoretical principles do not determine this scaling parameter.
Spin and Isospin in S-Matrix Theory
Authors: Taylor John R.
Published: January 2005

The Gunson-Olive technique for exploiting physical region poles and the unitarity equations to derive the analytic properties of S-matrix elements is applied to particles which belong to isospin multiplets and to particles with spin. In the former case it is shown that, in the framework of S-matrix theory, crossing symmetry leads to the requirement that particles and antiparticles transform according to complex conjugate representations; and it is possible to clarify the origin of the familiar isospin crossing matrices. In the case of particles with spin the construction of appropriate ``spinor'' basic states is discussed and the analytic properties of the corresponding spinor S-matrix elements or ``M-functions'' are derived.
On Series Expansion of a Determinant and Solution of the Secular Equation
Authors: Chan Yau W.
Published: January 2005

A determinant and its cofactors are expanded in terms of the cyclic products of its elements. With the aid of this expansion, an implicit equation for the eigenvalue and an explicit equation for amplitudes of the corresponding eigenfunction are obtained, respectively, in terms of a ratio of two simple series expansions. Comparisons with Feenberg's perturbation formula and with Sasakawa's perturbation method are also discussed.
Asymptotic Behavior of the Scattering Amplitude in the Left Half-Plane of Complex Angular Momentum
Authors: Dilley J. , Barut A. O.
Published: January 2005

It is shown, for a certain class of analytic potentials, that the previously reported result A(λ,k)→ lim ∼|λ|→∞1/k2V(±λ/k), Re λ≥0,where the sign ± is chosen such that - π/2 ≤ arg ± (λ/k) ≤ π/2, is valid also for Re λ < 0 except for a discrete set of points on the negative real axis. The result holds for arbitrary complex k ≠ 0, finite or infinite.
Mathematical Aspects of the Weyl Correspondence
Authors: Pool James C. T.
Published: January 2005

The Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *-operation are introduced to make the Hilbert space of Lebesgue square-integrable complex-valued functions on phase space into a H*-algebra. The Weyl correspondence is realized as a *-isomorphism f → W(f) of this H*-algebra onto the H*-algebra of Hilbert-Schmidt operators on the Hilbert space of Lebesgue square-integrable complex-valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier-Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.
Relation between the Onsager and Pfaffian Methods for Solving the Ising Problem. II. The General Lattice
Authors: Hurst C. A.
Published: January 2005

The considerations of a previous paper are extended to the case of a general lattice. It is shown algebraically why the Pfaffian and Onsager methods of solution of the Ising problem coincide only when the lattice is planar, and that the problem is then a linear one. When the lattice is nonplanar the Pfaffian method breaks down due to the appearance of unwanted negative signs, and it is shown how the Onsager method compensates for this at the expense of making the problem nonlinear.
Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition
Authors: Lebowitz J. L. , Penrose O.
Published: January 2005

Rigorous upper and lower bounds are obtained for the thermodynamic free-energy density a(ρ, γ) of a classical system of particles with two-body interaction potential q(r) + γνφ(γr) where ν is the number of space dimensions and ρ the density, in terms of the free-energy density a0(ρ) for the corresponding system with φ(x) ≡ 0. When φ(x) belongs to a class of functions, which includes those which are nonpositive and those whose ν-dimensional Fourier transforms are nonnegative, the upper and lower bounds coincide in the limit γ → 0 and limγ → 0 a(ρ, γ) is the maximal convex function of ρ not exceeding a0(ρ) + ½αρ2, where α ≡ ∫ φ(x) dx. The corresponding equation of state is given by Maxwell's equal-area rule applied to the function p0(ρ) + ½αρ2 where p0(ρ) is the pressure for φ(x) ≡ 0. If a0(ρ) + ½αρ2 is not convex the behavior of the limiting free energy indicates a first-order phase transition. These results are easily generalized to lattice gases and thus apply also to Ising spin systems.

The two-body distribution function is found, in the limit γ → 0, to be normally identical with that for φ(x) ≡ 0, but if the system has a phase transition it has the form appropriate to a two-phase system.

Some of the upper and lower bounds on a(ρ, γ) are simple enough to be useful for finite γ. Also, some of our results remain valid for quantum systems.
Linked Cluster Theorem and the Green's Function Equations of Motion for a Many-Fermion System
Authors: Kobe Donald H.
Published: January 2005

The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.
A Remark on a Reducible Quantum Field Theory with a One-Parameter Symmetry Group
Authors: Lopuszanski Jan , Reeh Helmut
Published: January 2005

The connection between certain symmetry properties of a reducible algebra generated by the field operator and the decomposition into irreducible algebras is exhibited.
Relativistic Spinor Formulation of Stokes Parameters with Application to the Inverse Compton Effect
Authors: De Young David S.
Published: January 2005

A formulation of the Stokes Parameters for light is found in terms of second-rank antisymmetric spinors, and arbitrary Lorentz transformations are made by using the unimodular representation of the Lorentz group. A three-component spinor form is established along with its accompanying representation of the Lorentz group, and the connection with the photon spin is explicitly shown. Differential Compton cross sections for any Lorentz frame are calculated. These are seen to be invariant only for particular polarization reference directions.
The Solution of a Stark-Effect Model as a Dynamical Group Calculation
Authors: Flamand G.
Published: January 2005

A model for the Stark mixing of orbital states in the annihilation of particles at rest is solved by using the dynamical group techniques. The Schrödinger equation for the problem is given meaning in a representation of the three-dimensional rotation group and reduced to a two-dimensional linear homogeneous differential equation.
On the Connection between External and Internal Symmetries of Strongly Interacting Particles
Authors: Flato Moshé , Sternheimer Daniel
Published: January 2005

The relation between the mass and spin-parity Jp, and internal quantum numbers of elementary particles, hints at a nontrivial connection between the external symmetry, namely the Poincaré group or its Lie algebra, and a so-called internal symmetry. Studying this connection mathematically, we find that any extension of (resp. by) the Poincaré Lie algebra 𝒫 by (resp. of) a semisimple Lie algebra 𝒳 is equivalent to the trivial one, 𝒫 ⊕ 𝒳. Moreover, if we are looking for a Lie algebra containing 𝒫 and 𝒳 in an economical and nontrivial way, namely what we call a (nontrivial) unification of 𝒫 and 𝒳, we find restrictions on the possibilities of choice of 𝒳, which exclude compact internal symmetries. An explicit treatment of SL(3, C) as internal symmetry, related to the external symmetry by the unification process of Lie algebras, gives a mass formula which is in very good accordance with the experimental data, and can be theoretically interpreted by means of a so-called ``classification principle''.
Characterizing Coherent States of the Radiation Field
Authors: Jordan Thomas F.
Published: January 2005

Simple proofs are given for two properties of a Bose field discovered recently in the quantum-theoretic description of optical coherence. The first is the theorem of Glauber and Titulaer that first-order coherence means that only one mode is excited. The second is the theorem of Aharanov, Falkoff, Lerner, and Pendleton that eigenstates of the annihilation operators are characterized by their ability to factor when the system is divided into two channels. The restriction of the latter to the case of a single excited mode is removed.
Theory of Paramagnetic Impurities in Semiconductors
Authors: Mattis Daniel C. , Lieb Elliott H.
Published: January 2005

In this paper, a model of a paramagnetic impurity in a semiconductor (or of an F′ center in an alkali halide) is proposed. It is an exactly soluble form of the quantum-mechanical 3-body problem. Specifically, we deal with 2 interacting particles in any number of dimensions in an attractive external potential, and present the qualitative features of the resulting eigenvalues and eigenfunctions. We find algebraically the conditions for a magnetic moment to appear (e.g., for an F′ center to become unstable with respect to an F center) and discover that even a large 2-body electronic repulsion U does not cause a moment to appear when the one-electron bound state orbits about the impurity are sufficiently great. Conversely, in the case of small, tightly bound orbits, beyond a certain value of U, the impurity does in fact become magnetic in the ground state. Using the exact ground-state solution, we show that a perturbation-theoretic expansion in powers of U has a finite radius of convergence.
Acoustic Scattering from an Interface between Media of Greatly Different Density
Authors: Mitzner K. M.
Published: January 2005

The problem of acoustic scattering from a curved interface between two homogeneous media is formulated as two integral equations relating the normal velocity and the velocity potential at the interface. The equations are so chosen as to minimize coupling. When γ, the ratio of the densities, goes to zero, the equations decouple, one becoming the equation of a soft-boundary problem, the other of a hard-boundary problem. For small γ, an approximate solution is constructed by perturbation methods from solutions to the related soft- and hard-boundary problems.
A Non-Hilbert-Space Formulation of Quantum Field Theory
Authors: Prugovečki Eduard
Published: January 2005

Based on earlier papers in which a non-Hilbert-space formalism for quantum physics, called the ℒ-formalism, was proposed, a new formalism for quantum field theory is developed by introducing the field concept into the ℒ-formalism. The present formalism is then compared with the Wightman approach and with the Haag-Araki approach to quantum field theory. The necessity of a global vacuum within the ℒ-framework and the alternative concept of the local vacuum are discussed.
A Soluble Fermi-Gas Model. Validity of Transformations of the Bogoliubov Type
Authors: Des Cloizeaux J.
Published: January 2005

A linear soluble Fermi-gas model is used to test the validity of perturbation theory and the criteria of instability obtained by applying transformations of the Bogoliubov type. The model Hamiltonian describing the interaction of the electrons in a simple band is H(ρ)=−Σ(cosk+ρ)ck+ck+ρΣcos q ck+q+cu−q+cuck. For a half-filled band, the energies of the low-lying states are calculated exactly, by using previous results concerning a linear chain of spins with interaction. It is shown that perturbation theory must be valid for −1 < ρ < 1, and, for values of ρ belonging to this range, the slope of the specific heat for T = 0 is calculated by direct application of Landau's theory. On the other hand, anomalous Hartree-Fock states can be built for all values of ρ, by using transformations of the Bogoliubov type, and these states are more stable than the normal Hartree-Fock state. Thus, it appears that the Bogoliubov method may lead to completely erroneous interpretations, when the coupling constants are small.
Peratization of the Logarithmically Singular Potential
Authors: Gale William A.
Published: January 2005

The potential of the form (gr−4ln2 r−g12r−3)ϑ(r−r0) is shown explicitly to be regularizable by an infinitesimal cutoff. It is also shown that, when a modified sense of the peratization technique is applied to the regulated scattering length, the correct answer is regained.
The Solution of the Nonrelativistic Quantum Scattering Problem without Exchange
Authors: Johnson B. Robert , Secrest Don
Published: January 2005

The method of ``amplitude density functions'' is a new formalism which allows a scattering problem to be broken up into problems with weaker interaction potentials. These simpler problems may be solved separately and be ``added'' together to give the total solution. A numerical method is discussed which takes advantage of this property. The formulas are given for the use of this method in the solution of the one-dimensional atom-molecule collision and the e+−H collision. A numerical example is discussed.
Properties of a Causal Green's Function for the Bethe-Salpeter Equation
Authors: Keam R. F.
Published: January 2005

The following properties of the free, but off-mass-shell, Bethe-Salpeter causal Green's function, for two spinless bosons whose masses may differ, are investigated: (a) its symmetries, (b) its expression in terms of the common higher transcendental functions, and (c) its forms in the asymptotic region and near the origin. An energy-spectral integral representation is also obtained.
A New Derivation of the Kinetic Equation for an Inhomogeneous Plasma
Authors: Kuszell A.
Published: January 2005

The kinetic equation for non-uniform plasma is derived from the Liouville equation by the Prigogine-Balescu diagram technique. The obtained equation is equivalent to the equation derived earlier by Balescu and this author, but has a little simpler form. The equation has a non-Markoffian form and is valid in both stable and unstable cases and for long and short times. Its explicit form is obtained by using the Resibois' summation procedure. The limiting form of the equation for long times in the stable case is derived.