# Showing papers in "Journal of Mathematical Physics in 1971"

••

[...]

TL;DR: In this paper, the number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

Abstract: The Einstein tensorGij is symmetric, divergence free, and a concomitant of the metric tensorgab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

2,450 citations

••

[...]

TL;DR: In this paper, the quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (harmonical) and/or inverse (centrifugal) potentials is solved.

Abstract: The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl], with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi st...

1,390 citations

••

[...]

TL;DR: In this article, the relation between the solutions of the timeindependent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables.

Abstract: The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.

1,154 citations

••

[...]

TL;DR: In this paper, the authors define a class of Ising models on d-dimensional lattices characterized by a number n = 1, 2, …, d (n = 1 corresponds to the Ising model with two-spin interaction).

Abstract: It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising modelsMdn on d‐dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two‐spin interaction). These models are related by two duality transformations. The models with 1 < n < d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long‐range order of the spins in the low temperature phase.

744 citations

••

[...]

TL;DR: In this paper, a method is described for constructing, from any source-free solution of Einstein's equations which possesses a Killing vector, a one-parameter family of new solutions.

Abstract: A method is described for constructing, from any source‐free solution of Einstein's equations which possesses a Killing vector, a one‐parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space‐times having a Killing vector.

661 citations

••

[...]

TL;DR: In this article, it was shown that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N) and discussed its unitary representation in quantum mechanics when the N coordinates are diagonal.

Abstract: We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well‐known dynamical group of the N‐dimensional harmonic oscillator. Finally, we study the case of n particles in a q‐dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)× O (q). An application to the calculation of matrix elements is given in a following paper.

655 citations

••

[...]

TL;DR: In this paper, the ground state of a system of either fermions or bosons interacting in one dimension by a 2-body potential V(r) = g/r2 was investigated.

Abstract: We investigate the ground state of a system of either fermions or bosons interacting in one dimension by a 2‐body potential V(r) = g/r2. In the thermodynamic limit, we determine the ground state energy and pair correlation function.

553 citations

••

[...]

TL;DR: In this article, it was shown that any integral invariants discussed in this series have a zero Poisson bracket, which is a bilinear antisymmetric operator on functionals.

Abstract: It is shown that if a function of x and t satisfies the Korteweg‐de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals. On a suitably restricted space of functionals, this operator satisfies the Jacobi identity. It is shown that any two of the integral invariants discussed in Paper II of this series have a zero Poisson bracket.

422 citations

••

[...]

TL;DR: In this article, an approximation for the eigenstate of a general 1-dimensional quantum many-body system with a 2-body potential V(r) = g/r2 was presented.

Abstract: We continue our investigation of a system of either fermions or bosons interacting in one dimension by a 2‐body potential V(r) = g/r2. We first present an approximation for the eigenstates of a general 1‐dimensional quantum many‐body system. We then apply this approximation to the g/r2 potential, allowing complete determination of the thermodynamic properties. Finally, comparing the results with those properties known exactly, we conjecture that the approximation is, in fact, exact for the g/r2 potential.

304 citations

••

[...]

TL;DR: In this article, the generalized Dirac equation is of a Heisenberg-Pauli type, and nonlinear terms induced by torsion express a universal spin-spin interaction of range zero.

Abstract: In order to take full account of spin in general relativity, it is necessary to consider space‐time as a metric space with torsion, as was shown elsewhere. We treat a Dirac particle in such a space. The generalized Dirac equation turns out to be of a Heisenberg‐Pauli type. The nonlinear terms induced by torsion express a universal spin‐spin interaction of range zero.

242 citations

••

[...]

TL;DR: In this article, a new scalar-tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance.

Abstract: A new scalar‐tensor theory of gravitation is formulated in a modified Riemannian manifold in which both the scalar and tensor fields have intrinsic geometrical significance. This is in contrast to the well‐known Brans‐Dicke theory where the tensor field alone is geometrized and the scalar field is alien to the geometry. The static spherically symmetric solution of the exterior field equations is worked out in detail.

••

[...]

TL;DR: In this paper, the authors describe geometrically differentiable solutions of partial differential equations using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups−vector fields in the space of forms.

Abstract: Methods are discussed for discovery of physically or mathematically special families of exact solutions of systems of partial differential equations. Such systems are described geometrically using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups‐vector fields in the space of forms. These isovectors then lead naturally to all the special solutions discussed, and it appears that other special ansatze must similarly be capable of geometric description. Application is made to the one‐dimensional heat equation, the vacuum Maxwell equations, the Korteweg‐de Vries equation, one‐dimensional compressible fluid dynamics, the Lambropoulos equation, and the cylindrically symmetric Einstein‐Maxwell equations.

••

[...]

TL;DR: In this article, the steady state heat flux J in a large harmonic crystal containing different masses whose ends are in contact with heat baths at different temperatures was investigated, and it was shown that J/ΔT approaches a fixed positive value as L→∞ corresponding to an infinite heat conductivity.

Abstract: We investigate the steady state heat flux J in a large harmonic crystal containing different masses whose ends are in contact with heat baths at different temperatures. Calling ΔT the temperature difference and L the distance between the ends, we are interested in the behavior of J/ΔT as L→∞. For a perfectly periodic harmonic crystal, J/ΔT approaches a fixed positive value as L→∞ corresponding to an infinite heat conductivity. We show that this will be true also for a general one‐dimensional harmonic chain (arbitrary distribution of different masses) if the spectral measure of the infinite chain contains an absolutely continuous part. We also show that for an infinite chain containing two different masses, the cumulative frequency distribution is continuous and that the spectrum is not exhausted by a denumerable number of points, i.e., the spectrum cannot consist entirely of point eigenvalues with a denumerable number of limit points. Using a theorem of Matsuda and Ishii, we show that for a random chain, ...

••

[...]

TL;DR: In this article, matched asymptotic expansions were used for radiating systems in the near zone and wave zone. But their results do not depend upon any definition of gravitational field energy.

Abstract: This paper treats the slow‐motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing‐wave boundary condition applied to the wave‐zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free‐fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.

••

[...]

TL;DR: In this paper, the Lee-Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many-spin interactions.

Abstract: The Lee‐Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many‐spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many‐spin interactions are present, all zeros lie on the imaginary Hz‐axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy) in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields Hz. The Ising theorem is also applied to hydrogen‐bonded ferroelectric models to prove, in particular, that the zeros for the KDP model lie on the imaginary electric field axis for all T below the transition temperature Tc.

••

[...]

TL;DR: In this paper, modification rules for nonstandard irreducible representations of the unitary, orthogonal, and symplectic groups in n dimensions were derived, expressible in terms of the removal of continuous boundary hooks.

Abstract: Modification rules, expressible in terms of the removal of continuous boundary hooks, are derived which relate nonstandard irreducible representations (IR's) of the unitary, orthogonal, and symplectic groups in n dimensions to standard IR's. Tensorial methods are used to derive procedures for reducing the outer products of IR's of U(n), O(n), and Sp(n), and for reducing general IR's of U(n) specified by composite Young tableaux with respect to the subgroups O(n) and Sp(n). In these derivations the conjugacy relationship between the orthogonal and the symplectic groups is fully exploited. The results taken in conjunction with the modification rules are valid for all n.

••

[...]

TL;DR: In this paper, it was shown that the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima.

Abstract: It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima...

••

[...]

TL;DR: In this paper, the authors presented a simplification of the path integral solution of the Schrodinger equation in terms of coordinates which need not be Cartesian and discussed the relationship between the distance and time differentials.

Abstract: In this paper we present a simplification of the path integral solution of the Schrodinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Ito integral.

••

[...]

TL;DR: In this article, a self-contained treatment of the technical parts of distribution theory needed in quantum field theory is presented, particularly suited for physicists since an absolute minimum of abstract functional analysis is used: in fact, only the Baire category theorem is needed.

Abstract: We present a self‐contained treatment of the technical parts of distribution theory needed in quantum field theory. The treatment is particularly suited for physicists since an absolute minimum of abstract functional analysis is used: In fact, only the Baire category theorem is needed. The simple nature of some proofs depends on extensive use of the expansion of a distribution as a sum of harmonic oscillator wave‐functions. While this Hermite expansion is not new, the fact that it provides elementary proofs of several theorems does appear to be new.

••

[...]

TL;DR: In this paper, the thermal conductivity of an infinite, one-dimensional harmonic crystal is investigated in a model system for which exact analytic results can be obtained, and it is shown that thermal conductivities approach infinity as least as fast as N 1/2.

Abstract: Energy transport is investigated in a model system for which exact analytic results can be obtained. The system is an infinite, one‐dimensional harmonic crystal which is perfect everywhere except in a finite segment which contains N isotopic defects. Initially, the momenta and displacements of all atoms to the left of the defect region are canonically distributed at a temperature T, and the right half of the crystal is at a lower temperature. This initial nonequilibrium state evolves according to the equations of motion, and ultimately a steady state is established in the vicinity of the region containing the defects. The thermal conductivity is calculated from exact expressions for the steady state energy flux and thermal gradient. For a crystal in which the N isotopic defects are distributed at random but in which the overall defect concentration is fixed, we demonstrate that the thermal conductivity approaches infinity as least as fast as N1/2. A Monte Carlo evaluation of the thermal conductivity for a given defect‐to‐host mass ratio and concentration is carried out for a series of random configurations of N defects for N in the range, 25 ≤ N ≤ 600. The thermal conductivity is proportional to N1/2 within the statistical uncertainty except for slight deviations at the smallest values of N.

••

[...]

TL;DR: In this article, physical, analytical, and numerical properties of the lattice Green's functions for the various lattices are described and various methods of evaluating the Green's function are discussed.

Abstract: Physical, analytical, and numerical properties of the lattice Green's functions for the various lattices are described. Various methods of evaluating the Green's functions, which will be developed in the subsequent papers, are mentioned.

••

[...]

TL;DR: In this paper, the motion of certain conserved particle-like structures is discussed in a model quantum theory of interacting mesons, and how collective coordinates may be introduced to describe them, leading, in lowest approximation, to a Dirac equation.

Abstract: In a model quantum theory of interacting mesons, the motion of certain conserved particlelike structures is discussed. It is shown how collective coordinates may be introduced to describe them, leading, in lowest approximation, to a Dirac equation.

••

[...]

TL;DR: In this paper, the relativistic Kepler problems in Dirac and Klein-Gordon forms are solved by dynamical group methods for particles having both electric and magnetic charges (dyons).

Abstract: The relativistic Kepler problems in Dirac and Klein‐Gordon forms are solved by dynamical group methods for particles having both electric and magnetic charges (dyons). The explicit forms of the O(4, 2)‐algebra and two special O(2, 1)‐algebras (which coincide in the symmetry limit) are given, and a new group‐theoretical form of the symmetry breaking is pointed out. The Klein‐Gordon O(2, 1)‐algebra also solves the dynamics in the case of very strong coupling constants (attractive singular potential), if the principal series of representations is used instead of the discrete series.

••

[...]

TL;DR: In this article, a hierarchy of tensor identities, satisfied by the generators of the general linear group GL(n), is obtained in terms of two different sets of invariants, which are applied to the identification of irreducible representations and the decomposition of reducible representations.

Abstract: A hierarchy of tensor identities, satisfied by the generators of the general linear group GL(n), is obtained in terms of two different sets of invariants. An application to the identification of irreducible representations and the decomposition of reducible representations is described. Similar results are obtained for the generators of orthogonal, pseudo‐orthogonal, and symplectic groups.

••

[...]

TL;DR: In this paper, the existence of phase transitions in several kinds of two-component lattice gases has been proved, some of which are isomorphic to spin systems and/or to fluids composed of asymmetrical molecules which can have different orientations.

Abstract: We prove the existence of phase transitions in several kinds of two‐component lattice gases: Some of these are isomorphic to spin systems and/or to fluids composed of asymmetrical molecules which can have different orientations. Among the models studied is one with infinite repulsion between particles of different species (hard cores), extending over arbitrarily many neighboring lattice sites. Some of these systems have been investigated previously in the mean field approximation and numerically.

••

[...]

TL;DR: In this article, dual transformations in many-component Ising models in two dimensions on a square lattice are studied from both a topological and an algebraic point of view.

Abstract: Dual transformations in many‐component Ising models in two dimensions on a square lattice are studied. The models considered include those of Ashkin and Teller and of Potts. In certain cases the dual transformation is a relation between the partition function of a lattice at high and low temperatures and can be used to determine a unique critical temperature if one exists. Dual transformations are considered both from a topological and an algebraic point of view. The topological arguments are a natural extension of those used by Onsager for the 2‐component Ising model. The transfer matrices for these models are constructed, and it is shown how the dual transformation arises in this formulation of the problem. The algebras generated by these models are investigated and provide a generalization of the spinor algebra introduced by Kaufman in the 2‐component Ising model.

••

[...]

TL;DR: In this paper, it was shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity.

Abstract: It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n‐vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.

••

[...]

TL;DR: In this paper, it was shown that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in the terms of field operators, which correspond in general to distributions of unbounded operators.

Abstract: It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand‐Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the p...

••

[...]

TL;DR: In this article, the Schrodinger equation of A nucleons is transformed to new coordinates, six of which have collective nature and 3A − 9 are single-particle coordinates, and the connection is given to the conventional collective model in which five collective coordinates are used.

Abstract: The Schrodinger equation of A nucleons is transformed to new coordinates. Six of them have collective nature and 3A − 9 are single‐particle coordinates. The connection is given to the conventional collective model in which five collective coordinates are used. The additional sixth collective coordinate of this paper gives a simple description of monopole vibrations. The new coordinates can also be used in the theory of nuclear reactions: By introducing a symmetrized distance vector for reaction partners, the antisymmetrization procedure is simplified considerably.

••

[...]

TL;DR: In this article, a dynamical group is used to generate the energy spectrum of a given Hamiltonian in the case of a superfluid Bose system, and the energy eigenvalues and eigenfunctions are obtained by means of the group.

Abstract: The method of using a dynamical group to generate the energy spectrum of a given Hamiltonian is applied to the case of a superfluid Bose system. Here the relevant group is found to be SU(1, 1) [or Πk⊗SU(1,1)k for a multilevel system]. The energy eigenvalues and eigenfunctions are obtained by means of the group.