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Nov 1960

Volume 1, Issue 6, pp. 453-559


Symmetry‐Adapted Functions Belonging to the Symmetric Groups

Harold V. McIntosh

J. Math. Phys. 1, 453 (1960); http://dx.doi.org/10.1063/1.1703681 (8 pages) | Cited 22 times

Online Publication Date: 22 December 2004

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Young's factorization of idempotents belonging to the symmetric groups is given a necessary and sufficient characterization, by means of a lemma due to Burrow. The use of these idempotents is contrasted with Yamanouchi's representation, and finally the equivalence of Löwdin's path diagram method to the group‐theoretical treatment of the angular momentum states arising from the coupling of an assemblage of spin ☒ particles is demonstrated.

On the Calculation of the Inverse of the Overlap Matrix in Cyclic Systems

P. O. Löwdin, R. Pauncz, and J. De Heer

J. Math. Phys. 1, 461 (1960); http://dx.doi.org/10.1063/1.1703682 (7 pages) | Cited 35 times

Online Publication Date: 22 December 2004

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The inverse of the overlap matrix of a cyclic system has been computed in three different ways, each of which throws light on a different aspect of the subject and has its own range of applicability to related problems. In all these derivations extensive use has been made of the properties of the Chebyshev polynomials. The results hold for every cyclic symmetric matrix.

Quantization of Nonlinear Systems

I. E. Segal

J. Math. Phys. 1, 468 (1960); http://dx.doi.org/10.1063/1.1703683 (21 pages) | Cited 45 times

Online Publication Date: 22 December 2004

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A direct method of quantization, applicable to a given nonlinear hyperbolic partial differential equation, is indicated. From such classical equations alone, without a given Lagrangian or Hamiltonian, or a priori linear reference system such as a bare or incoming field, a quantized field is constructed, satisfying the conventional commutation relations. While mathematically quite heuristic in part, local products of quantized fields do not intervene, and there are grounds for the belief that the formulation is free from nontrivial divergences.

Kemmer Wave Equation in Riemann Space

Adel Da Silveira

J. Math. Phys. 1, 489 (1960); http://dx.doi.org/10.1063/1.1703684 (3 pages) | Cited 2 times

Online Publication Date: 22 December 2004

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The Kemmer equation is written in Riemann space. The form of the equation, the supplementary relations and the form of the covariant derivatives of the operators βμ are considered. As a check of the equation it is shown that the equations for the values zero and one of the spin can be obtained from this generalized Kemmer equation with the help of generalized Fujiwara operators. In particular the equation for photons in interaction with the gravitational field is obtained.

Hamiltonian Formalism and the Canonical Commutation Relations in Quantum Field Theory

H. Araki

J. Math. Phys. 1, 492 (1960); http://dx.doi.org/10.1063/1.1703685 (13 pages) | Cited 114 times

Online Publication Date: 22 December 2004

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Cyclic representations of the canonical commutation relations and their connection with the Hamiltonian formalism are studied. The vacuum expectation functional E(f) = (Ψ0,ei[openphi](f)Ψ0) turns out to be a very convenient tool for the discussion. The uniqueness of a translationally invariant state (vacuum) is proved under the assumption of the cluster decomposition property for E(f). The existence and near uniqueness of the Hamiltonian in cyclic representations of the canonical commutation relations are established. The conditions for the relativistic invariance of the theory are stated in terms of vacuum expectation values at a fixed time. It is shown that E(f) is the Fourier transform of a quasi‐invariant nonnegative measure on the space of all linear functionals of the test functions.

Approximate Solutions of the Bethe‐Salpeter Equation

S. H. Vosko

J. Math. Phys. 1, 505 (1960); http://dx.doi.org/10.1063/1.1703686 (11 pages) | Cited 9 times

Online Publication Date: 22 December 2004

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A modified representation of the Bethe‐Salpeter wave function for scalar particles interacting via a massless scalar field is presented and related to Salpeter's approximate wave function. A procedure for obtaining approximate solutions of the Bethe‐Salpeter equation for arbitrary interactions is introduced. The method is based on a variational principle and is capable of high accuracy when used with the trial functions developed here. The choice of trial function is suggested by the important features of the exact solutions for the special case described above. The method is applied to a finite range potential which corresponds to the lowest order approximation to a simple field theory. The results of the calculation suggest that the effect of retardation is large when an interaction is transmitted by a field with mass.

Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension

M. Girardeau

J. Math. Phys. 1, 516 (1960); http://dx.doi.org/10.1063/1.1703687 (8 pages) | Cited 118 times

Online Publication Date: 22 December 2004

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A rigorous one‐one correspondence is established between one‐dimensional systems of bosons and of spinless fermions. This correspondence holds irrespective of the nature of the interparticle interactions, subject only to the restriction that the interaction have an impenetrable core. It is shown that the Bose and Fermi eigenfunctions are related by ψB=ψFA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one‐one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many‐boson problem.

Generalization of the ``Edge‐of‐the‐Wedge'' Theorem

H. Epstein

J. Math. Phys. 1, 524 (1960); http://dx.doi.org/10.1063/1.1703688 (8 pages) | Cited 19 times

Online Publication Date: 22 December 2004

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It is proved that two analytic functions of several complex variables, having the same boundary values when the imaginary parts of the variables tend to zero inside two arbitrary, but fixed, open cones, possess a common analytic continuation in a certain open set. This is a generalization of the ``edge‐of‐the‐wedge'' theorem, a proof of which is obtained in passing.

``Front'' Description in Relativistic Quantum Mechanics

R. Acharya and E. C. G. Sudarshan

J. Math. Phys. 1, 532 (1960); http://dx.doi.org/10.1063/1.1703689 (5 pages) | Cited 19 times

Online Publication Date: 22 December 2004

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The problem of introducing a Cartesian position operator canonically conjugate to the momentum operator into relativistic one‐particle theories is investigated independent of any particular relativistic wave equation. The known result that such a description is possible for particles with nonvanishing mass is rederived. The general problem of introduction of canonical variables into relativistic theories is formulated and solved. The configurational indices so obtained correspond to directed plane wavefronts rather than point particles.

On the Nonexistence of a Class of Static Einstein Spaces Asymptotic at Infinity to a Space of Constant Curvature

H. A. Buchdahl

J. Math. Phys. 1, 537 (1960); http://dx.doi.org/10.1063/1.1703690 (5 pages) | Cited 12 times

Online Publication Date: 22 December 2004

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It is known that there exist no nontrivial static regular solutions of the Einstein vacuum equations Rkl=0 which are asymptotically Galilean at infinity. One may ask correspondingly whether there exist static solutions of the equations Rkl=λgkl(λ<0) which are regular at all finite points and asymptotic (in a sense to be defined) to a space of constant curvature at infinity. The answer to this question is here shown to be in the negative. The proof rests upon the possibility of writing a certain quadratic invariant density of the Riemann tensor in the form of an ordinary divergence.

Galvanomagnetic and Thermomagnetic Effects in Isotropic Materials

A. C. Pipkin and R. S. Rivlin

J. Math. Phys. 1, 542 (1960); http://dx.doi.org/10.1063/1.1703691 (5 pages) | Cited 7 times

Online Publication Date: 22 December 2004

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Invariant‐theoretical considerations are employed to obtain constitutive equations for the current density vector, the heat flux vector, and the magnetic intensity field in isotropic materials (both holohedral and hemihedral) when an electric field, a magnetic induction field, and a temperature gradient are simultaneously present in the material. Certain of the interaction effects which are indicated by these constitutive equations are discussed.

Computation of Order Parameters in an Ising Lattice by the Monte Carlo Method

J. R. Ehrman, L. D. Fosdick, and D. C. Handscomb

J. Math. Phys. 1, 547 (1960); http://dx.doi.org/10.1063/1.1703692 (12 pages) | Cited 35 times

Online Publication Date: 22 December 2004

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The long‐range and short‐range order parameters are computed for the Ising lattice using a Monte Carlo sampling scheme. The square lattice, the simple cubic lattice, and the body‐centered cubic lattice are considered. In the three‐dimensional calculations both the antiferromagnetic and ferromagnetic cases are considered as well as the coupling to an external magnetic field of various strengths. Good agreement is found where the results can be compared with the exact two‐dimensional results, and in the three‐dimensional case the results agree well with those obtained from series approximations in the regions where the series approximations are valid. The present method appears to give good results for the short‐range order even very close to the critical temperature, but in this neighborhood the long‐range order estimate is crude. The computations were performed on the high‐speed computer ILLIAC, located at the University of Illinois.
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Linearized Plasma Oscillations in Arbitrary Electron Distributions

George E. Backus

J. Math. Phys. 1, 559 (1960); http://dx.doi.org/10.1063/1.1703693 (1 page) | Cited 1 time

Online Publication Date: 22 December 2004

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