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Nov 1961

Volume 2, Issue 6, pp. 743-894


A Criterion for the Free Character of Fields

Jan Lopuszanski

J. Math. Phys. 2, 743 (1961); http://dx.doi.org/10.1063/1.1724217 (5 pages) | Cited 2 times

Online Publication Date: 22 December 2004

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It is shown that a field is a free field as soon as the states generated by the Heisenberg field and the incoming field operators operating on the vacuum coincide [statement (i)]. Several conclusions are drawn from statement (i) concerning the strong convergence of the field for t tending to infinity [statement (ii)], the uselessness of the local clothed particle representation [statement (iii)], and the diagonalization of the Hamiltonian [statement (iv)], as well as the time behavior of the mathematical vacuum [statement (v)].

Analyticity of Wightman Functions

Christian Fronsdal

J. Math. Phys. 2, 748 (1961); http://dx.doi.org/10.1063/1.1724218 (11 pages) | Cited 2 times

Online Publication Date: 22 December 2004

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The problem of mapping certain domains in the space of complex four‐vectors onto the space of their inner products is solved by a novel method. The ``primitive'' domains of regularity of the three and four point Wightman functions are determined. The domains whose X‐space characterization was obtained by Streater, for the holomorphy envelope of the union of several primitive domains (both for the three and four point function) are also determined. The corresponding problem in perturbation theory is examined, and the analytic form of the perturbation theory boundary of the three and four point functions is obtained by the same method. The problem of the four‐point function is reduced to constructing the holomorphy envelope of the union of three domains. It is shown that the boundary of the domain of four‐point singularitites in perturbation theory bears no resemblance to any part of the boundary of the primitive domains, or the domains found or conjectured by Streater.

Support of a Field in p Space

G. F. Dell'Antonio

J. Math. Phys. 2, 759 (1961); http://dx.doi.org/10.1063/1.1724219 (8 pages) | Cited 16 times

Online Publication Date: 22 December 2004

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The notion of generalized free field is introduced, as an obvious extension of a discrete superposition of independent free fields with different masses. The following assumptions are also made: there is an underlying Hilbert space H (positive‐definite metric), the theory is Lorentz invariant, the vacuum belongs to H and is there unique, the spectrum of the energy‐momentum operator is—apart from the origin—completely contained within the region p2≥ϵ2, p0>0. It is then shown that a necessary condition for a cyclic field to have support in p2 only on a finite interval of the positive real axis, is that A(x) be a generalized free field. In the Appendix a similar result is derived under slightly weaker conditions.

Nature of the Axioms of Relativistic Quantum Field Theory. I

E. C. G. Sudarshan and K. Bardakci

J. Math. Phys. 2, 767 (1961); http://dx.doi.org/10.1063/1.1724220 (5 pages) | Cited 7 times

Online Publication Date: 22 December 2004

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The formulation of field theories by means of Wightman functions is studied. It is shown that, given two field theories that satisfy all the axioms, one can construct a family of Wightman fields with the same properties by a process of superposition of Wightman functions. The condition of unitarity is formulated without reference to asymptotic conditions, and it is proved that the Wightman fields constructed by the above superposition process (starting with ``unitary'' fields) fail to preserve unitarity, and a fortiori, the standard asymptotic condition.

Dynamical Mappings of Density Operators in Quantum Mechanics

Thomas F. Jordan and E. C. G. Sudarshan

J. Math. Phys. 2, 772 (1961); http://dx.doi.org/10.1063/1.1724221 (4 pages) | Cited 21 times

Online Publication Date: 22 December 2004

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The most general dynamical law for a quantum mechanical system is studied with particular reference to the necessary and sufficient conditions for such a law to represent Hamiltonian dynamics. The main results are stated in the form of three theorems.

A Note on k‐Commutative Matrices

D. W. Robinson

J. Math. Phys. 2, 776 (1961); http://dx.doi.org/10.1063/1.1724222 (2 pages)

Online Publication Date: 22 December 2004

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Let A and B be square matrices over a field in which the minimum polynomial of A is completely reducible. It is shown that A is k commutative with respect to B for some non‐negative integer k if and only if B commutes with every principal idempotent of A. The proof is brief, simplifying much of the previous study of k‐commutative matrices. The result is also used to generalize some well‐known theorems on finite matrix commutators that involve a complex matrix and its transposed complex conjugate.

New Possibilities for a Unified Field Theory

W. Israel and R. Trollope

J. Math. Phys. 2, 777 (1961); http://dx.doi.org/10.1063/1.1724223 (10 pages) | Cited 6 times

Online Publication Date: 22 December 2004

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The introduction of nonsymmetric gik in unified field theories of the Einstein‐Schrödinger type is open to the objection, on group‐theoretical grounds, that the symmetric and antisymmetric parts transform independently. This objection does not apply to the use of nonsymmetric Γikμ, since these quantities are irreducible under the ``extended group,'' consisting of the point transformations and the Einstein λ transformations.
We consider a theory based on symmetric gik and nonsymmetric Γikμ. The Lagrangian L is assumed to depend only on gik and the contracted curvature tensor Rik (this insures the λ invariance and transposition invariance of the theory). For simplicity, we suppose further that L involves Rik rationally and, at most, quadratically.
The resulting theory is able to account satisfactorily for the main feature of gravitation, electromagnetism, and their interaction. In particular, the theory yields the correct equations of motion for charged masses. The electromagnetic tensor is associated with the skew part of Rik, and the λ transformations correspond roughly to the gauge transformations of electrodynamics.

Approximate Stress Energy Tensor for Gravitational Fields

A. H. Taub

J. Math. Phys. 2, 787 (1961); http://dx.doi.org/10.1063/1.1724224 (7 pages) | Cited 11 times

Online Publication Date: 22 December 2004

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An invariant formulation in Minkowski space‐time of an approximation to the Einstein theory of gravitation is given. In this formulation a tensor is introduced which may be interpreted as the approximate stress energy tensor of the gravitational field. Conservation laws involving this tensor and the material stress energy tensor are formulated. The behavior of these tensors under ``gauge transformations'' of the weak gravitational fields is discussed. The classical limit of the conservation of energy equation is studied and the results are compared to some observations of Bondi on a possible analog of the Poynting vector for a gravitational field.

Principles of Limiting Absorption and Limiting Amplitude in Scattering Theory. I. Schrödinger's Equation

Farouk M. Odeh

J. Math. Phys. 2, 794 (1961); http://dx.doi.org/10.1063/1.1724225 (7 pages) | Cited 4 times

Online Publication Date: 22 December 2004

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The outgoing solution of the time‐independent Schrödinger equation, with a suitably restricted real potential, is shown to be the uniform limit of the square‐integrable solutions of the same equation with complex energy as the imaginary part of the energy tends to zero. Under further restrictions on the potential, it is also shown that the solution to the initial‐value problem for the time‐dependent Schrödinger equation tends to the outgoing solution as time increases indefinitely.

Principles of Limiting Absorption and Limiting Amplitude in Scattering Theory. II. The Wave Equation in an Inhomogeneous Medium

Farouk M. Odeh

J. Math. Phys. 2, 800 (1961); http://dx.doi.org/10.1063/1.1724226 (3 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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This is a continuation of Part I with special emphasis on wave equations in an inhomogeneous medium.

Foundations for a Treatment of the Scattering of Light by the Hydrodynamical and Statistical Atom Model

Masami Wakano

J. Math. Phys. 2, 803 (1961); http://dx.doi.org/10.1063/1.1724227 (22 pages) | Cited 3 times

Online Publication Date: 22 December 2004

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A study has been made on the scattering of light by the hydrodynamical and statistical atom model. Bloch's treatment of the hydrodynamical equations of motion for this model is supplemented here by inclusion of the interaction with the electromagnetic field. We limited attention to oscillations of small amplitude. By correspondence principle arguments, general expressions were derived for the cross sections for absorption, coherent and incoherent scattering.
The energy can be expressed—following Bloch—as the energy of a Thomas‐Fermi atom plus a Hamiltonian which is associated with departures from the Thomas‐Fermi distribution. Using Bloch's quantization of this Hamiltonian and applying the method of quantum field theory, we rederived the correspondence principle results for elementary cross sections.
Then applying the correspondence theoretical argument to the matrix element, we rederived Heisenberg's result for the total intensity of the Compton scattering. We also apply the method of stationary phase to the hydrodynamical treatment and show that this method gives the same result as does the linear term in the momentum transfer in Heisenberg's expression, except for a numerical factor 3☒/2—a point that was discussed by Bloch many years ago.
Application of the general formulas given here for angular distribution of Rayleigh and Compton scattering will require electronic machine calculations of higher modes of oscillation of the gas model of the atom analogous to those made by Wheeler and Fireman for l=1.

Symmetric Expansion of One‐ and Two‐Center Coulomb Potentials

Peter R. Fontana

J. Math. Phys. 2, 825 (1961); http://dx.doi.org/10.1063/1.1724228 (4 pages) | Cited 16 times

Online Publication Date: 22 December 2004

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A one‐center expansion of the electrostatic interaction energy of a discrete charge distribution is developed by making use of the algebra of irreducible tensors. The result is completely symmetric in the coordinates of the particles, and the relative magnitude of the vectors need not be specified. It is shown that a suitable interaction representation provides useful formulas for electrostatic and quantum mechanical applications. In addition, some transformation equations make it possible to refer any arbitrary number of vectors to a second origin, thus yielding general two‐center expansions for overlapping charge distributions.

Isoperimetric and Other Inequalities in the Theory of Neutron Transport

Lawrence Dresner

J. Math. Phys. 2, 829 (1961); http://dx.doi.org/10.1063/1.1724229 (19 pages) | Cited 3 times

Online Publication Date: 22 December 2004

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Some isoperimetric and other inequalities occurring in the one‐velocity theory of neutron transport are derived. The quantities involved in these inequalities all refer to bare solids with isotropic scattering and are: the critical multiplication, the first‐collision probability, the non‐escape probability, and the buckling. The inequalities proved provide upper and lower bounds for the quantities considered, and numerous examples of the estimation of these quantities in cases not readily amenable to direct calculation are given.

Probability Distribution for Classical Fluids

A. A. Broyles

J. Math. Phys. 2, 848 (1961); http://dx.doi.org/10.1063/1.1724230 (10 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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The probability distribution in phase space is obtained for particles in a small part of the total volume of a classical monatomic fluid in thermal equilibrium. It is shown that the distribution reduces to that obtained from a grand canonical ensemble as this part of the volume increases in size. The Debye‐Huckel pair distribution function is obtained in the proper limit for the Coulomb case. The distribution is in the form of a first approximation with an infinite series of correction terms.

Note on the Albertoni‐Bocchieri‐Loinger Theorem of Classical Statistical Mechanics

J. S. Lomont

J. Math. Phys. 2, 858 (1961); http://dx.doi.org/10.1063/1.1724231 (3 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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The A‐B‐L theorem states that for almost all density functions the time‐averaged probability of a system being in a region R of an energy shell in phase space is mR/m, where m is the volume of the energy shell and mR is the volume of R. In the present note a stronger form of the A‐B‐L theorem is proved. It is proved that for almost all density functions, the probability of the system being in R is mR/m. In particular, it is proved that the time averaging of the original A‐B‐L theorem is unnecessary.

A Lattice with an Unusual Frequency Spectrum

Robert J. Rubin and Robert Zwanzig

J. Math. Phys. 2, 861 (1961); http://dx.doi.org/10.1063/1.1724232 (4 pages) | Cited 5 times

Online Publication Date: 22 December 2004

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The lattice is a special rooted Cayley tree, generated by N successive m‐fold branchings. With each point of the tree are associated a mass M and a position coordinate xi. All end points are held fixed at xi=0. The potential energy is V=☒ Σi,j Kij(xi−xj)2, where Kij=K if i and j are connected neighbors and neither is an end point, Kij=αK if i and j are connected neighbors and either is a branch tip point, and Kij=0 if i and j are not connected neighbors. The allowed frequencies of vibration are obtained for two different cases: In the first case all springs are identical (α=1), and in the second case the springs connecting interior points to the branch tips are cut (α=0). In the case in which all force constants are the same, the allowed frequencies of vibration, in the limit of infinite N, are given by ω(r) = (K/M) ☒[m+1 − 2m☒ cosrπ]☒, where r is any rational number between zero and one. The fraction of all normal modes having precisely the value ω(r) is ρ[ω(r)] = (m − 1)2 / (mq − 1), where r is expressed as the ratio r=p/q of relatively prime integers p and q. The frequency spectrum is dense within the interval (m☒ − 1, m☒+1); and ρ[ω] is discontinuous at every ω for which it does not vanish.

Electrical Conduction in a Noncircular Rod

A. C. Pipkin and R. S. Rivlin

J. Math. Phys. 2, 865 (1961); http://dx.doi.org/10.1063/1.1724233 (4 pages) | Cited 1 time

Online Publication Date: 22 December 2004

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The general constitutive equations for galvanomagnetic effects in isotropic materials are applied to the study of electrical conduction in rods. It is shown that, in general, rectilinear current flow is not possible, unless the rod has a circular cross section or is an infinite parallel‐sided slab.

Method for Defining Principal Modes of Nonlinear Systems Utilizing Infinite Determinants

Demetrios G. Magiros

J. Math. Phys. 2, 869 (1961); http://dx.doi.org/10.1063/1.1724234 (7 pages)

Online Publication Date: 22 December 2004

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A method for calculation of ``principal modes'' of linear or nonlinear systems is discussed. The physical definition of ``principal modes'' is formulated mathematically in two ways. The trial solution of the differential equation of the motion of the system is taken in an appropriate structure. The calculation of principal modes leads to infinite determinants of Hill's and von Koch's type, which are analyzed. The above method yields the possibility of getting the ``principal modes'' in the form of a series, all the coefficients of which can be calculated.

Note on the Algebraic Aspect of the Integration of a System of Ordinary Linear Differential Equations

Eyvind H. Wichmann

J. Math. Phys. 2, 876 (1961); http://dx.doi.org/10.1063/1.1724235 (5 pages) | Cited 19 times

Online Publication Date: 22 December 2004

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In this note the Lie algebra generated by the coefficient matrix of a system of ordinary, linear, first‐order differential equations is considered. A systematic discussion, based on some well‐known results in the theory of Lie albegras, is given for the reduction of the problem of integration of such a system. For the purposes of this note the integration of a system of equations for which the coefficient matrix does not depend on the independent variable is regarded as ``elementary.'' It will be shown that the problem of integrating any system of linear ordinary differential equations can be reduced to the problem of integrating a set of such systems, each one of which has the property that the corresponding Lie algebra is simple, and in such a way that the sum of the dimensionalities of the Lie algebras of the reduced systems in the set does not exceed the dimensionality of the Lie algebra of the original system.
The application of the reduction principle to the equations of motion in classical mechanics and in quantum mechanics is considered. It is shown that the principle in question applies to a class of Hamiltonian equations of motion not customarily regarded as describing linear systems.

General Perturbational Solution of the Harmonically Forced van der Pol Equation

Raimond A. Struble and John E. Fletcher

J. Math. Phys. 2, 880 (1961); http://dx.doi.org/10.1063/1.1724236 (12 pages) | Cited 8 times

Online Publication Date: 22 December 2004

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Some formal techniques for the study of nonlinear oscillations are illustrated through a development of a general perturbational solution of the harmonically forced van der Pol equation. These techniques provide for the study of perturbations of nearly linear oscillations in almost complete generality. The intricate resonance problems associated with small divisors, and in this case leading to the entrainment phenomena, are treated with unusual ease and serve to illustrate both the versatility and the generality of the techniques.

Transient Response of a Dipole Antenna

Tai Tsun Wu

J. Math. Phys. 2, 892 (1961); http://dx.doi.org/10.1063/1.1724237 (3 pages) | Cited 7 times

Online Publication Date: 22 December 2004

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The current distributed for a dipole antenna driven by a step‐function voltage is found shortly after the switch‐on of the voltage.
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