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Dec 1971

Volume 12, Issue 12, pp. 2401-2543


On Some Properties of Physical Symmetries

Jan T. Lopuszanski

J. Math. Phys. 12, 2401 (1971); http://dx.doi.org/10.1063/1.1665551 (12 pages) | Cited 12 times

Online Publication Date: 28 October 2003

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It is shown that each symmetry of a field theory with interaction and with a mass gap, induced by a local current, is also induced by at least two other currents, local with respect to the asymptotic free fields and bilinear in these fields. The currents are asymptotic currents in the sense of Araki and Haag [H. Araki and R. Haag, Commun. Math. Phys. 4, 77 (1967)] of the original current. It is also shown that the charges, displaying spinorial transformation character with respect to the Lorentz group, vanish, while the tensorial charges are a linear combination of the scalar charges (internal symmetries) and the energy‐momentum vector (translational symmetry).

Computation of the Cluster Coefficients of the δ‐Function Gases through the U Functions

Sergio Servadio

J. Math. Phys. 12, 2413 (1971); http://dx.doi.org/10.1063/1.1665552 (3 pages) | Cited 2 times

Online Publication Date: 28 October 2003

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It is shown that the wavefunctions of the δ‐function l‐body problem are simple enough to allow an explicit evaluation of the cluster operator Ul. Upon integration one can then obtain the general cluster coefficient bl. The l = 2 case is explicitly solved for particles of any statistics.

Classical Lorentz Invariant Particles

Richard Arens

J. Math. Phys. 12, 2415 (1971); http://dx.doi.org/10.1063/1.1665553 (8 pages) | Cited 11 times

Online Publication Date: 28 October 2003

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A classical Lorentz invariant completely Hamiltonian elementary one‐particle system is defined as having a state space K in which the Poincaré group acts transitively, its infinitesimal actions having generating functions relative to some Poisson bracket, such that there can be associated with each state k a world line Γ(k) in Cartesian 4‐space. It is determined that there are nine families of such particles. Two have their speed in the usual range, three travel at the speed of eight, and four always faster. In each family the members are distinguished by one or two parameters such as mass and spin.

The Classical Moment Problem and the Calculation of Thermal Averages

A. Lonke

J. Math. Phys. 12, 2422 (1971); http://dx.doi.org/10.1063/1.1665554 (17 pages) | Cited 20 times

Online Publication Date: 28 October 2003

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The physical information contained in the first 2n moments of the single‐particle spectral weight function of a fermionic many‐body system is investigated. The approach is based on the mathematical theory of the classical moment problem. Under consideration are the thermal as well as dynamical properties of the system. Using this information, approximate n‐pole single‐particle thermal Green's functions and the corresponding spectral weight functions are constructed. It is shown that these approximations are not unique and depend on a real parameter. This dependence is used for the calculation of the rigorous error bounds of the approximate thermal averages.

A Technique for Solving Recurrence Relations Approximately and Its Application to the 3‐J and 6‐J Symbols

Donald Neville

J. Math. Phys. 12, 2438 (1971); http://dx.doi.org/10.1063/1.1665556 (16 pages) | Cited 6 times

Online Publication Date: 28 October 2003

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It is shown that not only some second‐order differential equations, but also some second‐order difference equations (three‐term recurrence relations) possess solutions of WKBJ type. WKBJ‐like solutions are derived and used to study the ``classical'' (large J) limits of both the 6‐J symbol and (in an appendix) the 3‐J symbol.

Irreducible Vector and Ray Representations of Some Cubic Crystal Point Groups in Four Dimensions

Li‐Ching Chen and Joseph L. Birman

J. Math. Phys. 12, 2454 (1971); http://dx.doi.org/10.1063/1.1665557 (8 pages) | Cited 7 times

Online Publication Date: 28 October 2003

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A new, and computationally simple, method is given for obtaining the characters of all the inequivalent irreducible vector representations of a finite group. The method was applied to determine the characters of the irreducible vector and ray representations of the four‐dimensional cubic crystal point groups: group 47 and group 45. These groups are of order 384 and 1152, respectively, and contain the cubic point group in three dimensions Oh[3] as a subgroup. Tables are given of the irreducible representations of Oh subduced by the irreducible representations of group 47 and group 45. These tables may be useful in testing the conjecture that accidental degeneracy in problems in solid state physics in three dimensions reflects a higher symmetry in four dimensions.

A Note on Asymptotically Flat Spaces

B. Aronson, R. Lind, J. Messmer, and E. Newman

J. Math. Phys. 12, 2462 (1971); http://dx.doi.org/10.1063/1.1665558 (6 pages) | Cited 12 times

Online Publication Date: 28 October 2003

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It is shown that there is a simple alternative to the Bondi type of coordinate condition in asymptotically flat spaces, which leads to a different asymptotic coordinate group whose structure is simpler than that of the BMS group. As a concomitant to this work, there was discovered a scalar function on null infinity. From an analysis of this function one can obtain, starting from the asymptotic shear of one family of hypersurface orthogonal rays, (1) all null asymptotically shear‐free congruences, (2) the asymptotic shear of all hypersurface orthogonal rays.

Many‐Body Point Transforms. II. An Exact Noncluster Approach to the Hard‐Core Many‐Body Problem

Norman M. Witriol

J. Math. Phys. 12, 2467 (1971); http://dx.doi.org/10.1063/1.1665559 (14 pages) | Cited 2 times

Online Publication Date: 28 October 2003

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The method of many‐body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard‐core many‐body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth‐order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.

One‐Particle Operators and Local Internal Symmetries

Ira W. Herbst

J. Math. Phys. 12, 2480 (1971); http://dx.doi.org/10.1063/1.1665560 (11 pages) | Cited 4 times

Online Publication Date: 28 October 2003

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The structure of local operators linear in the creation and destruction operators of a finite number of particles is investigated. It is shown that these operators determine a set of local and relatively local free fields relative to which the original operators are local. This result is used to study local internal symmetries in a theory of interactions and to show that these symmetries commute with the Lorentz group. The assumptions of Haag and Ruelle which lead to a complete particle interpretation are also discussed. Asymptotic many‐particle states are constructed under less restrictive assumptions.

Electromagnetic Radiation in Curved Spaces

Richard Sigal

J. Math. Phys. 12, 2490 (1971); http://dx.doi.org/10.1063/1.1665561 (5 pages) | Cited 3 times

Online Publication Date: 28 October 2003

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The classes of trajectories of charged particles, for which the radiation reaction force vanishes, in curved spaces are examined. The trajectories in the Robertson‐Walker cosmologies are given in detail and it is shown that the only reaction‐free geodesic trajectories are those at rest relative to the local matter density.

Combined Neutrino‐Gravitational Fields in General Relativity

D. Trim and J. Wainwright

J. Math. Phys. 12, 2494 (1971); http://dx.doi.org/10.1063/1.1665562 (5 pages) | Cited 20 times

Online Publication Date: 28 October 2003

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The combined gravitational‐neutrino field equations are solved, subject only to the restriction that the energy‐flow vector of the neutrino field be timelike or null. The principal null congruence (pnc) of the neutrino field is necessarily geodesic and shear‐free, and coincides with a repeated pnc of the gravitational field, which is thus algebraically special. The twist of the neutrino pnc plays an important role, and is zero if and only if the neutrino energy‐flow vector is null, in which case the neutrino field represents pure radiation.

Some Considerations of Entropy Change

S. Okubo and A. Isihara

J. Math. Phys. 12, 2498 (1971); http://dx.doi.org/10.1063/1.1665563 (6 pages) | Cited 2 times

Online Publication Date: 28 October 2003

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An inequality for two positive operators is used to discuss entropy theorems for time smoothing, mixing, and other processes. Especially, it is proved that neglect of nondiagonal matrix elements for the density matrix causes the entropy to increase.

Perturbation of Statistical Semigroups in Quantum Statistical Mechanics

D. A. Uhlenbrock

J. Math. Phys. 12, 2503 (1971); http://dx.doi.org/10.1063/1.1665564 (10 pages) | Cited 5 times

Online Publication Date: 28 October 2003

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The perturbation theory of Hille and Phillips for semigroups of bounded linear operators on a Banach space is modified to apply to the semigroups of positive traceclass operators encountered in quantum statistical mechanics.

Minimal Uncertainty States for Bounded Observables

Evangelos K. Ifantis

J. Math. Phys. 12, 2512 (1971); http://dx.doi.org/10.1063/1.1665565 (5 pages) | Cited 2 times

Online Publication Date: 28 October 2003

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The procedure of finding the states, which minimize the uncertainty product of two noncompatible observables, leads to the study of the approximate point spectrum of a non‐self‐adjoint operator. The spectrum of a large class of such bounded non‐self‐adjoint operators is studied. The results are applied to the theory of the oscillator phase operators.

On Clifford Numbers, Dirac and Relativistic Hamilton‐Jacobi Equations

Jerome K. Percus and Nikitas L. Petrakopoulos

J. Math. Phys. 12, 2516 (1971); http://dx.doi.org/10.1063/1.1665566 (5 pages) | Cited 1 time

Online Publication Date: 28 October 2003

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Conditions are established under which the Dirac equation for an electron in an electromagnetic field has an exact semiclassical solution. In other words, the phase is identified with a solution of the corresponding relativistic Hamilton‐Jacobi equation and the spinor amplitude has no explicit dependence upon ℏ. The complete set of admissible fields is determined for one‐dimensional and stationary three‐dimensional systems, while an extensive class is indicated for the general time‐dependent problem.

Path Integrals in Curved Spaces

David W. McLaughlin and L. S. Schulman

J. Math. Phys. 12, 2520 (1971); http://dx.doi.org/10.1063/1.1665567 (5 pages) | Cited 90 times

Online Publication Date: 28 October 2003

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In this paper we present a simplification of the path integral solution of the Schrödinger 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 Itô integral.

The Phase Shift. II. As a Continuous Functional

William M. Frank and David W. McLaughlin

J. Math. Phys. 12, 2525 (1971); http://dx.doi.org/10.1063/1.1665568 (5 pages) | Cited 1 time

Online Publication Date: 28 October 2003

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The continuity of the phase shift and S‐wave scattering length as a functional of the potential is considered. The question is investigated in the present article for sequences of square wells which converge pointwise to zero. The results show certain conditions to be sufficient for convergence of the phase shift and scattering length to zero. These results are generalized to certain integral norms.

The Phase Shift. III. As a Continuous Functional

William M. Frank

J. Math. Phys. 12, 2529 (1971); http://dx.doi.org/10.1063/1.1665569 (9 pages) | Cited 1 time

Online Publication Date: 28 October 2003

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The continuity of the phase shift and scattering length as a functional of the potential is considered. We find that the convergence of ∫dr∣Vn∣ to zero is a sufficient though not necessary condition for the vanishing of the phase shift, while the convergence of ∫dr r∣Vn(r)∣ or ∫dr∣Vn(r)∣1∕2 to zero would be a sharp condition. A number of theorems of this type are proven. Theorems are also proven demonstrating the continuity of the phase shift and scattering length as a sequence of potentials converging in the proper norms to a nonzero potential. Some implications of the existence of a Banach space of potentials are discussed briefly.

The Inverse Decay Problem

L. P. Horwitz, J. A. LaVita, and J.‐P. Marchand

J. Math. Phys. 12, 2537 (1971); http://dx.doi.org/10.1063/1.1665570 (7 pages) | Cited 32 times

Online Publication Date: 28 October 2003

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Let UK(t) be a one‐parameter operator family of positive type in a Hilbert space K and U(t) its minimal unitary dilation with infinitesimal generator H. If UK(t) is a contractive semigroup, then H is not positive. If in addition UK(t)→0 for t → ∞, then there exists a state ϕ ∊ K on which H is not defined. We interpret these and other results in the context of the quantum‐mechanical theory of unstable particles and the scattering theory of Lax and Phillips.
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