Volume 10, Issue 9, September 1969
Index of content:

Quantization of the Generalized Hamiltonian
View Description Hide DescriptionQuantization of systems described by Lagrangians with higher‐order derivatives is performed. It is shown that the usual Lagrangian with first‐order derivatives may be consistently replaced by one with second‐order derivatives, the resulting wavefunction being one of a mixed representation.

Rigorous Results for Ising Ferromagnets of Arbitrary Spin
View Description Hide DescriptionThe following results for spin‐½ Ising ferromagnets are extended to the case of arbitrary spin: (1) the theorem of Lee and Yang, that the zeros of the partition function lie on the unit circle in the complex fugacity plane; (2) inequalities of the form <AB> ≥ <A><B>, where A and B are products of spin operators; (3) the existence of spontaneous magnetization on suitable lattices. Results (2) and (3) are also extended to the infinite‐spin limit in which the spin variable is continuous on the interval −1 ≤ x ≤ 1.

Conserved Quantities in the Einstein‐Maxwell Theory
View Description Hide DescriptionIt is shown that the 10 gravitationally‐conserved quantities defined in asymptotically flat, empty, space‐times are, when suitably modified, also conserved in asymptotically flat Einstein‐Maxwell space‐times. Furthermore, the implied selection rules for transitions between stationary Einstein‐Maxwell states are the same as those in the pure gravitational case.

On the Wigner Supermultiplet Scheme
View Description Hide DescriptionCalculation of Wigner and Racah coefficients for the group make it possible to perform the spin—isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one‐ and two‐body operators in the Wigner supermultiplet scheme. The SU(4) coefficients needed to evaluate one‐ and two‐particle cfp's, the matrix elements of one‐body operators, and the diagonal matrix elements of two‐body operators are calculated in general algebraic form for many‐particle states characterized by the SU(4) irreducible representations [yy0], [y y − 1 0], [yy1], [y11], [y y − 1 y − 1], [y10], [yy y − 1], [y00], and [yyy], whose states are specified completely by the spin and isospin quantum numbers (y = arbitrary integer). Applications are made to the calculation of the matrix elements of the complete space‐scalar part of the Coulomb interaction and the space‐scalar part of the particle‐hole interaction for nucleons in different major oscillator shells.

On the Renormalization of the Susceptibility of a Fermi Liquid
View Description Hide DescriptionIt is shown that the susceptibility of a normal Fermi liquid can be renormalized without using Ward identities for the derivatives of the mass operator with respect to the magnetic field. The procedure is completely analogous to the renormalization of the compressibility. The result which expresses the susceptibility in terms of Landau parameters is correct only to lowest order in temperature. The behavior of higher‐order terms in temperature is discussed.

New Formulation of Stochastic Theory and Quantum Mechanics
View Description Hide DescriptionThe theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker‐Planck‐Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum‐mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non‐Markoffian terms are briefly studied.

Potentials for Three‐Body Systems
View Description Hide DescriptionSimple expressions for the Coulomb, Gaussian, and harmonic‐oscillator potentials acting between pairs of particles in a three‐body system are developed. Each expression consists of an expansion in the S‐wave generalized angular‐momentum eigenfunctions for three particles.

On the Combinatorial Structure of State Vectors in U(n). II. The Generalization of Hypergeometric Functions on U(n) States
View Description Hide DescriptionThe derivation of the explicit algebraic expressions of the SU(n) state vectors in the boson‐operator realization is shown to lead to a generalization of hypergeometric functions. The SU(3) state vectors are rederived by the combinatorial method‐propounded in Paper I [J. Math. Phys. 10, 221 (1969)] of this series of papers‐and are shown to be represented by a hypergeometric distribution function and an associated generalization of the Young tableaux calculus. The SU(4) state vectors are also derived to demonstrate the main features of the general U(n) state vectors. The SU(4) state vectors are expressed in terms of the constituents of Radon transforms.

Critical‐Point Singularities of the Perturbation Series for the Ground State of a Many‐Fermion System
View Description Hide DescriptionWe show that the many‐fermion ground‐state energy with an attractive potential has a critical singularity. This singularity destroys the validity of ``low‐density'' approximations. We also find that the K‐matrix formalism is, in principle, not applicable to attractive potentials because of the presence of ``Emery singularities.'' We introduce an R‐matrix formalism which is numerically very close to the K matrix and free from manifest ``Emery singularities.'' A model calculation is performed on the lattice gas to try to anticipate what quality of results can be expected from summing an R‐matrix expansion with fixed density.

Properties of ``Quadratic'' Canonical Commutation Relation Representations
View Description Hide DescriptionA class of representations of the canonical commutation relations is studied, each of which is characterized by an expectation functional that is the exponential of a Euclidean‐invariant quadratic form of the test functions. The underlying field operators are realized as the direct product of two Fock representations and the consequences of this realization are analyzed. Compatible Hamiltonians are constructed and an extensive study of the most general quadratic Hamiltonians is presented. In order to include thermodynamic examples, the analysis includes indefinite Hamiltonian spectra as well as the usual definite spectra. Finally, conditions are given for a theory to be local in the sense that all time derivatives of the field operator commute with one another at equal times but unequal spatial arguments.

Linear Physical Chains with Sturm‐Liouville Characteristic Polynomials
View Description Hide DescriptionThis paper deals with the existence and construction of linear physical chains whose characteristic (secular) polynomials are essentially one of four classical types: Hermite, generalized Laguerre, generalized Bessel, or Jacobi. Each of the results is useful for determining the natural frequencies (normal frequencies or eigenvalues) of the systems involved. A chain of coupled harmonic oscillators or the corresponding electrical analog can be regarded as a prototype of the systems under consideration. Chains of both arbitrary finite order and infinite order are considered. Let N be a prespecified positive integer. Consider a finite sequence of linear dissipationless spring‐mass chains in which S_{n} consists of masses m _{0}, m _{1}, …, m_{n}−1 and springs with spring constants k _{0}, k _{1}, …, k_{n} , and in which S_{n}+1 is obtained from S_{n} by attaching mass m_{n} and spring with spring constant k_{n}+1. S_{n} is connected to a wall by the spring having spring constant k_{n} , but the chain may be free at the other end. In this case S _{1} is to consist only of mass m _{0} and spring with spring constant k _{1}. Three major results relating to such a sequence are obtained. First, given a positive integer N, a procedure is developed whereby an existing Nth‐order spring‐mass system can readily be tested to determine whether the characteristic polynomials φ_{1}, φ_{2}, …, φ_{ N } associated with chains S _{1}, S _{2}, …, S_{N} are all classical polynomials of a single type; and if so, which type. The testing procedure can also be extended to the infinite‐order case. Secondly, by means of large classes of examples, it is demonstrated that physical systems of any preassigned order N can actually be constructed so that for 1 ≤ n ≤ N the characteristic polynomials of S_{n} are all a specified one of the four classical types. Finally, it is shown that of the four possible kinds, only Jacobi‐ and Laguerre‐type infinite‐order systems can be generated; and the latter type can occur only if relatively stringent conditions on the physical parameters are satisfied.

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. I. Representations with Cyclic Field
View Description Hide DescriptionWe investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct‐(or tensor‐) product representations of fields. For partial tensor‐product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis‐independent integral defined by a supremum over all bases of the test function space can be replaced by an ordinary limit over a kind of diagonal sequence through finite‐dimensional subspaces of . In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.

Group Analysis of Maxwell's Equations
View Description Hide DescriptionIn this paper we write down and solve Maxwell's equations without sources when the field variables are considered as functions over the group SU _{2}. A Hilbert space is then constructed out of the field functions. An expansion of the field functions in terms of the matrix elements of the irreducible representation of SU _{2} is shown to reduce the problem of solving Maxwell's equations to that of solving one partial differential equation with two variables. A Fourier transform reduces this equation into an ordinary differential equation which is identical to the partial‐wave equation obtained from the Schrödinger equation with zero potential. The analogy between the mathematical method used in this paper in relation to the group SU _{2} and the Fourier transform in relation to the additive group of real numbers is pointed out.

Generating Functions of Classical Groups and Evaluation of Partition Functions
View Description Hide DescriptionThe generating functions of classical groups are used to set up recursion relations for their partition functions. These are then used to find the internal multiplicity structure of the weights using Kostant's formula.

Representation Theory of SP(4) and SO(5)
View Description Hide DescriptionThe basis states for all the irreducible unitary representations of Sp(4) are constructed by means of a calculus of boson operators. The Gel'fand states are explicitly expanded in terms of their constituent Weyl patterns. In terms of these states the matrix elements of finite rotations in five dimensions are obtained.

Nonlinear Electrodynamics and General Relativity
View Description Hide DescriptionA generalization of Born‐Infeld nonlinear electrodynamics, due to Plebanski, is reformulated in the context of general relativitytheory. A class of nonsingular, static, spherically symmetric solutions of the modified Einstein—Maxwell equations are given, corresponding to a point‐charge source. The metric tensors of these solutions are shown to approach the Riesner‐Nordstrom metric tensor at large distances from the source, if one makes the proper identification of mass.

General Form of the Einstein Equations for a Bianchi Type IX Universe
View Description Hide DescriptionThe Einstein equations for a general Bianchi type IX universe are presented in a form suitable for numerical solution. As an example, the complete equations for a cosmology with a pure fluid stress tensorT _{μν} = εu _{μ} u _{ν} + P(g _{μν} + u _{μ} u _{ν}) are also given.

Concerning the Zeros of Some Functions Related to Bessel Functions
View Description Hide DescriptionThe real zeros of the Riccati‐Bessel functions of their derivatives ,and of their cross products are investigated. Expansions analogous to those provided by McMahon and Olver for the zeros of the Bessel functions are obtained for the zeros of the derivatives of the Riccati‐Bessel functions. The analysis of Kalähne for the zeros of the cross product of Bessel functions is considerably expanded and analogous results are obtained for the zeros of the cross product of the derivatives of the Riccati‐Bessel functions. Included are derivations of the expansions for large zeros at fixed v, of asymptotic expansions for large v at fixed number of the zero, and also asymptotic expansions for the zeros as and Figures illustrating the behavior of the zeros are provided for v = l + ½, where l is an integer. These zeros correspond to the TE and TM electromagnetic normal modes inside a conducting spherical shell and in the region between two concentric conducting shells.

Conserved Quantities of Newman and Penrose
View Description Hide DescriptionGreen's theorem is used to obtain the Newman‐Penrose constants in flat and asymptotically flat space‐time and to investigate the invariant transformations which these quantities generate. The zero‐rest‐mass free fields and the coupled Einstein‐Maxwell fields are considered and the relationship of this approach to Noether's theorems is discussed.

One‐Dimensional Bonded Fluids
View Description Hide DescriptionA study is made of a linear lattice fluid in which bonds can form between pairs of molecules on second‐neighbor sites with an empty site between them. At low temperatures and pressures there is, thus, competition between local open configurations and more close‐packed local configurations of higher energy. By using a matrix expression for the grand partition function, the density and effective coordination number are found as functions of the absolute temperature T and one‐dimensional pressurep. Equivalent results are shown to follow from a constant‐pressure partition function based on an explicit expression for the configuration number. It is found that a pressurep _{0} exists such that for any p < p _{0} the open configuration is stable at T = 0, while the density passes through a maximum as T increases. For any p > p _{0}, on the other hand, the close‐packed configuration is stable at T = 0 and the density decreases monotonically with T. Using a constant‐pressure partition function, we also consider continuous models with the ``bonding'' represented by a potential well separated from the hard core. With a well of parabolic shape, similar results to those of the lattice model are obtained, while with a square well the curve of density against T displays a minimum as well as a maximum for any p < p _{0}.