Volume 5, Issue 7, July 1964
Index of content:

Convergence of Virial Expansions
View Description Hide DescriptionSome bounds are obtained on R(V), the radius of convergence of the density expansion for the logarithm of the grand partition function of a system of interacting particles in a finite volume V, and on R, the radius of convergence of the corresponding infinite‐volume expansion (the virial expansion). A common lower bound on R(V) and R is 0.28952/(u+1)B, where [so that u ≥ 1, with equality for nonnegative φ(r)], , and φ(r) is the binary interaction potential; the irreducible Mayer cluster integrals have the related upper bounds , when φ(r) ≥ 0]. For potentials with hard cores the maximum density is an upper bound on R(V), though possibly not on R; an example shows how both R(V) and R can be less than the maximum density, even if there is no phase transition. A theorem is proved, analogous to Yang and Lee's theorem on uniform convergence in the complex z plane, defining a class of domains in the complex ρ plane within which the operations V → ∞ and d/dρ commute. This theorem is used to show that , and that there is no phase transition for 0 ≤ ρ < 0.28952/(u + 1)B.

An Algebraic Approach to Quantum Field Theory
View Description Hide DescriptionIt is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors H_{1} and H_{2} are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.

Note on Wigner's Theorem on Symmetry Operations
View Description Hide DescriptionWigner's theorem states that a symmetry operation of a quantum system is induced by a unitary or an anti‐unitary transformation. This note presents a detailed proof which closely follows Wigner's original exposition.

Zero‐Mass Representations of the Proper Inhomogeneous Lorentz Group
View Description Hide DescriptionThe representations of the proper inhomogeneous Lorentz group are investigated as a function of both real and imaginary mass, in the limit as the mass approaches zero. One obtains only the physical mass‐zero representations in either limit, the infinite spin representations being unique to zero mass. It is found that there exists a superselection rule which prohibits the position and spin operators from being physically observable in the mass‐zero limit.

Continuous‐Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics
View Description Hide DescriptionA rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase‐space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L ^{2}(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.

Uniqueness of 4‐ and 8‐Dimensional Spaces
View Description Hide DescriptionAmong rotation groups R_{n} , the cases n = 4 and 8 are unique in having two inequivalent n × n representations. Mathematically this is related to the uniqueness of quaternions and octonions; physically these groups seem to underlie the real and charge‐space symmetries of elementary particles. An attempt is made to interpret this fact by assuming a lack of inherent geometrical preference between Fermi‐Dirac and Bose‐Einstein statistics. Corollaries are the identity of real and charge‐space statistics and the complete disjointness of real and charge‐space coordinates.

Lower Bounds on the Lehmann Weights in Spin‐Zero Meson Theory
View Description Hide DescriptionIt is shown within the framework of conventional spin‐zero meson theory that, if all renormalizations are assumed finite, the Lehmann weights of the fermion and mesonGreen's functions cannot decrease arbitrarily rapidly as a function of energy.

Lagrangian and Hamiltonian Formalisms with Supplementary Conditions
View Description Hide DescriptionThe present note generalizes a transformation due to Takahashi. The Takahashi transformation introduces a Hamiltonian formalism in the presence of a single linear supplementary condition imposed on n + 1 coordinates. The Takahashi transformation is generalized to treat the case when there are N independent linear supplementary conditions imposed on the differentials of n coordinates. It is shown that the generalized transformation leads to true coordinates when the coefficients of the differentials are coordinate‐independent; otherwise the transformation generally leads to quasicoordinates. More general transformations are discussed, the relation with the method of Lagrange multipliers is established, and some problems which arise in connection with quantization are pointed out.

Singular Bethe‐Salpeter Scattering Amplitudes
View Description Hide DescriptionThe Bethe‐Salpeter equation for scattering is investigated in configuration space for the class of singular ``potentials'' (i.e., a λ ^{4}theory in the ladder approximation) which behave as r ^{−4} near the light cone. The discussion relies on the similarity between solutions of the Bethe‐Salpeter equation and the Schrödinger equation where the corresponding problem is scattering by a r ^{−2} potential. Through a consideration of the asymptotic properties of the two‐particle, free Green's function, the elasticscattering amplitude is shown to be the coefficient of the outgoing wave part of the wave‐function, just as it is in the nonrelativistic case. At zero total energy, it is just the coefficient of e ^{−mr }/r ^{½}. The differential form of the Bethe‐Salpeter equation is expanded in four‐dimensional spherical harmonics, and the singular part of the potential is incorporated into the differential operator. The resulting equation is formally solved by converting it into what is now a Fredholm integral equation. Care is taken to choose the proper asymptotic behavior for the solutions of the new equation. The discussion of the singular potential is carried out at zero total energy in order to obtain spherical symmetry. The technique for handling the singular potential and extracting the T matrix at zero energy is demonstrated by application to two examples. The exact scattering amplitude is found for exchange of two massless mesons. A first‐order solution is obtained for a phenomenological potential that approximate the exchange of two massive mesons. This solution exhibits many of the features expected from a truly physical potential.

On the Canonical Relativistic Kinematics of N‐Particle Systems
View Description Hide DescriptionThe ``canonical'' relativistic kinematics is discussed for n‐particle systems. Our aim is to derive the formulas in as simple and symmetrical a way as possible and thus to avoid the extra complications arising for n > 2 in the usual stepwise generalization of the two‐particle method. At first we derive the canonical form of the infinitesimal operators (N, M) in a highly symmetrical form. It is then shown that though the restrictions imposed by the condition that our states be also energy eigenstates compel us to sacrifice a part of the symmetry and simplicity of the formulas, we can indeed include the effect of the spins of the component particle in a completely symmetric way. This reduces to a minimum the additional complications introduced when the component particles have nonzero spins. The corresponding, relatively simple, generalized (C‐G) coefficients connecting the canonical and the direct‐product states are calculated. It is shown that the use of ``spinor'' representation for the individual particles simplifies the deductions considerably. Explicit results are given usually for n = 3 only, since the generalization to n > 3 introduces no essentially new features.

Momentum Distribution in the Ground State of the One‐Dimensional System of Impenetrable Bosons
View Description Hide DescriptionGirardeau has shown that an exact analytical formula may be given for the ground‐state wave‐function of a system of one‐dimensional impenetrable bosons. Starting with this formula, we give a mathematically rigorous analysis leading to the determination of major features of the momentum distribution in the limit of an infinitely large system.

Correlation Functions and the Critical Region of Simple Fluids
View Description Hide DescriptionThe ``classical'' (e.g. van der Waals) theories of the gas‐liquid critical point are reviewed briefly and the predictions concerning the nature of the singularities of the coexistence curve, the specific heat, and the compressibilities are compared critically with experiment and with the analytical and numerical results for lattice gas models.
The critical singularities are related to the behavior of the pair correlation functionG(r) = g(r) − 1 and the Ornstein‐Zernike theory of critical scattering is reviewed. Alternative derivations of the theory are discussed and its validity is assessed in relation to experiment and to more detailed theoretical calculations. The nature and magnitude of the expected deviations from the ``classical'' theory are described. The analogies with critical magnetic phenomena are mentioned briefly.

On the Stability of Flow of a Thermally Stratified Fluid under the Action of Gravity
View Description Hide DescriptionThe equation for the small disturbances from the plane‐parallel flow of a thermally stratified fluid under the influence of gravity acting perpendicular to the plane of stratification is derived. It was found necessary to include not only viscosity but also heat conductivity to preclude the resulting differential equation from having a singularity. Asymptotic solutions of the sixth‐order differential equation thus derived are obtained. They show the presence of a Stokes point. The limiting form of the differential equation near the Stokes point is next obtained and an exact solution of this equation is derived by means of a Laplace transformation. In the general case the integrand of the Laplace transformation involves Whittaker's confluent hypergeometric functions. In the special case of a Prandtl number of 1, the integrand is considerably simpler and for this case asymptotic representations of the solutions on both sides of the Stokes point have been derived from the Laplace transformation solution by the method of steepest descent. The connection formulas between the solutions are the same as that previously derived by Tollmien and Lin for the case when stratification, gravity, and heat conduction are neglected.

Bergman's Integral Operator Method in Generalized Axially Symmetric Potential Theory
View Description Hide DescriptionThis paper contains a study of properties of solutions to the equation of generalized axially symmetric potentials. These potentials play an important role in many aspects of mathematical physics, in particular to an understanding of compressible flow in the transonic region. The ideas that have been basic in this investigation are contained in the integral operator method of Bergman. This method allows one to transplant certain properties of analytic functions to the solutions of linear partial differential equations. Results are obtained concerning singularities, residues, bounds, and growth of entire solutions, which are analogous to those found in classical functiontheory.

Fluctuations in Multiple Capture Processes
View Description Hide DescriptionThe fluctuation in the number of many‐fold captures by an element exposed to a high neutron flux is computed. It is found that the fluctuations are identical to those that would be obtained by selecting nuclei at random from an infinite supply with the appropriate average composition.

Errata: Excitation Spectrum of an Impurity in the Hard‐Sphere Bose Gas
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