Volume 9, Issue 10, October 1968
Index of content:

Unitary Representations of the Generalized Poincaré Groups
View Description Hide DescriptionThe problem of the explicit construction of unitary representations is solved for the generalized inhomogeneous Poincaré groups ISL(n, C) and their subgroups. As a key to the solution, a method is developed to find the generalized Wigner operator, which transforms a given n ^{2} momentum to the standard one. Results are also specified for the U(m, m) subgroup of the group ISL(2m, C).

Note on Statistical Mechanics of Random Systems
View Description Hide DescriptionWe discuss some equilibrium properties of random systems, i.e., systems whose Hamiltonian depends on some random variables y with a distribution P(y) which is independent of the dynamic state of the system. For a system of noninteracting particles which interact with randomly distributed scattering centers, the important quantity is the average density of states of a single particle per unit volume, n(E). Feynman's path‐integral formulation of quantum statistics is used to derive some properties of the average partition function for one particle , which is the Laplace transform of n(E). In particular, it is shown that is an analytic function of the density of scatterers ρ for a wide class of particle‐scattering center potentials V(r), including those with hard cores and those with negative parts. The analyticity in ρ of the equilibrium properties of these systems is very remarkable and is in contrast to the conjectured nonanalytic behavior of their transport (i.e., diffusion) coefficients. We find also upper and lower bounds on for a particle acted upon by a random potential V(r) obeying Gaussian statistics with . In the limit of ``white noise,'', is shown to diverge in two and three dimensions but remains finite in one dimension. This agrees with approximate results on the density of states. In appendices we prove the existence, in the thermodynamic limit, of the free‐energy density of a system with random scatterers and also of the frequency distribution and, thus, the free‐energy density for a random harmonic crystal.

Quantum Corrections to the Second Virial Coefficient at High Temperatures
View Description Hide DescriptionThe Laplace transform of exp (−βH) is the Green's operator of the negative‐energy Schrödinger equation (H + W)^{−1}. Conditions are stated under which a large W asymptotic series for the Green's operator can be inverse‐Laplace‐transformed term‐by‐term to obtain a small β expansion for exp (−βH). This approach and the Watson transformation are used to calculate the first few terms of high‐temperature asymptotic expansions for the exchange second virial coefficient for hard spheres and for the Lennard‐Jones potential. The known results for the direct second virial coefficient for hard spheres are extended. The Wigner‐Kirkwood expansion is calculated to order ℏ^{6} and used to calculate the direct second virial coefficient for the Lennard‐Jones potential through order ℏ^{6}.

Crossing, Hermitian Analyticity, and the Connection between Spin and Statistics
View Description Hide DescriptionThe analytic S‐matrix framework is further developed. First some results of earlier works are collected and the physical‐region analyticity properties recently derived from macroscopic causality conditions are described. These entail that scattering functions are analytic at physical points except on positive‐α Landau surfaces, and that there they are iε limits of analytic functions from certain well‐defined directions, except possibly at certain points where four or more positive‐α surfaces intersect. A general iε rule that also covers these exceptional points is then stated. It is then shown that the scattering function defined by analytic continuation is either symmetric or antisymmetric under interchange of variables describing identical particles and that the sign induced by the interchange is independent of the particular scattering function in which the variables appear. The physical‐region analyticity properties of bubble‐diagram functions are then derived from the general iε rule. These functions are products of scattering functions and conjugate scattering functions integrated over physical internal‐particle variables, as in the terms of unitarity equations. They are shown to be analytic in the physical region except on Landau surfaces and, more specifically, except on those Landau surfaces that correspond to Landau diagrams that are supported by the bubble diagram in question, with the further restriction that the Landau α's must be positive or negative for lines lying within positive or negative bubbles, respectively. Also, the basic rule for continuation around these singularities is derived. A new general derivation of the pole‐factorization theorem is given, which is based on slightly weaker assumptions than earlier proofs. Particular attention is paid to the over‐all sign. A general derivation of the crossing and Hermitian analyticity properties of scattering functions is then given. On the basis of the deduced general rules for constructing the paths of continuation that connect the crossed and Hermitian conjugate points, the various related points are found to be boundary values of a single physical sheet. In particular, a certain sequence of continuations is shown to take one back to the original point. From this fact it follows that abnormal statistics are incompatible with simultaneous unitarity in both the direct and crossed channels. The proof given here does not depend on the notion of interchange of variables other than those of identical particles. Earlier proofs depended on the unphysical notion of interchange of variables representing conjugate particles. Finally it is shown that the analytically continued M functions with normal‐ordered variables are precisely the scattering functions: no extra signs are needed or permitted. Aside from the general i rule, the analyticity assumptions are these: (1) The discontinuity around a singularity of a bubble diagram B has no residue at a physical‐particle mass value (in an appropriate variable) unless the singularity corresponds to a diagram that is supported by B and has the single‐particle‐exchange form that corresponds to a pole at that mass value. (2) The residue just described has the pole‐factorization property. (3) Confluences of infinite numbers of singularity surfaces do not invalidate the results established by assuming that this number is locally finite. Assumption (1) entails that all relevant singularities of scattering functions lie on Landau surfaces. That is the basic assumption.

TCP Invariance and the Dimensionality of Space‐Time
View Description Hide DescriptionThe original proofs of the TCP theorem suggest that it may breakdown in an odd‐dimensional space‐time. An explicit example of this breakdown is given and a comment is made about the current × current interaction.

Null and Pseudonull Data for Scalar Fields
View Description Hide DescriptionIn this paper, a projective geometric formalism for the description of the conformal compactification of Minkowski space and of its invariance groups is developed. It is then modified to include the description of Minkowskian vectors, in particular, the momentum space. The d'Alembert equation is solved by constructing its solutions from global null data, completely arbitrary numbers assigned to the null cone at infinity. The Klein‐Gordon equation, solved by the same method, leads to the concept of pseudonull data. Pseudonull data are also arbitrary numbers, but they are assigned to hyperboloids at a suitably defined infinity outside the conformal compactification of the Minkowski space.

Green's Functions for the Ising Chain
View Description Hide DescriptionThe Ising chain is represented as a system of fermions interacting through two‐particle forces. All thermodynamicGreen's functions are calculated exactly in this model. They exhibit many properties expected on the basis of general theory for Green's functions of the interacting particles.

Possible Kinematics
View Description Hide DescriptionThe kinematical groups are classified; they include, besides space‐time translations and spatial rotations, ``inertial transformations'' connecting different inertial frames of reference. When parity and time‐reversal are required to be automorphisms of the groups, and when a weak hypothesis on causality is made, the only possible groups are found to consist of the de Sitter groups and their rotation‐invariant contractions. The scheme of the contractions connecting these groups enables one to discuss their physical meaning. Beside the de Sitter, Poincaré, and Galilei groups, two types of groups are found to present some interest. The ``static group'' applies to the static models, with infinitely massive particles. The other type, halfway between the de Sitter and the Galilei groups, contains two nonrelativistic cosmological groups describing a nonrelativistic curved space‐time.

Unitary Irreducible Representations of SU(2, 2). II
View Description Hide DescriptionPaper II of this series [Paper I in J. Math. Phys. 8, 1931 (1967)] is concerned with a general study of the degenerate representations. The explicit expressions for the ``raising'' and ``lowering'' functions , and , i = 1, 2, 3, 4 are found. The three Casimir operators C _{2}, C _{3}, and C _{4} depend on only two complex parameters A and B, a fact reflecting the degenerate nature of the representations under study here. The finite representations are studied first, and thus provide a proof for the degenerate part of Theorem 2, Paper I. The unitary representations are studied next, and we find that there are fourteen classes of degenerate unitary irreducible representations. There are two continuous series, ten discrete series, and two series which depend on one discrete and one continuous parameter. The degenerate part of the D ^{±} series is studied, and thus provides an explicit demonstration of Harish‐Chandra's theorem.

Inverse Functions of the Products of Two Bessel Functions and Applications to Potential Scattering
View Description Hide DescriptionInverse functions of products of two Bessel functions j_{l} (xy)j_{m} (xy) are determined for the cases m = l, l + 1, and l + 2. Integral representations for these inverse functions in terms of Neumann functions are given, and some of the simplest ones are expressed in terms of trigonometric functions.
We show how one may obtain an integral representation for any well‐behaved function in terms of products of two Bessel functions, with the help of these inverse functions and also outline some of their applications to potential scattering. In particular, we demonstrate the usefulness of the inverse functions in determining the potential explicitly from the phase shifts in the Born approximation.

High‐Energy Fixed‐Angle Potential Scattering
View Description Hide DescriptionWe continue the study of the asymptotic behavior of the scattering amplitude f(k, θ) at high‐momenta k for fixed (nonforward) scattering angles θ. The considerations here are limited to the class of potentials for which the momentum‐space representations V(q) decrease less rapidly than some inverse power of q as q → ∞ through positive real values. We place some additional relatively simple conditions upon V(q) for q > 0 which are sufficient to guarantee that the asymptotic limit of f for large k with fixed θ is, in fact, the first Born approximation f _{1}(k, θ).

Limitable Dynamical Groups in Quantum Mechanics. I. General Theory and a Spinless Model
View Description Hide DescriptionA pure group‐theoretical description of nonrelativistic interacting systems in terms of irreducible representations U(D) of a so‐called dynamical group D is investigated. The description is assumed to be complete in the sense that all observable quantities of the system can be calculated from U(D) in the same way as the nonrelativistic free particle can be identified with an irreducible representation U(G_{E} ) of the central extension of the inhomogeneous Galilei group G_{E}. D depends on the interaction. It is a noninvariance group and it contains a spectrum‐generating algebra. Our problem is to connect a representation of an arbitrary abstract group with a complete description of an interacting system. This needs some physically motivated principles. Some such principles are proposed. We assume that the interaction can be turned off, which implies that U(D) and the physical representation U(G_{E} ) of the free‐particle group G_{E} can be limited into each other. If this limitation can be formulated without violating super‐selection rules, i.e., mass and spin conservation in nonrelativistic systems, the group D^{t} is called a limitable group. Properties of these groups are derived. An explicit construction of a limitable D^{t} is given by embedding the free‐particle group G_{E} into a larger group. A discussion of all embeddings leads to the special choice. is the central extension of the pure inhomogeneous Galilei group in N dimensions and Sp(2N, R) the noncompact real form of the symplectic group. A representation theory for D^{t} is established using the technique of Nelson extensions, together with some properties of the universal enveloping algebra of the Lie algebra of . Our main success is that D^{t} is a limitable dynamical group and that the physical system described by D^{t} and the physical representation can be calculated uniquely from the proposed principles. The group‐theoretical description is equivalent to nonrelativistic quantum mechanics for a spinless particle in N dimensions with an arbitrary second‐order polynomial in P_{i}, Q_{i}, i = 1, ⋯, N as Hamiltonian. The possibility of further models is discussed.

Almost Symmetric Spaces and Gravitational Radiation
View Description Hide DescriptionA generalization of the idea of Killing fields to spaces which are not symmetric is given. The field so defined specifies coordinate lines along which the variation of the metric tensor is the slowest possible in a global sense. Thus it generalizes the Killing fields in spaces with a symmetry where the metric tensor does not change along Killing trajectories. Several examples are given, and the method is then applied to spaces containing gravitational radiation of the type considered by Issacson.
For spaces containing radiation, it is shown that a real functional λ[ξ], associated with every vector field ξ, measures some parameters associated with the radiation. In the simplest case this parameter is the ``energy density'' of the radiation, but, if a sufficient number of vector fields can be invariantly defined in the background, the average gravitational ``stress'' associated with the wave may also be measured. We conclude with some conjectures about further application of these ideas to the theory of gravitational radiation.

Multiplicity Problem for Compact Subgroups of Noncompact Groups
View Description Hide DescriptionA method is shown for calculating the multiplicities occurring in the reduction of a reducible representation of a compact group. The reducible representation is obtained by constructing induced irreducible representations of a noncompact group and then restricting the elements of this representation to elements of the compact subgroup.

Exact Occupation Kinetics for One‐Dimensional Arrays of Dumbbells
View Description Hide DescriptionBy utilizing a branch‐dependent counting technique, exact relationships are developed which describe the ensemble average of the kinetics of occupation by dumbbells of finite one‐dimensional arrays of compartments. It is shown that, as the number of compartments per array tends to infinity, , the ensemble average of the fraction of an array which is occupied, is given by,where v is the striking frequency and t is time. By contrasting these results with the statistics of one‐dimensional arrays of dumbbells, it is demonstrated that it is inappropriate to employ classical statistical‐mechanical methods (e.g., the Bethe approximation) to treat the kinetic aspects of occupation where configurational correlation exists. (Here we define configurational correlation to be the situation in which the occupation of a compartment precludes the occupation of one of its nearest neighbors.)

Crossing and Unitarity in a Multichannel Static Model. I
View Description Hide DescriptionThe infinite multichannel‐scattering process of the two‐particle scattering of mesons of arbitrary isospin off an isospin‐½ target, inelastic two‐particle channels being admitted, is considered in the static limit. The restrictions on the S matrix arising from crossing under SU(2) and two‐particle unitarity are expressed in terms of three equations alone in the limit of degenerate meson masses. These equations are solved by a perturbative technique, the expansion parameter being a measure of the strength of the inelastic scattering. The principal result of this paper is that the inelastic amplitudes are all related, so that if one is zero all must be zero. Comparisons are made with strong‐coupling theory, which also requires an infinite number of channels.

Poincaré Group and the Invariant Relativistic Equations for Massive Particles of Any Spin
View Description Hide DescriptionThe aim of this paper is to clarify some aspects of the connection between the Poincaré group and the invariant equations for nonzero‐mass particles of any spin (Bargmann‐Wigner equations). With this purpose we first make some general considerations about the representations of the Poincaré group and analyze the equivalence between the realizations corresponding to a given class and a selected ``canonical'' one which was given by Wigner. For the spin‐½ case the equivalence is given by the Chakrabarti transformation, and for higher spins we introduce a generalization of it; we also consider specifically the case S = 1. We give the form of the equations which provide the corresponding canonical realization; some comments about the equivalence of the theories provided by the different realizations are also made.

Retarded and Advanced Electromagnetic Fields in Friedmann Universes
View Description Hide DescriptionMaxwell's equations of classical electrodynamics are solved in the framework of the curved space‐time of the Friedmann universes. Explicit formulas are given for the retarded and advanced fields in terms of the four‐current.

Electro‐Optical Effects. II
View Description Hide DescriptionThe constitutive equations governing the propagation of electromagnetic waves of small amplitude in a centro‐symmetric isotropic material to which a static electric field is applied involve, in general, six scalar functions of the static field strength and wave frequency. It is shown that if the material is nondissipative, four of these functions are real, while the remaining two are complex conjugates. Conditions are also derived such that the material shall be nonabsorptive with respect to plane‐electromagnetic waves and it is seen that a nonabsorptive material is not necessarily nondissipative, but is necessarily stable. It is shown that for a stable, nondissipative material, there are four real velocities corresponding to any direction of propagation. If it is assumed that these are neither zero nor infinite for any direction of propagation, then for each direction of propagation two of these velocities are positive and two are negative.

Modified Admittance, Perpendicular Susceptibilities, and Transformation of Correlations of the Ising Model
View Description Hide DescriptionA modified admittance is introduced to give Kubo's admittance at nonzero frequencies and to give the isothermal static admittance at zero frequency within the scope of the Kubo linear‐response theory. The method is demonstrated by exact calculations of the frequency‐dependent perpendicular susceptibility at zero field and its modified susceptibility of the regular Ising model. The results appear as linear combinations of the equilibrium spin‐spin correlation functions of the lattice. The results are valid for all dimensions and all frequencies and temperatures. A (q + 1) × (q + 1) matrix a ^{(q)} describes the linear combinations explicitly, where q is the coordination number of the lattice. The properties of this matrix are extensively discussed as a special case of a matrix A ^{(q)}(ξ), which satisfies a simple quadratic equation of the form [A ^{(q)}(ξ)]^{2} = (1 + ξ)^{ q } for arbitrary values of ξ. Fisher's algebraic transformation of the spin‐spin correlation functions for the regular Ising lattice is derived from the linear relation which holds between the perpendicular susceptibility and the corresponding modified susceptibility. By means of the product rules of the matrix A ^{(q)}(ξ), the higher‐order spin‐spin correlation functions are expressed in terms of the lower‐order ones in complete generality.