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Volume 11, Issue 6, June 1970

Energy Spectrum According to Classical Mechanics
View Description Hide DescriptionThe phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ‐function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz‐Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spectrum of Balmer terms with appropriately reduced degeneracy. However, the orbits which penetrate the screening charge give discrete levels obeying the Bohr‐Sommerfeld conditions with the correct degeneracy. The continuous background in the approximate density of states can be discussed on the basis of the formulas derived in this paper. This is necessary as an introduction to the problem of a particle in a potential where the motion is not multiply periodic.

Further Generalization of the Generalized Master Equation
View Description Hide DescriptionFor a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced‐density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non‐Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time‐evolution equation for the reduced distribution function ρ_{ r }(t, w _{ r }, J), depending on the subset w _{ r } of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.

Generalized Group‐Theoretical Analysis of Spontaneous Symmetry Breaking
View Description Hide DescriptionA group‐theoretical analysis of spontaneous symmetry breaking is carried out in an extension of Glashow's work. A general theorem is proved which is then used to give several interesting results. Apart from a rederivation of Glashow's result, we are able to show, among other things, the following: If a hadronic triplet exists in nature which is nontrivially coupled to the rest of the hadrons, then octet enhancement immediately follows independently of any dynamical detail; in models containing the vector nonet, φ ‐ ω mixing can, in principle, occur as a particular form of spontaneous symmetry breaking in the octet pattern; and if octet enhancement holds, the very accurate mass formula is established without recourse to any dynamical detail. Under the same general assumption, a further relation is found relating the mass of the nonet with the mixing angle, which is .

Current Algebras, the Sugawara Model, and Differential Geometry
View Description Hide DescriptionThe Lie algebra defined by the currents in the Sugawara model is defined in a way that is natural from the point of view of Lie transformation theory and differential geometry. Previous remarks that the Sugawara model is associated with a field‐theoretical dynamical system on a Lie groupmanifold are made more precise and presented in a differential geometric setting.

Numerical Analysis of an Integral Equation with a δ Function in the Kernel
View Description Hide DescriptionAn integral equation which has a two‐part kernel, where one part contains a δ function, is analyzed with respect to its analytic structure in the λ plane and with respect to numerical approximations to it. The analytic structure of the approximate solution is also investigated. It is found that there is no difficulty in approximating the Dirac δ function by a Kronecker δ; even though the approximate kernel does not approach the true kernel, the solution corresponding to the approximate kernel does approach the true solution. In keeping with the fact that the kernel does not meet the Fredholm conditions, we find that the solution has branch cuts in the λ plane. A form for the solution analogous to Fredholm's solution which emphasizes the analytic structure (i.e., a branch cut) is obtained.

Interpretation of the Symmetry of the Clebsch‐Gordan Coefficients Discovered by Regge
View Description Hide DescriptionAn interpretation is proposed for the intriguing symmetry of the Clebsch‐Gordan coefficients discovered in 1958 by Regge. The interpretation is based on the observation that, in the reduction of the Kronecker product of two irreducible representations of an SU(2) group, there appears in a natural way another SU(2) group, which is almost independent of the original one. The Regge symmetry is interpreted as the symmetry under the interchange of these two SU(2) groups. In more picturesque language, the Regge symmetry is the symmetry under the interchange of the ``two‐ness'' of the two angular momenta being added with the ``two‐ness'' of SU(2). It follows from this interpretation that a symmetry of the same nature is present in the generalized Clebsch‐Gordan coefficient that appears in the reduction of the Kronecker product of n (and not two, except when n = 2) irreducible representations of SU(n).

Properties of a Simplified Liquid ^{4}He Ground State
View Description Hide DescriptionA many‐body wavefunction is constructed by antisymmetrization, with respect to the electron variables, of a product of ^{4}He‐atom wavefunctions, each in its individual ground state. This wavefunction is shown to have the following properties: (1) The pair distribution functionD(R_{ij} ) vanishes at zero separation R_{ij} = 0; part of the mutual repulsion of ^{4}He atoms is built into the wavefunction as a result of the exclusion principle acting between the electron shells on different atoms. (2) Neither the nuclei nor the whole atoms undergo Bose condensation, but there is a ``Fermi condensation'' similar to that shown by Yang to characterize superconductivity, arising from off‐diagonal long‐range order of the appropriate reduced density matrix.

Spin Representations of the Orthogonal Groups
View Description Hide DescriptionThe spin representations of the algebrasB_{n} and D_{n} are constructed from the spin representations of the canonical subalgebras B _{ n−1} and D _{ n−1}, respectively.

Spectrum of Casimir Invariants for the Simple Classical Lie Groups
View Description Hide DescriptionThe spectra of the Casimir invariants for the classical compact simple Lie groups are presented. It is proved that these invariants are irreducible and functionally independent. The highest weight of a representation is determined in terms of its invariant spectrum.

Transmission Properties of an Isotopically Disordered One‐Dimensional Harmonic Crystal. II. Solution of a Functional Equation
View Description Hide DescriptionThe amplitude of a wave of frequency ω which is transmitted by a disordered array of N isotopic defects in a 1‐dimensional harmonic crystal is investigated in the limit N → ∞. In particular, the ratio T̂_{N} (ω) of the amplitude of the Nth defect to the amplitude of the first defect is represented as , where {a_{n} }, n = 2, ⋯, N, is the sequence of nearest‐neighbor spacings and Q = (M − m)/m. It is known from earlier work that is the logarithm of the Nth root of the magnitude of a continuant determinant of order N. The value of the continuant is expressed formally as a product of N factors ĝ_{n} which are recursively related. In the present case, the ĝ_{n} happen to lie on a circle K _{0} in the complex ĝ plane. Assuming that the spacings between defects are independent identically distributed random variables with the mean value c ^{−1} and going to the limit N → ∞, a functional equation for the limiting distribution function of the ĝ_{n} on K _{0} is derived. The limiting value , as N → ∞, can be determined from the limiting distribution function of the ĝ_{n} . We determine the solution of the functional equation in three different ways for three different cases: (a) In the case of the special frequency of Matsuda, ω = 2^{−½} and Q = 1, we obtain exact values of the integral of the ĝdistribution function which are in excellent agreement with Monte Carlo estimates; (b) in the physically interesting case where the mean spacing between defects is small compared to the incident wavelength, i.e., c ^{−1}ω ≪ 1, we obtain the solution of the functional equation correct to first order in c ^{−1}ω and we calculate the lowest‐order nonzero value of α(ω, Q, c); (c) for the general case of moderate values of ω, Q, and c, we develop a numerical method for solving the functional equation and present the results of the numerical calculations in several representative cases. These numerical results are in good agreement with Monte Carlo estimates. One of the principal results, obtained by solving the functional equation, is that α(ω, Q, c) > 0 for ω ≪ c < 1 and ω(Q + c ^{−1}) ≪ 1 with Q ≠ 0.

Unbounded Local Observables in Quantum Statistical Mechanics
View Description Hide DescriptionThe algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*‐algebra of quasilocal bounded observables in Fock space , an arbitrary, locally normal state φ on , and a corresponding Gel'fand‐Naimark‐Segal (GNS) representation R _{φ} of in a Hilbert space. We then construct a set of local, closed operators whose domains are dense in , such that includes all the local observables of the system. The representation R _{φ} is then extended so as to provide a *‐homomorphism of into the closed, densely defined operators in . Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo‐Martin‐Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space‐correlation functions, and the temporally ergodic properties of time‐correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of are thermodynamical limits of the corresponding correlations for finite systems.

Short Proof of a Conjecture by Dyson
View Description Hide DescriptionDyson made a mathematical conjecture in his work on the distribution of energy levels in complex systems. A proof is given, which is much shorter than two that have been published before.

Degeneracy of the SU(3) Direct Product and the Symmetric Representations of
View Description Hide DescriptionThe relation between a state of U(n) associated with an m‐rowed Young diagram, m ≤ n, and a state of U(m) associated with an m‐rowed Young diagram provides the basis for the symmetric state of . As an application, the state vectors associated with the irreducible representations of SU(6) restricted to the subgroup are explicitly constructed for the symmetric representation, and generalized to the case of a 2‐rowed Young diagram. Expressions are given for the degeneracy of an state in SU(6), and the completeness of the obtained set of states is discussed. The direct product of symmetric states implies a direct product of SU(3) states; the operator which breaks the degeneracy of the ``2‐rowed'' state is shown to be Moshinsky's operator X which breaks the degeneracy of the SU(3) direct product.

Eikonal Approximation for Nonlinear Equations
View Description Hide DescriptionAn eikonal approximation for nonlinear equations is derived from an expansion in powers of space and time derivatives. For the special case of one dependent variable, the method is equivalent to an averaging method proposed by Whitham and derived by Luke. A general solution for each order of the expansion is obtained and discontinuous solutions are discussed.

Quantum Theory of the Generalized Wave Equations. I
View Description Hide DescriptionWe have made a systematic analysis of the quantum theory of the infinite‐component fields that transform under the combined representations of . A complete set of solutions of the wave equation includes solutions with timelike and spacelike momenta. We have explicitly calculated the mass spectra for the timelike and spacelike cases. Our method makes use of the decomposition of the product representation into reducible representations of the ``little'' groups SU(2) and SU(1, 1). Finally, the quantization of the generalized fields is presented.

Convolution Approximation for the n‐Particle Distribution Function
View Description Hide DescriptionThe convolution approximation introduced by Jackson and Feenberg for the 3‐particle distribution function is extended to the n‐particle distribution functiong_{n} . It is shown that the generalized convolution form satisfies the limiting and the recursion relations connecting g_{n} and g _{ n+1}.

Infinite‐Dimensional Representations of the Lorentz Group
View Description Hide DescriptionThe method used by Naimark to obtain symmetrical spinors and their transformation law from finite‐dimensional representations of the group SL(2, C) is extended to infinite‐dimensional representations. As an infinite‐dimensional representation, we use the principal series of representations realized by means of the special unitary groupSU _{2}. As a result another form of the principal series of representations of SL(2, C) is obtained which describes the transformation law of an infinite set of numbers under the group transformation in a way which is very similar, but as a generalization, to the way spinors appear in the finite‐dimensional case.

Instantaneous Interaction Relativistic Dynamics for Two Particles in One Dimension
View Description Hide DescriptionDifferential conditions which guarantee the Lorentz invariance of instantaneous action‐at‐a‐distance relativistic dynamics have been given by Currie and by Hill. The present paper obtains the general solution of these conditions for the special case of two particles in one dimension. The resulting equations of motion are integrated to obtain the world lines. World‐line invariance is explicitly demonstrated. The equations of motion are cast into Hamiltonian form with the transformations of the inhomogeneous Lorentz group canonical. The Hamiltonian formulation is made unique up to canonical transformation for those forces which fall off faster than the inverse square of the interparticle separation by the demand of asymptotic reduction to free particle form. The special case of the inverse‐square‐law forces of electrodynamics is considered; the constant of the motion associated with Lorentz invariance is found to have an interaction piece which survives asymptotically as in the relativistic mechanics of Van Dam and Wigner. The Poisson bracket [x _{1}, x _{2}] between the physical coordinates also has an interaction piece which survives asymptotically for electrodynamics.

Green's Function for the Nonlocal Wave Equation
View Description Hide DescriptionThe Green's function for the nonlocal wave equation in a semi‐infinite medium is calculated using the Wiener‐Hopf technique and shown to be given by an integral whose integrand contains only the source‐free solution of that same equation.

Scattering in General Higher Sectors of the Lee Model
View Description Hide DescriptionThe technique of an iterative expansion in terms of the 2‐body solution used in V−2θ sector of the Lee model has been extended to the general higher sector, i.e., V−nθ sector with n > 2 so that S matrices for the general cases of scattering (n − 1)θ particles off the (Vθ) bound state can be calculated. Again, as in V−2θ sector, each term of the expansion preserves the properties of the (Vθ) bound state, the analytic structure, and symmetries of the τ functions.