Volume 5, Issue 8, August 1964
Index of content:

Inversion of Cyclic Matrices
View Description Hide DescriptionThe inversion of the overlap matrix in LCAO calculations of crystals is discussed for the general case, when an arbitrary number of neighbors is taken into account. Of the several alternative methods that are possible in one dimension, one is singled out—which can be used as well in three dimensions—to calculate the inverse to any desired accuracy. Particular attention is paid to the numerical aspects of the problem.

Ising‐Model Spin Correlations on the Triangular Lattice
View Description Hide DescriptionA Pfaffian representation, of the partition function of the triangular lattice is used to derive expressions for various two, four, and six spin correlations in terms of Pfaffians. The pair correlations along a diagonal are expressed as a Toeplitz determinant whose limiting form yields the spontaneous magnetization. At the ferromagneticcritical point the correlations decay as with approximately radial symmetry. At the antiferromagnetic zero point the ground state is highly degenerate—it has finite entropy—and on a given sublattice the pair correlations along a row decay as , where on the sublattice containing the origin spin and on the other two sublattices. Finally, the perpendicular susceptibility, , which depends on a finite number of correlations, is calculated; its ferromagnetic behavior is similar to that of the perpendicular susceptibilities of the quadratic and honeycomb lattices, but for an antiferromagnet diverges as 1/T at low temperatures.

Nested Hilbert Spaces in Quantum Mechanics. I
View Description Hide DescriptionA nested Hilbert space is a pair of Hilbert spacesH _{0}, H _{1}, each of which is in a certain sense identified with a dense subset of the other. These structures are used here to study analytic continuation into ``unphysical sheets'' and to discuss nonnormalizable states of quantum‐mechanical systems.

Finite and Disconnected Subgroups of SU _{3} and their Application to the Elementary‐Particle Spectrum
View Description Hide DescriptionAn attempt is made to fit the symmetries of the currently observed elementary‐particle spectrum into the structure of finite or disconnected subgroups of SU _{3}. Surprisingly, the detailed properties of these subgroups have not been elucidated previously. As a first step, therefore, character tables and other relevant properties are derived for these groups. Next, the classification of elementary particles is made on the basis of the representations of the groups discussed. The techniques previously employed by Case, Karplus, and Yang for the application of finite subgroups of SU _{2} to isotopic spin are extended to the subgroups of SU _{3}. The structure of SU _{3} is utilized to suggest how charge and hypercharge operators are to be assigned in the subgroups. The results obtained are similar to those of SU _{2} and isotopic spin. There is an upper limit, for any given group, to the dimension of the irreducible representation. For some of the groups considered, these upper limits are eight and even ten. There exist finite groups which can accommodate the eight baryons in one of the irreducible representations. However, when one looks at scattering problems, use of the finite groups, as expected, gives charge or hypercharge conservation only modulo an integer determined by the group. Charge independence is also lost. In a representative group analyzed in detail, the imposition of exact charge conservation leads automatically to the exact conservation of hypercharge and to the full SU _{3} symmetry. Exact charge and hypercharge conservation can be maintained for the disconnected groups, but the maximum dimension of the irreducible representations is six, and only charge symmetry, not charge independence, is satisfied. A short discussion of the representations of the group SU _{3}/C is included in the appendix.

On Pais's Charge Correlation Coefficients
View Description Hide DescriptionAn alternative proof of a theorem of Pais which states that the charge correlation coefficients for an n‐π system are independent of the row of the representation of the symmetric group is presented in which no detailed knowledge of the representations is used.

Renormalization of Singlet Amplitude in Intermediate‐Vector‐Boson Theory of Weak Interactions
View Description Hide DescriptionA simple prescription for renormalizing the singlet amplitude in the intermediate‐vector‐boson theory of weak interactions is given.

Upper and Lower Bounds on Generalized Fourier Coefficients
View Description Hide DescriptionMethods are given for obtaining variational upper and lower bounds on the nth Fourier coefficient of a function g relative to a sequence of eigenfunctions U_{n} . The methods differ in their ease of application and in the amount of information required concerning the eigenvalues associated with the U_{n} . Some illustrative examples are given.

Upper and Lower Bounds on the Matrix Elements of an Arbitrary Bounded Operator
View Description Hide DescriptionMethods are given for constructing variational upper and lower bounds on the matrix element of an arbitrary bounded operator W between two eigenstates U_{m} , U_{n} of a Hamiltonian H. Numerical examples of the method are given.

Many‐Channel Bargmann Potentials
View Description Hide DescriptionA generalization of the Bargmann potentials to include the case of nonrelativistics‐wave scattering of a particle by a target having a finite number of discrete excited states is presented. This generalization allows the explicit construction of a large class of many‐channel S‐matrices meromorphic on their energy Riemann surfaces as well as the explicit construction of the corresponding potential matrices. A two‐channel example is treated in detail.

Existence of Scattering Solutions for the Schrödinger Equation
View Description Hide DescriptionThe Fredholm alternative for the Schrödinger (Lippmann‐Schwinger) equation is established for potentials with finite first moment, using the Ascoli selection theorem and Banach space methods.

On Trapped Trajectories in Brownian Motion
View Description Hide DescriptionThe Brownian motion of an ion in the attractive field of an infinite line charge is investigated with reference to the existence of trapped trajectories, i.e., trajectories which never depart from the neighborhood of the center of attraction. It is shown that there is a discontinuity in the nature of the Brownian motion as a function of the line charge density, marked by the appearance of trapped trajectories when the line charge density becomes greater than a critical value.

The Representation of Canonical Variables as the Limit of Infinite Space Volume: The Case of the BCS Model
View Description Hide DescriptionWe examine the possibility of obtaining the representation suitable for a given Hamiltonian through a limiting procedure. The model employed is that of the Bardeen‐Cooper‐Schrieffer's superconductivity theory, for which the infinite‐volume limit of the Wightman functions gives the right representation. This method is found to be helpful in analyzing the so‐called symmetry breaking solutions. In particular, it serves for throwing some light to the Haag's work on the gauge property of the BCSmodel as well as to the Nambu's dynamical model of elementary particles. What is meant by the limit of infinite volume is studied with regard to the convergence of the Hamiltonian operator.

Linear Heisenberg Model of Ferro‐ and Antiferromagnetism
View Description Hide DescriptionThe partition function of the one‐dimensional Heisenberg model is considered. Hamiltonian of the systemis expressed in terms of Fermi operators. The term which contains J _{∥}, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked‐cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J _{∥}. Numerical values of energy, specific heat, and susceptibility up to second order in J _{∥} are shown. The ground‐state energy is.E/NJ _{∥} for the antiferromagnetic case J _{∥} = −J _{∥} = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.

On the Theory of Randomly Dilute Ising and Heisenberg Ferromagnetics
View Description Hide DescriptionThe Brout expansion for the free energy of the Ising or Heisenberg model is formally summed over all interaction graphs with not more than m vertices: the result is expressed in terms of the partition functions of isolated physical clusters, again having not more than m vertices. These partition functions are multiplied by occurrence factors closely related to the occurrence factors for the corresponding isolated physical clusters in a randomly dilute ferromagnet at low concentrations of the magnetic elements. Comparison is made with earlier work on the randomly dilute Ising and Heisenberg models.

Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions
View Description Hide DescriptionAn exact method for determining the critical percolation probability, p _{c}, for a number of two‐dimensional site and bond problems is described. For the site problem on the plane triangular lattice p _{c} = ½. For the bond problem on the triangular, simple quadratic, and honeycomb lattices, respectively. A matching theorem for the mean number of finite clusters on certain two‐dimensional lattices, somewhat analogous to the duality transformation for the partition function of the Ising model, is described.

On the Number of Self‐Avoiding Walks. II
View Description Hide DescriptionLet χ_{ n }(d) [resp. γ_{2n−1}(d)] be the number of self‐avoiding walks (polygons) of n (2n − 1) steps on the integral points in d dimensions. It is known that In this paper β_{ d } is compared with where χ_{ n,2r }(d) is the number of n‐step walks on the integral points in d dimensions with no loops of 2r steps or less. In other words the walks counted in χ_{ n,2r }(d) may visit the same point more than once as long as there are more than 2r steps between consecutive visits. It turns out that and it follows in particular that 1 It is also shown that, for suitable constants α_{6} = α_{6}(d) and α_{7} = α_{7}(d),.

Product Property and Cluster Property Equivalence
View Description Hide DescriptionUhlenbeck and Ford in their presentation of the Ursell development express a sequence of symmetric functions W_{N} in terms of another sequence of symmetric functions U_{N} . They invert this sequence of equations and remark that from the first set of equations it follows that the product property of W_{N} sequence is equivalent to the cluster property for the U_{N} sequence. They ask for a simple proof of the equivalence. Here a modification of an algebra developed by Bohnenblust to prove Spitzer's formula on the fluctuations of the sums of independent, identically distributed, random variables is used systematically to invert the relations and to prove the equivalence under lighter assumptions than previously used.

Kinetic Equation for an Inhomogeneous Plasma far from Equilibrium
View Description Hide DescriptionA kinetic equation for a nonuniform plasma is derived from the Liouville equation by a general diagram technique. It describes the evolution of a small spatial inhomogeneity in a plasma whose velocity distribution is far from equilibrium (and hence time‐dependent). The equation is valid for short and long times, within the ring approximation. Its explicit form is obtained by the exact closed solution of a singular integral equation. The kinetic equation is non‐Markoffian and, contrary to the corresponding homogeneous equation, keeps a trace of this character even in the limit of long times. Only when the velocity distribution and the two‐body correlation function reach thermal equilibrium does the equation reduce to a Markoffian limit. The latter is identical with the kinetic equation derived earlier by Guernsey. The treatment of unstable inhomogeneous plasmas is briefly indicated.

Note on the Evaluation of Some Fermi Integrals
View Description Hide DescriptionA method is illustrated for evaluating Fermi integrals by means of an integral representation for the derivative of the Fermi function. Several examples from free‐electron diamagnetism are discussed. A convenient representation for the free energy of a fermi gas is also derived.

Equation of State for a Gas with a Weak, Long‐Range Positive Potential
View Description Hide DescriptionA one‐dimensional fluid model in which the pair interaction potential is exponential and repulsive is considered, and the equation of state in the ``long‐range'' limit is determined exactly. This model is complementary to one studied by Kac, in which the pair potential consists of a hard core and an exponential, attractive tail. In that model a phase transition occurs, but in the current model there is no phase transition. This fact lends support to a conjecture of Ruelle that no phase transition occurs if the potential is bounded. The possibility of applying the method of Kac to a wider class of potentials is suggested, and some of the mathematical difficulties yet to be overcome are outlined.