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Volume 6, Issue 9, September 1965

Reduction of Representations of SU_{mn} with respect to the Subgroup SU_{m} ⊗ SU_{n}
View Description Hide DescriptionWith a view to application to theories such as the Wigner supermultiplet theory of nuclear structure and its recent extension into the domain of particle physics, the reduction of representations of the special unitary groupSU_{mn} into representations of its SU_{m} ⊗ SU_{n} subgroup is considered. Proof is given of the result, that, in the case of a representation of SU_{mn} of plurality type p_{mn} , this reduction yields only those irreducible representations of SU_{m} ⊗ SU_{n} , whose SU_{m} and SU_{n} parts have plurality types p_{m} and p_{n} equal to p_{mn} modulo m and modulo n, respectively. A practical method of obtaining reductions is described and a tabulation of results in the case of SU _{6} and SU _{2} ⊗ SU _{3} is made. The method depends on the use of a very concise version of the Weyl character formula for unitary groups which does not seem to have been used elsewhere.

Reduction of Representations of SU_{m+n} with respect to the Subgroup SU_{m} ⊗ SU_{n}
View Description Hide DescriptionA direct and rapid method is described for the reduction of representations of the special unitary groupSU_{m+n} with respect to its SU_{m} ⊗ SU_{n} subgroup. The technique is illustrated by specific reference to the case of SU _{6} and its SU _{2} ⊗ SU _{4} subgroup in the context of the recent extension of Wigner's (SU _{4}) supermultiplet theory into the domain of particle physics. A tabulation of results is given for a slight generalization of this case where reduction is carried out with respect to U _{1} ⊗ SU _{2} ⊗ SU _{4}.

Symmetrical Coupling of Three Angular Momenta
View Description Hide DescriptionWe introduce here a new coupling scheme for three angular momenta. It relies on the properties of an operator which depends ``democratically'' upon the three individual angular momenta; this is in fact their mixed product. This operator, the total angular momentum and one of its components form a complete set of commuting observables. In the case where the three individual angular momenta are equal, the eigenstates of this set possess remarkable symmetry properties with respect to the permutation group S _{3}.

On the Possibility of Relating Internal Symmetries and Lorentz Invariance
View Description Hide DescriptionIn this note we investigate the possibility that the inhomogeneous Lorentz group is only a subgroup of a larger Lie groupG of symmetries for strong interaction physics. The discussion is restricted to the Lie algebraL of G. We make the assumption that the remaining generators A_{i} of G commute with the generators of translations P ^{λ} which build the ideal I in L. It is then shown that the A_{i} generate an ideal in L modulo I. If this ideal is semi‐simple then L breaks up in a direct sum L = P ⊕ a, where P is isomorphic with the Lie algebra of the inhomogeneous Lorentz group and a is semisimple.

The Gravitational Compass
View Description Hide DescriptionA purely covariant approach to general relativity, using the equation of geodesic deviation, is adopted. The physical interpretation is essentially that due to Pirani, but instead of using clouds of particles to analyze the gravitational field, a ``gravitational compass'' is proposed which fulfills the same purpose. Particular attention is focussed on the different roles played by the matter and the free gravitational field. The latter splits up conveniently into a super‐position of a transverse wave component, a longitudinal component, and a ``Coulomb'' field, all of which introduce ``shearing'' forces on the gravitational compass, while the matter contributes a general contraction. Applications to the Friedmann cosmological models and the problem of interactinggravitational waves are discussed.

Algebraic Derivation of the Partition Function of a Two‐Dimensional Ising Model
View Description Hide DescriptionIn the algebraic formulation of the Ising model, the partition function is expressed as the trace of a power V^{M} of the transfer operator V, or equivalently, as the sum of Mth powers of the eigenvalues of V. In the derivations of Kaufman, Onsager, and more recently of Schultz, Mattis, and Lieb (SML), the transfer operator is first reduced to a more amenable form for computation and then, in principle at least, diagonalized. For the infinite lattice, only the largest eigenvalue of V is needed, and this is all Onsager and SML compute. Kaufman finds all the eigenvalues and is thus able to write down the partition function for the finite lattice. In the present work we give an alternative derivation of the SML form for V, and show how the Kaufman result can be obtained from this form without actual diagonalization. Instead of diagonalizing V, the evaluation of the trace is done directly after assigning a simple representation to V.

Momentum‐Transfer Theorem for Inelastic Processes
View Description Hide DescriptionRecently it has been shown that for potential scattering, the well‐known optical theorem—relating the total cross section to the imaginary part of the forward scattering amplitude—can be generalized to yield a ``momentum‐transfer cross‐section theorem.'' The present paper further generalizes the previous potential scattering result. Specifically, it appears that the momentum‐transfer cross‐section theorem is valid also for many‐particle systems, wherein inelastic processes occur. Although this last assertion probably holds quite generally, a proof is given only for the collisions of electrons with atomic hydrogen. The proof takes into account electron indistinguishability, as well as the possibility that the incident electron ionizes the atom, but assumes the forces are not spin‐dependent.

On the Proof and Uniqueness of Wulff's Construction of the Shape of Minimum Surface Free Energy
View Description Hide DescriptionThe existing proof of Wulff's theorem shows that, among all convex bodies of fixed volume, the shape given by Wulff's construction has the least surface free energy. Here it is pointed out that the restriction to convexity is unnecessary: among all bodies of fixed volume, the shape given by Wulff's construction has (uniquely) the least surface free energy. Obvious generalizations are noted.

Theorem on Nonradiative Electromagnetic and Gravitational Fields
View Description Hide DescriptionA vacuum electromagnetic field is considered which is (a) stationary below a hypersurface Σ intersecting all parametric lines of the time coordinate t and (b) nonradiative above Σ. It is proved that such a field is stationary also above Σ. The same theorem is proved to be valid for the pure gravitational field in general relativity and for the combined gravitational and electromagnetic field in the Einstein—Maxwell theory.

Backward Superconducting Switching
View Description Hide DescriptionA discussion is given of the initial speeds at which superconducting material changes state according to the London electrody namics. These transitions are taken to occur in the form of phase boundary motions. Phase changes from the normal to the superconducting state and vice versa are considered for the cases in which the external magnetic field is radiated to the boundary of the superconducting material. A distinct difference is found in transition rates depending on whether the transition is from the superconducting or from the normal state. In another case considered, the transition from normal to super is studied when the superconducting material is bounded by a good conductor. In all cases, constant critical field is taken as the switching criterion. The mathematical treatment involves the approximate solution of free boundary problems and mixed hyperbolic—parabolic boundary value problems.

Structure of Fermion Density Matrices. II. Antisymmetrized Geminal Powers
View Description Hide DescriptionFunctions which can be expressed as an antisymmetrized power of a single two‐particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second‐order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one‐particle occupation numbers of the independent‐particle model. A necessary and sufficient condition is given that an alleged first‐order reduced density matrix be derivable from an N‐particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2‐matrix. The extreme 2‐matrix is diagonalized explicitly and provides a vivid example of the fact that one‐particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2‐matrix satisfies the assumption basic to current theories of superconductivity.

Reduced Density Matrices of Quantum Gases. III. Hard‐Core Potentials
View Description Hide DescriptionIn two previous papers, the reduced density matrices of quantum gases, in the grand canonical formalism and for suitably restricted interaction potentials, have been shown to be analytic vector‐valued functions of the activity in a neighborhood of the origin, to tend in some sense to well‐defined limits as the volume of the system becomes infinite, and to satisfy a cluster decomposition property. The same results are extended here by the same methods to a wider class of potentials, including hard‐core, and allowing attractive interactions, which were excluded previously.