On representations of the groups Sp(n, 1) and Sp(n)

New classes of unitary irreducible representations of Sp(n, 1) which can be useful for applications in physics are obtained. The infinitesimal operators of these representations of Sp(n, 1) and of irreducible representations of Sp(n+1) with highest weights (m, m, m₃,…,m_n+1) and (m₁, m₂, 0,…,0) are expressed in terms of the simple Clebsch–Gordancoefficients for Sp(n). For Sp(3) and Sp(2, 1) they are found in an explicit form.     

On the integrability of discrete representations of the Lie algebra so(p,q)

This note contains the proof that all discrete skew-symmetric irreducible representations of the Lie algebra so(p, q) described by Nikolov [5] are integrable to unitary representations of the corresponding connected and simply-connected covering group.     

Clebsch-Gordan coefficients of the groups SO(p, 1)

An explicit expression for the Clebsch-Gordan coefficients for the coupling of most degenerate unitary representations of SO(p, 1) is obtained.     

On the connection of matrix elements of the SU (1,1) discrete series representations with SU (2) matrix elements

The purpose of this note is to establish the connection between the matrix elements of representations of two simplest groups: representations of the discrete series of the non-compact SU (1,1) group and the corresponding representations of the compact SU (2) group.     

Invariants of the representations of the group of pseudo-orthogonal matrices,SO(p,q)

The irreducible unitary representations of the group SO(p,q) are investigated by a method based on a study of invariant bilinear functionals. The invariant polylinear functionals are constructed. Integral representations are obtained for the Wigner coefficients and for the crossing symmetrical conformal invariant four-point Green function.     

On a resolution of the multiplicity problem for U(n)

The Borel-Weil theory of holomorphic induction is used to provide a resolution of the multiplicity problem occuring in the tensor product decomposition of representations of the U(n) groups.     

Representations of the Sugawara SU(2) current algebra

The Sugawara SU(2) current algebra is exponentiated in order to obtain the group. Further, the representations of the current algebra are recovered from the representations of the group.Generators of the three-dimensional Euclidean group E, (space translations and space relations) are constructed. An example showing the relation with Yang-Mills theory is presented.     

On the representation of finite transformations of O2 in the canonical basis of the group On

The matrix elements of the orthogonal transformations of the two-coordinate subspace of the n-dimensional space in the canonical basis of the orthogonal (O_n) and the rotation (SO_n) groups are considered. The matrix elements of the projection operator of the representations of SO₂ as the non-canonical subgroup of SO_n in the canonical basis of SO_n have been used. The relations between basis states of O_n and SO_n representations are described, and it is shown how to use the substitution group and the Regge symmetries and other types of symmetries.     

Spinor-inverted solution to thirring model and its generalization to U(n) symmetry

Using the properties of massless free Fermi fields in (1-1) dimensions, it is shown that the spinor inverted form of Klaiber's operator solution to Thirring model is also a scale-invariant solution of the model. But unlike the former it admits a nonvanishing SU(n) current coupling in the generalization of the model to include U(n) symmetry. The value of this coupling constant is fixed and equals Dashen-Frishman number $\text{?4π}\text{(n + 1)}$. The general form of the 2m-point function is given and operates product expansions are exhibited in terms of composite local operators. Scale dimensions of all the bilinear and quadrilinear local operators with U(n) symmetry are computed and are found to depend on n. However, different parts of a composite local operator belonging to different irreducible U(n) representations have the same dimension.     

A systematic search for a realistic SU(n) tumbling gauge model

Within the framework of the dynamical symmetry breaking and the tumbling ideas, a systematic search was carried out in SU(n) groups with the ultimate aim of determining if a realistic and phenomenologically acceptable model exists which tumbles down to SU(3)_c ? U(1), or a suitable large group. To do so all the anomaly free and the asymptotically free fermion contents for any SU(n) were first determined. In order to have nontrivial tumbling the real and the pseudo-real representations have been eliminated, and the tumbling patterns of all the allowed complex ones in detail have been examined. No such realistic model has been found. These results combined with those of Srednicki's concerning the SO(4n + 2) and E₆ groups establish the fact that there cannot be any realistic tumbling gauge model within the context of the original tumbling hypotheses. Having thus established the need for a change of these hypotheses some suggestions and comment on various ways of remedying the problem are made.     

Entropy Bounds in R×S Geometries

Exact calculations are given for the Casimir energy for various fields in R×S³ geometry. The Green's function method naturally gives a result in a form convenient in the high-temperature limit, while the statistical-mechanical approach gives a form convenient for low temperatures. The equivalence of these two representations is demonstrated. Some discrepancies with previous work are noted. In no case, even for N=4 SUSY, is the ratio of entropy to energy found to be bounded. This deviation, however, occurs for low temperature, where the equilibrium approach may not be relevant. The same methods are used to calculate the energy and free energy for the transverse electric modes in a half-Einstein universe bounded by a perfectly conducting 2-sphere.     

The unitary representations of G̶L(4, R)

In the paper, nine series of unitary representations of the group $\text{G}\text{?}$L(4, R) are found. Moreover, there are obtained matrix elements of the generators of $\text{G}\text{?}$L(4, R) in an orthonormal basis. The elements of the basis are classified over the subgroup SU(2)?SU(2).     

Poincaré subalgebras in so(4, 2) and sl(5, R)

Injective homomorphisms ε which map the Poincaré Lie algebra P into so(4, 2)=C are considered. Two mappings ε₁ and ε₂ are said to be Int-equivalent iff there exists an inner automorphism ρⁱⁿ_c> of C such that ε₁=ρⁱⁿ_cε₂. The set of all ε splits into four Int- equivalence classes. Mappings in different classes are equivalent with respect to non-inner automorphisms of C. A corresponding theorem holds for homomorphisms of P into sl (5, R). These algebraic properties give strong limitations for those integrable representations of P which can be found by restriction of integrable representations of so(4, 2) and of sl(5, R) to ε(P). Physical applications of the results are discussed.     

The eikonal approximation and E(2) invariance

A group theoretic interpretation is given for the eikonal approximation in potential scattering. This is based upon the approximate invariance at high energies under translations and rotations in the transverse scattering plane; that is, symmetry under the group E(2). The Lippman-Schwinger equation is formulated in a set of basis states which transform invariantly under irreducible representations of E(2) and the solution for the eikonal Hamiltonian, together with lowest order (Saxon-Schiff) corrections is obtained within this basis. A formulation of unitarity in the impact-parameter representation, based upon the E(2) invariance is given. The “geometrical” interpretation of this representation, in connection with the eikonal approximation, is made clear by our approach.     

Representations of N = 4 supersymmetry on superfields

We determine all of the irreducible representations (irreps) of the extended supersymmetry algebra (for N = 2 and 4) on superfields with arbitrary external spin in terms of the associated eigenvalues of suitable Casimirs of enlarged algebras. Detailed results are presented for the scalar, spinor and vector superfield cases. The component content of the corresponding irreps are also discussed by means of suitable basis functions.     

A new approach to harmonic analysis on the group SL(2,R)

The Gelfand-Graev horospheric approach to the Fourier transform on SO(n,1) is confined to Class I representations; we extend their method to the regular representation of the group $\text{SL(2,}\text{R}\text{)}$. The role of the cone is played by the manifold of 2×2 real singular matrices, upon which the representation and decomposition theory is much simplified. The discrete series is connected with the appearance of associated homogeneous functions on this manifold; in the inversion formula expressing ?(g) in terms of its components this is related to the presence of double poles in the measure of the continuous series. We examine in some detail the relations between equivalent formulations of the discrete series.     

O(3, 1) expansion for asymptotically growing amplitudes

It is shown how to expand asymptotically growing scattering amplitudes which considered as functions defined on the group O(3, 1) do not satisfy the square-integrability criterion. We discuss the cases when subgroup chosen to be diagonal is either the rotation group or the Lorentz group O(3, 1). Application to inclusive reactions in the fragmentation and the triple Regge region is considered. There as no fixed poles at integer point representations. We comment on the relation of the present work to methods based on integral geometry.     

Global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on bipartite lattices

In this paper the global symmetry of the Hubbard model on a bipartite lattice is found to be larger than SO(4). The model is one of the most studied many-particle quantum problems, yet except in one dimension it has no exact solution, so that there remain many open questions about its properties. Symmetry plays an important role in physics and often can be used to extract useful information on unsolved non-perturbative quantum problems. Specifically, here it is found that for on-site interaction U ≠ 0 the local SU(2) × SU(2) × U(1) gauge symmetry of the Hubbard model on a bipartite lattice with N_a^D sites and vanishing transfer integral t = 0 can be lifted to a global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with t > 0. (Examples of a bipartite lattice are the D-dimensional cubic lattices of lattice constant a and edge length L = N_aa for which D = 1, 2, 3,... in the number N_a^D of sites.) The generator of the new found hidden independent charge global U(1) symmetry, which is not related to the ordinary U(1) gauge subgroup of electromagnetism, is one half the rotated-electron number of singly occupied sites operator. Although addition of chemical-potential and magnetic-field operator terms to the model Hamiltonian lowers its symmetry, such terms commute with it. Therefore, its 4^N_aD energy eigenstates refer to representations of the new found global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry. Consistently, we find that for the Hubbard model on a bipartite lattice the number of independent representations of the group SO(3) × SO(3) × U(1) equals the Hilbert-space dimension 4^N_aD. It is confirmed elsewhere that the new found symmetry has important physical consequences.     

Nonadiabatic effects in the B, C, B′, and D states of H2

The Kolos-Wolniewicz potentials for the H₂$\text{B}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and C¹Π_u states were used together with the hypothesis of pure precession for the rotation-electronic interaction, to calculate the nonadiabatic energy levels of these states for J = 1 to 5. The complete coupling matrix was computed using accurate numerical vibrational wavefunctions. The calculated Λ-doubling of the v = 0 to 12 C vibrational levels generally agrees well with experimental values, and the nonadiabatic shifts in the B rotational constants qualitatively explain the difference between the theoretical and RKR potentials for this state.The interaction of the B′${}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and D¹Π_u states was also investigated, but only qualitatively since adiabatic potentials of sufficient accuracy do not exist for these states. The Λ-doubling of the Dv = 0 rotational levels agrees well with the experimental values. An appreciable “background” nonadiabatic shift in the B′ rotational constants was found. This shift, which is nearly 5 percent of B_v, is in addition to that of strong local two-level interactions and was not “deperturbed” in constructing the B′ RKR potential. The result is that the RKR turning points differ by about 0.04 au from the “true” adiabatic turning points. This conclusion is supported by a Hartree-Fock calculation of the B′ potential to the left of R_e.