Irreducible Subspaces Research Articles

Symmetry properties of the zero sets of nil-theta functions

Let N N denote the three dimensional Heisenberg group, and let Γ \Gamma be a discrete two-generator subgroup of N N such that N / Γ N/\Gamma is compact. Then we may decompose L 2 ( N / Γ ) {L^2}(N/\Gamma ) into primary summands with respect to the right regular representation R R of N N on L 2 ( N / Γ ) {L^2}(N/\Gamma ) as follows: L 2 ( N / Γ ) = ⊕ ∑ m ∈ Z H m ( Γ ) {L^2}(N/\Gamma ) = \oplus \sum \nolimits _{m \in {\mathbf {Z}}} {{H_m}(\Gamma )} . It can be shown that for m ≠ 0 , H m ( Γ ) m \ne 0,{H_m}(\Gamma ) is a multiplicity space for the representation R R of multiplicity | m | \left | m \right | . The distinguished subspace theory of L {\text {L}} . Auslander and J {\text {J}} . Brezin singles out a finite number of the decompositions of H m ( Γ ) , m ≠ 0 {H_m}(\Gamma ),m \ne 0 , which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set Ω m {\Omega _m} of irreducible closed R R -invariant subspaces of H m ( Γ ) {H_m}(\Gamma ) such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in Ω m {\Omega _m} contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on N / Γ N/\Gamma . These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in Ω m {\Omega _m} using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in H m ( Γ ) {H_m}(\Gamma ) using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in Ω m {\Omega _m} equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function F G FG relates to the indices of the nil-theta functions F F and G G .

A proof of the boundary theorem

This note contains a simple proof of the following theorem of Arveson: If A \mathcal {A} is an irreducible subspace of B ( H ) \mathcal {B}(H) , then the identity map ϕ 0 ( A ) = A {\phi _0}(A) = A on A \mathcal {A} has a unique completely positive extension to B ( H ) \mathcal {B}(H) if and only if the quotient map q q by the compact operators is not completely isometric on S = [ A + A ∗ ] \mathcal {S} = [\mathcal {A} + \mathcal {A}^*] .

On the algebraic formulation of collective models. II. Collective and intrinsic submanifolds

A general procedure is given for decomposing N-particle configuration space into orbits of a kinematical collective group and a smooth transversal. Evaluation of the Laplace-Beltrami operator, in terms of vector fields tangent to the orbit and to the transversal, gives the decomposition of the kinetic energy into collective and intrinsic parts. By such geometric means, the essence of the various transformations of coordinates is revealed in a simple coordinate free way. Specifically considered are the kinematical collective groups SO(3) and GL +(3, R). It is also shown that, with center-of-mass motion removed, N-particle configuration space can be considered as an orbit of the group GL +(3, R) × SO( N − 1). This decomposition of the configuration space is observed to induce a corresponding decomposition of the N-particle Hilbert space into irreducible subspaces with respect to both interesting spectrum generating algebras for collective motion and of the symmetric group. Thus the connection is established with the algebraic approach to the microscopic realization of collective states, with full respect for the Pauli principle.

Koornwinder’s polynomials and representations of the conformal group

The tensor product of two massive, spinless, positive-energy ray representations of the conformal group of spacetime SO0(4,2)/Z2 is reduced in a momentum basis. The basis vectors for the irreducible subspaces (the ’’Clebsch–Gordan coefficients’’) are found to be intimately connected with Koornwinder’s polynomials in two dimensions.

Rational Subspaces of Induced Representations and Nilmanifolds

Recently, Auslander and Brezin developed a technique of distinguishing between certain unitarily equivalent irreducible subspaces of ${L^2}$ of the Heisenberg nilmanifold. In this paper we extend the Auslander-Brezin technique to arbitrary induced representations of arbitrary locally compact groups. We then return to nilmanifolds, showing that the existence of a “nice” theory of distinguished subspaces is equivalent to the existence of square integrable representations for the group.

Spherical distributions on nilmanifolds

Let N be a connected nilpotent Lie group and Γ be a discrete subgroup for which M = Γ\\ N is compact. Let R be the regular representation of N in L 2( M). Projections onto primary (irreducible) subspaces of R are given by convolution against distributions (the spherical distributions). In this paper we give formulas and several characterizations for these distributions. We apply these results in a specific case to the study of theta functions.

Rational subspaces of induced representations and nilmanifolds

Recently, Auslander and Brezin developed a technique of distinguishing between certain unitarily equivalent irreducible subspaces of L 2 {L^2} of the Heisenberg nilmanifold. In this paper we extend the Auslander-Brezin technique to arbitrary induced representations of arbitrary locally compact groups. We then return to nilmanifolds, showing that the existence of a “nice” theory of distinguished subspaces is equivalent to the existence of square integrable representations for the group.

New statistical measures of symmetry breaking in the ground state region of nuclei

We define two statistical measures of symmetry breaking which take all the internal moments of the irreducible subspaces into account, and can be used in the ground state region of nuclei. We test their validity in an exactly solvable model.

Spinors in Hilbert space

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

The structure of groups with index-3 subgroups and Landau’s second theorem

For any group G0 which contains an index-3 subgroup G, it is shown that either (a) G is an invariant subgroup or (b) G contains an index-2 subgroup GA, where GA is an invariant subgroup of G0. For case (a), G and its cosets give rise to three operators which span a stable three-dimensional subspace of the group algebra which further reduces to three one-dimensional stable subspaces. For case (b), GA and its cosets give rise to six operators which span a six-dimensional stable subspace of the group algebra which reduces to two one-dimensional and two two-dimensional irreducible stable subspaces of the group algebra. The irreducible representations and the corresponding basis elements of the group algebra are given for both cases. Landau’s second theorem pertaining to second order phase transitions is proven.

General properties of representations of local currents “at infinite momentum”

The results of an earlier work on the representations of currents on the physical Hilbert space are extended to “representations at infinite momentum.” A detailed construction of the infinite momentum representation space in a proper basis is given and it is shown that irreducible subspaces correspond uniquely to irreducible subspaces of the physical Hilbert space. The internal symmetry spectrum in each irreducible representation is unbounded and the mass spectrum is unbounded and continuous from some point onward if it is not strictly degenerate. Representations of free-field currents are given special attention.

SL(2, C) Representations in Explicitly ``Energy-Dependent'' Basis. I

Unitary and nonunitary representations of the SL(2, C) group are investigated in such a basis, in which the subgroup diagonalized is that one which in the four-dimensional representation leaves invariant the 4-vector pμ = (½(1 + v), 0, 0, ½(1 − v)) for an arbitrary real value of Pμ2=v. The split of the representation space into irreducible subspaces changes smoothly when varying the value of v. The formalism is of importance in physical theories which postulate analyticity requirements and Lorentz invariance simultaneously (e.g., Regge and Lorentz pole theory). In this paper we construct explicit basis functions of the representation spaces.

Level density for some partially random Hamiltonians

We study the level density of a random HamiltonianH such that the total configuration space can be split into irreducible subspaces using for example invariance properties. The level density is supposed to be known for each individual subspace and the matrix elements connecting the various subspaces are treated as random variables. A similar problem arises when the Hamiltonian may be written in the formH0 +V, whereH0 has a known level density andV, the residual interaction, is a random matrix. In both cases, we start from the relation of the level density to the average of the resolvent. We expand the latter in powers of the interaction, and by scaling considerations we get the dominant contributions. The final result is easily obtained when the matrix elements of the residual interaction are statistically independent. Some simple algebraic examples are given; in particular, we recover the « semi-circle » law and indicate some generalizations.

Zero-mass infinite spin representations of the Poincaré group and quantum field theory

It is shown that a local quantized field with a manifestly covariant transformation law under the Poincare group cannot have nonvanishing matrix elements between the vacuum and an irreducible subspace of zero mass and infinite spin.

Harmonic Analysis on Nilmanifolds

We compute, using a device of A. Weil, an explicit decomposition of ${L^2}$ of a nilmanifold into irreducible translation-invariant subspaces. The results refine previous work of C. C. Moore and L. Green.

On quantum mechanics of identical particles

The statement commonly made in the quantum mechanical theory of identical particles, that the characters of the symmetric group of particle permutations are constants of motion, is shown not to be always correct when paraparticles come into play. A general quantum mechanical theory, which can, as well, deal with paraparticles, is formulated on the basis of the corresponding parafield theory. It is shown that to describe such a general system, a multicomponent wavefunction is needed, with each component referring to each inequivalent irreducible subspace under particle permutations, and consequenlty that physical observables have the corresponding matrix-like structure. General properties of our formalism are studied in detail, and its connection with the usual formalism clarified. A further generalization of our formalism to include particles that do not obey parastatistics is also discussed.

Harmonic analysis on nilmanifolds

We compute, using a device of A. Weil, an explicit decomposition of L 2 {L^2} of a nilmanifold into irreducible translation-invariant subspaces. The results refine previous work of C. C. Moore and L. Green.

Wavefunctions of identical particles

Group-theoretical properties are studied of the space A n spanned by wavefunctions of n identical particles by decomposing it into irreducible subspaces invariant under the symmetric group S n . It is shown that when particles obey para-Fermi (Bose) statistics of order p, the space A n contains all irreducible subspaces which correspond to any Young diagrams with the first row (column) not longer than p, and that the multiplicity of such equivalent irreducible subspaces is always one. Two examples are given, which show that the corresponding multiplicity is not equal to one when particles do not obey para-statistics. It is conjectured that this property of the multiplicity yields a necessary and sufficient condition for particles to obey para-statistics.

Calculation of Relativistic Transition Probabilities and Form Factors from Noncompact Groups

Relativistic transition probabilities and form factors have been evaluated in closed form (in terms of hypergeometric functions) for a model of fundamental particles constructed on the unitary irreducible representation space of a noncompact dynamical group. Majorana-type equations have been used to project out irreducible subspaces of the Poincar\'e group and to specify the transformation properties of the electromagnetic current. The magnetic moment of the spin-\textonehalf{} ground state in the simplest representation is -\textonehalf{}, and one obtains the observed symmetry ${G}_{E}(t)=\frac{{G}_{M}(t)}{\ensuremath{\mu}}$.

Dominance theorem in crossing matrices in arbitrary compact group

Previously, the present author proved the new dominance theorem in crossing matrices in SU~ and SU B groups and discussed its physical implications in highenergy diffraction scattering theory (1). In this note we shall briefly show that this theorem holds generally in an arbitrary compact group. That is, the absolute magnitude of the nonvanishing crossing-matrix element between s-channel amplitude in the singlet state and t- or u-channel amplitude in a given arbitrary multiplet state is never smaller than that ~f the erossing-matrix~ element between s-channel amplitude in an arbitrary multiplet state and t- or re-channel amplitude in a given aa\bitrary multiptet st~.e. For saving space we shall use the definitions of the various quantities by A~AT~ et at. (~) and their notations with only brief explanations when there is no mathematical ambiguity. We assign the four particles of the scattering process to the four unitary irreducible representation spa~es for an ~rbitrary compact group which we designate by A, B, C, D. In each space we select an orthonormal basis and represent the vectors of these basis by IAa}, IBb), ICe}, IDd}. The spaces conjugate to A, B, C, D will be designated A, B, C, D and the basis of the e~njugate representation A, B, C, D will be designated by !Zg}, !B~>, IC~>, ID3~), respectively. If a paxtiele belongs to a representation, its antiparticle belongs to the conjugate one. We can decompose the product space such aM A  B into irreducible subspaces under an ~rbitrary compact group. V~e shall use the symbol X to represent both such invariant subspaces and the irreducible representations induced in it. In each such subspaee we may introduce an orthonormal basis 1X(~)x(A where ~ is introduced to distinguish the equivalent invariant subspaees. It is easily shown that the transformation coefficient (C.G. coefficient) (X(~) x 1Aa, Bb) has the usual properties such as orthonormality and symmetries as the C.G. coefficient in SU~ and SU s groups (s.4). Therefore we shatt use theme properties of C.G. coefficients without detailed mathems~ical pr(~of. The necessary properties of C.G. coefficients are listed Jn the