Unitary Representation Of SO Research Articles

The 𝒫(𝜑)₂ Model on de Sitter Space

In 1975 Figari, Høegh-Krohn and Nappi constructed theP(φ)2{\mathscr P}(\varphi )_2model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs ofSO0(1,2)SO_0(1,2)for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations ofSO(3)SO(3)on the Euclidean sphere, and discuss how theP(φ)2{\mathscr P}(\varphi )_2interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physicalinfrared problemsare absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for theP(φ)2{\mathscr P}(\varphi )_2model and we explain how thegeneratorsof the boosts and the rotations for the interacting quantum field theory arise from thestress-energy tensor. Finally, we show that the interacting quantum fields satisfy theequations of motionin their covariant form. In summary, we argue that the de SitterP(φ)2{\mathscr P}(\varphi )_2model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.

On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO ( 3 ) and the principal series of the unitary representation of SU ( 1 , 1 ) .

On application of the Snyder theory to macroscopic objects

Unitary representations of SO(4,1) are used for construction of quantum gravitational wave functions of macroscopic bodies. Feasible experimental verification of the theory are pointed out.

Massless conformal fields, AdS(d + 1)/CFTd higher spin algebras and their deformations

We extend our earlier work on the minimal unitary representation of SO(d,2) and its deformations for d=4,5 and 6 to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d,2) and its deformations and massless conformal fields in Minkowskian spacetimes in d dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS(d+1)/CFTd higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d−2) for massless representations.

2D fuzzy anti-de Sitter space from matrix models

We study the fuzzy hyperboloids AdS^2 and dS^2 as brane solutions in matrix models. The unitary representations of SO(2,1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS^2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.

Formula omitted] deformations of the minimal unitary representation of [formula omitted] as massless 6 D conformal supermultiplets

Minimal unitary representation of SO ⁎ ( 8 ) ≃ SO ( 6 , 2 ) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU ( 2 ) subgroup correspond to massless conformal fields in six dimensions as was shown in [S. Fernando, M. Gunaydin, arXiv:1005.3580]. In this paper we study the minimal unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) with the even subgroup SO ⁎ ( 8 ) × USp ( 2 N ) and its deformations using quasiconformal methods. We show that the minimal unitary supermultiplet of OSp ( 8 ⁎ | 2 N ) admits deformations labeled uniquely by the spin t of an SU ( 2 ) subgroup of the little group SO ( 4 ) of lightlike vectors in six dimensions. We construct the deformed minimal unitary representations and show that they correspond to massless 6 D conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ⁎ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 . We study its deformations in further detail and show that they are isomorphic to the doubleton supermultiplets constructed by using twistorial oscillators.

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations as massless 6 D conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO ( 6 , 2 ) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO ( 6 , 2 ) , which coincides with the minrep of SO ∗ ( 8 ) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of “deformations” of the minrep of SO ∗ ( 8 ) labeled by the spin t of an SU ( 2 ) T subgroup of the little group SO ( 4 ) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO ∗ ( 8 ) that describe massless conformal fields in six dimensions. The SU ( 2 ) T spin t is the six-dimensional counterpart of U ( 1 ) deformations of the minrep of 4D conformal group SU ( 2 , 2 ) labeled by helicity. We also construct the supersymmetric extensions of the minimal unitary representation of SO ∗ ( 8 ) to minimal unitary representations of OSp ( 8 ∗ | 2 N ) that describe massless six-dimensional conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ∗ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 .

Weyl’s law for the cuspidal spectrum of SLn

Let r be a principal congruence subgroup of SL n (Z) and let σ be an irreducible unitary representation of SO(n). Let N Γ cus (λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to a. In this paper we prove that the counting function N Γ cus (λ,σ) satisfies Weyl's law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SL n (Z).

Weyl's law for the cuspidal spectrum of SL n

Let Γ be a principal congruence subgroup of SL n( Z) and let σ be an irreducible unitary representation of SO( n). Let N cus Γ ( λ, σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO( n) according to σ. In this Note we prove that the counting function N cus Γ ( λ, σ) satisfies Weyl's law. In particular, this implies that there exist infinitely many cusp forms for the full modular group SL n( Z) . To cite this article: W. Müller, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Group theoretical quantization of a phase spaceS1×R+and the mass spectrum of Schwarzschild black holes inDspace-time dimensions

The symplectic reduction of pure spherically symmetric (Schwarzschild) classical gravity in D space-time dimensions yields a 2-dimensional phase space of observables consisting of the Mass M (>0) and a canonically conjugate (Killing) time variable T. Imposing (mass-dependent) periodic boundary conditions in time on the associated quantum mechanical plane waves which represent the Schwarzschild system in the period just before or during the formation of a black hole, yields an energy spectrum of the hole which realizes the old Bekenstein postulate that the quanta of the horizon A_{D-2} are multiples of a basic area quantum. In the present paper it is shown that the phase space of such a Schwarzschild black hole in D space-time dimensions is symplectomorphic to a symplectic manifold S={(phi in R mod 2 pi, p = A_{D-2} >0)} with the symplectic form d phi wedge d p. As the action of the group SO_+(1,2) on that manifold is transitive, effective and Hamiltonian, it can be used for a group theoretical quantization of the system. The area operator p for the horizon corresponds to the generator of the compact subgroup SO(2) and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of SO_+(1,2) yields an (horizon) area spectrum proportional k+n, where k =1,2,... characterizes the representation and n = 0,1,2,... the number of area quanta. If one employs the unitary representations of the universal covering group of SO_+(1,2) the number k can take any fixed positive real value (theta-parameter). The unitary representations of the positive discrete series provide concrete Hilbert spaces for quantum Schwarzschild black holes.

Particle dynamics on hyperboloid and unitary representation of SO(1, N) group

We analyze particle dynamics on N dimensional one-sheet hyperboloid embedded in N+1 dimensional Minkowski space. The dynamical integrals constructed by SO ↑(1, N) symmetry of spacetime are used for the gauge-invariant Hamiltonian reduction. The physical phase-space parametrizes the set of all classical trajectories on the hyperboloid. In quantum case the operator ordering problem for the symmetry generators is solved by transformation to asymptotic variables. Canonical quantization leads to unitary irreducible representation of SO ↑(1, N) group on Hilbert space L 2( S N−1 ).

Harmoniques Sphériques et Représentations Irréductibles de O (

We build an irreducible unitary representation of SO( ∞ ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ℝn. We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ∞ ), we thus get some results of Ol‘shanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).

Representations of the Poincaré Lie algebra from unitary representations of SO(1,4)

An algebraic isomorphism of an extension of the Lie field of so(1,4) and an extension of the Poincaré Lie field was given in Havlicek and Moylan [J. Math Phys. 34, 5320 (1993)]. Here we use the isomorphism described in that paper in order to obtain matrix elements of the Poincaré Lie algebra generators in a normalizable basis for an arbitrary unitary spherical principal series representation of SO0(1,4).

Spinors over a Cone, Dirac Operator, and Representations of Spin(4, 4)

The representations of Spin(4, 2) e seem to be of particular physical interest since its quotient SO(4, 2) e is the conformal group of the spacetime. Kostant [7] has considered the Laplacian on a projective cone in R 8 and has shown that the kernel H of the Laplacian is an irreducible unitary representation of SO(4, 4) e . Moreover, it is known [9] that there is a Howe pair ( SO(2), SO(4, 2) e ) in SO(4, 4) e and that the weight spaces of SO(2) in H are irreducible representations of SO(4, 2) e . He conjectured that the analogous facts should hold for the Dirac operator acting on the spinor fields on S 3 × S 3 ⊆ R 8. We introduce a hitherto unknown embedding of the bundle of spinors on S 3 × S 3 in the bundle of spinors on R 8. This embedding makes it possible to obtain a particularly convenient expression for the Dirac operator on S 3 × S 3 in terms of spinors and vector fields on the underlying R 8 in particular the conformal invariance is manifest. We explicitly describe the kernel F of the restriction of the Dirac operator to even spinor fields and show that it is an irreducible representation of Spin(4, 4) e . We also show that it has an invariant sesquilinear scalar product (with is not, however, positive definite). The existence of this scalar product can also be established by showing that F ⊆ S + ⊗ H where S + the vector space of even spinors. We identify within Spin(4, 4) e a Howe pair consisting of SO(2) and Spin(4, 2) e and show that the weight spaces of SO(2) in F are irreducible representations of Spin(4, 2) e .

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.

Spectrum generating algebras in Kaluza-Klein theories

We describe the action of the euclidean conformal group on spheres. Using the example of S 7 compactification of the eleven-dimensional supergravity we show how the full spectrum provides unitary representations of SO(8,1). Our methods can be applied to compactification on spheres, products of spheres or any other manifold on which there is an action of a conformal group. We also make some conjectures concerning the relationship between the conformal group and supersymmetry.

Infinitesimal operators and structure of the representations of the groups SO*(2n) and SO(2n) in a U(n) basis and of the groups SU*(2n) and SU(2n) in an Sp(n) basis

The infinitesimal operators of the most degenerate representations of the groups SO*(2n) and SU*(2n) are found in a discrete basis. The structure (composition series) of these representations is studied. The classification of unitary irreducible representations of these groups which belong to most degenerate series is given. The infinitesimal operators of irredicuble unitary representations of SO(2n) in a U(n) basis and of SU(2n) in an Sp(n) basis are found for the cases of highest weights (M, M, 0, ..., 0), M≥0.

A duality between massless particles and a charge-monopole system

It is pointed out that the same realisation of a unitary representation of SO(4, 2) from the ladder series is used in the coordinate-space description of a charge-monopole system, (or more generally a dyon-dyon system) and in the canonical momentum-space description of a massless particle. Therefore, in the latter case a momentum-space analogue appears for the monopole vector potential, complete with its Dirac string singularity. Analogues of gauge transformations relate equivalent realisations with different locations of the momentum-space string. Quantisation of helicity replaces quantisation for the product of electric and magnetic charge. The problem of localising a charge on a monopole string is related to recent work by Flato et al. [5] on the localisability of a massless particle in momentum-space. Further, the multi-component form of the generators of SO(4, 2) for a massless particle has a dual, in coordinate space, corresponding to a charge-monopole system for a monopole of the Yang-Mills type. The question of the interpretation of the momentum analogue of the monopole field is raised.

Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements

The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.

A vanishing theorem for fermion fields

Spaces of zero eigenvalue fermions (harmonic spinors) are constructed over S 4. A vanishing theorem is obtained, and it follows that these spaces carry irreducible unitary representations of SO(5).