Projective unitary antiunitary representations of locally compact groups

We give here a systematic presentation of the theory of projective representations when antiunitary operators are present. In particular the imprimitivity theorem of Mackey is proved in this situation and all the unitary antiunitary representations of the extended Poincaré group are derived.     

Dynamical group quantization

The projective unitary irreducible representations U(G) of the space-time symmetry group G provides a unique quantization scheme for elementary particles. By extension a direct method of quantization for more general systems by the projective unitary representations U^$\text{G}$(G) induced from a dynamical group $\text{G}$ is outlined. Reducible relativistic composite systems are defined and the geometry of $\text{G}$ is discussed.     

On locally continuous cocycles

Cohomology groups of locally continuous cocycles of a topological group G with values in a topological G-module A_Ψ are considered and the role of those of degree 2 in the theory of topological extensions of G by A_Ψ is analyzed. The exactness of long sequence is proven. It is shown that, if G and A_Ψ are Polish and if all topological extensions of G by A_Ψ are fibered, the cohomology groups of Borel cocycles and those of locally continuous Borel cocycles are isomorphic for degree 0,1,2.     

On the irreducible representations of groups that contain a subgroup of index 2

The objects under consideration are a groupG containing a subgroupN of index 2 and an irreducible multiplier representationU ofG by semiunitary (=unitary or antiunitary) operators on a complex Hilbert space of arbitrary dimension. It is assumed thatU(g) is unitary for allg belonging toN. Then the following assertion is proved. The representation ofN that is obtained by restrictingU toN is either irreducible or an orthogonal sum of two irreducible representations.     

Unitary representations of some infinite dimensional groups

We construct projective unitary representations of (a) Map(S ¹;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S ¹;T), whereT is a maximal torus ofG, can be extended to representations of Map(S ¹;G),     

On Gapped Phases with a Continuous Symmetry and Boundary Operators

We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different methods, for G=SU(2) in one, and G?SU(d), in the other we construct explicitly an ‘excess spin’ operator that implements rotations of half of the infinite chain on the GNS Hilbert space of the ground state of the full chain. Since this operator is constructed as the limit of a sequence of observables, the representation itself is, in principle, experimentally observable. We claim that the corresponding unitary representation of G is closely related to the representation found at the boundary of half-infinite chains. We conclude with determining the precise relation between the two representations for the class of frustration-free models with matrix product ground states.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

Some general properties of representations of local currents

Some properties of representations of local current operators are studied. The currents are assumed to be conserved and to have charge densities transforming like the regular representation of any internal symmetry group G containing the isospin SU₂. The representation space is the “physical” Hilbert space, having a positive definite metric and carrying time-like positive-energy representations of the Poincaré group. The main results are that in every irreducible representation space, (A) arbitrarily large irreducible representations of G must occur, and (B) the mass spectrum is unbounded and continuous from some point onwards if it is not strictly degenerate. These results have strong implications for current algebra saturation schemes, both at finite and infinite momentum.     

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.     

1- Products in the method of orbits for nilpotent groups

On each orbit W of the coadjoint representation of any nilpotent (connected, simply connected) Lie group G, we construct 1-products and associated Von Neumann algebras $\text{G}$. G acts canonically on $\text{G}$ by automorphisms. In the unique faithful, irreducible representation of $\text{G}$, this action is implemented by the unitary irreducible representation of G corresponding to W by the Kirillov method. This construction is uniquely determined by W and gives the classification of all unitary irreducible representations of G.     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Representations of the infinite direct product of universal coverings of isometry groups of the complex ball

In this paper we consider the group G?_M consisting of measurable functions on the set M with values in the group G? which is the universal covering of the group G of isometries of the n-dimensional complex ball. We construct a family of irreducible unitary representations of this group. The main elements of the construction are irreducible representations of the group G? obtained by interpolation from the representations of the discrete series of the group G.Suppose G? is a group, M a set with measure. Denote by G?_M the group of G?-valued functions on M. (The group G?_M does not necessarily contain all the functions). It is natural to consider the group G?_M as the direct product of ? copies of the group G?, where ? is the cardinality of the set M. In the case where the set M has no points of non-zero measure, it is natural to call G?_M the continuous tensor product of ? copies of G?. In the case where G? is the Heisenberg–Weyl group $\text{1}\text{a}\text{c}\text{0}\text{1}\text{b}\text{0}\text{0}\text{1}$ the group G?_M has been known since the end of the 1920, in view of the fact that the Lie algebra of this group is the Lie algebra of the Bose commutation relations.¹ The case where the group G? is semi-simple apparently first occurred in the theory ofcalibrated fields. A number of physical and mathematical papers (see [1], [2], [8]–[11]) are concerned with the groups G?_M and their linear representations. The present paper deals with the case where the group G? is the universal covering of the isometry group G of the complex ball (i.e. the domain in Cⁿ determined by the inequality ∑ z_kz?_k < 1). No conditions are imposed on the set M. By comparison with the papers [1], [2], [8]–[10], the explicit construction of the representations, copying the Fock representation, is new, as well as the proof of their irreducibility.The construction proposed below may be generalized to the case where G is the isometry group of any homogeneous domain Ω in Cⁿ, G? is the one-dimensional central extension of G. However, if Ω is a classical symmetrical domain (which is not a ball) and the set M contains no points of non-zero measure, it follows from [6] that the representation thus obtained is unitary in the indefinite sense. In the case where Ω = Cⁿ and G is the group of parallel translations G?—the Heisenberg-Weyl group, the proposed construction becomes the Fock construction of the representation of commutation relations. The other cases have not been studied yet. The main results of this paper have been (briefly) published in [4].     

On the representations of deformed Galilei group

The projective representations of k-Galilei group G _k are found by contracting the relevant representations of –Poincare group. The projective multiplier is found. It is shown that it is not possible to replace the projective representations of G _k by vector representations of some its extension.     

Extremal vectors for Verma-type representations of su(2, 2)

Starting from the Verma modules of the algebra sl(4, ?) we explicitly construct factor representations of the algebra su(2, 2) which are connected with unitary representation of group SU(2, 2). We find a full set of extremal vectors for this kind of representations, so we can solve explicitly the problem of irreducibility of these representations.     

A note on coherent states

Given a connected Lie groupG with an Abelian invariant Lie subgroup and a continuous unitary representation ofG on the Hilbert space ?, we investigate a relationship between the first cohomology groupH ¹(G, ?) and classes of sectors, determined by coherent states with a projectivelyG-covariant Weyl system. This result is applied to calculateH ¹(G, ?), if the groupG has in addition a compact subgroup with certain properties.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

The metaplectic semigroup and related topics

There exists an extension of the projective representation of the real symplectic group associated with the CCR's to a projective representation of a semigroup S such that $\text{Sp(2n,}\text{R}\text{) ? S ? Sp(2n, C)}$. It is proven that S is connected, and the universal covering semigroup S? of S is then constructed. It is also shown that the projective representation V? of S can be lifted to a strongly continuous representation V of S. The representation operators are studied; it is shown that they are injective contractions, and that they leave a subspace isomorphic to the Schwartz space invariant. Finally, the one- parameter contraction subsemigroups of the representation are discussed and their generators are computed.     

The representations of the oscillator group

Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force.