Projective Unitary Representations Research Articles

On quasifree representations of infinite dimensional symplectic group

We consider an infinite dimensional generalization of metaplectic representations (Weil representations) for the (double covering of) symplectic group. Given quasifree states of an infinite dimensional CCR algebra, projective unitary representations of the infinite dimensional symplectic group are constructed via unitary implementors of Bogoliubov automorphisms. Complete classification of these representations up to quasi-equivalence is obtained.

Teleportation and entanglement distillation in the presence of correlation among bipartite mixed states

The teleportation channel associated with an arbitrary bipartite state denotes the map that represents the change suffered by a teleported state when the bipartite state is used instead of the ideal maximally entangled state for teleportation. This work presents and proves an explicit expression of the teleportation channel for teleportation using Weyl's projective unitary representation of $(Z/dZ{)}^{2n}$ for integers $d>~2,$ $n>~1,$ which has been known for $n=1.$ This formula allows any correlation among the n bipartite mixed states, and an application shows the existence of reliable schemes for distillation of entanglement from a sequence of mixed states with correlation.

Approximations for Gabor and wavelet frames

Let ψ \psi be a frame vector under the action of a collection of unitary operators U \mathcal U . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.

Tensor products of Minimal Holomorphic Representations

LetD=G/KD=G/Kbe an irreducible bounded symmetric domain with genusppandHν(D)H^{\nu }(D)the weighted Bergman spaces of holomorphic functions forν>p−1\nu >p-1. The spacesHν(D)H^\nu (D)form unitary (projective) representations of the groupGGand have analytic continuation inν\nu; they give also unitary representations whenν\nuin the Wallach set, which consists of a continuous part and a discrete part ofrrpoints. The first non-trivial discrete pointν=a2\nu =\frac a2gives the minimal highest weight representation ofGG. We give the irreducible decomposition of tensor productHa2⊗Ha2¯H^{\frac a2}\otimes \overline {H^{\frac a2}}. As a consequence we discover some new spherical unitary representations ofGGand find the expansion of the corresponding spherical functions in terms of theKK-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.

On the projective representations of the Bondi-Metzner-Sachs group

With the Bondi‐Metzner‐Sachs (BMS) group in general relativity as the main motivation and example, a theorem is proved which may be described as follows. Let G be a complex semisimple, connected and simply connected Lie group with compact real form K , and let A be a metrizable, complete and locally convex real topological vector space on which there is a continuous G action. Consider the semidirect product topological group Gx s A (which is, in general, infinite‐dimensional) constructed naturally out of G and A . If the set of equivalence classes of irreducible representations of K in A satisfies certain hypotheses, then the second cohomology group of G x s A in the sense of continuous group cohomology is trivial. When G = SL (2, C ) and A is an appropriate function space of real‐valued functions of the 2‐sphere endowed with a specific G action (e.g. A may consist of C k , k ⩾ 3, real‐valued functions defined on the 2‐sphere), the semidirect product group is the universal cover of the BMS group. The theorem implies the existence of lifting of the projective unitary representations of the BMS group to the linear unitary representations of its universal cover. In the quantum context when we consider massless quantum fields at null infinity of a non‐stationary, asymptotically Minkowskian space‐time, in place of the projective unitary representations of the BMS group, there is no loss of generality in considering the linear unitary representations of its universal cover instead.

The concept of a noncommutative Riemann surface

We consider the compactification M(atrix) theory on a Riemann surface Σ of genus g>1. A natural generalization of the case of the torus leads to construct a projective unitary representation of π 1( Σ), realized on the Hilbert space of square integrable functions on the upper half-plane. A uniquely determined gauge connection, which in turn defines a gauged sl 2( R) algebra, provides the central extension. This has a geometric interpretation as the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the 2nd Hochschild cohomology group of the Fuchsian group uniformizing Σ. Our construction can be seen as a suitable double-scaling limit N→∞, k→−∞ of a U( N) representation of π 1( Σ), where k is the degree of the associated holomorphic vector bundle, which can be seen as the higher-genus analog of 't Hooft's clock and shift matrices of QCD. We compare the above mentioned uniqueness of the connection with the one considered in the differential-geometric approach to the Narasimhan–Seshadri theorem provided by Donaldson. We then use our infinite dimensional representation to construct a C ★-algebra which can be interpreted as a noncommutative Riemann surface Σ θ . Finally, we comment on the extension to higher genus of the concept of Morita equivalence.

From Endomorphisms to Automorphisms and Back: Dilations and Full Corners

When S is a discrete subsemigroup of a discrete group G such that G = S−1S, it is possible to extend circle-valued multipliers from S to G, to dilate (projective) isometric representations of S to (projective) unitary representations of G, and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost–Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product C0(Af)⋊Q*+, corresponding to the multiplicative action of the positive rationals on the additive group Af of finite adeles.

Integrating Unitary Representations of Infinite-Dimensional Lie Groups

We show that in the presence of suitable commutator estimates, a projective unitary representation of the Lie algebra of a connected and simply connected Lie group G exponentiates to G. Our proof does not assume G to be finite-dimensional or of Banach–Lie type and therefore encompasses the diffeomorphism groups of compact manifolds. We obtain as corollaries short proofs of Goodman and Wallach's results on the integration of positive energy representations of loop groups and Diff(S1) and of Nelson's criterion for the exponentiation of unitary representations of finite-dimensional Lie algebras.

Quantum Theory as the Representation Theory of Symmetries

It is proposed that an unconstrained quantum system is completely specified by its configuration space C and its group of symmetries G . It is then shown that the theory of projective unitary representations of G on a certain Hilbert space determined by C and G leads to a definitive resolution of supposed ambiguities relating to superselection rules, choice of wave functions, topological properties of G and C , anomalies, etc. {copyright} {ital 1997} {ital The American Physical Society}

Projective representations of the 1+1-dimensional Poincaré group

The unitary irreducible representations of the central extension of the Poincaré group in 1+1 dimensions are constructed by an application of the Kirillov theory. These are then lifted to projective unitary irreducible representations of the Poincaré group. The 1+1 Galilean group is treated separately in an appendix.

Projective Systems of Imprimitivity and Applications for the Galilei Group in 1+2 and 1+3 Dimensions

The study of the projective unitary irreducible representations of the Galilei group (in 1+3 and 1+2 dimensions) is usually done using firstly some group extensions techniques (in this way one is reduced to the study of true unitary representations) and then Mackey induction procedure. In this paper we reobtain these results using a different approach based on the notion of projective systems of imprimitivity due also to Mackey. This extension of the usual Mackey procedure is presented rather extensively and illustrated by detailed computations concerning the classification of the projective unitary irreducible representations.

QUANTUM THEORY OF LANDAU AND PEIERLS ELECTRONS FROM THE CENTRAL EXTENSIONS OF THEIR SYMMETRY GROUPS

By Wigner’s theorem on symmetries, the total state space of a quantum system whose symmetries form the group G is the collection of all projective unitary representations of G; these are, in turn, realised as certain unitary representations of the set of all central extensions of G by U(1). Exploiting this relationship, we present in this paper a new approach to the quantum mechanics of an electron in a uniform magnetic field B, in the plane (the Landau electron) and on the 2-torus in the presence of a periodic potential V whose periodicity is that of an N×N lattice (the Peierls electron). For the Landau electron, G is the Euclidean group E(2) whose central extensions arise from the Heisenberg Lie group central extensions, determined by B, of the translation subgroup. The state space is a unitary representation of the direct product of two such groups corresponding to B and -B and the Hamiltonian is a unique element of the universal enveloping algebra of the centrally-extended E(2). The complete quantum theory of the Landau electron follows directly. For the Peierls electron, lattice translation-invariance is possible only if the flux per unit cell Φ takes rational values with denominator N. The state space is a unitary representation of the direct product of a finite Heisenberg group, which is a central extension of the translation group, and a Heisenberg Lie group, both characterised by Φ. The following new results are rigorous consequences. In the empty lattice limit V=0, the energy spectrum is the Landau spectrum with degeneracy equal to the total flux through the sample. As V moves away from zero, every Landau level splits into NΦ discrete sublevels, each of degeneracy N. More generally, for V≠0 of any strength and (periodic) form, and B such that Φ is nonintegral, every point in the spectrum has multiplicity N. The degeneracy is thus proportional to the linear size rather than the area of the sample. Throughout the paper, vector potentials and gauges are dispensed with and many misconceptions thereby removed.

SYMMETRIES AND QUANTIZATION: STRUCTURE OF THE STATE SPACE

This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.

A realization of the Kac-Moody algebra on holomorphic fields

We discuss the loop group G(S1,G) of mappings of the circle S1 into a compact Lie group G. We show that there exists a realization of projective unitary representation of G in the Hilbert space of functionals of holomorphic fields with values in G/sup /C (the complexification of G). We show that this representation belongs to the discrete series of positive energy representations of G We discuss the quantum field theory of holomorphic fields.

Berry phase from symmetry groups

We show that the change of the phase of any quantum state evolving cyclically under the action of the Poincaré or Galilei group is described by a natural affine connection on the symmetry group and by a projective unitary representation of that group. In the special case of a massive particle with spin in a rotating magnetic field and also in the case of photon passing through a helically wound optical fibre, our method reproduces the formulae calculated by Berry and by Chiao and Wu, respectively. Thus the relativistic and nonrelativistic approaches are considered simultaneously and we acquire a more general view of the origin of Berry's phase in quantum systems.

The projective unitary irreducible representations of the Poincaré group in 1+2 dimensions

A complete analysis is given of the projective unitary irreducible representations of the Poincaré group in 1+2 dimensions applying the Mackey theorem and using an explicit formula for the universal covering group of the Lorentz group in 1+2 dimensions. Explicit formulas are given for all representations.

The projective unitary irreducible representations of the Galilei group in 1+2 dimensions

We give an elementary analysis of the multiplicator group of the Galilei group in 1+2 dimensions 𝒢+↑. For a nontrivial multiplicator we give a list of all the corresponding projective unitary irreducible representations of 𝒢+↑.

Superconductivity in layered materials: Pairing theory for projectively translation invariant states

A mechanism for the superconducting transition in layered bulk substances, based on the existence of (non-plane wave) projective unitary representations (PURs) of their symmetry group and the generalisation of the theory of Cooper pairs to such representations, is proposed and investigated. In the continuum limit, the symmetry group appropriate to layering is a subgroup of the three-dimensional Euclidean group. It has an infinite family of PURs which can be characterised by an arbitrary real central charge in the commutators of momentum components within the layer. Quantum kinematics/dynamics in a sector with nonzero central charge (nontrivial PUR) is studied and the invariant one-particle Hamiltonian and equation of motion are shown to be formally those of a spinless charged particle in a magnetic field normal to the layers (pseudocharge in a pseudomagnetic field) irrespective of the actual charge or spin of the particle. The dielectric response of a gas of electrons in a sector with nonzero central charge is then worked out by standard Green function methods and the nature of the dynamically screened two-particle interaction determined. The influence of the pseudocyclotron frequency makes the effective potential attractive in a regime of wave number and frequency whose extent increases with increasing central charge, thus signalling instability towards the formation of (generalised) Cooper pairs. The general gap equation is then set up and, after a standard simplification, reduced to a form whose solution is known to lead to a critical temperature increasing monotonically with central charge. This model of high-temperature superconductivity is parity and time-reversal invariant and is in qualitative agreement with observations including those on the isotope effect. The fundamental differences between this model and the anion model are commented upon and speculative remarks offered on the physical origin and detection of the central charge.

Nontransititive imprimitivity systems for the Galilei group

A class of nontransitive imprimitivity systems and the corresponding projective unitary representations for the inhomogeneous Galilei group are worked out with the Mackey–Varadarajan method of group representations.

HOLOMORPHIC EXTENSIONS OF REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE

This paper gives the construction of a semigroup ? which could be thought of as the complexincation of the group Diff of analytic diffeomorphisms of the circle, and it is shown that any unitary projective representation of Diff with highest weight has a holomorphic extension to ?: For this, ? is embedded in the semigroup of endomorphisms of canonical commutation relations (this is a certain part of the Lagrange Grassmannian in complex symplectic Hilbert space). Bibliography: 25 titles.