Projective Unitary Representations Research Articles

Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.

Operator valued frames and structured quantum channels

Operator-valued frames are natural generalization of frames that have been used in many applied areas such as quantum computing, packets encoding and sensor networks. We focus on developing the theory about operator-valued frame generators for projective unitary representations of finite or countable groups which can be viewed as the theory of quantum channels with group structures. We present new results for operator-valued frames concerning (general and structured) dilation property, orthogonal frames, frame representation and dual frames. Our results are complementary to some of the recent work of Kaftal, Larson and Zhang, and in some cases our treatment is more elementary and transparent.

Groups with faithful irreducible projective unitary representations

For a countable group Γ and a multiplier , we study the property of Γ having a unitary projective -representation which is both irreducible and projectively faithful. We show that this property is equivalent to Γ being the quotient of an appropriate group by its centre. Moreover, we give a criterion in terms of the minisocle of Γ. Several examples are described to show the existence of various behaviours.

Orthonormal dilations of non-tight frames

We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson, and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets due to Dutkay, Han, Picioroaga, and Sun by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.

Structure, classification, and conformal symmetry of elementary particles over non-archimedean space-time

It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the p-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over p-adic fields and discuss the imbedding of the p-adic Poincaré group into the p-adic conformal group. Finally, we show that the massive and the so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.

Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean Space–Time

It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of spacetime may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the p-adic numbers, is explored. Both the Poincaré and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over p-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved.

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4 k + 1 , generalizing positive energy representations of the loop group of the circle at level 2. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations being finite is determined by the cohomology of the manifold.

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time–frequency lattice is a frame for L 2 ( R d ) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II 1 factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Let K and L be two full-rank lattices in R d . We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time–frequency lattice K × L . Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K × L is less than or equal to 1 2 . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume v ( K × L ) ⩽ 1 2 or v ( K × L ) ⩾ 2 . Moreover, if K = α Z d , L = β Z d with α β = 1 , then a subspace Gabor frame G ( g , L , K ) has a tight Gabor pseudo-dual only when G ( g , L , K ) itself is already tight.

The quantum dilogarithm and representations of quantum cluster varieties

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

Frame duality properties for projective unitary representations

Let π be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set B π of Bessel vectors for π is dense in H, then for any vector x ∈ H the analysis operator Θ x makes sense as a densely defined operator from B π to l 2 (G)-space. Two vectors x and y are called π-orthogonal if the range spaces of Θ x and Θ y are orthogonal, and they are π-weakly equivalent if the closures of the ranges of Θ x and Θ y are the same. These properties are characterized in terms of the commutant of the representation. It is proved that a natural geometric invariant (the orthogonality index) of the representation agrees with the cyclic multiplicity of the commutant of π(G). These results are then applied to Gabor systems. A sample result is an alternate proof of the known theorem that a Gabor sequence is complete in L 2 (ℝ d ) if and only if the corresponding adjoint Gabor sequence is l 2 -linearly independent. Some other applications are also discussed.

Frame representations and Parseval duals with applications to Gabor frames

Let {x n } be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {x n } which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether (x n ) can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame {π(g)ξ: g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π(g)ξ: g ∈ G} can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame {π(g)ξ:g ∈ G} (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame G(g, L, K) (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of L × K is less than or equal to 1/2.

Projective Representations and Pure Pseudorepresentations of Hermitian Symmetric Simple Lie Groups

It is shown that an arbitrary irreducible continuous unitary projective representation of a simple Hermitian symmetric Lie group is generated by a strongly continuous pure unitary pseudorepresentation of the adjoint group of the Lie group.

LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES

A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

Extremal covariant positive operator valued measures

We consider the convex set of positive operator valued measures (POVM) which are covariant under a finite dimensional unitary projective representation of a group. We derive a general characterization for the extremal points, and provide bounds for the ranks of the corresponding POVM densities, also relating extremality to uniqueness and stability of optimized measurements. Examples of applications are given.

The uniqueness of the dual of Weyl–Heisenberg subspace frames

From the Weyl–Heisenberg (WH) density theorem, it follows that a WH-frame (g mα,nβ) m,n∈ Z for L 2( R) has a unique WH-dual if and only if αβ=1. However, the same argument does not apply to the subspace WH-frame case and it is not clear how to use standard methods of Fourier analysis to deal with this situation. In this paper, we apply operator algebra theory to obtain a very simple necessary and sufficient condition for a given frame (induced by a projective unitary representation of a discrete group) to admit a unique dual (induced by the same system). As a special case, we obtain a characterization for the subspace WH-frames that have unique WH-duals (within the subspace). Using this characterization and the Zak transform, we are able to prove that if (g mα,nβ) m,n∈ Z is a WH-frame for a subspace M of L 2( R) , then, (i) (g mα,nβ) m,n∈ Z has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (g mα,nβ) m,n∈ Z has a unique WH-dual in M if and only if (g mα,nβ) m,n∈ Z is a Riesz sequence; (iii) if αβ<1, then the WH-dual for (g mα,nβ) m,n∈ Z in M is not unique.

Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space

A systematic approach to the C*-Weyl algebra W(E,σ) over a possibly degenerate pre-symplectic form σ on a real vector space E of possibly infinite dimension is elaborated in an almost self-contained manner. The construction is based on the theory of Kolmogorov decompositions for σ-positive-definite functions on involutive semigroups and their associated projective unitary group representations. The σ-positive-definite functions provide also the C*-norm of W(E,σ), the latter being shown to be *-isomorphic to the twisted group C*-algebra of the discrete vector group E. The connections to related constructions are indicated. The treatment of the fundamental symmetries is outlined for arbitrary pre-symplectic σ. The construction method is especially applied to the trivial symplectic form σ=0, leading to the commutative Weyl algebra over E, which is shown to be isomorphic to the C*-algebra of the almost periodic continuous function on the topological dual Eτ′ of E with respect to an arbitrary locally convex Hausdorff topology τ on E. It is demonstrated that the almost periodic compactification aEτ′ of Eτ′ is independent of the chosen locally convex τ on E, and that aEτ′ is continuously group isomorphic to the character group Ê of E. Applications of the results to the procedures of strict and continuous deformation quantizations are mentioned in the outlook.

Tight frame approximation for multi-frames and super-frames

We consider a generator Φ=(φ 1, … ,φ N) for either a multi-frame or a super-frame generated under the action of a projective unitary representation for a discrete countable group. Examples of such frames include Gabor multi-frames, Gabor super-frames and frames for shift-invariant subspaces. We show that there exists a unique normalized tight multi-frame (resp. super-frame) generator Ψ=( ψ 1,…, ψ N ) such that ∑ j=1 N || φ j − ψ j || 2⩽∑ j=1 N || φ j − ψ j || 2 holds for all the normalized tight multi-frame (resp. super-frame) generators η=( η 1,…, η N ). We also investigate the similar problems for dual frames and discuss a few applications to Gabor frames and some other frames.

On Infinite Tensor Products of Projective Unitary Representations

We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.