Projective Unitary Representations Research Articles

Characterising the Haar measure on the [formula omitted]-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact p-adic special orthogonal groups of rotations SO(d)p in dimension d=2,3, by exploiting the machinery of inverse limits of measure spaces, for every prime p>2. We characterise the groups SO(d)p as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each SO(d)p. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on SO(d)p. Our results pave the way towards the study of the irreducible projective unitary representations of the p-adic rotation groups, with potential applications to the recently proposed p-adic quantum information theory.

The Role of Covariancy in Calculating Capacity of Quantum Channels Generated by Projective Unitary Representations of Non-Abelian Groups

The Role of Covariancy in Calculating Capacity of Quantum Channels Generated by Projective Unitary Representations of Non-Abelian Groups

SICs and the Triangle Group (3,3,3)

The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group $(3,3,3)$. We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.

G-frame generator sets for projective unitary representations

<abstract><p>Frames with special structures play a crucial role in various industrial applications, such as medical imaging, quantum communication, recognition and identification software, and so on. In this paper, we will discuss a more general setting, i.e., a g-frame induced by the projective unitary representation. We show some new results on the dilation property for g-frame generator sets for unitary groups and projective unitary representations. In particular, by using complete wandering operators, several properties of g-frame generators for projective unitary representations have been obtained. Moreover, we explore some characterizations of the g-frame generator dual pairs.</p></abstract>

Generalized positive energy representations of groups of jets

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra \mathfrak{k} . Consider the Fréchet–Lie group G := J_0^\infty(V; K) of \infty -jets at 0\in V of smooth maps V \to K , with Lie algebra \mathfrak{g}=J_0^\infty(V; \mathfrak{k}) . Let P be a Lie group and write \mathfrak{p}:=\operatorname{Lie}(P) . Let \alpha be a smooth P -action on G . We study smooth projective unitary representations \bar{\rho} of G\rtimes_\alpha P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by \bar{\rho}(G) . We show that this condition imposes severe restrictions on the derived representation d\bar{\rho} of \mathfrak{g}\rtimes \mathfrak{p} , leading in particular to sufficient conditions for {\bar{\rho}}|_{G} to factor through J_0^2(V; K) , or even through K .

Ʌ-Hypernuclear States as Dihadronic Molecules

The study of exotic hypernuclei attracts a great deal of interest in nuclear physics. The reality of heavy hyperon hypernuclei is the subject of intense concern among theoreticians and experimenters in recent years. The core-hyperon model uses to explain abnormal nuclei spectra, recent observations of new exotic heavy hyperon hypernuclei cannot be explained or predicted by ordinary heavy core nuclei. These exotic hypernuclei states are a two-cluster bound states. We calculate the mass spectrum and constituent mass of particles in hypernuclei using the relativistic Schrödinger equation with molecular pseudoharmonic-type potential between particles inside the core and hyperon. Such calculations represent the interaction between the hyperon and the nuclei core. I review recent theoretical studies on the ground states and the excited states of hypernuclei bound states. Finally, we present explicit predictions of the exotic bound states based on the interactions obtained from quantum field theory and the projective unitary representation model. Studies have shown that by increasing the mass number of hyperon-core states, the value of the constituent mass and energy eigenvalue of Ʌ-hypernucleus increases. Also, by growing and increasing the proton number in the (Ʌ-N) states the value of the constituent mass of Ʌ-hyperon increases.

On Bergman space zero sets. Editors’ comments on the article by Vaughan Jones

This article has two purposes. The first one is to describe what the Editors know on how and when Vaughan Jones worked on the subject of the article published in the same volume. The starting point, in the late 1980’s, was his fascination for a formula giving Murray–von Neumann dimensions of Hilbert spaces of unitary representations of Fuchsian groups. Over the years, he discovered surprising relations of these dimensions with other domains of mathematics. The second purpose is to expose with some details a subject which plays an important role in Jones’ article: the irreducible projective unitary representations of \operatorname{PSL}_2(\mathbf{R}) , which constitute a continuous family known as the discrete series, and which have interesting restrictions to various discrete subgroups of \operatorname{PSL}_2(\mathbf{R}) .

A Product Formula for Homogeneous Characteristic Functions

A bounded linear operator T on a Hilbert space is said to be homogeneous if $$\varphi (T)$$ is unitarily equivalent to T for all $$\varphi $$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $$\sigma $$ of Möb is said to be associated with an operator T if $$\varphi (T)= \sigma (\varphi )^* T \sigma (\varphi )$$ for all $$\varphi $$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy–Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $$\sigma $$ , then there is a unique projective unitary representation $${\hat{\sigma }}$$ , extending $$\sigma $$ , associated with the minimal unitary dilation of T. The representation $${\hat{\sigma }}$$ is given in terms of $$\sigma $$ by the formula $$\begin{aligned} {\hat{\sigma }} = (\pi \otimes D_1^+) \oplus \sigma \oplus (\pi _*\otimes D_1^-), \end{aligned}$$ where $$D_1^\pm $$ are two unitary representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $$H^2({\mathbb {D}})$$ , and $$\pi , \pi _*$$ are representations of Möb living on the two defect spaces of T defined explicitly in terms of $$\sigma $$ . Moreover, a cnu contraction T has an associated representation if and only if its Sz.-Nagy–Foias characteristic function $$\theta _T$$ has the product form $$\theta _T(z) = \pi _*(\varphi _z)^* \theta _T(0) \pi (\varphi _z),$$ $$z\in {\mathbb {D}}$$ , where $$\varphi _z$$ is the involution in Möb mapping z to 0. We obtain a concrete realization of this product formula for a large subclass of homogeneous cnu contractions from the Cowen–Douglas class.

Asymptotic representations of Hamiltonian diffeomorphisms and quantization

We show that for a special class of geometric quantizations with “small” quantum errors, the quantum classical correspondence gives rise to an asymptotic projective unitary representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.

On quantum channels generated by covariant positive operator-valued measures on a locally compact group

We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group G with its dual \({\hat{G}}\). The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert–Schmidt operators in \(L^2(G)\) and the Hilbert space \(L^2({\hat{G}}\times G)\). Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble of quantum states on the group. It is shown that the second channel can be called a complementary channel to the measurement channel.

An approach to p-adic qubits from irreducible representations of SO(3)p

We introduce the notion of a p-adic quantum bit (p-qubit) in the context of the p-adic quantum mechanics initiated and developed after the seminal paper of Volovich [Theor. Math. Phys. 71, 574 (1987)]. In this approach, physics takes place in a three-dimensional p-adic space rather than Euclidean space. Based on our prior work describing the p-adic special orthogonal group [Di Martino et al., arXiv:2104.06228 [math.NT] (2021)], we outline a program to classify its continuous unitary projective representations, which can be interpreted as a theory of p-adic angular momentum. The p-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes p.

On quantum tomography on locally compact groups

We introduce quantum tomography on locally compact Abelian groups G. A linear map from the set of quantum states on the C⁎-algebra A(G) generated by the projective unitary representation of G to the space of characteristic functions is constructed. The dual map determining symbols of quantum observables from A(G) is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which G=R (the optical tomography), G=Zn (corresponding to measurements in mutually unbiased bases) and G=T (the tomography of the phase). As an application we have calculated the quantum tomogram for the output states of quantum Weyl channels.

On capacity of quantum channels generated by irreducible projective unitary representations of finite groups

We study mixed unitary quantum channels generated by irreducible projective unitary representations of finite groups. Under some assumptions on the probability distribution determining a mixture the classical capacity of the channel is found. Examples illustrating the techniques are given.

Inhomogeneous Conformal Field Theory Out of Equilibrium

We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity v(x) explicitly breaking translation invariance. Such inhomogeneous CFT is argued to effectively describe 1+1-dimensional quantum many-body systems with certain inhomogeneities varying on mesoscopic scales. Both heat and charge transport are studied, where, for concreteness, we suppose that our CFT has a conserved U(1) current. Based on projective unitary representations of diffeomorphisms and smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states defined by smooth inverse-temperature and chemical-potential profiles β(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (x)$$\\end{document} and μ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (x)$$\\end{document}. Using this recipe, the following exact analytical results are obtained: (i) the full time evolution of densities and currents for heat and charge transport, (ii) correlation functions for components of the energy–momentum tensor and the U(1) current as well as for any primary field, and (iii) the thermal and electrical conductivities. The latter are computed by direct dynamical considerations and alternatively using a Green–Kubo formula. Both give the same explicit expressions for the conductivities, which reveal how inhomogeneous dynamics opens up the possibility for diffusion as well as implies a generalization of the Wiedemann–Franz law to finite times within CFT.

Mass Spectra of Mesonic Molecules at Finite Temperature

At a finite temperature, exotic hadronic systems, mass spectrum was studied using the quark structure and the radial Schrödinger equation with the Cornell interaction potential representing the hadronic interaction. The projective unitary representation has been used in the study of hadronic ground states in physics. A relativistic bound state's mass spectrum with temperature dependence is shown to have a unique feature. The resulting values are compared to experimental and theoretical values, and the study showed a high level of conformity with additional values wherever feasible.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of mathrm{Diff}_+(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms mathcal {D}^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms mathrm{Diff}_+^k(S^1) with kge 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to mathcal {D}^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also mathcal {D}^s(S^1)-covariant.

A Duality Principle for Groups II: Multi-frames Meet Super-Frames

The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that {pi _{Lambda }(m, n)g_{1} oplus cdots oplus pi _{Lambda }(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d})oplus cdots oplus L^{2}({mathbb {R}},^{d}) if and only if cup _{i=1}^{k}{pi _{Lambda ^{o}}(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a Riesz sequence, and cup _{i=1}^{k} {pi _{Lambda }(m, n)g_{i}}_{m, nin {mathbb {Z}}^{d}} is a frame for L^{2}({mathbb {R}},^{d}) if and only if {pi _{Lambda ^{o}}(m, n)g_{1} oplus cdots oplus pi _{Lambda ^{o}}(m, n)g_{k}}_{m, n in {mathbb {Z}}^{d}} is a Riesz sequence, where pi _{Lambda } and pi _{Lambda ^{o}} is a pair of Gabor representations restricted to a time–frequency lattice Lambda and its adjoint lattice Lambda ^{o} in {mathbb {R}},^{d}times {mathbb {R}},^{d}.

Quantization of Integrable Systems with Spectral Parameter on a Riemann Surface

Given an integrable system defined by a Lax representation with spectral parameter on a Riemann surface, we construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields by means of operators of covariant derivatives with respect to the Knizhnik–Zamolodchikov connection. It is a Dirac-type prequantization of the integrable system from a physical point of view. Simultaneously, it establishes a correspondence between integrable systems in question and conformal field theories. In the present paper, we focus on systems whose spectral curves possess a holomorphic involution. Examples are presented by Hitchin systems of the types Bn, $${{C}_{n}}$$ , $${{D}_{n}}$$ , and also of the type An on hyperelliptic curves.

On the Lévy-Leblond–Newton equation and its symmetries: a geometric view

The Lévy-Leblond–Newton (LLN) equation for non-relativistic fermions with a gravitational self-interaction is reformulated within the framework of a Bargmann structure over a -dimensional Newton–Cartan (NC) spacetime. The Schrödinger–Newton (SN) group introduced in Duval and Lazzarini (2015 Class. Quantum Grav. 32 175006) as the maximal group of invariance of the SN equation, turns out to be also the group of conformal Bargmann automorphisms preserving the coupled Lévy-Leblond and NC gravitational field equations. Within the Bargmann geometry a generalization of the LLN equation is provided as well. The canonical projective unitary representation of the SN group on four-component spinors is also presented. In particular, when restricted to dilations, the value of the dynamical exponent is recovered as previously derived in Duval and Lazzarini (2015 Class. Quantum Grav. 32 175006) for the SN equation. Subsequently, conserved quantities associated to the (generalized) LLN equation are also exhibited.

The operator algebra content of the Ramanujan–Petersson problem

Let G be a discrete countable group, and let \Gamma be an almost normal subgroup. In this paper we investigate the classification of (projective, with 2-cocycle \varepsilon\in H^2(G,\mathbb T) ) unitary representations \pi of G into the unitary group of the Hilbert space l^2(\Gamma, \varepsilon) that extend the (projective, with 2-cocycle \varepsilon ) unitary left regular representation of \Gamma . Representations with this property are obtained by restricting to G (projective) unitary square integrable representations of a larger semisimple Lie group \bar{G} , containing G as dense subgroup and such that \Gamma is a lattice in \bar{G} . This type of unitary representations of of G appear in the study of automorphic forms. We obtain a classification of such (projective) unitary representations and hence we obtain that the Ramanujan–Petersson problem regarding the action of the Hecke algebra on the Hilbert space of \Gamma -invariant vectors for the unitary representation \pi\otimes \bar{\pi} is an intrinsic problem on the outer automorphism group of the skewed, crossed product von Neumann algebra {\mathcal L(G \rtimes_{\varepsilon} L^{\infty}(\mathcal G,\mu))} , where \mathcal G is the Schlichting completion of G and \mu is the canonical Haar measure on \mathcal G .