Continuous Unitary Representation Research Articles

Error bounds for Lie group representations in quantum mechanics

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated with the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group, and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy-constrained diamond norm distance of any suitably continuous channel representation of the group.

Fourier transform on compact Hausdorff groups

This article deals with the generalization of the abstract Fourier analysis on the compact Hausdorff group. In this paper, the generalized Fourier transform F is defined as F (?)(?) = R ?(h)M? (h?1) d? (h) for all ? ? L2 (G) ? L1 (G), where M? is a continuous unitary representation M? : G ? UC (Cn(?)) of the group G in Cn(?), and its properties are studied. Also, we define the symplectic Fourier transform and the generalized Wigner function WA (?, ?) and establish the Moyal equality for the Wigner function. We show that the homomorphism ? : G ? U (L2 (G/K,H1)) induced by ? : G ? (G/K) ? U(H1) by (? (?)) (g, h) = (? (h?1, g))?1 (? (h?1g)), g ? G/K, h ? G, ? ? L2 (G/K,H1) is a unitary representation of the group G, assuming the mapping h 7? (?(?)) (g, h) is continuous as morphism G ? U (L2 (G/K,H1)). We study the unitary representation ?? : G ? H induced by the unitary representation V : K ? U(H1) given by ??g (?) (t) = ? (g?1t) for all t ? G/K.

A covariant Stinespring theorem

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.

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.

An amenability-like property of finite energy path and loop groups

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B(ℋ π ).

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.

Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak {g}}} = {{\mathfrak {h}}} \oplus {{\mathfrak {q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal {H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak {q}}},{{\mathfrak {q}}}] \oplus i{{\mathfrak {q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak {h}}} \oplus i{{\mathfrak {q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

Almost Connected DMAP Groups Are FIR

Let DMAP be the class of locally compact groups that admit a (not necessarily continuous) embedding in a compact topological group, and let FIR be the class of locally compact groups all of whose continuous irreducible unitary representations are finite-dimensional. We prove that every almost connected DMAP group is an FIR group.

Unitary representations of infinite wreath products

Using C∗-algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.

An exotic group as limit of finite special linear groups

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.

ASYMPTOTIC SYMMETRIES IN 3-DIM GENERAL RELATIVITY: THE B(2,1) CASE

The ordinary Bondi—Metzner—Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space—times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). Strongly continuous unitary irreducible representations (IRs) of B(2, 1), the analogue of B in three space—time dimensions, are analysed in the Hilbert topology. It is proved that all IRs of B(2, 1) are induced from IRs of compact ‘little groups’, which are the closed subgroups of SO(2). It is proved that all IRs of B(2, 1) are obtained by Wigner—Mackey’s inducing construction notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups

This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and $$\mu $$ be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let $$(\pi ,\mathcal {H}_\pi )$$ be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to $$\pi $$ on the Hilbert function $$L^2(G/H,\mu )$$ . We then study basic analytic properties of these transforms.

On the representation theory of the Bondi–Metzner–Sachs group and its variants in three space–time dimensions

The original Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, McCarthy classified all relativistic B-invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here we introduce the analogue B(2, 1) of the BMS group B in 3 space–time dimensions. B(2, 1) itself admits thirty-four analogues both real in all signatures and in complex space–times. In order to find the IRS of both B(2, 1) and its analogues, we need to extend Wigner–Mackey’s theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2, 1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space–time dimensions, d≥3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2, 1). The main results of the representation theory are as follows: The IRS are induced from “little groups” which are compact. The finite “little groups” are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

Smoothing operators and C∗-algebras for infinite dimensional Lie groups

A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the derived representation, for [Formula: see text] in a neighborhood of [Formula: see text], are uniformly bounded from above, we show that [Formula: see text] coincides with the space of smooth vectors for the one-parameter group [Formula: see text]. As the main application of our results on smoothing operators, we present a new approach to host [Formula: see text]-algebras for infinite dimensional Lie groups, i.e. [Formula: see text]-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of [Formula: see text]. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

Factorial Unitary Representations of the Translational Group, Invariant Pure States and the Super Symmetric Graviton

Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define a representation of T as a group of automorphisms of the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. Provided this group representation α is weakly measurable, then we have previously proved that it is also norm continuous, and is implemented by a norm, hence weakly and strongly, continuous unitary representation, by unitaries in the algebra. We discuss the dependence of this key result on the strange properties of Stonean spaces and extend our exploitation of Mackey Theory to give a more straightforward group theoretic proof. From such unitary representations a minimal form of Supersymmetry naturally emerges, which predicts the existence of both the mass zero graviton, and its gravitino partner.

Representations of the Bondi—Metzner—Sachs group in three space—time dimensions

The original Bondi–Metzner–Sachs group B is the common asymptotic symmetry group of all asymptotically at Lorentzian 4–dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B–invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here, we construct the IRS of B(2, 1), the analogue of B, in 3 space–time dimensions. The IRS are induced from ‘little groups’ which are compact. The finite ‘little groups’ are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

First results on the representation theory of the Ultrahyperbolic BMS group UHB(2, 2)

The Bondi–Metzner–Sachs (BMS) group B is the common asymptotic group of all asymptotically flat (lorentzian) space–times, and is the best candidate for the universal symmetry group of General Relativity (G.R.). B admits generalizations to real space–times of any signature, to complex space–times, and supersymmetric generalizations for any space— time dimension. With this motivation McCarthy constructed the strongly continuous unitary irreducible representations (IRs) of B some time ago, and he identified B(2,2) as the generalization of B appropriate to the to the ultrahyperbolic signature (+,+,−,−) and asymptotic flatness in null directions. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). We report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner–Mackeys inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology.

Lp-Fourier and Fourier–Stieltjes algebras for locally compact groups

Let G be a locally compact group and 1 ≤ p < ∞ . A continuous unitary representation π : G → B ( H ) of G is an L p -representation if the matrix coefficient functions s ↦ 〈 π ( s ) x , x 〉 lie in L p ( G ) for sufficiently many x ∈ H . The L p -Fourier algebra A L p ( G ) is defined to be the set of matrix coefficient functions of L p -representations. Similarly, the L p -Fourier–Stieltjes algebra B L p ( G ) is defined to be the weak*-closure of A L p ( G ) in the Fourier–Stieltjes algebra B ( G ) . These are always ideals in the Fourier–Stieltjes algebra containing the Fourier algebra A ( G ) . In this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras. As an application of this theory, we characterize the Fourier–Stieltjes ideals of SL ( 2 , R ) .

On an invariance property of the space of smooth vectors

Let $(\pi, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $\pi^\#(g,t) = \pi(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_\gamma \mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ of smooth vectors of $\pi$ under the operators $U_f = \int_\mathbb R f(t)U_t\, dt$ for $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$. Using this theorem we show that, for suitably defined spectral subspaces $\mathfrak g_{\mathbb C}(E)$, $E \subseteq \mathbb R$, in the complexified Lie algebra $\mathfrak g_{\mathbb C}$, and $\mathcal H^\infty(F)$, $F\subseteq \mathbb R$, for $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}\pi(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

On the Representation Theory of the Ultrahyperbolic BMS group UHB(2, 2). I. General Results

The Bondi-Metzner-Sachs (BMS) group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of General Relativity (G.R.). B admits generalizations to real space-times of any signature, to complex space-times, and supersymmetric generalizations for any space- time dimension. With this motivation McCarthy constructed the strongly continuous unitary irreducible representations (IRs) of B some time ago, and he identified B(2,2) as the generalization of B appropriate to the to the 'ultrahyperbolic signature' (+,+,−,−) and asymptotic flatness in null directions. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). In this short paper we report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner-Mackey's inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology. At the end of the paper we comment on the significance of these results.