Unitary Representations Research Articles

A simplified approach to the holomorphic discrete series

Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.

Realization of unitary representations of the Lorentz group on de Sitter space

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups G acting on a non-compactly causal symmetric space M=G/H, every irreducible unitary representation of G can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over M. In the present paper we discuss this procedure for the connected Lorentz group G=SO1,d(R)e acting on de Sitter space M=dSd. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain Gℂcut in the complexified group Gℂ=SO1,d(ℂ), which for d=1 specializes to the complex cut plane ℂ∖(−∞,0]. A number of special situations is discussed specifically: (a) The case d=1, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for d>2 is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case d=3, where we obtain rather explicit formulas for the matrix-valued spherical functions.

A universal lower bound for certain quadratic integrals of automorphic L–functions

Let π be a cuspidal unitary representation od GL(m,A) where A denotes the ring of adèles of Q. Let L(s,π) be its L-function. We introduce a universal lower bound for the integral ∫−∞+∞|L(12+it,π)12+it−s|2dt where s is equal to 0 or is a zero of L(s) on the critical line. In the main text, the proof is given for m≤2 and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series L(s,π). In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.

Balancing the Senses: Electrophysiological Responses Reveal the Interplay between Somatosensory and Visual Processing During Body-Related Multisensory Conflict.

In the study of bodily awareness, the predictive coding theory has revealed that our brain continuously modulates sensory experiences to integrate them into a unitary body representation. Indeed, during multisensory illusions (e.g., the rubber hand illusion, RHI), the synchronous stroking of the participant's concealed hand and a fake visible one creates a visuotactile conflict, generating a prediction error. Within the predictive coding framework, through sensory processing modulation, prediction errors are solved, inducing participants to feel as if touches originated from the fake hand, thus ascribing the fake hand to their own body. Here, we aimed to address sensory processing modulation under multisensory conflict, by disentangling somatosensory and visual stimuli processing that are intrinsically associated during the illusion induction. To this aim, we designed two EEG experiments, in which somatosensory- (SEPs; Experiment 1; N = 18; F = 10) and visual-evoked potentials (VEPs; Experiment 2; N = 18; F = 9) were recorded in human males and females following the RHI. Our results show that, in both experiments, ERP amplitude is significantly modulated in the illusion as compared with both control and baseline conditions, with a modality-dependent diametrical pattern showing decreased SEP amplitude and increased VEP amplitude. Importantly, both somatosensory and visual modulations occur in long-latency time windows previously associated with tactile and visual awareness, thus explaining the illusion of perceiving touch at the sight location. In conclusion, we describe a diametrical modulation of somatosensory and visual processing as the neural mechanism that allows maintaining a stable body representation, by restoring visuotactile congruency under the occurrence of multisensory conflicts.

The hidden Lorentz covariance of quantum mechanics

This paper introduces a systematic algorithm for deriving a new unitary representation of the Lorentz algebra (so(1,3)) and an irreducible unitary representation of the extended (anti) de-Sitter algebra (so(2,4)) on L2(R3,1r). This representation is equivalent to a representation on L2(R3), and the corresponding similarity transformation is identified. An explicit representation in terms of differential operators is given, and it is shown that the inner product is Lorentz invariant. Ensuring Lorentz covariance demands a modification of the Heisenberg algebra, recognized as a phase space algebra at the interface of gravitational and quantum realms (IGQR), which we consider subordinate to Lorentz covariance. It is also demonstrated that time evolution can be cast in a manifestly covariant form. Each mass sector of the Hilbert space carries a representation of the Lorentz algebra, and the (anti) de-Sitter algebra on each mass sector contracts to the Poincare algebra in the flat configuration and momentum space limits. Finally, we show that three-dimensional fuzzy space also carries a unitary representation of these algebras, algebraically equivalent to the L2(R3,1r) representation but not necessarily equivalent as representations. Several outstanding issues are identified for future exploration.

3d super hyperbolic geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSpC(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The invariant quadratic form is x1x2−xx¯+ϕψ+ϕ¯ψ¯ in Wigner's coordinates x1,x2∈R and x∈C, where ϕ,ψ are Dirac fermions. The extended action is linear in the super space variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie super group with its Lie algebra. This extension of Wigner's representation opens the door to studying geometry in super hyperbolic three-space with this metric, initiated here with foundations on super geodesics, triangles and tetrahedra, and culminating in a proof of divergence of the volume of a typical ideal tetrahedron and a discussion of the non-zero fermionic correction to the Schläfli formula.

Dissipative dynamics for infinite lattice systems

We study dissipative dynamics constructed by means of non-commutative Dirichlet forms for various lattice systems with multiparticle interactions associated to CCR algebras. We give a number of explicit examples of such models. Using an idea of quasi-invariance of a state, we show how one can construct unitary representations of various groups. Moreover in models with locally conserved quantities associated to an infinite lattice we show that there is no spectral gap and the corresponding dissipative dynamics decay to equilibrium polynomially in time.

Gain total graphs and their spectra via G-phases and group representations

We introduce a definition for the total graph of a gain graph (Γ,ψ) on a group G by using G-phases, which are a generalization of the notion of orientation to gain graphs. Our construction is well-defined in the sense that gain graphs that are switching isomorphic have switching isomorphic total graphs. In particular, the switching equivalence class of the total graph does not depend on the particular choice of the G-phase associated with ψ. More precisely, we consider the left-right multiplication action of G on the space of all G-phases, proving that its orbits consist of sets of G-phases inducing the same switching equivalence class of gain functions on Γ or, equivalently, on its total graphs. Moreover, we prove that the stabilizers of this action are isomorphic to the centralizers of the sets of gains of closed walks in the associated gain graphs. Our construction is consistent with the existing notions of total graph for signed graphs and, in analogy with the signed case, we are able to explicitly compute the spectrum of the total graph of a regular gain graph over an arbitrary group G, where the spectrum is defined by means of a unitary representation of the gain group G.

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.

Scalar two-point functions at the late-time boundary of de Sitter

We calculate two-point functions of scalar fields of mass m and their conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies boundary conditions, in general d + 1 spacetime dimensions. We perform the calculation using the wavefunction picture and using canonical quantization. With the latter one clearly sees how the late-time field and conjugate momentum operators are linear combinations of the normalized late-time operators αN and βN that correspond to unitary irreducible representations of the de Sitter group with well-defined inner products. The two-point functions resulting from these two different methods are equal and we find that both the autocorrelations of αN and βN and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars m<d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m<\\frac{d}{2}H\\right) $$\\end{document}, corresponding to complementary series representations, and heavy scalars m>d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m>\\frac{d}{2}H\\right) $$\\end{document}, corresponding to principal series representations of the de Sitter group, where H is the Hubble scale of de Sitter. In the special case m = 0, only the βN autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest αN to correspond to discrete series representations for this case at d = 3.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

The Relative Contribution of Facial and Body Information to the Perception of Cuteness.

Faces and bodies both provide cues to age and cuteness, but little work has explored their interaction in cuteness perception. This study examines the interplay of facial and bodily cues in the perception of cuteness, particularly when these cues convey conflicting age information. Participants rated the cuteness of face-body composites that combined either a child or adult face with an age-congruent or incongruent body alongside manipulations of the head-to-body height ratio (HBR). The findings from two experiments indicated that child-like facial features enhanced the perceived cuteness of adult bodies, while child-like bodily features generally had negative impacts. Furthermore, the results showed that an increased head size significantly boosted the perceived cuteness for child faces more than for adult faces. Lastly, the influence of the HBR was more pronounced when the outline of a body's silhouette was the only available information compared to when detailed facial and bodily features were presented. This study suggests that body proportion information, derived from the body's outline, and facial and bodily features, derived from the interior surface, are integrated to form a unitary representation of a whole person in cuteness perception. Our findings highlight the dominance of facial features over bodily information in cuteness perception, with facial attributes serving as key references for evaluating face-body relationships and body proportions. This research offers significant insights into social cognition and character design, particularly in how people perceive entities with mixed features of different social categories, underlining the importance of congruency in perceptual elements.

Carrollian conformal correlators and massless scattering amplitudes

The theory of particle scattering is concerned with transition amplitudes between states that belong to unitary representations of the Poincaré group. The latter acts as the isometry group of Minkowski spacetime \U0001d544, making natural the introduction of relativistic tensor fields encoding the particles of interest. Since the Poincaré group also acts as a group of conformal isometries of null infinity ℐ, massless particles can also be very naturally encoded into Carrollian conformal fields living on ℐ. In this work we classify the two- and three-point correlation functions such Carrollian conformal fields can have in any consistent quantum theory of massless particles and arbitrary dimension. We then show that bulk correlators of massless fields in \U0001d544 explicitly reduce to these Carrollian conformal correlators when evaluated on ℐ, although in the case of time-ordered bulk correlators this procedure appears singular at first sight. However we show that the Carrollian correlators of the descendant fields are perfectly regular and precisely carry the information about the corresponding S-matrix elements.

Manifestly covariant worldline actions from coadjoint orbits. Part I. Generalities and vectorial descriptions

We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence à la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincaré and (A)dS symmetry. In the Poincaré case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincaré symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in “bitemporal” AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.

G-frame generator sets for projective unitary representations

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

The discreet charm of the discrete series in dS2

We study quantum field theories placed on a two-dimensional de Sitter spacetime (dS2) with an eye on the group-theoretic organization of single and multi-particle states. We explore the distinguished role of the discrete series unitary irreducible representation (UIR) in the Hilbert space. By employing previous attempts to realize these states in free tachyonic scalar field theories, we propose how the discrete series may contribute to the Källén–Lehmann decomposition of an interacting scalar two-point function. We also study BF gauge theories with SL(N,R) gauge group in dS2 and establish a relation between the discrete series UIRs and the operator content of these theories. Although present at the level of the operators, states carrying discrete series quantum numbers are projected out of the gauge-invariant Hilbert space. This projection is reminiscent of what happens for quantum field theories coupled to semiclassical de Sitter gravity, where we must project onto the subspace of de Sitter invariant states. We discuss how to impose the diffeomorphism constraints on local field-theory operators coupled to two-dimensional gravity in de Sitter, with particular emphasis on the role of contact terms. Finally, we discuss an SYK-type model with a random two-body interaction that encodes an infinite tower of discrete series operators. We speculate on its potential microscopic connection to the SL(N,R) BF theory in the large-N limit.

The Rough with the Smooth of the Light Cone String

The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D$$\\end{document} to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators -iX1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}X^1$$\\end{document} and P-=(P0-Pz)/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^-=(P^0 - P^z)/2$$\\end{document} and as their commutator the multiplicative operator R=P1/(P0+Pz)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=P^1/(P^0 + P^z)$$\\end{document}. This is not smooth but rough on the negative z-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z-$$\\end{document}axis of massless momentum. Using only the commutation relations of Pm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^m$$\\end{document} with the generators -iMiz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ extrm{i}M_{iz}$$\\end{document} of rotations in the Pi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^i$$\\end{document}-Pz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^{z}$$\\end{document}-plane we show that on massless states the operator R is inconsistent with a unitary representation of SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document}. This makes the algebraic determination of the critical dimension, D=26\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=26$$\\end{document}, of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO(D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(D-1)$$\\end{document} of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{X}$$\\end{document}.

A novel holographic framework preserving reflection positivity in dSd spacetime

This manuscript introduces a novel holographic correspondence in d-dimensional de Sitter (dSd) spacetime, connecting bulk dSd scalar unitary irreducible representations (UIRs) with their counterparts at the dSd boundary I±, all while preserving reflection positivity. The proposed approach, with potential applicability to diverse dSd UIRs, is rooted in the geometry of the complex dSd spacetime and leverages the inherent properties of the (global) dSd plane waves, as defined within their designated tube domains.

Scattering theory with unitary twists

AbstractWe study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of 1/2.

A symmetry algebra in double-scaled SYK

The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the “chord Hilbert space”. We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the \mathfrak{sl}_2𝔰𝔩2 near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to \mathfrak{sl}_2𝔰𝔩2, but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2𝔰𝔩2 algebra more standard at the cost of extending the boundary circle to include some “fake” portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.