We prove that the extended Poincare group in (1+1) dimensions is non-nilpotent solvable exponential, and therefore that it belongs to type I. We determine its first and second cohomology groups in order to work out a classification of the two-dimensional relativistic elementary systems. Moreover, all irreducible unitary representations of the extended Poincare group are constructed by the orbit method. The most physically interesting class of irreducible representations corresponds to the anomaly-free relativistic particle in (1+1) dimensions, which cannot be fully quantized. However, we show that the corresponding coadjoint orbit of the extended Poincare group determines a covariant maximal polynomial quantization by unbounded operators, which is enough to ensure that the associated quantum dynamical problem can be consistently solved, thus providing a physical interpretation for this particular class of representations.
We classify the unitary representations of the extended Poincaré supergroups in three dimensions. Irreducible unitary representations of any spin can appear, which correspond to supersymmetric anyons. Our results also show that all irreducible unitary representations necessarily have physical momenta. This is in sharp contrast to the ordinary Poincaré group in three dimensions, that admits in addition irreducible unitary representations with non-physical momenta, which are discarded on physical grounds.
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.
We give a pedagogical presentation of the irreducible unitary representations of $\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$, that is, of the universal cover of the complexified Poincaré group $\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner-Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero "complex mass", we give a detailed constru…
Multidimensional contractions of irreducible representations of the Cayley-Klein unitary algebras in the Gel'fand-Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be isomorphic. In this case method of transitions describes the same reducible representations in different basises, say, discrete and continuous ones.
Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the…
There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way. Moreover, they are the only states with a finite number of non-zero quantum numbers with respect to a certain set of maximally commuting linearly independent quantum observables. Any other state is a tensor product of a multiple meson state and a state coming from a representation of a quotient algebra that extends and generalizes the Virasoro algebra. We expect the…
The capabilities of some approaches to the relativistic description of hadronic states with any rest spin are analysed. The key feature in the Wigner's construction of irreducible representations of the Poincare group which makes this construction fruitless in the particle physics is picked out. A realization of unitary irreducible representations of the Poincare group of the standard type, which yet have never been considered, is discussed. The viability of the description of hadrons by the Poincare group representations of the standard type in the space of the infinite-component ISFIR-class …
The formulation of quantum mechanics with a complex Hilbert space is equivalent to a formulation with a real Hilbert space and particular density matrix and observables. We study the real representations of the Poincare group, motivated by the fact that the localization of complex unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity. We review the map from the complex to the real irreducible representations---finite-dimensional or unitary---of a Lie group on a Hilbert space. Then we show that all the finite-dimensional real rep…
Irreducible representations (irreps) of a finite group $G$ are equivalent if there exists a similarity transformation between them. In this paper, we describe an explicit algorithm for constructing this transformation between a pair of equivalent irreps, assuming we are given an algorithm to compute the matrix elements of these irreps. Along the way, we derive a generalization of the classical orthogonality relations for matrix elements of irreps of finite groups. We give an explicit form of such unitary matrices for the important case of conjugated Young-Yamanouchi representations, when our g…
Though the irreducible representations of the Poincare' group form the groundwork for the formulation of relativistic quantum theories of a particle, robust classes of such representations are missed in current formulations of these theories. In this work the extended class of irreducible representations with positive `mass' parameter is explicitly determined. Several new representations in such extension, so far excluded, give rise to consistent theories for Klein-Gordon particles and also to new species of particle theories.
The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors.
It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant constructi…
As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternion types with torsion number 1,2,4 respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition of irreducibility for projective Reps of anti-unitary groups. Then a practical procedure is provided to reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is to const…
While Wigner functions forming phase space representation of quantum states is a well-known fact, their construction for noncommutative quantum mechanics (NCQM) remains relatively lesser known, in particular with respect to gauge dependencies. This paper deals with the construction of Wigner functions of NCQM for a system of 2-degrees of freedom using 2-parameter families of gauge equivalence classes of unitary irreducible representations (UIRs) of the Lie group $\g$ which has been identified as the kinematical symmetry group of NCQM in an earlier paper. This general construction of Wigner fun…
The explicit form of the unitary representation of the Poincaré group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators
We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fréchet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, bounded unitary irreducible (factor) representations are possibl…
The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.
We study the structure of the coadjoint orbits of the 2+1 Poincaré group, using a matricial representation of the group. We also obtain the orbits connected to irreducible representations of the group. Finally we obtain coherent states for the hyperboloidal and conical orbits.
The unitary irreducible representations of a Lie group defines the Hilbert space on which the representations act. If this Lie group is a physical quantum dynamical symmetry group, this Hilbert space is identified with the physical quantum state space. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. The Poincare group is the archetypical example with the unitary representations defining the Hilbert space of relativistic particle states and the Klein-Gordon, Dirac, Maxwell equations are obtained from the represe…
We study the massless irreducible representations of the Poincaré group in the six-dimensional Minkowski space. The Casimir operators are constructed and their eigenvalues are found. It is shown that the finite spin (helicity) representation is defined by two integer or half-integer numbers while the infinite spin representation is defined by the real parameter $μ^2$ and one integer or half-integer number.
Let $G$ and $T$ be topological groups, $α: T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_αT$ the corresponding semidirect product group. In this paper we address several issues concerning irreducible continuous unitary representations $(π^\sharp, \cH)$ of $G^\sharp$ whose restriction to $G$ remains irreducible. First we prove that, for $T = \R$, this is the case for any irreducible positive energy representation of $G^\sharp$, i.e., for which the one-parameter group $U_t := π^\sharp(\1,t)$ has non-negative spectrum. The passage from irreduc…
In standard quantum field theory, the one-particle states are classified by unitary representations of the Poincaré group, whereas the causal fields' classification employs the finite dimensional (non-unitary) representations of the (homogeneous) Lorentz group. A natural question arises - why the fields are not allowed to transform nontrivially under translations? We investigate this issue by considering the fields that transform under the full representation of the Poincaré group. It follows that such fields can be consistently constructed, although the Lagrangians that describe them necessar…
Let $V$ be a simple vertex operator algebra, and $G$ a finite automorphism group of $V$ such that $V^G$ is regular. The definition of entries in $S$-matrix on $V^G$ is discussed, and then is extended. The set of $V^G$-modules can be considered as a unitary space. In this paper, we obtain some connections between $V$-modules and $V^G$-modules over that unitary space. As an application, we determine the fusion rules for irreducible $V^G$-modules which occur as submodules of irreducible $V$-modules by the fusion rules for irreducible $V$-modules and by the structure of $G$.
Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R). We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way.
A general technique is presented for constructing a quantum theory of a finite number of interacting particles satisfying Poincaré invariance, cluster separability, and the spectral condition. Irreducible representations and Clebsch-Gordan coefficients of the Poincaré group are the central elements of the construction. A different realization of the dynamics is obtained for each basis of an irreducible representation of the Poincaré group. Unitary operators that relate the different realizations of the dynamis are constructed. This technique is distinguished from other solutions of this proble…
All unitary irreducible representations of centrally extended (N-odd) N-conformal Galilei group are constructed. The "on-shell" action of the group is derived and shown to coincide, in special but most important case, with that obtained in: J. Gomis, K. Kamimura, Phys. Rev. {\bf D85} (2012), 045023.
We study asymptotics of representations of the unitary groups U(n) in the limit as n tends to infinity and we show that in many aspects they behave like large random matrices. In particular, we prove that the highest weight of a random irreducible component in the Kronecker tensor product of two irreducible representations behaves asymptotically in the same way as the spectrum of the sum of two large random matrices with prescribed eigenvalues. This agreement happens not only on the level of the mean values (and thus can be described within Voiculescu's free probability theory) but also on the…
We lift any (infinitesimal) unitary irreducible representation of $GL_n(\mathbb{R})$ to a family of representations that strongly contracts to a certain type of (infinitesimal) unitary irreducible representations of $\mathbb{R}^n\rtimes {M}_n$, with $M_n$ being the mirabolic subgroup of $GL_n(\mathbb{R})$. For the case of $n=2$ we obtain the full unitary dual of $\mathbb{R}^2\rtimes {M}_2$ as a strong contraction. We demonstrate the role of the Kirillov conjecture and Kirillov model for these contractions.
There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields are defined and related to the linear and angular momenta of a spin one-half representation of the Poincare group. We show that the Majorana spinor field with finite mass is an unitary…
When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by the Dirac form for spin ${1 \over 2}$ systems. The general spinor formulation for arbitrary spins given in a previous paper is extended to include momentum operators. Unitary quantum mechanical representations are developed for these operators, and physical interpretations are suggested.
In this paper we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott, and Goldman. Let $Σ_{g}$ denote a topological surface of genus $g\geq2$. We establish the existence of a large $n$ asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $π_{1}(Σ_{g})$ under a random representation of $π_{1}(Σ_{g})$ into $\mathsf{SU}(n)$. Each such expected value involves a contribution from all irreducible representations of $\mathsf…
Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.
