Representations Of The Poincare Group Research Articles

Worldsheet of a continuous helicity particle

We consider the class of spinning particle theories, whose quantization corresponds to the continuous helicity representation of the Poincare group. The classical trajectories of the particle are shown to lie on the parabolic cylinder with a lightlike axis irrespectively to any specifics of the model. The space-time position of the cylinder is determined by the values of momentum and total angular momentum. The value of helicity determines the focal distance of parabolic cylinder. Assuming that all the world lines lying on one and the same cylinder are connected by gauge transformations, we derive the geometrical equations of motion for the particle. The timelike world paths are shown to be solutions to a single relation involving the invariants of trajectory up to fourth order in derivatives. Geometrical equation of motion is non-Lagrangian, but it admits equivalent variational principle in the extended set of dynamical variables. The lightlike paths are also admissible on the cylinder, but they do not represent the classical trajectories of this spinning particle. The classical trajectories of massless particle (with zero helicity) are shown to lie on hyperplanes, whose space-time position depends on momentum and total angular momentum.

New representations of Poincare group for consistent Relativistic Particle Theories

The class of representations of the Poincare group for the formulation of relativistic quantum theories of single particle is consistently extended. In so doing new species of such representations are singled out. By making use of the enlarged class quantum theories of Klein-Gordon free particle are formulated without the problems suffered by the early theory. Furthermore, new species of consistent theories can be developed.

Space Localization of the Photon

Starting from the phase space representation of quantum mechanics we provide an Euclidean system of covariance for the photon. In particular, we consider systems with the Poincare group as the symmetry group and use a standard procedure in order to build a phase space and a localization observable on the phase space. Then we focus on the massless representations of the Poincare group that we use to build a space localization observable for the photon.

Group Theoretical Settling of Spin Zero Relativistic Particle Theories

In order to avoid the difficulties encountered by relativistic quantum theory of single particles, we pursue a deductive development of the theory from physical principles, without canonical quantization, by making use of group-theoretical methods. Our work has pointed out the necessity of new classes of irreducible representations of the Poincare group the quantum theory of a particle can be based on. For spin 0 particle, four inequivalent theories are completely determined, with fundamental differences with respect to Klein-Gordon theory.

World Sheet of a Massive Spinning Particle in d = 4 Minkowski Space

A geometrical model of a massive spinning particle, whose quantization corresponds to the unitary irreducible representation of the Poincare group, is considered. It is shown that the class of gauge-equivalent trajectories of the particle forms a cylindrical surface in d = 4 Minkowski space. Basing ourselves on the fact that the trajectories of the particle are cylindrical lines, we have obtained the equations of motion of the particle on a world sheet along timelike and isotropic trajectories.

Null infinity and unitary representation of the Poincare group

Following Pasterski-Shao-Strominger we construct a new basis of states in the single-particle Hilbert space of massless particles as a linear combination of standard Wigner states. Under Lorentz transformation the new basis states transform in the Unitary Principal Continuous Series representation. These states are obtained if we consider the little group of a null momentum direction rather than a null momentum. The definition of the states in terms of the Wigner states makes it easier to study the action of space-time translation in this basis. We show by taking into account the effect of space-time translation that the dynamics of massless particles described by these states takes place completely on the null-infinity of the Minkowski space. We then second quantize the theory in this basis and obtain a manifestly Poincare invariant (field) theory of free massless particles living on null-infinity. This theory has unitary time evolution. The null-infinity arises in this case purely group-theoretically without any reference to bulk space-time. Action of BMS is particularly natural in this picture.As a by-product we generalize the conformal primary wave-functions for massless particles in a way which makes the action of space-time translation simple. Using these wave-functions we write down a modified Mellin(-Fourier) transformation of the S-matrix elements. The resulting amplitude is Poincare covariant. Under Poincare transformation it transforms like products of primaries of inhomogeneous SL(2, ℂ) (ISL(2, ℂ)) inserted at various points of null-infinity. ISL(2, ℂ) primaries are defined in the paper.

Constrained BRST-BFV and BRST-BV Lagrangians for Half-Integer HS Fields on $${{R}^{{1,d - 1}}}$$

Gauge invariant Lagrangian descriptions of irreducible and reducible half-integer higher-spin mixed-symmetric massless and massive representations of the Poincare group with off-shell algebraic constraints are constructed within a metric-like formulation in a $d$-dimensional flat space-time on the basis of a suggested constrained BRST approach. A Lorentz-invariant resolution of the BRST complex within the constrained BRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely in terms of the initial triple gamma-traceless spin-tensor field $\Psi_{(\mu)_{n}}$ with gamma-traceless gauge parameter. The triplet and quartet formulations are derived. The minimal (un)constrained BRST--BV actions for above formulations are obtained, from proposed constrained BRST--BV approach to be by appropriate tools to construct interacting constrained Lagrangians.

English

To form the basis spin states of a field, vector states  of the multicomponent wave function to the four coordinates of Minkowski space that determine the position of the local observer have been added by angular extra dimensions that determine the orientation of the local observer. The generators of the Poincare group in the angular representation have been obtained. The Dirac equation (generalized for any spin) and Maxwell’s equation have been designed from these generators. In the framework of transformations of the Lorentz group in angular representation united with its transpose representation, the transformations of the (generalized) Dirac equations that is similar to the Heaviside-Larmor transformations for Maxwell’s equations have been performed. As a result, the Dirac equation for the Dirac monopole, which corresponds to a particle with mirror symmetry have been obtained. Indication of a low probability of the existence of the Dirac monopole had been obtained.   Key words: Spinor representation of the Poincare group, seve¬n dimensions of space-time, mirror symmetry.

Gauge fields and infinite chains of dualities

We show that the particle states of Maxwell’s theory, in D dimensions, can be represented in an infinite number of ways by using different gauge fields. Using this result we formulate the dynamics in terms of an infinite set of duality relations which are first order in space-time derivatives. We derive a similar result for the three form in eleven dimensions where such a possibility was first observed in the context of E 11. We also give an action formulation for some of the gauge fields. In this paper we give a pedagogical account of the Lorentz and gauge covariant formulation of the irreducible representations of the Poincare group, used previously in higher spin theories, as this plays a key role in our constructions. It is clear that our results can be generalised to any particle.

Momentum entanglement in relativistic quantum mechanics

I present a new group-theoretical approach to the interaction mechanism of elementary particle physics. Within an irreducible unitary two-particle representation of the Poincare group, the commutation relations of the Poincare group require that the two-particle states be momentum entangled. As in gauge theories, momentum entanglement defines a correlation between two particles that can be described as an interaction provided by the exchange of virtual (gauge) quanta. The coupling constant of this interaction is uniquely determined by the structure of the irreducible two-particle state space. For two massive spin one-half particles, the coupling constant matches the empirical value of the electromagnetic coupling constant.

Wavefunctions of a prolate spheroid and multiplexing in relativistic quantum cryptography on orthogonal states

Relativistic quantum cryptography involves not only the geometric properties of the state vectors of a quantum system in the Hilbert space but also the properties of carriers of quantum states in the Minkowski spacetime. A physical type of quantum object carrying information with extremely high velocity in spacetime is of fundamental importance. The structure of spacetime, more precisely, irreducible representations of the Poincare group, in the Hilbert space is responsible for the existence of massless particles, photons. In this sense, the structure of spacetime is in fact used in relativistic quantum cryptography systems for cryptographic key distribution. This makes it possible to guarantee the security of keys even with the use of orthogonal states.

Reflection Positive One-Parameter Groups and Dilations

The concept of reflection positivity has its origins in the work of Osterwalder–Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincare group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary one-parameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the $$ax+b$$ -group, the Heisenberg group, the euclidean motion group and the euclidean conformal group.

Emergence of Space-Time and Gravitation

In relativistic quantum mechanics, elementary particles are described by irreducible unitary representations of the Poincare group. The same applies to the center-of-mass kinematics of a multi-particle system that is not subject to external forces. As shown in a previous article, for spin-1/2 particles, irreducibility leads to a correlation between the particles that has the structure of the electromagnetic interaction, as described by the perturbation algorithm of quantum electrodynamics. The present article examines the consequences of irreducibility for a multi-particle system of spinless particles. In this case, irreducibility causes a gravitational force, which in the classical limit is described by the field equations of conformal gravity. The strength of this force has the same order of magnitude as the strength of the empirical gravitational force.

General Lagrangian formulation for higher spin fields with arbitrary index symmetry. 2. Fermionic fields

We continue the construction of a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having k rows, on a basis of the BRST–BFV approach suggested for bosonic fields in our first article [I.L. Buchbinder, A. Reshetnyak, Nucl. Phys. B 862 (2012) 270, arXiv:1110.5044 [hep-th]]. Starting from a description of fermionic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a system of first-class constraints. To do this, we find, in first time, by means of generalized Verma module the auxiliary representations of the constraint subsuperalgebra, to be isomorphic due to Howe duality to osp(k|2k) superalgebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We suggest a universal procedure of finding unconstrained gauge-invariant Lagrangians with reducible gauge symmetries, describing the dynamics of both massless and massive fermionic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by constraints corresponding to an irreducible Poincare-group representation. As examples of the general approach, we propose a method of Lagrangian construction for fermionic fields subject to an arbitrary Young tableaux having 3 rows, and obtain a gauge-invariant Lagrangian for a new model of a massless rank-3 spin-tensor field of spin (5/2,3/2) with first-stage reducible gauge symmetries and a non-gauge Lagrangian for a massive rank-3 spin-tensor field of spin (5/2,3/2).

Noninteraction of Waves in Two-dimensional Conformal Field Theory

In higher dimensional quantum field theory, irreducible representations of the Poincare group are associated with particles. Their counterpart in two-dimensional massless models are "waves" introduced by Buchholz. In this paper we show that waves do not interact in two-dimensional Moebius covariant theories and in- and out-asymptotic fields coincide. We identify the set of the collision states of waves with the subspace generated by the chiral components of the Moebius covariant net from the vacuum. It is also shown that Bisognano-Wichmann property, dilation covariance and asymptotic completeness (with respect to waves) imply Moebius symmetry. Under natural assumptions, we observe that the maps which give asymptotic fields in Poincare covariant theory are conditional expectations between appropriate algebras. We show that a two-dimensional massless theory is asymptotically complete and noninteracting if and only if it is a chiral Moebius covariant theory.

Constraints of Cluster Separability and Covariance on Current Operators

Realistic models of hadronic systems should be defined by a dynamical unitary representation of the Poincare group that is also consistent with cluster properties and a spectral condition. All three of these requirements constrain the structure of the interactions. These conditions can be satisfied in light-front quantum mechanics, maintaining the advantage of having a kinematic subgroup of boosts and translations tangent to a light front. The most straightforward construction of dynamical unitary representations of the Poincare group due to Bakamjian and Thomas fails to satisfy the cluster condition for more than two particles. Cluster properties can be restored, at significant computational expense, using a recursive method due to Sokolov. In this work we report on an investigation of the size of the corrections needed to restore cluster properties in Bakamjian-Thomas models with a light-front kinematic symmetry. Our results suggest that for models based on nucleon and meson degrees of freedom these corrections are too small to be experimentally observed.

The physical meaning of the de Sitter invariants

We study the Lie algebras of the covariant representations transforming the matter fields under the de Sitter isometries. We point out that the Casimir operators of these representations can be written in closed forms and we deduce how their eigenvalues depend on the field's rest energy and spin. For the scalar, vector and Dirac fields, which have well-defined field equations, we express these eigenvalues in terms of mass and spin obtaining thus the principal invariants of the theory of free fields on the de Sitter spacetime. We show that in the flat limit we recover the corresponding invariants of the Wigner irreducible representations of the Poincare group.

The Poincaré Group in a Demisemidirect Product with a Non-associative Algebra with Representations that Include Particles and Quarks—II

The quarks and particles’ mass and mass/spin relations are provided with coordinates in configuration space and/or momentum space by means of the marriage of ordinary Poincare group representations with a non-associative algebra made through a demisemidirect product, in the notation of Leibniz algebras. Thus, we circumvent the restriction that the Poincare group cannot be extended to a larger group by any means (including the (semi)direct product) to get even the mass relations. Finally, we will discuss a connection between the phase space representations of the Poincare group and the phase space representations of the associated Leibniz algebra.

Current exchanges for reducible higher spin multiplets and gauge fixing

We compute the current exchanges between triplets of higher spin fields which describe reducible representations of the Poincare group. Through this computation we can extract the propagator of the reducible higher spin fields which compose the triplet. We show how to decompose the triplet fields into irreducible HS fields which obey Fronsdal equations, and how to compute the current-current interaction for the cubic couplings which appear in ArXiv:0708.1399 [hep-th] using the decomposition into irreducible modes. We compare this result with the same computation using a gauge fixed (Feynman) version of the triplet Lagrangian which allows us to write very simple HS propagators for the triplet fields.

Relativistic diffusion of elementary particles with spin

We obtain a generalization of the relativistic diffusion of Schay and Dudley for particles with spin. The diffusion equation is a classical version of an equation for the Wigner function of an elementary particle. The elementary particle is described by a unitary irreducible representation of the Poincare group realized in the Hilbert space of wavefunctions in the momentum space. The arbitrariness of the Wigner rotation appears as a gauge freedom of the diffusion equation. The spin is described by an SU(2) connection of a fiber bundle over the momentum hyperbolic space (the mass shell). Motion in an electromagnetic field, transport equations and equilibrium states are discussed.