Poincare Group Research Articles

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.

Auxiliary Conformally Invariant Higher-Spin Field in de Sitter Space

We employ de Sitter isometry to study a mixed symmetric rank-3 tensor field in de Sitter space by finding the field equation, solution and two-point function which are conformally invariant. It is proved that such a tensor field plays a key role in conformal theory of linear gravity (Binegar et al., Phys. Rev. D 27, 2249, 1983) . In de Sitter space from the group theoretical point of view this kind of tensor could associate with two unitary irreducible representations (UIR) of the de Sitter group (Takook et al., J.Math. Phys. 51, 032503, 2010), which one representation has a flat limit, namely, in zero curvature coincides to the UIR of Poincare group, however, the second one which is named as auxiliary field, becomes significant in the study of conformal gravity in de Sitter background. We show that the rank-3 tensor solution can be written in terms of a massless minimally coupled scalar field and also the related two-point function is a function of a massless minimally coupled scalar two-point function.

Note on the phase space of asymptotically flat gravity in Ashtekar–Barbero variables

We describe the canonical phase space of asymptotically flat gravity in Ashtekar–Barbero (AB) variables. We show that the Gauss constraint multiplier must fall off slower than previously considered in order to recover ADM phase space. The generators of the asymptotic Poincare group are derived within the AB phase space without reference to the ADM generators. The resulting expressions are shown to agree, modulo Gauss constraint terms, with those obtained from the ADM generators. A payoff of this procedure is a new expression for the generator of asymptotic rotations, which is polynomial in the triad and hence better suited for quantum theory. Our treatment complements an earlier description by Thiemann in the context of self-dual variables.

Chiral fermions, massless particles and Poincare covariance

The coadjoint orbit method is applied to the construction of Hamiltonian dynamics of massless particles of arbitrary helicity. The unusual transformation properties of canonical variables are interpreted in terms of nonlinear realizations of Poincare group. The action principle is formulated in terms of new space–time variables with standard transformation properties.

Canonical formulation of Poincaré BFCG theory and its quantization

We find the canonical formulation of the Poincare BFCG theory in terms of the spatial 2-connection and its canonically conjugate momenta. We show that the Poincare BFCG action is dynamically equivalent to the BF action for the Poincare group and we find the canonical transformation relating the two. We study the canonical quantization of the Poincare BFCG theory by passing to the Poincare-connection basis. The quantization in the 2-connection basis can be then achieved by performing a Fourier transform. We also briefly discuss how to approach the problem of constructing a basis of spin-foam states, which are the categorical generalization of the spin-network states from Loop Quantum Gravity.

Nonabelian T-duals of the flat background

Classification of nonabelian T-duals of the flat matric in D=4 with respect to the four-dimensional continuous subgroups of the Poincare group is given. After dualizing the flat background we identify majority of dual models as conformal sigma models in plane- parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. Besides the pp-waves we find several diagonalizable curved metrics with nontrivial scalar curvature and torsion.Due to the nonabelian T-duality, we find general solution of classical field equations of all the sigma models in terms of d'Alembert solutions of wave equation.

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.

Dynamical representation of the operators for the Dirac particle in the field of a plane wave

We find an explicit form of the integrals of motion for a Dirac particle placed in a plane-wave field. These operators are a realization of the Lie algebra of the Poincare group in the case where the representation space consists of solutions of the Dirac-Pauli equation for the particle in a plane-wave field.

Quantization of almost-circular orbits in the Fokker action formalism

The time-non-local action principle of Fokker type determining a two-particle dynamics is considered. The system is assumed to be general but invariant with respect to the Aristotle group, which is a common subgroup of the Galileo and Poincare groups. It is shown that integral-differential equations of motion of such system admit circular orbit solutions. The dynamics of perturbations of these solutions is derived and studied. On this ground, the Hamiltonian description of the time-non-local two-particle system is built in the almost circular orbit approximation. The Aristotle invariance of the system is exploited to select physical degrees of freedom. The quantization procedure and a construction of energy spectrum of the system is proposed. The application in meson spectroscopy is meant.

Teleparallel Gravity as a Higher Gauge Theory

We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincare 2-group as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way if M is simply connected and has vanishing 2nd deRham cohomology. Extending from the Poincare 2-group to the teleparallel 2-group, a 2-connection includes an additional piece: a coframe field. Taking advantage of the teleparallel reformulation of general relativity, which uses a coframe field, a flat connection and its torsion, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.

On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

Quantum deformations of (anti-)de Sitter (A)dS algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincare algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the K-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.

Geometry and physics of null infinity

In asymptotically Minkowski space-times, one finds a surprisingly rich interplay between geometry and physics in both the classical and quantum regimes. On the mathematical side it involves null geometry, infinite dimensional groups, symplectic geometry on the space of gravitational connections and geometric quantization via Kahler structures. On the physical side, null infinity provides a natural home to study gravitational radiation and its structure leads to several interesting effects such as an infinite dimensional enlargement of the Poincare group, geometrical expressions of energy and momentum carried by gravitational waves, emergence of non-trivial ‘vacuum configurations’ and an unforeseen interplay between infrared properties of the quantum gravitational field and the enlargement of the asymptotic symmetry group. The goal of this article is to present a succinct summary of this subtle and beautiful interplay.

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.

The deformation of Poincaré subgroups concerning very special relativity

We investigate here various kinds of semi-product subgroups of Poincare group in the scheme of Cohen-Glashows very special relativity along the deformation approach by Gibbons- Gomis-Pope. For each proper Poincare subgroup which is a semi-product of proper lorentz group with the spacetime translation group T (4), we investigate all possible deformations and obtain all the possible natural representations inherited from the 5- d representation of Poincare group. We find from the obtained natural representation that rotation operation may have additional accompanied scale transformation when the original Lorentz subgroup is deformed and the boost operation gets the additional accompanied scale transformation in all the deformation cases. The additional accompanied scale transformation has a strong constrain on the possible invariant metric function of the corresponding geometry and the field theories in the spacetime with the corresponding geometry.

Dimension four wins the same game as the standard model group

In a previous article Don Bennett and I looked for,found and proposed a game in which the Standard Model group S(U(2)XU(3)) gets singled out as the "winner". Here I propose to extend this "game" to construct a corresponding game between different potential dimensions for space time. The idea is to formulate how the same competition as the one between the potential gauge groups would run out, if restricted to the potential Lorentz or Poincare groups achievable for different dimensions of space time d. The remarkable point is that it is the experimental dimension of space time 4 which wins. So the same function defined over Lie groups seems to single out both the gauge group and the space time dimension in nature. This seems a rather strange coincidence unless there is really some similar physics behind.

Hamiltonian anomalies of bound states in QED

The Bound State in QED is described in systematic way by means of nonlocal irreducible representations of the nonhomogeneous Poincare group and Dirac's method of quantization. As an example of application of this method we calculate triangle diagram $Para-Positronium \to \gamma\gamma$. We show that the Hamiltonian approach to Bound State in QED leads to anomaly-type contribution to creation of pair of parapositronium by two photon.

Non-linear realizations, Goldstone bosons of broken Lorentz rotations and effective actions for p-branes

We consider the non-linear realizations of the Poincare group for p-branes with local subgroup SO(1,p)×SO(D−(p+1)). The Nambu–Goto p-brane action is constructed using the Maurer–Cartan forms of the unbroken translations. We perform a throughout phase space analysis of the action and show that it leads to the canonical action of a p-brane. We also construct some higher order derivative terms of the effective p-brane action using the MC forms of the broken Lorentz transformations.

Duality Invariance Implies Poincaré Invariance

We consider all possible dynamical theories which evolve two transverse vector fields out of a three-dimensional Euclidean hyperplane, subject to only two assumptions: (i) the evolution is local in space, and (ii) the theory is invariant under "duality rotations" of the vector fields into one another. The commutators of the Hamiltonian and momentum densities are shown to be necessarily those of the Poincaré group or its zero signature contraction. Space-time structure thus emerges out of the principle of duality.

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.