Poincare Group Research Articles

A Very Fast Reasoning in Favor of the Poincare Group, or of Minkowskian Rather than Euclidean Line Elements

1. We present a very fast derivation of Minkowski’s line element which is the geometric foundation not only of the special Lorentz transformation but of the whole Poincar’e group aka inhomogeneous Lorentz group [1]. The Euclidean line element is excluded using the wave equation and Einstein’s special principle of relativity (Einstein’s first postulate) [2]. For doing so, it is not necessary, 1. to assume the speed of light to be independent of the motion of the observer (Einstein’s second postulate [2], 2. to synchronize watches, 3. to interrelate the coordinate systems of the inertial frames involved; their relative velocity (and hence reciprocity [3]) even plays not any role in the derivation. 4. to make any assumption about the properties of the underlying coordinate transformation such as linearity, uniqueness etc

Matter Is the Representation of Space-Time

The continuous flowing spacetime forms a spacetime group _G_, one of its fundamental groups is the Poincare group _PO(1,3)_, and the matter and interaction fields are representations of its intrinsic spacetime group. As a kind of quantized space-time unit, space-time group elements generate both space-time manifolds and matter. Matter and fields are aggregates of the quantum space-time within them and determine their type: the Lorentz group _SO(1,3)_ represents a rotating spacetime corresponding to visible matter and the translation group _P1,3_ represents a translational spacetime corresponding to dark matter and dark energy. The aggregation mode of space-time degrees of freedom in matter reveals that charge conjugation symmetry is the inverse symmetry of internal time of charged particles, and the external parity breaking of neutrinos is also due to their internal spatial aggregation mode.

The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents.

On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

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.

Algebra of discrete symmetries in the extended Poincaré group

We begin with the comprehensible review of the basics of the Lorentz, (extended) Poincare Groups and O(3,2) and O(4,1). On the basis of the Gelfand-Tsetlin-Sokolik-Silagadze research [1-3], we investigate the definitions of the discrete symmetry operators both on the classical level, and in the secondary-quantization scheme. We studied the physical content within several bases: light-front form formulation, helicity basis, angular momentum basis, on several practical examples. The conclusion is that we have ambiguities in the definitions of the the corresponding operators P, C; T, which lead to different physical consequences.

Spinning particles, coadjoint orbits and Hamiltonian formalism

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

Covariant Ergodic Quantum Markov Semigroups via Systems of Imprimitivity

We construct relativistic quantum Markov semigroups from covariant completely positive maps. We proceed by generalizing a step in Stinespring's dilation to a general system of imprimitivity and basing it on Poincare group. The resulting noise channels are relativistically consistent and the method is applicable to any fundamental particle, though we demonstrate it for the case of light-like particles. The Krauss decomposition of the relativistically consistent completely positive identity preserving maps (our set up is in Heisenberg picture) enables us to construct the covariant quantum Markov semigroups that are uniformly continuous. We induce representations from the little groups to ensure the quantum Markov semigroups that are ergodic due to transitive systems imprimitivity.

Covariant Quantum White Noise from Light-like Quantum Fields

We derive covariant Weyl operators for light-like fields, with the massless Weyl fermion as an illustrative example, in such a way that they correspond to quantum white noises in vacuum state of a symmetric Fock space. First, we build a representation of a light-like little group in terms of Weyl operators. We then use this construction to induce a representation of Poincare group to construct relativistic quantum white noises from the fields via Mackey's systems of imprimitivity (SI) machinery. Our construction proceeds by fashioning the fermionic processes on a symmetric Fock space using re ection and identifying the corresponding processes on the isomorphic white noise space.

One Particle States in Curved Spacetime

The traditional particle interpretation of states of quantum field theory in Minkowski spacetime relies on the Minkowski spacetime symmetry, i.e., the Poincare group uplifted to the Hilbert space. We argue that Weinberg’s approach to particles and interactions in Minkowski spacetime utilizes the representations of the inhomogeneous Lorentz transformations in the Hilbert space dissociated from the spacetime symmetry. Thus it provides a particle interpretation of states in general nonstationary curved spacetimes straightaway. We give Lagrangians for scalars, spinors and vector fields supporting the argument and show that these classical fields contribute to the cosmological constant.

Extension of relativistic mechanics by maximum symmetry group of Maxwell equations

Relativistic mechanics historically arose as a generalization of classical mechanics, which is consistent with the symmetries of the equations of electrodynamics by the use of the Lorentz and Poincare transforms instead of Galileo transformations. However, the Lorentz and Poincare groups are not maximal symmetry groups of the Maxwell equations, even if we restrict ourselves to local transformations of coordinates and time. In this paper, we propose a formulation of relativistic mechanics that is consistent with the group of maximal local symmetries of the Maxwell equations. We take into account dilations and geometric inversions in Minkowski space in addition to the Poincare and Lorentz transformations. The proposed most general form of relativistic mechanics, which is consistent with electrodynamics, is constructed using coordinate and time transformations from the conformal group C(1, 3). Equations of extended relativistic dynamics are proposed.

Gravity and Nonlinear Symmetry Realization

Application of nonlinear symmetry realization technique to gravity is studied. We identify the simplest extensions of the Poincare group suitable for nonlinear realization at the level of physical fields. Two simple models are proposed. The first one introduces additional scalar degrees of freedom that may be suitable for driving inflation. The second one describes states with well-defined mass that lack a linear interaction with matter states. We argue that this phenomenon points out a necessity to draw a distinction between gravitational states with well-defined masses and states that participate in interaction with matter.

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.

Geometry of the non-compact G(2)

Geometrical applications of the non-compact form of Cartan’s exceptional Lie group G(2) are considered. This group comprises specific rotations of 7-dimensional Minkowski-like space with three extra time-like coordinates and in some limiting cases imitates standard Poincare transformations. In this model space–time translations are non-commutative and are represented by the rotations towards the extra time-like coordinates. The second order Casimir element of non-compact G(2) and its expression by the Casimir operators of the Lorentz and Poincare groups are found.

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.

Polarization Tensors for Massive Arbitrary-Spin Particles and the Behrends–Fronsdal Projection Operator

Based on theWigner unitary representations for the covering Poincare group ISL(2,ℂ), we construct spin–tensor wave functions of free massive arbitrary-spin particles satisfying the Dirac–Pauli–Fierz equations. We obtain polarization spin–tensors and indicate conditions that fix the density matrices (Behrends–Fronsdal projection operators), which determine the numerators in the propagators of the fields of such particles. Using such conditions extended to the multidimensional case, we construct a generalization of Behrends–Fronsdal projection operators (for any number D >2 of space–time dimensions) corresponding to a symmetric representation of the D-dimensional Poincare group.

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.

Localizability, gauge symmetry and Newton–Wigner operator for massless particles

The notion of position operator for massless spinning particles is discussed in some detail. The noncommutativity of coordinates is related to the gauge symmetry following from the freedom in definition of standard state in Wigner’s construction of induced representations of Poincare group. The counterpart of Newton–Wigner operator is discussed. It is explained why the Newton–Wigner construction works only for helicity |λ|≤1∕2.