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Register a new userLifting Spacetimes Poincare Symmetries
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Added by
Alexander S. Glasser
Publication date
2019
fields
Physics
and research's language is
English
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In the following work, we pedagogically develop 5-vector theory, an evolution of scalar field theory that provides a stepping stone toward a Poincare-invariant lattice gauge theory. Defining a continuous flat background via the four-dimensional Cartesian coordinates ${x^a}$, we `lift the generators of the Poincare group so that they transform only the fields existing upon ${x^a}$, and do not transform the background ${x^a}$ itself. To facilitate this effort, we develop a non-unitary particle representation of the Poincare group, replacing the classical scalar field with a 5-vector matter field. We further augment the vierbein into a new $5times5$ funfbein, which `solders the 5-vector field to ${x^a}$. In so doing, we form a new intuition for the Poincare symmetries of scalar field theory. This effort recasts `spacetime data, stored in the derivatives of the scalar field, as `matter field data, stored in the 5-vector field itself. We discuss the physical implications of this `Poincare lift, including the readmittance of an absolute reference frame into relativistic field theory. In a companion paper, we demonstrate that this theoretical development, here construed in a continuous universe, enables the description of a discrete universe that preserves the 10 infinitesimal Poincare symmetries and their conservation laws.
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The following work demonstrates the viability of Poincare symmetry in a discrete universe. We develop the technology of the discrete principal Poincare bundle to describe the pairing of (1) a hypercubic lattice `base manifold labeled by integer vertices-denoted ${mathbf{n}}={(n_t,n_x,n_y,n_z)}$-with (2) a Poincare structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincare group whose dynamics and gauge transformations on the lattice closely resemble those of a scalar field in spacetime. We demonstrate that such a theory generates discrete dynamics with the complete infinitesimal symmetry-and associated invariants-of the Poincare group. Following our companion paper, we `lift the Poincare gauge symmetries to act only on vertical matter and solder fields, and recast `spacetime data--stored in the $partial_muphi(x)$ kinetic terms of a free scalar field theory--as `matter field data-stored in the $phi^mu[mathbf{n}]$ components of the 5-vector field itself. We gauge 5-vector theory to describe a lattice gauge theory of gravity, and discuss the physical implications of a discrete, Poincare-invariant theory.
We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $kappa$-Poincare dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use this general construction to build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the $kappa$-Poincare dispersion relation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that the photon sphere around a black hole becomes a thick shell since photons of different energy will orbit the black hole on circular orbits at different altitudes. We also compute the redshift of a photon between different observers at rest, finding that there is a Planck-scale correction to the usual redshift only if the observers detecting the photon have different masses.
In this paper we excavate, for the first time, the most general class of conformal Killing vectors, that lies in the two dimensional subspace described by the null and radial co-ordinates, that are admitted by the generalised Vaidya geometry. Subsequently we find the most general class of generalised Vaidya mass functions that give rise to such conformal symmetry. From our analysis it is clear that why some well known subclasses of generalised Vaidya geometry, like pure Vaidya or charged Vaidya solutions, admit only homothetic Killing vectors but no proper conformal Killing vectors with non constant conformal factors. We also study the gravitational collapse of generalised Vaidya spacetimes that posses proper conformal symmetry to show that if the central singularity is naked then in the vicinity of the central singularity the spacetime becomes almost self similar. This study definitely sheds new light on the geometrical properties of generalised Vaidya spacetimes.
A Z3 symmetric generalization of the Dirac equation was proposed in recent series of papers, where its properties and solutions discussed. The generalized Dirac operator acts on coloured spinors composed out of six Pauli spinors, describing three colours and particle-antiparticle degrees of freedom characterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetries of 12-component generalized wave functions. Spinorial representation of the Z3-graded generalized Lorentz algebra was introduced, leading to the appearance of extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin, flavors and families. The present article proposes a construction of Z3-graded extension of the Poincare algebra. It turns out that such a generalization requires introduction of extended 12-dimensional Minkowskian space-time containing the usual 4-dimensional space-time as a subspace, and two other mutually conjugate replicas with complex-valued vectors and metric tensors. Representation in terms of differential operators and generalized Casimir operators are introduced and their symmetry properties are briefly discussed.
Within the electroweak theory, it is shown that the form of the total Lagrangian is invariant, under local phase changes of the basis states for leptons and under local changes of the mathematical spaces employed for the description of left-handed spinor states of leptons. In doing this, a contribution from vacuum fluctuations of the leptonic fields, which causes no experimentally-observable effect, is added to the total connection field. Accompanying the above-mentioned changes of basis states, the leptonic and connection fields are found to undergo changes whose forms are similar to $U(1)$ and $SU(2)$ gauge transformations, respectively. These results suggest a simple physical interpretation to gauge symmetries in the electroweak theory.
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