Conformal invariance in noncommutative geometry and mutually interacting Snyder particles

Relativistic particles and commutator algebras with twisted Poincaré transformation

This paper explores conformal Killing vectors in teleparallel theory for the plane-symmetric static spacetime. Ten non-linear coupled teleparallel conformal Killing equations are solved to get the general form of teleparallel conformal Killing vectors and the conformal factor along with the integrability conditions. Three different possibilities are considered where the spacetime may exhibit a teleparallel conformal Killing vector. In two cases the integrability conditions are solved completely and teleparallel conformal Killing vectors are obtained along with the conformal factor for the special choice of metric functions. Another possibility reveals that no teleparallel proper conformal Killing vector exists and the spacetime admits only a teleparallel homothetic vector.

The present analysis deals with the wormhole (WH) solutions in [Formula: see text] gravity, where [Formula: see text], [Formula: see text] and [Formula: see text] represent the Ricci scalar, kinetic expression and potential field, respectively. To complete this analysis, we use the WH geometry via spherical spacetime with the anisotropic matter distribution. Further, we consider the Gaussian distribution as non-commutative geometry to complete the analysis under conformal symmetry. We calculate the exact WH shape function by plugging the possible conformal Killing vectors. Further, we have discussed the embedded surface to understand the WH geometry. Furthermore, the Tolman–Oppenheimer–Volkoff equation is considered to discuss the stability of WH configuration with the Gaussian energy density source.

The classical Einstein's gravity can be reformulated from the constrained\nU(2,2) gauge theory on the ordinary (commutative) four-dimensional spacetime.\nHere we consider a noncommutative manifold with a symplectic structure and\nconstruct a U(2,2) gauge theory on such a manifold by using the covariant\ncoordinate method. Then we use the Seiberg-Witten map to express noncommutative\nquantities in terms of their commutative counterparts up to the first-order in\nnoncommutative parameters. After imposing constraints we obtain a\nnoncommutative gravity theory described by the Lagrangian with up to\nnonvanishing first order corrections in noncommutative parameters. This result\ncoincides with our previous one obtained for the noncommutative SL(2,C)\ngravity.\n

Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

Perfect fluid spherically symmetric spacetimes admitting a proper inheriting conformal Killing vector (CKV) are studied. It is found that, other than Friedmann-Robertson-Walker (FRW) spacetimes, the only known examples of proper CKV perfect fluid spherically symmetric spacetimes in which the CKV is inheriting are conformal FRW spacetimes, static Schwarzschild interior spacetimes or generalized Gutman-Be'spalko-Wesson spacetimes and all of these spacetimes are either conformally flat or have a CKV which is either parallel to or orthogonal to the fluid 4-velocity. A general theorem is proven in which the (restricted) form of the CKV (and conformal factor) is given should a perfect fluid spherically symmetric spacetime admit a proper inheriting CKV. Various results on the non-existence of perfect fluid spherically symmetric spacetimes admitting a proper inheriting CKV are then derived, thereby providing further validity of a general non-existence conjecture (at least in the perfect fluid case).

Non-commutative Maxwell–Chern–Simons theory in three dimensions and its dual

Full field algebras, operads and tensor categories

An equilibrium system of N relativistic spinless particles with chargee and mass/« interacting with the electromagnetic field is considered. We show that after averaging over the field variables the action integral of the system of relativistic particles with direct action can be calculated A certain procedure of renormalization for the particle mass is proposed. The action integral for a single particle with the account of the influence of its own field is obtained.

New solution for the Einstein field equations satisfying a neutral anisotropic quark star object is generated. The quark linear equation of state together with the conformal Killing vector (CKV) are used to investigate the behavior and properties of quark stars. The CKV plays an important role in providing the relationship between the two gravitational potentials. The process of combining the CKV and an equation of state has currently led to new realistic solutions. In obtaining the matter variables, one of the gravitational potentials is specified on physical grounds to generate compact star model with physical significance. The generated quark star model undergoes several physical tests for the validity and acceptability. Several realistic physical conditions are found to be satisfied. The model stability in terms of the adiabatic index and equilibrium of the physical forces is satisfied. The behavior of the mass-radius relationship and the surface red shift are well obeyed. Their parameters values are found to be compatible with observations. The gravitational potentials are continuous throughout the interior of the star, and the model's energy conditions are satisfied as well. Quark star models that admit conformal symmetry in the absence of charge are missing in the existing literature.

The bipartite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative space (NCS), is investigated with the help of Simon’s separability condition (generalized Peres-Horodecki criterion). It turns out that, to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy a unique constraint equation. We have considered the most general form of an anisotropic oscillator in NCS, with both spatial and momentum noncommutativity. The system is transformed to the usual commutative space (with usual Heisenberg algebra) by a well-known Bopp shift. The system is transformed into an equivalent simple system by a unitary transformation, keeping the intrinsic symplectic structure (\(Sp(4,{\mathbb {R}})\)) intact. Wigner quasiprobability distribution is constructed for the bipartite Gaussian state with the help of a Fourier transformation of the characteristic function. It is shown that the identification of the entangled degrees of freedom is possible by studying the Wigner quasiprobability distribution in phase space. We have shown that the co-ordinates are entangled only with the conjugate momentum corresponding to other co-ordinates.

In this note, we investigate conformal Killing vectors (CKVs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes. Subject to some integrability conditions, CKVs up to implicit functions of (t,x) are obtained. Solving these integrability conditions in some particular cases, the CKVs are completely determined, obtaining a classification of LRS Bianchi type V spacetimes. The inheriting conformal Killing vectors of LRS Bianchi type V spacetimes are also discussed.

An example from a perfect fluid FRW space-time is presented to show that a conformal Killing vector (CKV) need not map fluid flow lines into fluid flow lines. Kinematic properties of the Lie derivative along a CKV of timelike and spacelike unit vectors are derived and applied to the fluid unit four-velocity vector. Dynamic properties of special conformal Killing vectors (SCKV) in a fluid with anisotropic pressure and vanishing energy flux are obtained using Einstein’s field equations. It is shown that a SCKV maps both fluid flow lines and integral curves of na into themselves, where na is the unit spacelike vector of anisotropy. The relation between the anisotropic pressure components and the energy density is considered. By means of an example from a radiationlike viscous fluid FRW space-time it is shown that the dynamic results depend crucially on the vanishing of the energy flux vector. The extension of the dynamic results to a fluid with arbitrary stress tensor and zero energy flux vector is examined.

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.

We find a class of solutions to the Einstein field equations with constant vacuum energy density (''cosmological constant'') that has a similarity symmetry of the second kind. We show this symmetry to be a global conformal symmetry. Nontrivial analytic solutions are given and one in particular (exhibiting intrinsic symmetry) is shown to evolve to a nonempty Robertson-Walker space-time with ''steady-state'' metric. This is found to be due to particle production associated with the negative matter pressure that is required by the assumed symmetry. These models can describe, classically, an origin of the Universe in terms of particle production from the vacuum, driving an exponential (de Sitter) expansion. This solution is inhomogeneous and anisotropic, but tends to homogeneity and isotropy at early times and large distances, and at late times and small distances. The solution therefore corresponds to the outward motion of a spherical disturbance which distorts the local homogeneity and isotropy in an asymptotically homogeneous and isotropic universe. The limiting homogeneous and isotropic forms are discussed.