The Boosts in the Noncommutative Special Relativity

Observables and Dispersion Relations in k-Minkowski Spacetime

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic …

Asymptotically flat boundary conditions for the $U(1)^3$ model for Euclidean Quantum Gravity

A generally covariant $U(1)^3$ gauge theory describing the $G_N \to 0$ limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we study boundary conditions and asymptotic symmetries of the $U(1)^3$ model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativit…

Kappa-Minkowski spacetime: mathematical formalism and applications in Planck scale physics

The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, wh…

The beauty of curved momentum space

In this manuscript, we will discuss the notion of curved momentum space, as it arises in the discussion of noncommutative or doubly special relativity theories. We will illustrate it with two simple examples, the Casimir effect in anti-Snyder space and the introduction of fermions in doubly special relativity. We will point out the existence of intriguing results, which suggest nontrivial connections with spectral geometry and Hopf algebras.

伽利略相对性及其不变双线性形式

Galilean relativity and its invariant bilinear forms

本文构造了 R3+1 上满足伽利略变换和空间旋转等距同构的双线性形式族 gG，该形式由物理意义尚不明确的伽利略不变向量参数化，并进一步构造了与其非退化反对称部分相关的 Poisson bracket。

On some consequences of the Snyder-Sidharth deformation of Special Relativity

The hypothesis on a minimal scale existence in the Universe leads to noncommutative geometry of Spacetime and thence to a modification of the Special Relativity constraint. Sidharth has deduced that this is equivalent to the Lorentz symmetry violation. This latter consideration was also used by Glashow, Coleman and other scholars though based on purely phenomenological models that have been suggested by the observation of Ultra High Energy Cosmic Rays and Gamma Bursts. On the other hand a parallel development has been the proposal of a small but nonzero photon mass $m_γ>0$ by some authors incl…

The Deformation of Poincaré Subgroups Concerning Very Special Relativity

We investigate here various kinds of semi-product subgroups of Poincaré group in the scheme of Cohen-Glashow's very special relativity along the deformation approach by Gibbons- Gomis-Pope. For each proper Poincaré 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 which inherit from the $5-d$ representation of Poincaré group. We find from the obtained natural representation that rotation operation may have additional accompanied scale transformation in …

Energy nonconservation as a link between $f(R,T)$ gravity and noncommutative quantum theory

$f(R,T)$ gravity was proposed as an extension of the $f(R)$ theories, containing not just geometrical correction terms to the General Relativity equations, but also material correction terms, dependent on the trace of the energy-momentum tensor $T$. These material extra terms prevent the energy-momentum tensor of the theory to be conserved, even in a flat background. Energy nonconservation is a prediction of quantum theory with time-space noncommutativity. If time is considered as an operator and there are compact spatial coordinates which do not commute with time, then the time evolution gets…

Geometry and General Relativity in the Groupoid Model with a Finite Structure Group

In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_Γ$ defined on a transformation groupoid $Γ$ determined by the action of the Lorentz group on the frame bundle $(E, π_M, M)$ over space-time $M$. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group $G$ and the frame bundle is trivial $E=M\times G$. The model is fully computable. We define the Ein…

Relativity implications of the quantum phase

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic …