Dimensional Deception from Noncommutative Tori: An alternative to Horava-Lifschitz

We study the dimensional aspect of the geometry of quantum spaces. Introducing a physically motivated notion of the scaling dimension, we study in detail the model based on a fuzzy torus. We show that for a natural choice of a deformed Laplace operator, this model demonstrates quite non-trivial behaviour: the scaling dimension flows from 2 in IR to 1 in UV. Unlike another model with the similar property, the so-called Horava-Lifshitz model, our construction does not have any preferred direction. The dimension flow is rather achieved by a rearrangement of the degrees of freedom. In this respect the number of dimensions is deceptive. Some physical consequences are discussed.Comment: 20 pages + extensive appendix. 3 figure

On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries

We explore the consequences of time-space noncommutativity in the quantum mechanics of atoms and molecules, focusing on the Moyal plane with just time-space noncommutativity ($[\hat{x}_\mu ,\hat{x}_\nu]=i\theta_{\mu\nu}$, \theta_{0i}\neqq 0, $\theta_{ij}=0$). Space rotations and parity are not automorphisms of this algebra and are not symmetries of quantum physics. Still, when there are spectral degeneracies of a time-independent Hamiltonian on a commutative space-time which are due to symmetries, they persist when \theta_{0i}\neqq 0; they do not depend at all on $\theta_{0i}$. They give no clue about rotation and parity violation when \theta_{0i}\neqq 0. The persistence of degeneracies for \theta_{0i}\neqq 0 can be understood in terms of invariance under deformed noncommutative ``rotations'' and ``parity''. They are not spatial rotations and reflection. We explain such deformed symmetries. We emphasize the significance of time-dependent perturbations (for example, due to time-dependent electromagnetic fields) to observe noncommutativity. The formalism for treating transition processes is illustrated by the example of nonrelativistic hydrogen atom interacting with quantized electromagnetic field. In the tree approximation, the $2s\to 1s +\gamma$ transition for hydrogen is zero in the commutative case. As an example, we show that it is zero in the same approximation for $\theta_{0i}\ne 0$. The importance of the deformed rotational symmetry is commented upon further using the decay $Z^0 \to 2\gamma$ as an example.Comment: 13 pages, revised version, references adde

Multimetric Finsler Geometry

Motivated in part by the bi-gravity approach to massive gravity, we introduce and study the multimetric Finsler geometry. For the case of an arbitrary number of dimensions, we study some general properties of the geometry in terms of its Riemannian ingredients, while in the 2-dimensional case, we derive all the Cartan equations as well as explicitly find the Holmes-Thompson measure.Comment: 27 page

Spectral action approach to higher derivative gravity

We study the spectral action approach to higher derivative gravity. The work focuses on the classical aspects. We derive the complete and simplified form of the purely gravitational action up to the 6-derivative terms. We also derive the equivalent forms of the action, which might prove useful in different applications, namely Riemann– and Weyl–dominated representations. The spectral action provides a rather rigid structure of the higher derivative part of the theory. We discuss the possible consequences of this rigidness. As one of the applications, we check whether the conformal backgrounds are preferred in some way on the classical level, with the conclusion that at this level, there is no obvious reason for such a preference, the space $$S^1 \times S^3$$ studied in earlier works being a special case. Some other possible properties of the higher derivative gravity given by the spectral action are briefly discussed

On S-matrix factorization of the Landau-Lifshitz model

We consider the three-particle scattering S-matrix for the Landau-Lifshitz model by directly computing the set of the Feynman diagrams up to the second order. We show, following the analogous computations for the non-linear Schrdinger model [1, 2], that the three-particle S-matrix is factorizable in the first non-trivial order.FAPESP[05/05147-3]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[06/56056-0]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[06/02939-9]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPqConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)PROSUL[490134/2006-8]PROSU