International Journal of Geometric Methods in Modern Physics

I stress that spacetime is a redundant abstraction, since describing the physical content of all so-called "spacetime measurements" only requires timing (by a physical/material clock) of particle detections (at a physical/material detector). It is interesting then to establish which aspects of our current theories afford us the convenient abstraction of a spacetime. I emphasize the role played by the assumed triviality of the geometry of momentum space, which makes room for an observer-independent notion of locality. This is relevant for some recent studies of the quantum-gravity problem that stumbled upon hints of a nontrivial geometry of momentum space, something which had been strikingly envisaged for quantum gravity already in 1938 by Max Born. If indeed momentum space has nontrivial geometry then the abstraction of a spacetime becomes more evidently redundant and less convenient: one may still abstract a spacetime but only allowing for the possibility of a relativity of spacetime locality. I also provide some examples of how all this could affect our attitude toward the quantum-gravity problem, including some for the program of emergent gravity and emergent spacetime. And in order to give an illustrative example of possible logical path for the "disappearance of spacetime" I rely on formulas inspired by the κ-Poincaré framework.

Relativistic particles with momentum space described by a group manifold provide a very interesting link between gravity, quantum group symmetries and non-commutative field theories. We discuss how group valued momenta emerge in the context of three-dimensional Einstein gravity and describe the related non-commutative field theory. As an application we introduce a non-commutative heat-kernel, calculate the associated spectral dimension and comment on its non-trivial behavior. In four spacetime dimensions the only known example of momenta living on a group manifold is encountered in the context of the κ-Poincaré algebra introduced by Lukierski et al. 20 years ago. I will discuss the construction of a one-particle Hilbert space from the classical κ-deformed phase space and show how the group manifold structure of momentum space leads to an ambiguity in the quantization procedure. The tools introduced in the discussion of field quantization lead to a natural definition of deformed two-point function.

We compare two ways of force terms generating in the model of nonrelativistic particle moving in the presence of constant field force . First of them uses the twist-deformed acceleration-enlarged Newton–Hooke quantum space-times while the second one incorporates the doubly enlarged Newton–Hooke transformations of classical space. Particularly, we find the conditions for which both the treatments provide the same force terms.

In this paper we discuss some ideas on how to define the concept of quasi-integrability. Our ideas stem from the observation that many field theory models are "almost" integrable; i.e. they possess a large number of "almost" conserved quantities. Most of our discussion will involve a certain class of models which generalize the sine-Gordon model in (1 + 1) dimensions. As will be mentioned many field configurations of these models look like those of the integrable systems and so appear to be close to those in integrable model. We will then attempt to quantify these claims looking in particular, both analytically and numerically, at field configurations with scattering solitons. We will also discuss some preliminary results obtained in other models.

The formalism of functions of commuting nilpotent variables allows to describe multi-qubit pure states and their entanglement. The family of states defined by the generalized trigonometric η-functions is specially interesting from mathematical and physical point of view (it covers the set of physically interesting states, including: Werner states, cluster Werner states, GHZ states etc.). We analyze the behavior of two recently proposed symmetric entanglement monotones on the trigonometric states.

This is an overview of applications of the harmonic variables in supersymmetric mechanics, with focusing on some latest results.

We investigate certain Z₃-graded associative algebras with cubic Z₃ invariant constitutive relations, introduced by one of us some time ago. The invariant forms on finite algebras of this type are given in the cases with two and three generators. We show how the Lorentz symmetry represented by the SL(2, C) group can be introduced without any notion of metric, just as the symmetry of Z₃-graded cubic algebra and its constitutive relations. Its representation is found in terms of the Pauli matrices. The relationship of such algebraic constructions with quark states is also considered.

I briefly discuss the construction of the theory of particles with curved momentum space and its consequence, the principle of relative locality.

The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left–right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.

Cubic extensions of the Poincaré algebra can be constructed in consistency with physical assumptions, and provide new insights for the description of phenomena. In this paper we review some recent results associated to these structures: algebraic construction, construction of invariant Lagrangians and construction of their associated group.

We present a brief outline of recent and new results on the mathematical structure underlying the κ-deformed space. We suggest to turn attention to the observable C*-algebra of κ-deformed coordinates and its Galilean symmetries.

In this contribution I would like to review some topics in which Jerzy Lukierski has been one of the main players and contributed over decades with many interesting and important results. We shall consider various ways in which relativistic particles can be endowed with spin degrees of freedom with the use of twistor-like variables and supersymmetrization, which upon quantization lead either to higher-spin fields (in D > 3) or to anions (in D = 3).

An overview is given of the extraordinary resistence of maximal supergravity and maximal super-Yang–Mills theories to the onset of ultraviolet divergences, in various spacetime dimensions. The ultraviolet cancellations found to date can be understood by some rather subtle applications of supersymmetric nonrenormalization theorems.

We introduce darkons as fluid particles of a Galilean massless self-gravitating fluid. This fluid exhibits anisotropic scaling with . The minimal gravitational coupling dynamically generates a gravitational mass density of either sign. Hence such fluid may serve as a model for the dark sector of the Universe. Its cosmological solutions give a deceleration phase for the early Universe and an acceleration phase for the late Universe. Will the steady flow solutions lead to a confining potential and so a possible model for halos?

A general construction of the Holst action is discussed. Based on this, the N = 1, 2, 4, 8 supergravities and N = 1 supergravities with matter coupling are presented. It is shown that in all these cases the Immirzi parameter does not influence the field equations. The construction ensures that the theory is invariant under supersymmetry as well as gauge transformations, but the Holst extension breaks parity.

We construct the theta-exact covariant noncommutative model and obtain various closed constraints on the noncommutative scale from inelastic neutrino-nucleon scatterings, from plasmon decay into neutrino pair, from the Big Bang Nucleosynthesis and from the reheating phase after inflation, respectively. We find neutrino two-point function in a closed form and decoupling of the UV divergent term from softened UV/IR mixing term. Deformed dispersion relations at low energies are capable to account for the recent results from the OPERA collaboration on the superluminal speed of the muon neutrinos.

Nonlinear equations of p-branes in D = (2p + 1)-dimensional Minkowski space are discussed. Presented are new exact solutions for a set of spinning p-branes with the Abelian symmetries U(1) × U(1) × ⋯ ×U(1) of their shapes.

The dynamical realizations of the Schrödinger group are analyzed. Under the assumption that the "mass" parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes the standard Newton equations.