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Abstract:In this paper we have studied the nature of kinematical and dynamical laws in κ-Minkowski spacetime from a new perspective: the canonical phase space approach. We discuss a particular form of κ-Minkowski phase space algebra that yields the κ-extended finite Lorentz transformations derived in [16]. This is a particular form of a Deformed Special Relativity model that admits a modified energy-momentum dispersion law as well as noncommutative κ-Minkowski phase space. We show that this system can be completely mapped to a set of phase space variables that obey canonical (and not κ-Minkowski) phase space algebra and Special Relativity Lorentz transformation (and not κ-extended Lorentz transformation). The complete set of deformed symmetry generators are constructed that obeys an unmodified closed algebra but induce deformations in the symmetry transformations of the physical κ-Minkowski phase space variables. Furthermore, we demonstrate the usefulness and simplicity of this approach through a number of phenomenological applications both in classical and quantum mechanics. We also construct a Lagrangian for the κ-particle.

We show that the Equivalence Principle (EP) is violated by Quantum Gravity (QG) effects. The predicted violations are compared to experimental observations for Gravitational Redshift, Law of Reciprocal Action and Universality of Free Fall. This allows us to derive explicit bounds for β -the QG scale.In our approach, there appears a deviation in the geodesic motion of a particle. This deviation is induced by a non-commutative spacetime, consistent with a Generalized Uncertainty Principle (GUP). Gup admits the presence of a minimum length scale, that is advocated by QG theories. Remarkably, the GUP induced corrections are quite robust since the bound on β obtained by us, in General Relativity scenario in an essentially classical setting of modified geodesic motion, is closely comparable to similar bounds in recent literature [10]. The latter are computed in purely quantum physics domain in flat spacetime.

We quantise the O(N ) nonlinear sigma model using the Batalin Tyutin (BT) approach of converting a second class system into first class. It is a nontrivial application of the BT method since the quantisation of this model by the conventional Dirac procedure suffers from operator ordering ambiguities. The first class constraints, the BRST Hamiltonian and the BRST charge are explicitly computed. The partition function is constructed and evaluated in the unitary gauge and a multiplier (ghost) dependent gauge.Over the last few years a method of generalised canonical quantisation of constrained dynamical systems has been developed by Fradkin and collaborators [1,2] as an alternative to the pioneering formulation of Dirac [3]. This method [1,2], which has been reasonably well established for systems with first class constraints only, has been very recently extended to include systems with second class constraints [4,5]. We shall henceforth refer to this later method as the Batalin-Fradkin (BF) [4] and Batalin-Tyutin (BT) [5] schemes. The basic idea of this method is to convert the second class system into first class by extending the phase space and then to use the familiar machinary valid for first class systems [1,2]. While the BT [5] method remains unexplored (as far as quantisation of specific models is concerned), Some applications of the BF [4] formalism have been reported recently [6][7][8][9]. It is noteworthy, however, that these applications are confined to examples like the chiral gauge theories [6,7], the chiral boson theory [8], the massive Maxwell [7] and the massive Yang-Mills [9] theories. In all these models the Dirac brackets among the canonical variables are very simple (i.e. there are no operator ordering problems) and the quantisation can be, and indeed has earlier been [10,11], just carried out by the classical method of Dirac [3]. Such examples are, therefore, pedagogic exercises and do not reveal the complete power or flexibility of either the BF [4] or BT [5] approaches. Furthermore, the quantisation presented in ref. [7] is not a systematic application of the BF or BT method [4,5].The motivation of this paper is to provide a non-trivial application of the BT procedure [5]. We shall consider the quantisation of the O(N) invariant nonlinear sigma model. This model, which is a second class system, is known to have (quadratic) field dependent Dirac brackets among the canonical variables. Consequently quantisation by Dirac's [3] procedure is riddled with operator ordering ambiguities. The conventional approach is to work in the configuration space functional integral formalism [11]. In this paper we show that an ambiguity free operator quantisation of the model can be performed by using the BT method [5] of converting the second class system into first class. The involutive (i.e. first class) Hamiltonian contains an infinite number of terms, although the number of additional (unphysical) fields introduced to extend the phase space is finite. A remarkable series of cancellations allows us to expres...

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.Comment: New references added, Published online in Foundations of Physic