Κ-Deformed DIRAC EQUATION

Abstract

In this paper we study the quantisation of Dirac field theory in the κ-deformed space-time. We adopt a quantisation method that uses only equations of motion for quantising the field. Starting from κ-deformed Dirac equation, valid up to first order in the deformation parameter, we derive deformed unequal time anti-commutation relation between deformed field and its adjoint, leading to undeformed oscillator algebra. Exploiting the freedom of imposing a deformed unequal time anti-commutation relations between κ-deformed spinor and its conjugate, we also derive a deformed oscillator algebra. We show that deformed number operator is the conserved charge corresponding to global phase transformation symmetry. We construct the κ-deformed conserved currents, valid up to first order in a, corresponding to parity and time-reversal symmetries of κ-deformed Dirac equation. a fundamental length scale. DSR is also known to modify the energy-momentum dispersion relation. It has been shown in [8], that the symmetry algebra of the κ-deformed space-time is defined by the κ-Poincare algebra, which is a Hopf algebra. The symmetry algebra of the κ-deformed space-time can also be realised using undeformed κ-Poincare algebra, where the defining relations are same as that of the usual Poincare algebra, but then the explicit form of the generators are deformed [9][10][11][12][13].Various studies have been carried out analysing field theory models defined in κ-deformed spacetime [14][15][16][17][18][19][20][21][22][23][24]. In most of these studies, field equations, which are invariant under the symmetry algebra, defined in κ-deformed space-times are set up from the κ-deformed quadratic Casimir [9] of the algebra. Since the quadratic Casimir has higher-order derivatives, the Lagrangian corresponding to these deformed field equations also contains higher-order derivative terms in it, signalling the non-local nature of the non-commutative field theories and make canonical quantisation difficult.The knowledge of the explicit form of the Lagrangian is necessary for applying the canonical scheme to quantise field theory, but there exists an alternate quantisation procedure which does not require the Lagrangian for quantising field theories. Instead, this method uses the equations of motion of the field theory as the starting point for the quantisation [25][26][27]. The quantisation of massive spin-one field has been studied using this method [28]. In [29] covariant commutation relation for field describing an arbitrary spin had been obtained using this method. In this method, every free field equation of motion is converted to Klein-Gordon equation with the help of an operator called Klein-Gordon divisor [25][26][27]. This Klein-Gordon divisor is then used to define an unequal time anti-commutation relation between the spinor field operator and its adjoint operator such that the field equations are consistent with the Heisenberg's equations of motion and it leads to the usual (anti-) commutation relations between the creation and annihilation ...

Twisted fermionic oscillator algebra in κ-minkowski space-time

“…(20), as required. This κ-Dirac equation is invariant under the action of parity operator as well as time reversal operator [28]. The κ-Dirac equation for charged particle interacting with external electromagnetic field is obtained by replacing p µ by p µ − eA µ , where e is the electric charge of the particle and it was shown that this equation is not invariant under the charge conjugation [28].…”

Noncommutative space–time and Hausdorff dimension

“…These effects of deformation incorporated in our analysis is through the parameter 'a' by using most general κ-deformed wave function in calculating the position expectation value of the particle moving in κ-space-time. We then generalise this method to study the Hausdorff dimension of path of a relativistic particle, governed by κ-deformed Dirac equation [29]. Applying the self-similarity criterion on the path of non-relativistic and relativistic particle on non-commutative space-time, we then derive generalised uncertainty relation valid up to first order in the deformation parameter.…”

Effect of noncommutativity of space-time on Zitterbewegung

“…κ-deformed electrodynamics was analysed using this approach in [30] and deformed geodesic equation was derived in [31]. In [11], κ-deformed Dirac equation was constructed and its non-relativistics limit was analysed. Modification to Newton's equation for a central force was studied in [32].…”