[Paper Review] Dirac Equation in $\kappa$-Minkowski space-time

Ravikant Verma|arXiv (Cornell University)|Oct 31, 2014

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper derives the $κ$-deformed Dirac equation in $κ$-Minkowski spacetime valid to all orders in the deformation parameter $a$, using undeformed $κ$-Lorentz algebra and spinor transformations. The deformation effects are fully encoded in the boost factor and momentum, while $γ$-matrices and generator realizations remain undeformed, ensuring consistency with the commutative limit as $a \to 0$. The resulting equation reproduces the $κ$-deformed Klein-Gordon equation upon squaring.

ABSTRACT

In this paper, we derive the Dirac equation in the $\\kappa$-deformed Minkowski space-time. We start with $\\kappa$-deformed Minkowski space-time and investigate the undeformed $\\kappa$-Lorentz transformation valid to all order in the deformation parameter $a$. Using the undeformed $\\kappa$-Lorentz algebra, we obtain the $\\kappa$-deformed Dirac equation, valid to all order in the deformation parameter $a$. In limit $a\ ightarrow$0, we get back the correct commutative result.

Motivation & Objective

• To derive a $κ$-deformed Dirac equation that is exact to all orders in the deformation parameter $a$.
• To ensure the equation remains consistent with the undeformed $κ$-Poincaré algebra and Lorentz symmetry.
• To demonstrate that the $γ$-matrices and generator realizations are independent of the deformation parameter $a$, with all $a$-dependence confined to the boost factor and momentum.
• To establish a direct mapping between the momentum-space Dirac equation and the coordinate-space $κ$-Dirac equation from prior work [12].

Proposed method

• Start from the $κ$-Minkowski spacetime with Lie algebra-type commutation relations: $[x^0, x^i] = ia x^i$, $[x^i, x^j] = 0$, where $a = 1/\kappa$.
• Derive the exact boost factor $\gamma'$ valid to all orders in $a$ from the deformed dispersion relation.
• Construct finite boost transformations using the undeformed $κ$-Lorentz algebra and obtain matrix realizations of boost generators independent of $a$.
• Re-express the undeformed $κ$-Lorentz algebra as a direct product of $su(2)$ algebras to define left and right spinors.
• Derive the transformation rule between left and right spinors with non-zero three-momentum $\vec{P}$, leading to the $κ$-deformed Dirac equation in momentum space.
• Map the momentum-space equation to position space using operator realizations of the Dirac derivatives $D_\mu$, yielding the final $κ$-deformed Dirac equation.

Experimental results

Research questions

• RQ1 How can the $κ$-deformed Dirac equation be derived to all orders in the deformation parameter $a$ without approximations?
• RQ2 What is the role of the undeformed $κ$-Lorentz algebra in constructing the $κ$-deformed Dirac equation?
• RQ3 Why are the $γ$-matrices and generator realizations independent of the deformation parameter $a$?
• RQ4 How does the resulting $κ$-deformed Dirac equation map to the coordinate-space equation from [12]?
• RQ5 Does squaring the $κ$-deformed Dirac equation reproduce the known $κ$-deformed Klein-Gordon equation?

Key findings

• The $κ$-deformed Dirac equation is derived in momentum space using undeformed $su(2)$ spinor transformations and exact boost factors valid to all orders in $a$.
• The matrix realizations of the $κ$-Lorentz generators and the $γ$-matrices are independent of the deformation parameter $a$, with all $a$-dependence encoded in the boost factor and momentum.
• The derived $κ$-deformed Dirac equation in position space matches the equation obtained in [12] when mapped via the operator realizations of $D_\mu$.
• Squaring the $κ$-deformed Dirac equation yields the $κ$-deformed Klein-Gordon equation, confirming consistency with the known relativistic dispersion relation.
• In the limit $a \to 0$, the equation reduces to the standard commutative Dirac equation, ensuring physical consistency.
• The entire deformation effect is contained in the modified boost parameter and deformed three-momentum, while the algebraic structure of the Lorentz group remains undeformed.