[Paper Review] Deformation of the Dirac Equation

Mir Faizal, S. I. Kruglov|arXiv (Cornell University)|Jun 8, 2014

Noncommutative and Quantum Gravity Theories4 references4 citations

TL;DR

This paper proposes a unified deformation of the Dirac equation by combining doubly special relativity with minimum measurable length and time scales, leading to non-local fractional derivative terms and a modified spacetime geometry that deforms Einstein's equations. The stochastic quantization of this deformed Dirac equation on curved spacetime is performed using a fictitious time and fermionic noise, revealing BRST symmetry and potential for further modifications via Lifshitz scaling or non-anticommutativity.

ABSTRACT

In this paper, we will first clarify the physical meaning of having a minimum measurable time. Then we will combine the deformation of the Dirac equation due to the existence of minimum measurable length and time scales with its deformation due to the doubly special relativity. We will also analyse this deformed Dirac equation in curved spacetime, and observe that this deformation of the Dirac equation also leads to a non-trivial modification of general relativity. Finally, we will analyse the stochastic quantization of this deformed Dirac equation on curved spacetime.

Motivation & Objective

• To clarify the physical meaning of minimum measurable time in quantum gravity.
• To unify deformations from doubly special relativity and minimum length/time scales in the Heisenberg algebra.
• To derive a covariantly deformed Dirac equation incorporating non-local fractional derivatives.
• To analyze the resulting deformation of general relativity and corrections to Einstein's equations.
• To perform stochastic quantization of the deformed Dirac action on curved spacetime using Langevin dynamics and superspace formalism.

Proposed method

• Define minimum measurable time via event-based measurement of dynamical quantities, ensuring covariance with minimum length.
• Combine deformations from doubly special relativity and minimal length/time scales into a single deformed Heisenberg algebra with covariant temporal component.
• Construct a deformed Dirac equation containing non-local fractional derivative terms arising from the deformed algebra.
• Use harmonic extension of functions to assign formal meaning to non-local terms in the deformed Dirac equation.
• Analyze the deformed Dirac equation in curved spacetime to derive corrections to Einstein's equations.
• Perform stochastic quantization using a fictitious time coordinate and anticommuting fermionic Gaussian noise, leading to a Langevin equation for the deformed action.

Experimental results

Research questions

• RQ1 What is the physical interpretation of a minimum measurable time in a relativistic quantum theory?
• RQ2 How does the combination of doubly special relativity and minimal length/time scales modify the Heisenberg algebra?
• RQ3 What are the implications of the resulting deformed Dirac equation for spacetime geometry and general relativity?
• RQ4 Can the stochastic quantization of the deformed Dirac equation be consistently formulated on curved spacetime?
• RQ5 What are the potential consequences of combining this deformation with Lifshitz scaling or non-anticommutative spacetime?

Key findings

• The deformed Heisenberg algebra leads to a discretization of space and is expected to induce time discretization when the full temporal component is included.
• The deformed Dirac equation contains non-local fractional derivative terms, formally interpreted via harmonic extension of functions.
• The deformation of the Dirac equation induces corrections to Einstein's equations, modifying the geometry of spacetime.
• Stochastic quantization of the deformed Dirac action on curved spacetime is achieved using a Langevin equation with fermionic noise and a fictitious time coordinate.
• The stochastic quantization procedure is compatible with the superspace formalism, suggesting a consistent quantum framework for the deformed theory.
• The resulting effective action for the modified gravity theory requires gauge fixing and ghost terms to restore BRST symmetry, indicating a need for further quantization analysis.