[Paper Review] Higher Dimensional Elko Theory

J. A. Nieto|arXiv (Cornell University)|Jul 3, 2013

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper demonstrates that the 4-dimensional Elko equation—describing mass dimension one fermions relevant for dark matter—can be derived from a 5-dimensional Dirac equation via Kaluza-Klein dimensional reduction. The key result is that the Elko equation emerges when compactification conditions are imposed on the fifth dimension, generalizing to higher dimensions and suggesting a deeper connection between Elko theory and higher-dimensional field theories in brane-world scenarios.

ABSTRACT

We show that the so called Elko equation can be derived from a 5-dimensional Dirac equation. We argue that this result can be relevant for dark matter and cosmological scenarios. We generalize our procedure to higher dimensions.

Motivation & Objective

• To explore the origin of the Elko equation in higher-dimensional field theories.
• To establish a connection between Elko fermions and 5D Dirac theory as a possible framework for dark matter.
• To generalize the construction to D-dimensional spacetimes and analyze the implications for spinor structure and mass dimension.
• To examine the role of compactification and Kaluza-Klein modes in the emergence of 4D Elko states.
• To investigate the consistency of the Majorana condition and charge conjugation in higher dimensions.

Proposed method

• Derive the 4D Elko equation from a 5D Dirac equation by imposing the condition that momentum in the fifth dimension vanishes.
• Use the 5D Klein-Gordon equation as a starting point, decomposing it into left- and right-handed components via gamma matrices.
• Introduce a splitting of indices into 4D spacetime and one internal dimension, with gamma matrices satisfying Clifford algebras in both sectors.
• Assume that the fifth-dimensional momentum operator annihilates the spinor fields, mimicking Kaluza-Klein zero modes.
• Generalize the procedure to D dimensions by introducing additional compactified dimensions with orthogonal gamma matrices.
• Apply the Majorana condition via charge conjugation to reduce the number of physical degrees of freedom, ensuring consistency with Elko's self-conjugate structure.

Experimental results

Research questions

• RQ1 Can the 4D Elko equation be derived from a higher-dimensional Dirac equation?
• RQ2 What is the role of Kaluza-Klein compactification in generating Elko fermions from higher-dimensional spinors?
• RQ3 How does the mass dimension of the spinor field change when moving from 4D to higher-dimensional theories?
• RQ4 What are the implications of the Majorana condition in higher-dimensional Elko theories?
• RQ5 How does signature change affect the behavior of Elko fermions in higher dimensions?

Key findings

• The 4D Elko equation is derived from a 5D Dirac equation under the assumption that the fifth-dimensional momentum operator vanishes on the spinor fields.
• The resulting 8-component spinor in 4D arises from a 5D Dirac spinor with two chiral components, each transforming under a 2×2 representation.
• The mass dimension of the spinor field increases from 3/2 in 4D to 2 in 5D, reflecting the higher-dimensional action's dimensionality.
• The Elko equation in 4D is shown to be equivalent to a generalized 5D Dirac equation with specific projection conditions.
• The construction generalizes to D dimensions, where the spinor field transforms under a Clifford algebra in the internal space, preserving the Elko structure.
• The Majorana condition in higher dimensions requires a modified charge conjugation operator and leads to a reduction to 4 physical complex components, consistent with the original Elko theory.