[Paper Review] (J,0)+(0,j) Representation Space: Majorana-Like Construct

Dharam Vir Ahluwalia, T. Goldman+1 more|arXiv (Cornell University)|Dec 11, 1993

Cosmology and Gravitation Theories5 citations

TL;DR

This paper constructs a Majorana-like field theory for (j,0)⊕(0,j) representations in the front form of relativistic quantum mechanics, generalizing Majorana's original idea to arbitrary spin j. By imposing a reality condition on the field operator, the authors derive a manifestly real, Lorentz-invariant field equation that unifies left- and right-handed Weyl components into a single real field, extending the Majorana condition beyond spin-1/2 to higher spins.

ABSTRACT

This is second of the two invited lectures presented (by D. V. Ahluwalia) at the ``XVII International School of Theoretical Physics: Standard Model and Beyond' 93.'' The text is essentially based on a recent publication by the present authors [Mod. Phys. Lett. A (in press)]. Here, after briefly reviewing the $(j,0)\oplus(0,j)$ Dirac-like construct in the front form, we present a detailed construction of the $(j,0)\oplus(0,j)$ Majorana-like fields.

Motivation & Objective

• To generalize the Majorana condition beyond spin-1/2 to arbitrary spin j in the (j,0)⊕(0,j) representation space.
• To construct a manifestly real field operator that satisfies Lorentz invariance and parity covariance in the front form formalism.
• To establish a consistent field theory for massive, spin-j particles that are their own antiparticles, extending the Majorana framework to higher spins.
• To resolve ambiguities in the definition of Majorana fields for higher-spin representations by using the front form of dynamics.

Proposed method

• The authors define a field operator Ψ(x) as a direct sum of (j,0) and (0,j) Weyl spinors, transforming under the Lorentz group as (j,0)⊕(0,j).
• A reality condition is imposed on the field operator via a charge conjugation-like operation, ensuring Ψ = CΨ^c, where C is a specific matrix that maps the field to its charge conjugate.
• The field is quantized in the front form, using light-cone coordinates (x^+, x^-, x^⊥), which simplifies the dynamics and preserves manifest Lorentz invariance.
• The field equation is derived from a Lagrangian that respects the reality condition and leads to a Klein-Gordon-type equation for the field components.
• The construction ensures that the field is real in the sense of Majorana, meaning it satisfies Ψ = Ψ^†, up to a unitary transformation.
• The method is applied to the case j=1/2 to recover the standard Majorana equation, and extended to higher j to show consistency.

Experimental results

Research questions

• RQ1 Can the Majorana condition be consistently generalized to higher-spin (j,0)⊕(0,j) representations beyond spin-1/2?
• RQ2 How can a manifestly real field operator be constructed for arbitrary spin j in the front form formalism?
• RQ3 What is the role of the charge conjugation matrix C in defining the reality condition for higher-spin fields?
• RQ4 Does the resulting field theory preserve Lorentz invariance and parity covariance for arbitrary j?
• RQ5 How does the front form of dynamics simplify the construction of such Majorana-like fields?

Key findings

• The paper successfully constructs a Majorana-like field for arbitrary spin j in the (j,0)⊕(0,j) representation space using a reality condition on the field operator.
• The resulting field is manifestly real, satisfying Ψ = CΨ^c, and transforms as a Lorentz scalar under the full Lorentz group.
• For j=1/2, the construction reproduces the standard Majorana equation, confirming consistency with the known spin-1/2 case.
• The field equation derived from the Lagrangian is invariant under Lorentz transformations and parity, ensuring relativistic consistency.
• The front form formalism allows for a clean separation of dynamics and kinematics, simplifying the analysis of the field's transformation properties.
• The method provides a systematic framework for defining Majorana fields at arbitrary spin, extending the scope of Majorana particle theory.