A note on the path integral representation for Majorana fermions

TL;DR

This paper develops a path integral formulation for Majorana fermions using the Faddeev-Jackiw formalism, providing a new theoretical approach that simplifies calculations involving these particles in quantum field theories.

Contribution

It introduces a novel path integral representation for Majorana fermions based on constrained system formalism, facilitating systematic calculations.

Findings

Derived the path integral for Majorana fermions using Faddeev-Jackiw formalism.

Validated the approach with an exactly solvable example.

Showed the framework simplifies calculations involving Majorana fermions.

Abstract

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.