Question

In this paper(An Exact Cosmological Solution of the Coupled Einstein-Majorana Fermion-Scalar Field Equations), the Majorana field Lagrangian has been stated as

LM=iψ¯(γ∧∗∇)ψ−imψ¯ψ∗1

where ψ¯
is the charge conjugated spinor field ψ¯=ψ†C
with charge conjugation matrix, C=γ0
.

In contrast to this, the same Majorana field Lagrangian has been written in this paper (Variational Field Equations of a Majorana Neutrino Coupled to Einstein’s Theory of General Relativity) as,

LM=i2ψ¯∗γ∧∇ψ+i2mψ¯ψ∗1

Now even if the mass term is neglected, the rest parts of two Lagrangians aren't seem to be same in both the cases!

What am I missing and what is the correct Lagrangian for a general Majorana field in curved spacetime? Are these two Lagrangians similar (or somehow same) in some sense?

Disclaimer: I am not very much habituated in the exterior calculus, maybe that's why such confusion is arising.
A) Both Lagrangians are equivalent, representing different notations or conventions in the formalism of Majorana fields in curved spacetime.

B) The correct Lagrangian for a general Majorana field in curved spacetime cannot be deduced from the given expressions due to ambiguities in notation and presentation.

C) There might be a mathematical equivalence between the two Lagrangians, incorporating specific transformations or relations that reconcile the seemingly different formulations.

D) The expressions indicate fundamental conceptual differences in describing the Majorana field in curved spacetime, necessitating further analysis to establish the correct formulation.

Answer 1

The two Lagrangians represent different notations or conventions in the formalism of Majorana fields in curved spacetime and there is no way to deduce the correct Lagrangian from the given expressions.

Step-by-step explanation:

The two Lagrangians you mentioned represent different notations or conventions in the formalism of Majorana fields in curved spacetime. The two Lagrangians represent different notations or conventions in the formalism of Majorana fields in curved spacetime and there is no way to deduce the correct Lagrangian from the given expressions.

Both Lagrangians describe the Majorana field, but they are written in slightly different ways. The correct Lagrangian for a general Majorana field in curved spacetime cannot be deduced from the given expressions due to ambiguities in notation and presentation.

So, option B) The correct Lagrangian for a general Majorana field in curved spacetime cannot be deduced from the given expressions due to ambiguities in notation and presentation.