# A New Route to the Majorana Equation

## Abstract

**:**

## 1. Introduction

## 2. A New Route to the Majorana Equation

**σ**, leads to its inversion, i.e., the operation $\tau \mathit{\sigma}{\tau}^{-1}=-\mathit{\sigma}$ produces a spin flip. Equivalently, one has the anticommutator $\{\tau ,\mathit{\sigma}\}=\mathbf{0}$. Trivially, $\tau \mathrm{i}=-\mathrm{i}\tau $, because of the complex conjugation, $\mathrm{C}$.

## 3. Symmetries and Lorentz Transformation of the Majorana Equation

**σ**. With these preparations in mind, it is easy to see which operators provide the requested symmetry operations. The operator of time reversal is $\mathcal{T}=\mathrm{i}\mathbb{T}$, of parity $\mathcal{P}=\tau \mathbb{P}$, and of chiral conjugation $\mathcal{C}=\tau $.

**σ**and $\mathbf{x}$ both change signs together, as τ anticommutes with the sigmas. Therefore, also ${\varphi}^{\mathcal{P}}={\sigma}_{\mathrm{y}}{\varphi}^{*}(-\mathbf{x},t)$ solves the Majorana equation. Finally, we consider the chiral conjugation defined as $\mathcal{C}={\rho}^{5}=\tau $. It changes the signs of the spatial derivative term in Equation (8), but has no effect on the time-derivate term and mass term. Therefore, ${\varphi}^{\mathcal{C}}={\sigma}_{\mathrm{y}}{\varphi}^{*}(\mathbf{x},t)$ does not solve the Majorana equation, but, actually, its chirality-conjugated version Equation (10). Apparently, the Majorana equation obeys parity and time reversal, yet maximally violates chiral symmetry.

## 4. Derivation of the Real Four-Component Majorana Equation

## 5. Eigenfunctions and Quantum Fields of the Complex Two-Component Majorana Equation

## 6. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Kaku, M. Quantum Field Theory, A Modern Introduction; Oxford University Press: New York, NY, USA, 1993. [Google Scholar]
- Dirac, P.M.A. The quantum theory of the electron. Proc. R. Soc. Lond. Math. Phys. Sci. A
**1928**, A117, 610–624. [Google Scholar] [CrossRef] - Case, K.M. Reformulation of the Majorana theory of the neutrino. Phys. Rev.
**1957**, 107, 307–316. [Google Scholar] [CrossRef] - Mannheim, P.D. Introduction to Majorana masses. Int. J. Theor. Phys.
**1984**, 23, 643–674. [Google Scholar] [CrossRef] - Majorana, E. Teoria simmetrica dell’ elettrone e del positrone. Nuovo Cim.
**1937**, 14, 171–184. [Google Scholar] [CrossRef] - Mohapatra, R.N.; Pal, P.B. Massive Neutrinos in Physics and Astrophysics; World Scientific: Singapore, 2004. [Google Scholar]
- Fukugita, M.; Yanagida, T. Physics of Neutrinos and Applications to Astrophysics; Springer: Berlin, Germany, 2003. [Google Scholar]
- Marsch, E. The two-component Majorana equation—Novel derivations and known symmetries. J. Modern Phys. A
**2011**, 56, 1109–1114. [Google Scholar] [CrossRef] - Marsch, E. On the Majorana equation: Relations between its complex two-component and real four-component eigenfunctions. Int. Sch. Res. Netw. ISRN Math. Phys.
**2012**. [Google Scholar] [CrossRef] - Aste, A. A direct road to majorana fields. Symmetry
**2010**, 2, 1776–1809. [Google Scholar] [CrossRef] - Pal, P.B. Dirac, Majorana, and Weyl fermions. Am. J. Phys.
**2011**, 79, 485–498. [Google Scholar] [CrossRef] - Weyl, H. Elektron und Graviton. I. Z. Phys.
**1929**, 56, 330–352. [Google Scholar] [CrossRef] - Dreiner, H.K.; Haber, H.E.; Martin, S.P. Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry. Phys. Rep.
**2010**, 494, 1–196. [Google Scholar] [CrossRef] - Pauli, W. Zur Quantenmechanik des magnetischen Elektrons. Z. Phys.
**1927**, 43, 601–623. [Google Scholar] [CrossRef] - Jehle, H. Two-component wave equations. Phys. Rev.
**1949**, 75, 1609. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Marsch, E.
A New Route to the Majorana Equation. *Symmetry* **2013**, *5*, 271-286.
https://doi.org/10.3390/sym5040271

**AMA Style**

Marsch E.
A New Route to the Majorana Equation. *Symmetry*. 2013; 5(4):271-286.
https://doi.org/10.3390/sym5040271

**Chicago/Turabian Style**

Marsch, Eckart.
2013. "A New Route to the Majorana Equation" *Symmetry* 5, no. 4: 271-286.
https://doi.org/10.3390/sym5040271