ISSN: 2689-7636
##### Annals of Mathematics and Physics
Short Communication       Open Access      Peer-Reviewed

# Dirac spinor’s transformation under Lorentz mappings

### J Yaljá Montiel-Pérez1, J López-Bonilla2* and VM Salazar del Moral2

1Centro de Investigación en Computación, Instituto Politécnico Nacional, México
2ESIME-Zacatenco, Instituto Politécnico Nacional, México
*Corresponding author: J López-Bonilla, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX, México, E-mail: jlopezb@ipn.mx
Received: 30 June, 2020 | Accepted: 13 July, 2021 | Published: 15 July, 2021
Keywords: Dirac 4-spinor, Lorentz transformation, Dirac equation, Pauli and dirac matrices

Cite this as

Montiel-Pérez JY, López-Bonilla J, Salazar del Moral VM (2021) Dirac spinor’s transformation under Lorentz mappings. Ann Math Phys 4(1): 028-031. DOI: 10.17352/amp.000022

For a given Lorentz matrix, we deduce the Dirac spinor’s transformation in terms of four complex quantities.

### Introduction

We have the Dirac equation for spin-1/2 particles [1-5]

where $\psi$ is a 4-spinor with the ${\gamma }^{\mu }$ matrices verifying the anticommutator [6-8]:

Here we shall use the Dirac-Pauli (or standard) representation [2,9]:

with the Cayley [10]-Sylvester [11]-Pauli [12] matrices:

to analyze the transformation law of under the orthochronic and proper Lorentz group [13-19]:

which implies the existence [2, 7, 20, 21] of a non-singular matrix S such that:

and we deduce the relativistic invariance of (1) if the Dirac 4-spinor obeys the transformation rule:

Here we exhibit a method to findS for a given Lorentz matrix.

##### Construction of the matrix S for a given Lorentz mapping.

The arbitrary complex quantities α, β, γ, δ verifying the constraint αδ – βγ = 1, generate a Lorentz matrix $L=\left({L}^{\mu }{}_{\nu }\right)$ via the relations [13-15, 17, 22-26]:

where cc means the complex conjugate of all the previous terms.

The inverse problem is to obtain α, β, γ, δ if we know L, and the answer is [26-29]:

where

From (6) are immediate the expressions [3, 30]:

that is, if we know S then with (10) we can determine the Lorentz matrix; (10) generates the relations:

which allow to obtain L if we have the expansion [31]:

However, here we have the inverse problem, that is, to obtain verifying (11) for a given Lorentz matrix. Our answer is the following:

hence the expressions (8) are deduced if we apply (13) into (11). Besides, with (13) the matrix (12) acquires the structure:

Therefore, for a given Lorentz transformation first we employ (9) to determine α,β,γ,δ, then S is immediate via (14); this approach is an alternative to the process showed in [31] and to the explicit general formula obtained by Macfarlane [30]:

however, the possible physical applications are not evident in it. In our procedure, for example, the relations (9) are of great interest for the physicists working on supersymmetry [29], and the expressions (14) are very useful to study the relativistic motion of a classical point particle [28].

### Conclusion

The Dirac equation is relativistic if the corresponding 4-spinor verifies the transformation (7) under Lorentz mappings, with the matrix S satisfying the condition (6). Here we showed a procedure to construct S for a given Lorentz matrix.

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