CPT Invariance of Quaternion Dirac equation

Abstract

In this paper the invariance of Quaternion Dirac equation under Lorentz Transformation, Charge conjugation, Parity transformation and Time reversal operation has been discussed successfully. The invariance under the combined operation of Charge conjugation, Parity and Time reversal (CPT) has also been discussed and expression for C, P, T and combined CPT operators have been obtained in terms of quaternions. Invariance condition for electric and magnetic field has also been obtained. It has been concluded that the Quaternion Dirac equation dominates over ordinary Dirac equation because of the advantage of algebra of quaternions.

Introduction

I. INTRODUCTION

Quaternions have the same properties as complex numbers but differs in the way that commutative law is not valid for quaternions, which has many advantages over complex numbers. They were first invented by Sir W.R. Hamilton [1] and several authors worked on quaternions [2-5]. Quaternions are the example of hypercomplex numbers. Quaternions means set of four and introduces new methods in Physics and Mathematics. Quaternions are important mathematical tools which are very useful in construction of four-dimensional world. Quaternion Quantum Mechanics is a new type of Mechanics having inner products, matrix elements and Quaternion coefficients. Adler [6-8] gave the idea that Quaternion Quantum Mechanics offers elegant substructure well for quarks and leptons.

Relativistic Quantum Mechanics is one of the important theories of Quantum Mechanics. Since Relativistic Quantum Mechanics includes 3+1 space-time dimension and it becomes difficult to explain since space and time are different coordinates. If we use Quaternions, then they have unique advantage that space and time are treated equal and can be defined by four component Quaternionic function. Quaternion provides four-dimensional structure to relativistic quantum mechanics, also Quaternion structure is compact and theory becomes simplified. Quaternions in matrix form can also be represented in terms of Pauli spin matrices [9], so spin is natural outcome of Quaternion Dirac equation, while in ordinary Dirac equation spin has to introduce by hand. The main contribution in Quaternion Quantum Mechanics was made by Finkelstein [10] A.J. Davies [11], Rotelli [12], Leo et. al. [13-14] and Rawat et. al. [16-19]. Most important use of Quaternion is in Relativistic Quantum Mechanics.

Lorentz invariance and CPT invariance are two of the most fundamental symmetry of nature. For any equation to be successful in physics, it is necessary that it should remain invariant in all frames of references. Although violation of individual C, P, T has been observed in some interaction but the violation of combined CPT has never been observed so far. CPT theorems were introduced by Luders and Pauli [20]. Quaternion Dirac equation should also be tested for invariance under proper Lorentz transformation, Charge conjugation, Parity and Time reversal operations.

In this paper the invariance of Quaternion Dirac equation under Lorentz Transformation, Charge conjugation, Parity transformation and Time reversal operation has been discussed separately. The invariance under the combined operation of Charge conjugation, Parity and Time reversal (CPT) has also been discussed. It has been concluded that the Quaternion Dirac equation dominates over ordinary Dirac equation because of the advantage of algebra of quaternions.

II. DEFINITIONS

Quaternionic function M is defined as

III. INVARIANCE OF QUATERNION DIRAC EQUATION UNDER PROPER LORENTZ TRANSFORMATION

Lorentz covariance of an equation means it has same form in all frame of reference and can be built up from succession of infinitesimal transformations. To check the consistency of Quaternion Dirac equation with theory of relativity, we will check its invariance under Lorentz transformation. The four-space component is defined as

A. Invariance of Quaternion Dirac Equation under Charge Conjugation

Charge conjugation operator changes a particle into its antiparticle or vice versa. According to hole theory absence of charge e with negative energy is equivalent to presence of positive energy operation, charge conjugation operation is

B. Invariance of Quaternion Dirac Equation under Parity operation

Under Parity operation only space coordinate changes sign but time coordinate does not change sign

Which suggests that Parity operator has two eigen values. P = +1 suggests positive energy states and P = -1 suggests negative energy states which represents positron state and shows that e+ and e- have opposite parity states.

C. Invariance of Quaternion Dirac Equation under Time Reversal

Time reversal operator changes sign of time only and leaves space coordinate unchanged. Invariance under time reversal is defined as

IV. INVARIANCE OF QUATERNION DIRAC EQUATION UNDER COMBINED CPT OPERATION

Combined CPT operator is defined as combined operation of charge, parity and time reversal operator, which is given as follow

V. DISCUSSION

Quaternion Dirac equation in electro-magnetic field must remain invariant in all frame of references and it has been observed that Dirac equation remains invariant under Proper Lorentz Transformation, Parity transformation, Time reversal, Charge conjugation and under combined operation of Charge conjugation, Parity transformation and Time reversal (CPT) in the same manner as for the case of usual Dirac equation. Expressions for C,P,T and CPT have been obtained in terms of quaternions. Invariance condition for electric and magnetic field has also been obtained.  It has been observed that Quaternionic Dirac equation consists of all the transformations as usual Dirac equation does, so we have developed Quaternion formalism of Dirac equation in simple, compact and consistent manner because of the advance algebraic structure of Quaternions and spin is natural outcome of quaternionic components.

References

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Copyright

Copyright © 2022 Seema Rawat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.