7/31/2019 A Comparative Study of Klein Gordon & Dirac Equation With General Relativistic Potential

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A comparative study of Klein Gordon & Dirac

equation with general relativistic potential

Md. Farhad Hossain

We know that the relativistic quantum mechanics obey relativistic Schrdinger equationthat is also known famously as Klein Gordon equation what is set for spin zero field

theoretical particles like photon. On the other hand the spin half field theoretical equation

is famous Dirac equation where the state function of K-G equation replaces by some

column matrix. We study the comparative relationship of these relativistic quantum

mechanical equations from general relativistic entity.

The famous problems of theoretical physics lay on the problems of entering

gravitational field in quantum mechanics. However the first attempt ofrelativistic entity in Schrdinger equation is called Klein Gordon equation

what arise from relativistic particle having both momentum and velocity.This equation is very popular in context

In quantum mechanics all the physical observables are related to someoperator and they operate on some state function and give their eigen value.

So our famous equation of mass-energy relation is replaced by Klein Gordon

(in short K-G) equation for quantum particles.

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This equation becomes a wave equation by entering time derivative operatoras Hamiltonian and space derivative as momentum operator and some

mathematical tricks.

This equation is easily fixed up to all spin zero particles like photon. Onecan called is as bosonic equation. But it is not applied for spin half particles

like electron. That is studied by entering a new equation called diracequation that replace wavefunction by spinor and that is the first derivative

of time.

The spin of electron can easily find from this equation. The equation

contains Dirac matrix what is a 4x4 matrix. Now we introduce the covariantformulation of both Klein Gordon equation and Dirac equation.

Here first equation is K-G equation and second is Dirac equation.

Dirac matrix of gamma representation obey Clifford algebra rule.

The two equations allow all the particles of both integer and half integer spincontains equation. And the quantum field theory starts from here. Though

the system contain the free electron equation we can entire external field andthat will be used as additional operator. But for gravitational field what will

happen? Then we impute the covariant derivative on the normal partialderivative of the equation.

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or

The middle term of second line of new Klein Gordon equation holds

Christoffel connection that is christoffel symbol notation of Generalrelativity. And this is equivalent to gravitational momentum correction that

adds new momentum to old special relativistic momentum with geometric

assumption.

And new Dirac equation is

Here last term of second equation contain spin connection. This equation

says that the spin is dynamical with curvature of space where particles aremoving.

So there will some new potential representation of gravitational field asspace-time curvature of general relativistic formulation for both Klein

Gordon and Dirac equation. The Klein-Gordon equation hold Christoffel

connection and Dirac equation hold spin connection. The entity of newoperators in two equations is the gravitational gauge transformation.

Christoffel connection is equivalent to gravitational four-momentum gaugetransformation. And new Dirac equation says the dynamicity of spin

depending on the curvature of space-time.

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REFERENCE

[1]Quantum theory of fields vol-1 by S Weinberg[2] Thaller, B., The Dirac Equation, Texts and Monographs in Physics

(Springer, 1992)[3] Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co.