A Comparative Study of Klein Gordon & Dirac Equation With General Relativistic Potential
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Transcript of A Comparative Study of Klein Gordon & Dirac Equation With General Relativistic Potential
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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.