Why does Dirac Equation describe spin 1/2 particles?

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SUMMARY

The Dirac Equation describes spin 1/2 particles due to its inherent angular momentum structure and the properties of its eigenstates. Specifically, the 4x4 dimensionality of the alpha, beta, and gamma matrices is crucial in establishing this relationship. The connection between the Dirac Equation and spin 1/2 arises from the linearization of the Klein-Gordon equation, which is designed for spin 0 particles. This transformation leads to the emergence of spin 1/2 characteristics, as detailed in the work of B. Carter in "Generalized total Angular Momentum Operator for the Dirac Equation in curved Spaced Time."

PREREQUISITES
  • Understanding of the Dirac Equation and its mathematical formulation
  • Familiarity with angular momentum in quantum mechanics
  • Knowledge of 4x4 matrices and their applications in quantum field theory
  • Basic concepts of spin in particle physics
NEXT STEPS
  • Study the mathematical structure of the Dirac Equation in detail
  • Explore the implications of angular momentum in quantum mechanics
  • Read "Generalized total Angular Momentum Operator for the Dirac Equation in curved Spaced Time" by B. Carter
  • Investigate the relationship between the Klein-Gordon equation and the Dirac Equation
USEFUL FOR

This discussion is beneficial for theoretical physicists, quantum mechanics students, and anyone interested in the mathematical foundations of particle physics, particularly those studying spin and the Dirac Equation.

bigbigtheory
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Hi, Everybody!

Currently, I am reading the book "Lectures on Quantum Field Theory" (by Ashok Das)

But I am a bit confusing. Why does Dirac Equation describe spin 1/2 particles?
I have already known that Dirac Equation bears some angular momentum structure, but why it just describe spin 1/2 not others?

Thanks for answering.

Best wishes.
 
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Please see: "Generalized total Angular Momentum Operator for the Dirac Equation in curved Spaced Time" by B. Carter in Physical Review, Vol. 19, No. 4, p. 1093-1097. The 1/2 spin relates to Dirac's eigenstructures. Feynman has a diagram about it you may find interesting.
 
It's one of the wonders and puzzles of theoretical physics the fact that linearizing the K-G equation which properly describes spin 0 particles leads to an equation which describes spin 1/2 particles.

The answer to your "why" is simply <study the whole Dirac's argument>. Spin 1/2 follows naturally from angular momentum considerations, once the 4x4 dimensionality of the alpha/beta/gamma matrices has been proven.
 
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