The Dirac equation as a linear tensor equation for one component

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SUMMARY

The Dirac equation, a cornerstone of modern physics, has been reformulated as a linear tensor equation for a single component of the Dirac spinor, as presented in the article published in Eur. Phys. J. C 84, 488 (2024). Previous tensor equivalents were either nonlinear or multi-component, but this new approach simplifies the equation significantly. The equivalency established with a fourth-order equation in an electromagnetic field allows for broader applications in fields such as general relativity and lattice approximations. This advancement enhances the understanding and utility of the Dirac equation in theoretical physics.

PREREQUISITES
  • Understanding of the Dirac equation and its significance in quantum mechanics.
  • Familiarity with spinor mathematics and tensor calculus.
  • Knowledge of electromagnetic field theory.
  • Basic concepts of general relativity and lattice field theory.
NEXT STEPS
  • Research the implications of linear tensor equations in quantum field theory.
  • Study the applications of the Dirac equation in general relativity.
  • Explore lattice approximations of quantum fields and their computational methods.
  • Investigate previous nonlinear tensor equivalents of the Dirac equation for comparative analysis.
USEFUL FOR

The discussion is beneficial for theoretical physicists, researchers in quantum mechanics, and students studying advanced topics in field theory and general relativity.

akhmeteli
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TL;DR
The Dirac equation is a spinor equation. Tensor equivalents of the equation proposed previously were nonlinear or involved several components of the Dirac field. I derived a linear tensor equivalent of the Dirac equation for just one component.
The abstract of my new article (Eur. Phys. J. C 84, 488 (2024)):

The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several components of the Dirac field. On the other hand, the author showed previously that the Dirac equation in electromagnetic field is equivalent to a fourth-order equation for one component of the Dirac spinor. The equivalency is used in this work to derive a linear tensor equivalent of the Dirac equation for just one component. This surprising result can be used in applications of the Dirac equation, for example, in general relativity or for lattice approximation of the Dirac field, and can improve our understanding of the Dirac equation.
 

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