Note to the derivation of Dirac equation

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Discussion Overview

The discussion centers on the derivation and interpretation of the Dirac equation in relation to the Pauli equation and the Klein-Gordon equation. Participants explore the historical context, the necessity of linearization, and the implications for electron spin, addressing both theoretical and conceptual aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants reference Feynman's assertion that the Dirac equation is a relativistic form of the Pauli equation, while others challenge the characterization of the Dirac equation as a correct form of the Klein-Gordon equation.
  • One participant suggests that the Dirac equation derives the concept of electron spin, contrasting it with the Pauli equation, which assumes spin.
  • Another participant notes that the Dirac equation was developed as a first-order matrix equation to address issues with the Klein-Gordon equation, particularly concerning negative energy states and the interpretation of conserved quantities.
  • Questions are raised about the necessity of linearization in deriving the Dirac equation and the role of gamma matrices in this context.
  • There is a discussion about whether the Pauli equation is sufficient for deriving spin or if the relativistic nature of the Dirac equation is required.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the Dirac and Pauli equations, particularly regarding the derivation of spin and the interpretation of the Klein-Gordon equation. There is no consensus on these points, and multiple competing perspectives remain.

Contextual Notes

Participants highlight the historical context of the Dirac equation's development, including the challenges posed by negative energy states and the interpretation of conserved quantities. There are unresolved questions about the necessity of linearization and the role of specific mathematical structures in the derivation process.

exponent137
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In book Quantum Electrodynamics, Feynman wrote that the Dirac equation is a relativistic form of the Pauli equation, not a correct form of Klein-Gordon equation. But, I think that the electron spin is only assumed in Pauli equation, but Dirac equation derives it?
I went through derivation in Feynman book, but somewhere I found something better. I do not find it now. It tries to linearize Klein-Gordon equation and so it obtains Dirac equations. Do anyone know any link? Why linearization is necessary? Why \gamma matrices are the only option? (I need to think to ask more concretely, because I think that there is more simple explanation.)
 
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exponent, The discovery of the Dirac equation was a lucky accident. Historically, people tried first to use the Klein-Gordon equation, and encountered two surprises. One, the equation seemed to imply the existence of negative energy states. Second, it came with a continuity equation, but the conserved quantity was not always positive. Dirac tried to find an alternative equation, guessing it would be a first-order matrix equation. Imposing Lorentz invariance, he was led to the Dirac equation. (Notice I did not say "derive"!) He was not expecting it to describe a particle with spin.

Since then, we've realized that the Klein-Gordon equation, properly interpreted, is perfectly valid. The negative energy states must be replaced by antiparticles, and the conserved quantity is a charge density, not a probability density.
 
And how it is with derivation of spin with the Pauli equation? Is it enough for derivation, or we need relativistic, Dirac equation that all parameters are fullfiled?

Pauli equation is generalized Schrödinger equation, including spin.
 
exponent137 said:
In book Quantum Electrodynamics, Feynman wrote that the Dirac equation is a relativistic form of the Pauli equation, not a correct form of Klein-Gordon equation.

It's true.

exponent137 said:
But, I think that the electron spin is only assumed in Pauli equation, but Dirac equation derives it?

In both equations, the properties of uncommuting objects (matrices) describing spin 1/2 are derived, not assumed.
 

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