Note to the derivation of Dirac equation

In summary: The derivation is necessary because the spin operator is not Hermitian. 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.)Linearization is necessary because the equation is nonlinear. \gamma matrices are the only option because they are invariant under the transformation that relates the commutators of the fields.
  • #1
exponent137
561
33
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.)
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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 Schrodinger equation, including spin.
 
  • #4
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.
 

Related to Note to the derivation of Dirac equation

1. What is the Dirac equation?

The Dirac equation is a quantum mechanical equation that describes the behavior of spin-1/2 particles, such as electrons. It was developed by physicist Paul Dirac in 1928 and is considered a fundamental equation in the field of quantum mechanics.

2. Why was the Dirac equation developed?

The Dirac equation was developed in order to incorporate the principles of special relativity into quantum mechanics. It also successfully predicted the existence of antimatter, which was later experimentally confirmed.

3. How is the Dirac equation derived?

The Dirac equation is derived using a combination of special relativity and quantum mechanics principles. It involves using the Klein-Gordon equation, which describes relativistic particles with no spin, and modifying it to include the spin of the particle.

4. What are the applications of the Dirac equation?

The Dirac equation has many applications in modern physics, including in the fields of particle physics, quantum field theory, and condensed matter physics. It has also been used in the development of technologies such as transistors and magnetic resonance imaging (MRI).

5. Are there any limitations to the Dirac equation?

While the Dirac equation is a powerful tool in understanding the behavior of spin-1/2 particles, it is limited in its applicability to particles with no spin. It also does not take into account the effects of gravity, and therefore cannot be used to describe particles with massless particles, such as photons.

Similar threads

I Why is the SE better than the KG equation for a hydrogen atom?
    • Quantum Physics
Replies
18
Views
2K
A Is the Klein-Gordon equation a quantization of classical particles?
    • Quantum Physics
2
Replies
45
Views
3K
I Relativistic quantum mechanics
    • Quantum Physics
Replies
32
Views
2K
I Dirac equation as one equation for one function
    • Quantum Physics
2
Replies
40
Views
3K
A Getting particle/antiparticle solutions from the Dirac Equation
    • Quantum Physics
Replies
6
Views
1K
A Dirac's solution to the Klein-Gordon equation
    • Quantum Physics
Replies
5
Views
2K
I Dirac equation in the hydrogen atom
    • Quantum Physics
Replies
1
Views
939
I The 4 solutions to the Dirac equation
    • Quantum Physics
Replies
6
Views
2K
I How Does the Dirac Equation Utilize Component Functions in Its Derivation?
    • Quantum Physics
Replies
2
Views
985
A The Lagrangian Density and Equations of Motion
    • Quantum Physics
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top