Do higher spin particles obey Klein-Gordon or Dirac equations?

In summary: No, I meant we could change a gauge for KG equation, and the form of KG eqn will look different, however it's still KG eqn in a loose sense because nothing physical has changed, just like in electromagnetism, different gauge fixing gives differentIn summary, the higher half integer spin particles obey Dirac equation.
  • #1
ndung200790
519
0
Please teach me this:
We know that 0-spin particles obey Klein-Gordon equation and 1/2spin particles obey Dirac equation.But I do not know whether higher integer spin particles obey Klein-Gordon equation or not.Similarly,do higher half integer spin particles obey Dirac equation?Because if we can not demontrate the higher spin particles obey Klein-Gordon(for Boson) or Dirac(for Fermion) equations,how can we say about the commutation relations of field operator and momentum operator of field for Bose fields and also how about the anticommutation relations for Fermi fields.Therefore,how can we make the quantization of fields, and how about the exclusion principle of Pauli...etc.
Thank you very much in advanced.
 
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  • #2
That depends on what you really mean by "obey", solutions of Dirac eqn obey Klein-Gordon equation automatically, so is Maxwell's equation for photon. This is because Klein-Gordon really tells you nothing but that the theory is relativistic. In this sense all relativistic particles obey Klein-Gordon equation.
However Klein-Gordan does not describe electrons appropriately, although electrons "obey" Klein-Gordan.
 
  • #3
I think the analysis made by Gelfand and Yaglom in 1948 remains the best when it comes to differential equations for <objects> transforming under the Lorentz group. It is exposed in the book by Gelfand, Minlos and Shapiro, 2nd Chapter of Section II.
 
  • #4
Spin 3/2 obeys the Rarita-Schwinger equation. The spin 3/2 field has a mixture of vector and spinor indices.

The spin 2 field should obey the Einstein equation.
 
  • #5
So,the higher half integer spin may be considered as a mixture of vector indices and spinor indices.Therefore, all half integer spin particles obey Dirac equation?
 
  • #6
ndung200790 said:
So,the higher half integer spin may be considered as a mixture of vector indices and spinor indices.Therefore, all half integer spin particles obey Dirac equation?

I'm not that familiar with the spin 3/2 equation.

But does the electromagnetic potential obey the Klein-Gordan equation? I don't think it does unless you choose the Lorenz gauge or the Coloumb gauge.

addendum:

it goes something like this:

[tex]\Box A^\mu-\partial^\mu \partial_\nu A^\nu=0 [/tex]

The Lorenz gauge obviously makes the 2nd term go away. The Coloumb gauge makes the 4-divergence produce the timed derivative of the scalar potential, which is zero (just set mu equal to zero in the equation to see that the scalar potential obeys Laplace's equation in free space).
 
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  • #7
I think we need to set the gauge fix to eliminate redundant configurations.So photons still obey Klein-Gordon equation.
 
  • #8
dextercioby said:
I think the analysis made by Gelfand and Yaglom in 1948 remains the best when it comes to differential equations for <objects> transforming under the Lorentz group. It is exposed in the book by Gelfand, Minlos and Shapiro, 2nd Chapter of Section II.

Do you have the title of the text?
 
  • #9
homology said:
Do you have the title of the text?

Sure, it's an old book from the 1960's (English translation): <Gelfand, Minlos, Shapiro - Representations of rotation and Lorentz groups>.
 
  • #10
ndung200790 said:
I think we need to set the gauge fix to eliminate redundant configurations.So photons still obey Klein-Gordon equation.

Doesn't it depend on which gauge you choose? What if I chose the axial gauge: A3=0 ?

Would:

[tex]
\Box A^\mu-\partial^\mu \partial_\nu A^\nu=0
[/tex]

reduce to the Klein-Gordan equation?
 
  • #11
RedX said:
Doesn't it depend on which gauge you choose? What if I chose the axial gauge: A3=0 ?

Would:

[tex]
\Box A^\mu-\partial^\mu \partial_\nu A^\nu=0
[/tex]

reduce to the Klein-Gordan equation?

But the thing is, for massless Klein-Gordan eqn you'll also have the freedom of choosing a gauge, without changing any physical content. So I think in a loose sense it still obeys KG eqn.
 
  • #12
kof9595995 said:
But the thing is, for massless Klein-Gordan eqn you'll also have the freedom of choosing a gauge, without changing any physical content. So I think in a loose sense it still obeys KG eqn.

I was under the impression that discussion was over the free, non-charged KG equation.

The free, charged KG equation has a U(1) symmetry but not a gauge symmetry.

The solution to the EOMs of free KG fields are unique, so I don't believe there is any freedom to impose gauge conditions.
 
  • #13
RedX said:
The solution to the EOMs of free KG fields are unique, so I don't believe there is any freedom to impose gauge conditions.

No, I meant we could change a gauge for KG equation, and the form of KG eqn will look different, however it's still KG eqn in a loose sense because nothing physical has changed, just like in electromagnetism, different gauge fixing gives different [tex]

\Box A^\mu-\partial^\mu \partial_\nu A^\nu=0

[/tex] but does not change the physics. (Actually I don't know formulate and prove " nothing physical has changed" in this quantum context, could somebody enlighten me?)
 
  • #14
RedX said:
I was under the impression that discussion was over the free, non-charged KG equation.
Of course not necessarily non-charged, or else I wouldn't say Dirac field also obeys KG eqn. Anyway what I tried to say was simply asking what equation a field "obeys" is not enough, there is missing information in how the field transforms.
RedX said:
The free, charged KG equation has a U(1) symmetry but not a gauge symmetry.
I'm confused, why not U(1) a gauge symmetry group?
 
  • #15
kof9595995 said:
No, I meant we could change a gauge for KG equation, and the form of KG eqn will look different, however it's still KG eqn in a loose sense because nothing physical has changed, just like in electromagnetism, different gauge fixing gives different [tex]

\Box A^\mu-\partial^\mu \partial_\nu A^\nu=0

[/tex] but does not change the physics. (Actually I don't know formulate and prove " nothing physical has changed" in this quantum context, could somebody enlighten me?)

O I see what you're saying. So for the charged KG equation:

[tex]\Box \phi + m^2 \phi=0 [/tex]

and it's conjugate equation, you can add a term to get a new equation:

[tex]\Box \phi + m^2 \phi+\partial^\mu(\phi^*\partial_\mu\phi-\phi \partial_\mu \phi^*)=0 [/tex]

whose solution is the KG-field, since the term in parenthesis is zero since it is the conserved 4-current of the KG-equation without this extra term, so that a solution of the KG-field without this extra term is also a solution to the KG-equation with this extra term.

This only works for the charged KG-field though, or else that extra term is zero.
 

1. What is the difference between the Klein-Gordon and Dirac equations?

The Klein-Gordon equation describes the behavior of spinless particles, while the Dirac equation describes the behavior of particles with spin. This means that the Dirac equation takes into account the intrinsic angular momentum of particles, while the Klein-Gordon equation does not.

2. Do all higher spin particles obey the Klein-Gordon or Dirac equations?

No, particles with spin greater than 1/2 cannot be described by either the Klein-Gordon or Dirac equations. These particles require more complex equations, such as the Proca equation or the Rarita-Schwinger equation.

3. How do the Klein-Gordon and Dirac equations relate to special relativity?

Both equations are relativistic in nature, meaning they take into account the effects of special relativity on the behavior of particles. The Dirac equation was specifically developed to be consistent with special relativity, while the Klein-Gordon equation was later modified to also be relativistic.

4. Can the Klein-Gordon and Dirac equations be used for particles with mass?

Yes, both equations can be used for particles with mass. However, the Dirac equation is more commonly used for particles with mass, as it was specifically developed to describe massive particles.

5. How are the Klein-Gordon and Dirac equations related to quantum mechanics?

The Klein-Gordon and Dirac equations are both examples of relativistic wave equations, which are fundamental in quantum mechanics. These equations describe the behavior of particles at the quantum level, taking into account the probabilistic nature of quantum mechanics.

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