Klein-Gordon field is spin-0: really?

However, for higher spins you need to consider also the spinorial representation of the Poincare group. This is what the book by Fushchych & Nikitin does, putting together the relativistic theory of fields with the theory of representations of the Poincare group.
  • #1
jjustinn
164
3
Usually in the first sentence of the definition of the Klein Gordon equation is the statement that it describes spin-0 particles.

Similarly, in the first sentence of the definition of the Dirac equation is the statement that it describes spin-1/2 particles.

But then comes the bit that got me to write this post -- a few sentences later in that latter definition, you're bound to run into the fact that the Dirac equation is the formal square root of the Klein-Gordon equation, and that therefore any solution to the Dirac equation is also solution of the Klein-Gordon equation (though obviously not vice-versa; [itex]x = y \rightarrow x^{2} = y^{2}[/itex], but [itex]x^{2} = y^{2} !\rightarrow x = y[/itex]).

So...what gives? How can you say that "the Klein-Gordon equation describes spin-0 particles" when every spin-1/2 particle (or at least all of those described by the Dirac equation) is also a solution?

Looking at the Klein-Gordon equation, it seems like it should be valid for *any* relativistic particle -- [itex]E^{2} = p^{2} + m^{2}[/itex] is generally true, right?

My gut is telling me that the answer is that while the Klein-Gordon has many non-scalar-particle solutions, it only provides a *complete* description of spin-0 particles...but that seems like a big enough distinction to warrant at least an asterisk in that ever-present first sentence.

Also, feel free to school me on spin, field quantization, or anything else that I obviously have only a really shallow understanding of.

Thanks,
Justin
 
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  • #2
I think there was a similar topic here on this issue in June, IIRC. Anyways, there's a general theory of <relativistic field equations> first set up in the 1930's (Dirac, Fierz and Pauli). One of its conclusions is that essentially p^2 = m^2 and p^2 = 0, the classical contraints for massive & massless particles, translate into second order partial diff. equations for fields. So essentially in the PDE

[tex] \left(\partial_{\mu}\partial^{\mu} + m ^2\right) \mbox{object}(x,t) = 0 [/tex]

the <object> can be any spin field (0,1/2,1,3/2) for a fundamental particle of mass m.

Thus, in the statement you mention below there's a mistake by omission. Other fields, too, obey the K-G equation.

<Usually in the first sentence of the definition of the Klein Gordon equation is the statement that it describes spin-0 particles.>
 
  • #3
If the "object" in the dextercioby´s equation is a one-component object, THEN this Klein-Gordon equation describes a spin-0 particle. Originally, Klein and Gordon assumed that this object IS a one-component object, which is why it is said that their equation describes a spin-0 particle.
 
  • #4
Thus, in the statement you mention below there's a mistake by omission. Other fields, too, obey the K-G equation.

<Usually in the first sentence of the definition of the Klein Gordon equation is the statement that it describes spin-0 particles.>

If the "object" in the dextercioby´s equation is a one-component object, THEN this Klein-Gordon equation describes a spin-0 particle. Originally, Klein and Gordon assumed that this object IS a one-component object, which is why it is said that their equation describes a spin-0 particle.

Both very helpful! I get it. So it's the number of components of the wavefunction.

Now since you got that one so quickly :devil:, I guess that brings up another question -- is there a first-order equation that describes spin-0 particles? It seems equally weird to me that the formal square root of the K-G equation describes only multi-component fields. I totally get that the way the Dirac equation achieves that square root necessitates that it operate on a multi-component field, but it still seems like there should be some other way.

And while we're at it, by analogy with putting multi-component wavefunctions into the K-G equation, does the Dirac equation also then describe higher-spin fields, e.g. with a higher-dimensional representation of the Dirac matrices? I realize that there's no lower-dimensional representation that would let you do scalars (since scalars commute), so I guess this is sort of a companion to the prior question.
 
  • #5
I find it difficult to summerize, but the issues raised in your post find an answer in the wonderful book by Fushchych & Nikitin <The symmetries of equations of Quantum Mechanics>, especially Chapter 2, Sections 6 & 8, anyway from page 75 onwards.

Ammendment to post 2: Even spin 2 can be brought to KG form (but necessarily with mass 0), as the field equations for gravitational waves prove it.
 
  • #6
dextercioby said:
I find it difficult to summerize, but the issues raised in your post find an answer in the wonderful book by Fushchych & Nikitin <The symmetries of equations of Quantum Mechanics>, especially Chapter 2, Sections 6 & 8, anyway from page 75 onwards.

Awesome, thanks for the recommend. I just got some Amazon credit, so if it's as interesting as it sounds I'll probably be flipping through it soon.

Correction: Looks like it's out-of-print, and no one is selling for any price on Amazon. sigh.

Ammendment to post 2: Even spin 2 can be brought to KG form (but necessarily with mass 0), as the field equations for gravitational waves prove it.
The way I read the post (and my gut feeling since it seems to be a simple re-statement of the relativistic kinetic energy relation) I took it to mean that anything would need to satisfy the Klein-Gordon equation, since not doing so would mean violation of relativity (off-shell virtual quanta or whatever obviously excluded -- though the reasons for that are well over my head).
 
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  • #7
jjustinn said:
[...]The way I read the post (and my gut feeling since it seems to be a simple re-statement of the relativistic kinetic energy relation) I took it to mean that anything would need to satisfy the Klein-Gordon equation, since not doing so would mean violation of relativity (off-shell virtual quanta or whatever obviously excluded -- though the reasons for that are well over my head).

That is the right understanding, indeed. Poincare invariant field equations obey special relativity principles.
 

FAQ: Klein-Gordon field is spin-0: really?

What is the Klein-Gordon field and why is it considered to have spin-0?

The Klein-Gordon field is a theoretical field used in quantum field theory to describe relativistic particles with spin-0, meaning they have no intrinsic angular momentum. This is in contrast to particles with half-integer spin, such as electrons, which are described by other fields like the Dirac field.

How does the Klein-Gordon equation describe spin-0 particles?

The Klein-Gordon equation is a relativistic wave equation that describes the behavior of spin-0 particles. It is a second-order differential equation that takes into account both the particle's mass and energy, and it has solutions that represent the particle's wave function in space and time.

Is the Klein-Gordon field used to describe any real-world particles?

No, the Klein-Gordon field is a theoretical construct used in quantum field theory. While it has been successful in predicting the behavior of certain particles, such as the Higgs boson, it is not a direct description of any known particle in our universe.

What are the limitations of the Klein-Gordon field in describing particles?

One major limitation of the Klein-Gordon field is that it does not take into account the effects of quantum spin, which is a critical aspect of many particles in our universe. It also does not account for interactions between particles, which are important in understanding their behavior.

Are there any alternative theories to the Klein-Gordon field for describing spin-0 particles?

Yes, there are alternative theories, such as the Proca field and the Klein-Gordon-Proca field, which attempt to incorporate the effects of spin and interactions into the description of spin-0 particles. However, the Klein-Gordon field remains a useful and widely accepted tool in quantum field theory.

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