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newphy
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I am a newbie to QM. Why can't the Klein Gordon equation be used to describe particles with spin?
Thanks
Thanks
newphy said:I am a newbie to QM. Why can't the Klein Gordon equation be used to describe particles with spin?
Thanks
This statement may be too strong. Indeed, in a general case, the Dirac equation is equivalent to a fourth-order partial differential equation for just one complex function. Furthermore, the latter can be made real by a gauge transform (at least locally). Source: http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (published in the Journal of Mathematical Physics).tom.stoer said:So essentially spin always requires a spinor (a multi-component object)...
akhmeteli said:This statement may be too strong. Indeed, in a general case, the Dirac equation is equivalent to a fourth-order partial differential equation for just one complex function. Furthermore, the latter can be made real by a gauge transform (at least locally). Source: http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (published in the Journal of Mathematical Physics).
Why is it exotic? Indeed, the physics is the same as for the Dirac equation. It's just a new form of the Dirac equation, which may have its strong and weak points. But it is usually good to know different forms of the same theory, as some of them may be better suited for a specific problem.tom.stoer said:This seems to be rather exotic
I don't know what I can do about that. It is definitely confusing for me as well, although I'm not exactly a beginner. However, OP asked:tom.stoer said:is is definately confusing for a beginner.
And it seems that the correct answer is "Actually, after some significant modifications, it can." I believe that might be interesting for OP, judging by the question (s)he asked. And it may be interesting for other readers, and maybe even for you:-) So I believe my information is relevant and appropriate here.newphy said:Why can't the Klein Gordon equation be used to describe particles with spin?
I don' have a clear idea right now. But as the equation is generally equivalent to the Dirac equation, there is no doubt it does adequately describe spin.newphy said:How does one represent spin and rotations? How does one represent helicity and chirality?
In my post 4 in this thread, I just commented on one specific statement of post 3. For reasons outlined in post 6, I still believe the comment was appropriate.tom.stoer said:I hope I am not too harsh, but the original question was, why the Klein-Gordon-equation cannot explain spin. I think that has been answered in posts #2 and #3.
The Klein-Gordon-equation does not contain enough degrees of freedom to describe spin. The paper presented by akhmeteli is rather exotic and I doubt that along these lines something like QFT can be constructed; and even this paper has to start with the Dirac equation. Of course one can introduce spin in some way into scalar equations (the well known Pauli-equation is the best example) but this means a) modifying the equation and b) guesswork b/c not everything can be constructed rigorously (g-factor) w/o referring to the original Dirac equation.
So it is correct that other equations are able to describe certain aspects of spin, but this is not an explanation for its origin rooted in the spacetime symmetry.
tom.stoer said:the reduction described in the article still relates to the original Dirac spinor; the effective 4th order equation alone is not able to describe rotations in spinor-space as far as I can see;
tom.stoer said:it's like the Kepler problem: a certain solution can be described using a special ellipsis, but in order to describe the whole SO(3) symmetry you still need the full setup in R³
tom.stoer said:anyway, I doubt that anybody here will agree that for spin 1/2, 1, ... a single wave equation for a scalar field is sufficient
akhmeteli said:So these two equations contain pretty much the same information about spin.
...
Then why does spin always require a multi-component object?
...
as this "one-component field" has more complex transformation properties than the scalar field.
tom.stoer said:Please show me how your single wave function transforms w.r.t a general Lorentz transformation; what is the transformation corresponding to
[tex]\Lambda_\mu^\nu \to S(\Lambda)[/tex]
[tex]x^\prime = \Lambda x;\;\psi^\prime(x^\prime) = S(\Lambda)\,\psi(x)[/tex]
tom.stoer said:Your claim is that a single wave function is able to fully describe spin 1/2 fields including its Lorentz transformation properties, so there is a burden of proof on you - not me ;-)
tom.stoer said:Sorry, this is totally confusing!
You claim that "spin does not always require a spinor and that the equation does adequately describe spin". I interpreted this as "a single wave function is able to fully describe spin 1/2 fields including its Lorentz transformation properties". Now you disagree?
tom.stoer said:So your equation does not describe spin 1/2 fields including Lorentz transformation?
tom.stoer said:What else do you then mean by "the equation does adequately describe spin"? How can your equation describe spin if it can't describe the Lorentz transformation of spin?
But generally no physical solutions are excluded in the fourth-order PDE (remember, there is a one-to-one correspondence between the sets of solutions).tom.stoer said:btw.: gauge symmetry is an 'unphysical symmetry' whereas Lorentz invariance (in SR) is a physical symmetry; therefore fixing a gauge is not the same as reducing e.g. rotational symmetry; fixing a gauge does not reduce the physical content of a theory whereas reducing e.g. rotational symmetry explicitly excludes physical solutions.
"so your equation does not describe spin 1/2 fields including Lorentz transformation?"akhmeteli said:not ready to answer the question, ...
tom.stoer said:You (or we ;-) should try to find how to relate your equations to the standard Lorentz transformation for spinors. That's definately required in order to support your claims.
tom.stoer said:In order to prove full equivalence of your equations with standard Dirac's theory you should provide a proof, not only a claim. I don't think that this is 'baseless'; a (your) claim requires a proof, and a (your) claim w/o a proof IS baseless.
But now I'll stop this discussion; our positions are now obvious to everybody, but it may be boring for the reader to find numerous repetitions.
The Klein Gordon equation is a relativistic wave equation that describes the behavior of spinless particles, such as mesons, in quantum field theory. It was first proposed by physicist Oskar Klein and Walter Gordon in 1926.
The Klein Gordon equation does not explicitly incorporate spin. It describes the behavior of spinless particles, so it is not applicable to particles with intrinsic spin, such as electrons. However, it can be modified to include spin by adding terms from the Dirac equation.
The Klein Gordon equation is significant in particle physics because it was the first relativistic wave equation proposed. It played a crucial role in the development of quantum field theory and helped pave the way for the development of the more comprehensive Standard Model of particle physics.
The Klein Gordon equation is a relativistic version of the Schrödinger equation. It was derived by applying the principles of special relativity to the non-relativistic Schrödinger equation. However, the two equations have different interpretations and are applicable to different types of particles.
The Klein Gordon equation has been used in various areas of physics, including quantum field theory, condensed matter physics, and cosmology. It has also been used to describe the behavior of particles in strong gravitational fields, such as black holes. In addition, it has been used in the development of theoretical models for high-energy particle collisions, such as those occurring in particle accelerators.