This question might seem basic, but how does one conclude that the Klein-Gordon equation describes spin zero particles but Dirac equation describes spin half particles. Thanks.

EDIT: Adding more details

One can derive both equations from the relativistic energy-momentum relation.Moreover, Dirac equation is just the linearization of the Klein-Gordon equation, and since both equation come from the same "source" how can both describe different things? is there something special about the linearization process? Also how does one know that a certain equation describes a certain type of particles is there a theoretical justification for this conclusion or just experimental? I am just starting to learn about these things and I am confused to how people came to these conclusions.

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    $\begingroup$ This question needs more details. KG is a scalar and Dirac a spinor equation, but it is a trivial answer. $\endgroup$ – my2cts May 22 at 14:09
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    $\begingroup$ I don't know what you mean by "the Dirac equation is just the linearization of the Klein-Gordon equation". The Klein-Gordon equation is for a $\mathbb{R}$- or $\mathbb{C}$-valued field, the Dirac equation for a $\mathbb{C}^4$-valued field. $\endgroup$ – ACuriousMind May 22 at 15:01

The short answer would be this: Klein-Gordon equation was written assuming that a particle(field) follows a relativistic energy conservation law. Much like Schrodinger's equation, the derivation of KG equation is not possible. You cannot "conclude" that this equation defines spin-0 particle, rather what you see is that the solution of this equation are complex functions $\phi(r)$ which are Lorentz invariant. Thus they are in a 1-dim representation of Lorentz group. Hence, they are spin-0 or scalar.

On the other hand you can take a "square-root" of the KG equation to reach Dirac equations. Basically, try and write the same equation in first order in space and time. You'll immediately see that scalar complex functions can no longer solve the equation. Hence you require to interpret it as an eigenvalue equation and see that the solution space is actually spin-$\frac{1}{2}$ representation of Lorentz group.

P.S. I haven't included the equations themselves in my answer because my hope is, it's not necessary for the present discussion.

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  • $\begingroup$ One more comment: each component of a solution of dirac equation solves KG equation. Which is understandable from above discussion. $\endgroup$ – Ari May 22 at 15:02
  • $\begingroup$ I have seen the solution of Dirac equation is the case of a free particle, but I can't see how would one conclude from the solution that the particle has spin half. $\endgroup$ – Joel May 22 at 15:22
  • $\begingroup$ The general solution of dirac equation is a matrix, more specifically a column vector. Once you derive it, you can see how that column vector transforms under Lorentz transformations. That tells you it's spin 1/2 $\endgroup$ – Ari May 22 at 15:28
  • $\begingroup$ The free particle solution you refer to actually involves making the dirac equation a second order differential equation i.e. KG equation. In that case each components of Dirac eqn is separated and they satisfy KG eqn as I mentioned above. $\endgroup$ – Ari May 22 at 15:31
  • $\begingroup$ Thanks for clearing that up. $\endgroup$ – Joel May 22 at 16:00

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