Why would Klein-Gordon describe spin-0 scalar field while Dirac describe spin-1/2?

Spin is a property of the representation of the rotation group $SO(3)$ that describes how a field transforms under a rotation. This can be worked out for each kind of field or field equation.

The Klein-Gordon field gives a spin 0 representation, while the Dirac equation gives two spin 1/2 representations (which merge to a single representation if one also accounts for discrete symmetries).

The components of every free field satistfy the Klein-Gordon equation, irrespective of their spin. In particular, every component of the Dirac equations solves the Klein-Gordon equation. Indeed, the Klein-Gordon equation only expresses the mass shell constraint and nothing else. Spin comes in when one looks at what happens to the components.

A rotation (and more generally a Lorentz transformation) mixes the components of the Dirac field (or any other field not composed of spin 0 fields only), while on a $k$-component spin 0 field, it will transform each component separately.

In general, a Lorentz transformation given as a $4\times 4$ matrix $\Lambda$ changes a $k$-component field $F(x)$ into $F_\Lambda(\Lambda x)$, where $F_\Lambda=D(\Lambda)F$ with a $k\times k$ matrix $D(\Lambda)$ that depends on the representation. The components are spin 0 fields if and only if $D(\Lambda)$ is always the identity.

If we say:

"A field has a spin 0, spin 1/2 or spin 1 representation"

then we in fact say something about how the field parameters transform if we go from one reference frame to another.

spin 0: The values of the field do not change if we go from one reference frame to another

spin 1: We have to apply the Lorentz transform matrix $\Lambda$ on the field parameters.

spin 1/2: We have to apply $\Lambda^{1/2}$ on the field parameters.

Remark: The use of an expression like $\Lambda^{1/2}$ should be interpreted in a somewhat symbolic way because vectors and bispinors are different objects. There is an extra factor 1/2 though in the exponent of the $\Lambda^{1/2}$ matrix.

The spin (associated with rotation) gets in here because the transformation matrix $\Lambda$ handles both boosts as well as rotations. The peculiar factor 1/2 however arises also in the 1 dimensional version of the Dirac equation where there is no such thing as spin (or rotation) and the corresponding 1 space + 1 time dimension version of $\Lambda$ only describes boosts.

The deeper reason for the factor 1/2 is that the Dirac equation relates two field components $\psi_R$ and $\psi_L$ which are equal to each other in the rest frame. In the 1 dimensional case these are the right-moving and left-moving components. The ratio of the two transforms as follows

$(\psi_R:\psi_L)\longrightarrow\Lambda~(\psi_R:\psi_L)$

In the normalization of the plane wave eigen functions this then ends up like

$\psi_R\longrightarrow\Lambda^{+1/2}\psi_R$

$\psi_L\longrightarrow\Lambda^{-1/2}\psi_L$

If we now go back to 3 spatial dimensions then $\Lambda$ includes both boosts and rotations and the factor 1/2 as an exponent on the rotation generation matrices leads two what we call spin 1/2 particles.

Hans.

Let's review how the KG equation is recovered from the Dirac: (in natural units where $\hbar=c_0=1)$

$$(i\gamma^\mu \partial_\mu - m)\Psi = 0$$ $$(-i \gamma^\mu \partial_\mu - m)(i \gamma^\mu \partial_\mu - m) = 0$$ $$(\gamma^\nu \gamma^\mu \partial_\nu \partial_\mu + m^2) \Psi = 0$$ $$(\partial^2+m^2)\Psi = 0.$$

In order for us to recover KG, we had to assume $\gamma^\nu \gamma^\mu = \eta^{\mu\nu}$. In other words, you can think of gamma's as doing the dot product of delta's. To take a lousy example, it's like we had an equation describing a velocity "spinor" and then squared it, so now it describes the speed "scalar" which has one less degree of freedom. This doesn't explain much more than how an equation describing a spinor can reduce into an equation describing a scalar.

The reason why the Dirac equation requires spinors and not scalars is because of special relativity. If it weren't for the pesky minus sign in $\eta^{\mu\nu}$, the algebra of the gamma's would be much simpler and we wouldn't need them to be 4x4 matrices. Then $\Psi$ could describe a scalar field.