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

The derivation of both Klein-Gordon equation and Dirac equation is due the need of quantum mechanics (or to say more correctly, quantum field theory) to adhere to special relativity. However, excpet that Klein-Gordon has negative probability issue, I do not see difference between these two. What makes Klein-Gordon describe scalar field while Dirac describe spin-1/2 field? Edit: oops. Klein-Gordon does not have non-locality issue. Sorry for writing wrongly.

Edit: Can anyone tell me in detail why $\psi$ field is scalar in Klein-Gordon while $\psi$ in Dirac is spin-1/2? I mean, if solution to Dirac is solution to Klein-Gordon, how does this make sense?

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4 Answers

Let's review how the KG equation is recovered from the Dirac:

$$(i \gamma \delta - m)\Psi = 0$$ $$(-i \gamma \delta - m)(i \gamma \delta - m) = 0$$ $$(\gamma^\nu \gamma^\mu \delta_\nu \delta_\mu + m^2) \Psi = 0$$ $$(\delta^2+m^2)\Psi = 0$$

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.

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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.

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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.

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The spin 1/2 description is wrong. One has to apply $\Lambda$ and not its square root, but as it is applied to a spinor rather than a vector, the action of $\Lambda$ is different. – Arnold Neumaier Oct 11 '12 at 10:47
Each and every text book uses the expression $\Lambda^{1/2}$ as applied to the spinor components. Spinors are for the same reason also normalized to the square of the mass $\sqrt{m}$ – Hans de Vries Oct 11 '12 at 11:06
You are confusing observables such as the current $\bar{\psi}\gamma^\mu\psi$ which transforms like a vector with the field $\psi$ itself. – Hans de Vries Oct 11 '12 at 11:10
Rather than looking at the specific form of $\Lambda$, Look at the general scaling of a general boost: Vectors: $\cosh(\vartheta)+\Gamma\sinh(\vartheta)$ versus Spinors:  $\cosh(\vartheta/2)+\Gamma\sinh(\vartheta/2)$ – Hans de Vries Oct 11 '12 at 11:35
I was not talking about currents. - ''Each and every text book uses the expression $Λ^{1/2}$''?? Then please show me where it is used in this form in Weinberg's Volume 1. – Arnold Neumaier Oct 11 '12 at 14:16
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Spin is part of what a field IS. The data for two fields of different spins are very different. The KG equation doesn't even make sense for a spin 1/2 field and likewise for the Dirac equation and spin 0 fields.

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