- Summary
- Why does the derivation of the Dirac equation naturally lead to spin ½ particles? The equation is derived from very general starting assumptions, so which of these assumptions has to be wrong to give us a spin-0 or spin-1 particle?

Why does the derivation of the Dirac equation naturally lead to spin ½ particles? The equation is derived from very general starting assumptions, so which of these assumptions has to be wrong to give us a spin-0 or spin-1 particle?

I have tried to search for an answer and got as far as this quote from Dirac himself from the Principles of Quantum Mechanics:

"We are led to the value 1/2 h-cross for the spin of the electron by an argument depending simply on general principles of quantum theory and relativity. One could apply the same argument to other kinds of elementary particles and one would be led to the same conclusion, that the spin angular momentum is a half quantum. This would be satisfactory for the proton and the neutron, but there are some kinds of elementary particle (e.g. the photon and certain kinds of meson) whose spins are known experimentally to be different from 1/2 h-cross, so we have a discrepancy between our theory and experiment. The answer is to be found in a hidden assumption in our work. Our argument is valid only provided the position of the particle is an observable. If this assumption holds, the particle must have a spin angular momentum of half a quantum. For those particles that have a different spin the assumption must be false and any dynamical variables x1, x2, x3 that may be introduced to describe the position of the particle cannot be observables in accordance with our general theory. For such particles there is no true Schrodinger representation. One might be able to introduce a quasi wave function involving the dynamical variables x1, x2, x3, but it would not have the correct physical interpretation of a wave function - that the square of its modulus gives the probability density. For such particles there is still a momentum representation, which is sufficient for practical purposes."

I don’t understand what this means though. The Schrodinger equation produces wave functions where the position is an observable, yet it is only for spin-0 particles. What is Dirac trying to say here? Seems like it is quite a deep and fundamental point about the difference between spin ½ and other spin particles.

I have tried to search for an answer and got as far as this quote from Dirac himself from the Principles of Quantum Mechanics:

"We are led to the value 1/2 h-cross for the spin of the electron by an argument depending simply on general principles of quantum theory and relativity. One could apply the same argument to other kinds of elementary particles and one would be led to the same conclusion, that the spin angular momentum is a half quantum. This would be satisfactory for the proton and the neutron, but there are some kinds of elementary particle (e.g. the photon and certain kinds of meson) whose spins are known experimentally to be different from 1/2 h-cross, so we have a discrepancy between our theory and experiment. The answer is to be found in a hidden assumption in our work. Our argument is valid only provided the position of the particle is an observable. If this assumption holds, the particle must have a spin angular momentum of half a quantum. For those particles that have a different spin the assumption must be false and any dynamical variables x1, x2, x3 that may be introduced to describe the position of the particle cannot be observables in accordance with our general theory. For such particles there is no true Schrodinger representation. One might be able to introduce a quasi wave function involving the dynamical variables x1, x2, x3, but it would not have the correct physical interpretation of a wave function - that the square of its modulus gives the probability density. For such particles there is still a momentum representation, which is sufficient for practical purposes."

I don’t understand what this means though. The Schrodinger equation produces wave functions where the position is an observable, yet it is only for spin-0 particles. What is Dirac trying to say here? Seems like it is quite a deep and fundamental point about the difference between spin ½ and other spin particles.