Confused about Dirac particles

[metroplex021]

I'm really confused! The Dirac equation describes spin -1/2 particles - i.e. particles of definite spin. And yet the spin operator does not commute with the Dirac Hamiltonian!

The reason I'm confused is because I thought if you were going to describe particles of a given kind - that is, particles identified by a property P - that satisfy a given Hamiltonian, it has to be the case that the operator corresponding to P commutes with that Hamiltonian. So, for example, I thought that if you want to say that a kind of particle with charge q obeys a given Hamiltonian, then the charge operator Q had to commute with that Hamiltonian.

Where have I gone wrong?! Any help massively appreciated.

 

[Bill_K]

No, but H commutes with the total angular momentum J. Must remember to include the orbital part.

H = α.p + β m
J = r x p + ½ σ

[H, J] = [α.p, r x p] + [β m, r x p] + [α.p, ½ σ] + [β m, ½ σ]
= -i α x p + 0 + i α x p + 0 = 0

 

[metroplex021]

Thank you for that, *but* is it not the case that when we say 'electrons are spin 1/2 particles', we're staying silent about their orbital momentum?

It still seems really weird to me to say that spin 1/2 particles satisfy the Dirac equation, and yet that the Dirac Hamiltonian does not conserve the spin of those particles. (For if the spin was measured as 1/2 at t=0 and something different at t=t', then how could we say that there's a spin 1/2 particle evolving in time according to the Dirac equation?!)

 

[Bill_K]

But wait! There's more! What about σ.p? What if I told you that the helicity σ.p also was conserved? Now what would you say?

 

[metroplex021]

But wait! There's more! What about σ.p? What if I told you that the helicity σ.p also was conserved? Now what would you say?

Well, I guess I'd say that unlike s(s+1) it's not Lorentz invariant (where there is non-zero mass), so we can't use it to classify particles as being of one type or another...

 

[dextercioby]

Total spin is not a conserved observable, because it's not a central element in the Poincare algebra, nor a Casimir operator in the universal envelopping one. Pauli-Liubanskii vector operator squared is a valid conserved observable for massive uniparticle quantum states.

 

[Bill_K]

metroplex,

Seriously, the form of the total angular momentum, J = r x p + ½ σ shows explicitly that the particle being described has spin 1/2. I think you want S to commute with the Hamiltonian separately, but in general it does not. This is not a paradox, because S *does* commute with H in the particle's rest frame, where p = 0.

 

[metroplex021]

thanks people (esp bill_k). that's really helpful. crisis over.

 

[gpran]

Wikipidea says (http://en.wikipedia.org/wiki/Spin_(physics ) : Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle.

Dirac analysis gives spin-1/2 value of 'elementary' point particles like electron. In that case, are composite particles with spin-1/2 also Dirac particles? If not, then why? Can anyone please explain how to define a Dirac particle?