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Why in every textbook when deriving Dirac's equation the smallest possible matrices ($2 \times 2$) are used? I wonder why one couldn't use spin 1 matrices ($3 \times 3$) and get relativistic equation for spin 1 particle?

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One can write down equations for spin-one particles but they won't be the Dirac equation because the Dirac equation, by definition, describes spin-1/2 particles (i.e. fermions such as leptons and quarks). The equations for free fields associated with spin-one particles are the Maxwell equations (for photons etc.) and/or Proca equations (for W-bosons or Z-bosons).

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but i think you would agree that its still an interesting exercise to try to obtain the bulk maxwell equations from such a field equation – lurscher Apr 17 '12 at 14:18
@Luboš Motl "by definition, describes spin-1/2" - could you elaborate? Wherever I see Dirac's equation derivation there is only considered the case of 2x2 matrices which obviously leads to spin-1/2 but what about other cases? – John Apr 17 '12 at 17:53
Dear @lurscher, the spin-one equation is Maxwell's equation. The work needed to get from one to another is zero. Well, at most, you may change a basis or something like that. John: electron has $j=1/2$ because the eigenvalue of $J_z=J_{12}=\gamma_1\gamma_2/2$ acting on the electron state is $1/2$ times the original state. Photon has $J=1$ because when you act with $J_{12}$ on the photon state moving in the z-direction, you get $1$ times the original state. One-half is something else than one. For example, if people only eat 1/2 of a bread, 1 bread may feed 2 of them, not one. – Luboš Motl Apr 18 '12 at 4:57

Particles in the standard model are in principle massless and acquire mass through interactions via the Lagrangian. This poses a problem if you want to use the 3x3 matrix generators of angular momentum because massless particles have different lie group representations as particles with mass.

The 3x3 generators for particles of spin 1 acts on three states, -1, 0 +1. In the massless case only two states remain.

With fermions and the 2x2 SU(2) matrices this issue doesn't exist. The two chiral components $\psi_L$ and $\psi_R$ propagate lightlike and the two states of each chiral component correspond with spin up and down where the spin is parallel or anti parallel to the direction of propagation.


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so long story short if I get you right: when it's spin-1/2 both $m=\pm 1/2$ values are ok for particles wheather or not they has a mass, while for spin-1 particles there's a problem with massless particles $m \neq 0$? – John Apr 17 '12 at 17:58

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