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To establish the Dirac equation we use Clifford algebra to find the spinor representation of the Lorentz group.

A four dimensional representation tells us that the Dirac spinor describes a quantum theory involving two particles and two spins or four degrees of freedom.

Now the question is what happens if we find a higher dimension representation, say 10 or more, of the Lorentz group?

Will the theory involve more particles with new more characteristics due to the increased degrees of freedom per spacetime points?

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    $\begingroup$ Do you know about the spin of an irreducible representation? Related: physics.stackexchange.com/q/73593/2451 , physics.stackexchange.com/q/277986/2451 and links therein. $\endgroup$ – Qmechanic Dec 4 '16 at 12:07
  • $\begingroup$ What do you mean by "if we find a higher dimension representation"? We are the ones putting the fields and their representations into the QFT, how could we suddenly "find" such a representation? While it's not exactly clear to me what you're asking, note that there are several theorem (sometimes called "Weinberg-Witten theorem") that forbid higher-spin particles in a large class of theories. $\endgroup$ – ACuriousMind Dec 4 '16 at 13:51
  • $\begingroup$ Hadrons may and do fit into higher dimensional reps of the Lorentz group. You may peruse the List of baryons to find examples of J=3/2, so 4-spin states, the Ω-. Or spin 5/2, 6-dim, the N(1675), etc... the sky is the limit. They are composite states, of course, but they illustrate higher-dimensional reps. Doesn't your text clarify that? $\endgroup$ – Cosmas Zachos Dec 4 '16 at 16:05
  • $\begingroup$ Look at the nifty meson $f_4(2050)$ on page 19. It has J=4, so it is 9-dim, close to your gambit. $\endgroup$ – Cosmas Zachos Dec 4 '16 at 20:00

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