The relation of colour charge to electric charge

[PAMD1958]

Dirac has shown how Einstein's expression for the relation of energy to momentum in Special Relativity can be factored into two linear parts using 4x4 Dirac matrices. [Dirac, P.A.M., The Principles of Quantum Mechanics, 4th edition (Oxford University Press) ISBN 0-19-852011-5]

This can also be done using 2x2 Pauli matrices (labelled K,L,M) because two inertial observers agree on the component of momentum Q orthogonal to the component of momentum P in the direction of a Lorentz boost.

(E/c)^2 - P^2 - Q^2 - (mc)^2I = (E/c + rKP + gLQ + mcbM)(E/c - rKP - gLQ - mcbM)

This is true for all 3! = 6 permutations of K,L,M where r,g,b equal +1 or -1. The set {I,K,L,M} forms the basis of a 4-dimensional real vector space.

For leptons r,g,b all equal -1 and for quarks two of r,g,b are equal to +1 and the third equals -1. The signs are all negated for anti-particles as in the equation above.

The 3 cyclic permutations KLM = MKL = LMK count the number of plus signs (say) for r,g,b which is 0 for leptons and 2 for quarks.

The 3 cyclic permutations MLK = LKM = KML count the number of minus signs (say) for r,g,b which is 3 for leptons and 1 for quarks.

For material particles r,g,b all equal -1 which is always true for leptons and true for three distinct quarks with r,g,b equal to -1 separately or a quark and an appropriate anti-quark.

Similarly, for material gauge bosons, the 3 cyclic permutations of KLM squared (which all equal -I) count the number of plus signs (say) for r,g,b which is 0 for the Z boson and the 3 cyclic permutations of MLK squared (which all equal -I) count the number of minus signs (say) for r,g,b which is 3 for the W boson.

The photon, having zero rest mass, carries no electric charge.

Edited August 17, 2013 by PAMD1958