It is well known that Antimatter was first predicted by interpreting the matrices that show up in the Dirac Equation as indicating its existence.

Dirac factorizes $E^2=p^2+m^2$ ($c=1,\hbar=1$) into $E=\alpha_x \hat p_x+\alpha_y \hat p_y+\alpha_z \hat p_z+\beta m$, such that $$i\partial_t \psi=-i\alpha_x \partial_x\psi-i\alpha_y \partial_y\psi-i\alpha_z \partial_z\psi+\beta m$$ This is only possible if the $\alpha$'s and the $\beta$ are matrices, so the wavefunction is a vector, which implies spin is integrated into the wavefunction, but the matrices have to be $4\times4$ matrices, so the wavefunction is a $4$-component vector, with two solutions of negative energy.

This all happens because we try to find the wavefunction of a relativistic particle. Matrix Mechanics, Heisenberg's formulation, instead of wavefunctions and differential operators has the observables represented by matrices, and the state represented by a state vector. Can spin and antimatter be predicted from this formulation of Quantum mechanics, without starting by accounting for the states of different spin and antimatter in the Hamiltonian a priori?


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