Quaternionic Quantum Mechanics and Noncommutative Dynamics

In this talk I shall first make some brief remarks on quaternionic quantum mechanics, and then describe recent work with A.C. Millard in which we show that standard complex quantum field theory can arise as the statistical mechanics of an underlying noncommutative dynamics. In quaternionic quantum mechanics, the Dirac transition amplitudes 〈ψ|φ〉 are quaternion valued, that is, they have the form r0 + r1i+ r2j + r3k, where r0,1,2,3 are real numbers and i, j, k are quaternion imaginary units obeying i = j = k = −1 , ij = −ji = k, jk = −kj = i, ki = −ik = j. The Schrödinger equation takes the form ∂|ψ〉 ∂t =− H̃|ψ〉 , H̃ =− H̃ . (1)