Quaternionic Quantum Mechanics and Noncommutative Dynamics.pdfVIP

IASSNS-HEP-96/72 Quaternionic Quantum Mechanics and Noncommutative Dynamics Stephen L. Adler 6 Institute for Advanced Study 9 9 1 Princeton, NJ 08540 l u J 1 1 v In this talk I shall first make some brief remarks on quaternionic quantum mechanics, 8 0 0 and then describe recent work with A.C. Millard in which we show that standard complex 7 0 6 quantum field theory can arise as the statistical mechanics of an underlying noncommutative 9 / h dynamics. t - p In quaternionic quantum mechanics, the Dirac transition amplitudes |φ are quater- e h : nion valued, that is, they have the form r + r i + r j + r k, where r are real numbers v 0 1 2 3 0,1,2,3 i X 2 2 2 r and i, j, k are quaternion imaginary units obeying i = j = k = −1 , ij = −ji = k, jk = a −kj = i, ki = −ik = j . The Schr¨odinger equation takes the form ∂ | ˜ = − H | , ∂t (1) ˜ ˜ † H = − H . A systematic study of quaternionic quantum mechanics is given in my recent book [1] on “Quaternionic Quantum Mechanics and Quantum Fields.” For information on how to obtain the book and updates on more recent work, see my web home page (http:// /∼adler/Html/quate