Hypercomplex Quantum Mechanics

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module. 1 1. Introduction In his discussion of the development of the theory of matrices in the middle of the nineteenth century, in which he remarked that " it seems almost uncanny how mathematics now prepared itself for its future service in quantum mechanics, " Max Jammer 1 recounted how the natural generalization of the real numbers to complex numbers and quaternions played a central role. He cites Tait 2 as attributing to Hamilton the discovery of matrices in a letter to A. Cayley, who discovered that quaternions could be represented as 2 × 2 matrices over complex elements. Hamilton 3 invented the quaternions in 1844; Tait referred to Hamilton's " linear and vector operators, " and called Cayley's discovery only a modification of Hamilton's ideas. Taber 4 , in 1890, renewed the claim that Hamilton had indeed originated the theory of matrices. Gibbs 5 " regarded 1844 as a 'memorable' year in the annals of mathematics because it was the year of the appearance of Hamilton's first paper 3 on quaternions. " 1. John von Neumann, in fact, emphasized the result of Hurwitz 6 , that there are just four normed division algebras, the real (R), complex (C), quaternion (H), and octonion, or Cayley, algebra (O), and that " nature must make use of them. " 7. Herman Goldstine and I set out to investigate the possibility of constructing a Hilbert space, with application to a more general form of the quantum theory, using the Cayley numbers O as coefficients on the vector space (they are not commutative or associative) 8. In fact, the octonions arise in a natural way as a result of the attempt of Jordan, von Neumann and Wigner 9 to set up a quantum theory in which the products of observables …