Quaternionic Quantum Interferometry

If scattering amplitudes are ordinary complex numbers (not quaternions) there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some possible experimental tests involve neutron interferometry, KS-meson regeneration, and low energy proton-proton scattering. When we progress in the hierarchy of numbers, we encounter integers, real numbers, complex numbers, and then quaternions. The latter are hypercomplex numbers which can be written as a + ib + jc + kd, where i = j = k = −1 and ij = −ji = k, etc. They are the only generalization of complex numbers that satisfies the associative and distributive laws, and for which division is possible and unique (Chevalley, 1946). They were originally introduced in classical physics by Hamilton, in order to describe 3-dimensional rotations. When we further progress from classical physics to quantum theory, we learn that the states of a physical system can be represented by a linear manifold (Peres, 1993). Namely, if ψ1 and ψ2 are two possible states of a quantum system, and c1 and c2 are arbitrary numbers, then the expression c1ψ1 + c2ψ2 also represents a possible state of that system. It is usually taken for granted that the coefficients c1 and c2 are complex numbers. However, it is possible to imagine a real quantum theory (Stueckelberg, 1960) or one based on quaternions (Finkelstein, Jauch, Schiminovich and Speiser, 1962–3; Emch, 1963; Wolff, 1981; Sharma and Coulson, 1987). The purpose of this article is to show how interferometric experiments can distinguish between these various quantum theories. Real quantum theory, although logically consistent, can be easily ruled out for our world: e.g., complex coefficients are needed in order to combine linearly polarized photons into circularly polarized ones. More generally, correspondence with classical physics leads to the commutation relation [q, p] = ih̄. [Here, it may be pointed out that Stueckelberg’s “real” quantum theory requires the introduction