The properties of wave tensors

1. In wave-mechanics there occur in addition to the ordinary physical vectors and tensors the four-valued quantities Ψ, Ψ introduced by Dirac. In this combination certain relations of invariance in regard to transformation appear which had escaped the ordinary tensor calculus. If we call the new type of quantity involving Ψ and its combinations Ψ-tensor, the position is that we cannot reach Ψ-tensors from the calculus of ordinary space tensors, but we can reach space tensors from a calculus of Ψ-tensors. I have shown that although Ψ-vectors cannot be expressed in terms of space vectors, mixed Ψ-tensors can be immediately resolved into space vectors. The present paper contains a simplification and systematisation of my earlier work on the Ψ-tensor calculus as well as a number of new results. The particular point round which the new results centre is as follows. The Ψ-tensors occurring in wave-mechanics are the product of two Ψ-vectors (Ψ, Ψ), and apparently the primary reason for introducing the unobservable quantity Ψ rather than working with physical tensors throughout is to impose this condition. I therefore examine the question: If we impose the condition that a wave tensor is the product of two Ψ-vectors, what is the corresponding restriction on the physical tensors equivalent to it ? The answer is (§7) that whatever is described by two Ψ-vectors Ψ, Ψ can equivalently be described by two space vectors of equal length at right angles to one another. One of these is the momentum vector; the other (generally ignored in current investigations) presumably represents positional relations (co-ordinates or distance); or rather I would regard it as the source of positional relations, which can only become explicit in more complicated developments involving many particles. The ordinary wave equation for one particle is obtained as anidentity .