Spinors in Hilbert Space

1. Hilbert Space The words "Hilbert space" here will always denote what math- ematicians call a separable Hilbert space. It is composed of vectors each with a denumerable infinity of coordinates ql q2 Q3, .... Usually the coordinates are considered to be complex numbers and each vector has a squared length rIQrI2. This squared length must converge in order that the qs may specify a Hilbert vector. Let us express qr in terms of real and imaginary parts, qr = Xr ] iYr Then the squared length is l: .r(x; + y;). The xs and ys may be looked upon as the coordinates of a vector. It is again a Hilbert vector, but it is a real Hilbert vector, with only real coordinates. Thus a complex Hilbert vector uniquely determines a real Hilbert vector. The second vector has, at first sight, twice as many coordinates as the first one. But twice a denumerable in- finity is again a denumerable infinity, so the second vector has the same number of coordinates as the first. Thus a complex Hilbert vector is not a more general kind of quantity than a real one.