OBSERVABILITY OF THE S MATRIX
David N. Williams
November 17, 2011
In a recent discussion involving cross sections, I asserted
the following well-known fact:
The S matrix is observable, up to an overall phase factor in
each superselection sector.
Upon realizing that I remembered neither a reference nor the
proof, I figured I'd better work it out and write it down, as a
crutch for my aging brain. Here's the result.
Assume that all absolute squares,
||² =
are observable, where |f> and |g> are normalized, scattering
state vectors in the same superselection sector, and where S is
unitary on each sector. By polarization, all expressions of the
form
=
are then observable, because |h> in the sector.
Now let S' be another operator, unitary on the same super-
selection sectors, such that S'|g> in each sector. Then
S*S'|g>