  The brief works of Peierls on the role of the observer in quantum mechanics
are examined, interpreted and expanded to widen accessibility and understanding
of these works. The approach followed here is very much in the spirit adopted
by Peierls who eschewed a `rigorous axiomatic' and aimed at using `logic at the
level of the working physicist'. The fundamental tenet of his work is that the
wavefunction or density matrix represents the knowledge of an observer and that
two observers of the same system may well have different knowledge and will use
different density matrices to describe it. Essential to the understanding of
Peierls's approach is the demonstration given here that the density matrix
generally can be expressed entirely in terms of probabilities of observable
outcomes and that such probabilities are subjective in the first instance.The
process by which the knowledge of two observers may be combined is discussed in
detail for some simple cases and, by using Bayes's theorem, the progress to a
common understanding of the correct density matrix for a given situation
demonstrated. Peierls gave a criterion to ensure that the amalgamation of data
from two observers can never lead to violation of the uncertainty principle :
that the density matrices of two observers must commute. It is pointed out that
it is essential that the density matrices of two observers commute under all
circumstances no matter how inaccurate or even fanciful the data or beliefs
used to construct them. Once this is appreciated it can be shown that Peierls's
density matrix commutation criterion is fully equivalent to the standard result
in quantum mechanics that two observables must commute if they are to be known
precisely and simultaneously,but now extended to imperfectly known mixed states
rather than a single eigenstate.
