  Complex phase factors are viewed not only as redundancies of the quantum
formalism but instead as remnants of unitary transformations under which the
probabilistic properties of observables are invariant. It is postulated that a
quantum observable corresponds to a unitary representation of an abelian Lie
group, the irreducible subrepresentations of which correspond to the
observable's outcomes. It is shown that this identification agrees with the
conventional identification as self-adjoint operators. The upshot of this
formalism is that one may 'second quantize' the representation to which an
observable corresponds, thus obtaining the corresponding Fock space
representation. This Fock space representation is then also identifiable as an
observable in the same sense, the outcomes of which are naturally interpretable
as ensembles of outcomes of the corresponding non-second quantized observable.
The frequency interpretation of probability is adopted, i.e. probability as the
average occurrence, from which Born's rule is deduced by enforcing the notion
of 'average' to such that are invariant under the second quantized unitary
representation which defines the quantum observable to which the initial state
is an outcome. The enforcement of this invariance is an application the
principle referred to as the 'principle of unitary equivalence'.
