  Starting with the quantum Liouville equation, we write the density operator
as the product of elements respectively in the left and right ideals of an
operator algebra and find that the Schrodinger picture may be expressed through
two representation independent algebraic forms in terms of the density and
phase operators. These forms are respectively the continuity equation, which
involves the commutator of the Hamiltonian with the density operator, and an
equation for the time development of the phase operator that involves the
anti-commutator of the Hamiltonian with this density operator. We show that
this latter equation plays two important roles: (i) it expresses the
conservation of energy in a system where energy is well defined and (ii) it
provides a simple way to evaluate the gauge changes that occur in the
Aharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these
operator (i.e. purely algebraic) equations also allow us to re-examine the Bohm
interpretation, showing that it is in fact possible to construct Bohm
interpretations in representations other than the x-representation. We discuss
the meaning of the Bohm interpretation in the light of these new results in
terms of non-commutative structures and this enables us to clarify its relation
to standard quantum mechanics.
