Zusammenfassung
Projection operators are central to the algebraic formulation of quantum
theory because both wavefunction and hermitian operators(observables) have
spectral decomposition in terms of the spectral projections. Projection
operators are hermitian operators which are idempotents also. We call them
quantum idempotents. They are also important for the conceptual understanding
of quantum theory because projection operators also represent observation
process on quantum system. In this paper we explore the algebra of quantum
idempotents and show that they generate Iterant algebra (defined in the paper),
Lie algebra, Grassmann algebra and Clifford algebra which is very interesting
because these later algebras were introduced for the geometry of spaces and
hence are called geometric algebras. Thus the projection operator
representation gives a new meaning to these geometric algebras in that they are
also underlying algebras of quantum processes and also they bring geometry
closer to the quantum theory. It should be noted that projection operators not
only make lattices of quantum logic but they also span projective geometry. We
will give iterant representations of framed braid group algebras, parafermion
algebras and the $su(3)$ algebra of quarks. These representations are very
striking because iterant algebra encodes the spatial and temporal aspects of
recursive processes. In that regard our representation of these algebras for
physics opens up entirely new perspectives of looking at fermions,spins and
parafermions(anyons).
Nutzer