The topology of the Moyal *-algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the *-algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. The equivalence of the three topologies thereby obtained is proved. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, new sufficient conditions are given for a phase-space function to correspond to a trace-class operator via the Weyl correspondence rule.
Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function
ψKvN ψ KvN as well as the quantum-mechanical one
ψQM ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular,
ψKvN ψ KvN and
ψQM ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle
E E follows from this.
ψKvN ψ KvN and
ψQM ψ QM are shown to be sections in
E E . The difference between
ψKvN ψ KvN and
ψQM ψ QM originates from a quantization map interpreted as “twisting” of sections over
E E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.
We develop a geometrical approach to Schrödinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kähler structure possessed by the set of the pure states of a quantum system. The Kähler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kähler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrödinger quantum mechanics.