Conventional approaches to quantum mechanics are essentially dualistic. This is reflected in the fact that their mathematical formulation is based on two distinct mathematical structures: the algebra of dynamical variables (observables) and the vector space of state vectors. In contrast, coherent interpretations of quantum mechanics highlight the fact that quantum phenomena must be considered as undivided wholes. Here, we discuss a purely algebraic formulation of quantum mechanics. This formulation does not require the specification of a space of state vectors; rather, the required vector spaces can be identified as substructures in the algebra of dynamical variables (suitably extended for bosonic systems). This formulation of quantum mechanics captures the undivided wholeness characteristic of quantum phenomena, and provides insight into their characteristic nonseparability and nonlocality. The interpretation of the algebraic formulation in terms of quantum process is discussed.
quantum algebraquantum pre-spacediscrete Weyl algebraClifford algebraimplicate order