Bohm's Ontological Interpretation and its Relations to Three Formulations of Quantum Mechanics

The standard mathematical formulation of quantum mechanics is specified. Bohm s ontological interpretation of quantum mechanics is then shown to be incapable of providing a suitable interpretation of that formulation. It is also shown that Bohm’s interpretation may well be viable for two alternative mathematical formulations of quantum mechanics, meaning that the negative result is a significant though not a devastating criticism of Bohm’s interpretation. A preliminary case is made for preferring one alternative formulation over the other. Introduction. To facilitate developing the central arguments of this essay, four layers of theorizing are distinguished: the formal, mathematical, physical, and metaphysical. The formal and mathematical are primarily syntactic categories; the physical and metaphysical are primarily semantic. The four layers are characterized here by way of examples using notions that are connected with QM (quantum mechanics). Dirac’s bra-ket notation and his delta-function are formal elements. Spaces of finite or infinite sequences of complex numbers, more abstract notions such as a linear vector space over a field, and linear functionals and operators on such spaces are mathematical. The association of the square modulus of an inner product of two vectors (normalized to unity) with the probability of a measurement outcome, or the association of a self-adjoint operator with an observable physical quantity are physical. The interpretation of a state vector as corresponding to a quantum potential or of the elements of a superposition as corresponding to states of a system in different worlds are metaphysical. Hidden variables are also regarded as metaphysical elements, as are the assumptions of determinism and of the continuity of physical entities and processes. The distinctions drawn above are pragmatic, as opposed to being hard distinctions, because the status of a notion can change over time. For example, Dirac’s delta function was a purely * This paper was originally submitted on 28 July 1998, and it was published in Synthese 117: 31-52, 1999.