From Hamilton-Jacobi to Bohm: Why the Wave Function Isn't Just Another Action

This paper examines the physical meaning of the wave function in Bohmian mechanics (BM), addressing the debate between causal and nomological interpretations. While BM postulates particles with definite trajectories guided by the wave function, the ontological status of the wave function itself remains contested. Critics of the causal interpretation argue that the wave function's high-dimensionality and lack of back-reaction disqualify it as a physical entity. Proponents of the nomological interpretation, drawing parallels to the classical Hamiltonian, propose that the wave function is a "law-like" entity. However, this view faces challenges, including reliance on speculative quantum gravity frameworks (e.g., the Wheeler-DeWitt equation) and conceptual ambiguities about the nature of "nomological entities". By systematically comparing BM to Hamilton-Jacobi theory, this paper highlights disanalogies between the wave function and the classical action function. These differences, particularly the wave function's dynamical necessity and irreducibility, support a sui generis interpretation, where the wave function represents a novel ontological category unique to quantum theory. The paper concludes that the wave function's role in BM resists classical analogies, demanding a metaphysical framework that accommodates its non-local, high-dimensional, and dynamically irreducible nature.