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Constrained Least Action and Quantum Mechanics

The wave-particle duality and its probabilistic interpretation are at the heart of quantum mechanics. Here we show that, in some standard contexts like the double slit experiment, a deterministic interpretation can be provided. This interpretation arises from the fact that if one mininizes a deterministic action {\it subject to spatial inequality contraints}, in general the solution is not unique. Thus, the probabilistic setting is replaced in part by the non-uniqueness of solutions of certain constrained optimization problems, itself often a result of the non-Lipschitzness of the constrained dynamics. While the approach leaves the results of associated Feynman integrals unchanged, it may considerably simplify their computation as the problems grow more complex, since only specific deterministic paths computed from the constrained minimization need to be included in the integral.