This is a subtle error in quantum mechanics that occasionally has consequences, but usually does not. I started to write this 11/2011, but it just trailed off.

According to von Neumann, a quantum system is a Hilbert space of states, a unitary time flow of the Hilbert space, and a (von Neumann) algebra of observables acting on the Hilbert space. Typically, people build quantum systems through canonical quantization, starting with a classical system, that is a symplectic manifold and a hamiltonian function on it. (the physics term “canonical” is roughly equivalent to the math term “symplectic”) Using the Moyal star or other explicit method, one forms a one parameter deformation of the commutative algebra of functions on the symplectic manifold. The parameter is called h-bar and eventually one fixes its value to be the observed physical value. I’m not sure how one forms the Hilbert space in general. In the specific case in which the symplectic manifold is the cotangent bundle of another manifold called space, then the Hilbert space is the functions on the smaller manifold and the algebra are the (pseudo-)differential operators, which may be thought of as functions on the symplectic manifold by the symbol.

Thinking of our observables as functions on the phase space or of the states as functions on space gives them more structure than von Neumann allows and can lead to error.

In particular, in the many worlds interpretation, people often say that the squared amplitude of the wave function at a particular point is the degree of reality of that world. But the wave function is not a function and thus one cannot ask about its amplitude at a particular point. At least, one cannot do so without imposing additional structure and the answer depends on that extra structure. (eg, one could express the symplectic manifold as the cotangent bundle of a different manifold, such as the graph of a 1-form) When h=0, the algebra of observables knows about the points of the symplectic manifold and there is no ambiguity. Maybe the ambiguity when h is not zero can be controlled by h. We can’t evaluate the wave function at a point, but maybe we can evaluate it at an h-fuzzy point. The uncertainty principle is relevant.

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