By William Elwood Byerly
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Our own preference is to proceed in a somewhat different manner, avoiding any use, even in the 24 DETLEF DURR ET AL. motivation for the theory, of probabilistic notions, which are after all somewhat subtle, and see what symmetry considerations alone might suggest. Assume for simplicity that we are dealing with spinless particles. Then one finds (Diirr et al. , by the fact that the wave function should be understood projectively, an understanding required for the Galilean invariance of SchrOdinger's equation alone - and the "Im" by time-reversal invariance, since timereversal is implemented on 'If by complex conjugation, again as demanded by Schrodinger's equation.
12 It is important to realize that an appeal to typicality is unavoidable if we are to explain why the universe is at present in quantum equilibrium. This is because our analysis also demonstrates that there is a set B of initial configurations, a set of nonvanishing Lebesgue measure, that evolve to present configurations violating the quantum equilibrium hypothesis and hence the quantum formalism. This set cannot be wished away by any sort of mixing argument. Indeed, if, as is expected, mixing holds on the universal level, then this set B should be so convoluted as to be indescribable without a specific reference to the universal dynamics and hence cannot be dismissed as unphysical without circularity.
7 In fact, it would appear that the argument against naive realism about operators provided by such an analysis has even greater force from an orthodox perspective: Given the initial wave function, at least in Bohmian mechanics the outcome of the particular experiment is determined by the initial configuration of system and apparatus, while for orthodox quantum theory there is nothing in the initial state which completely determines the outcome. Indeed, we find it rather surprising that most proponents of the von Neumann analysis of measurement, beginning with von Neumann, nonetheless seem to retain their naive realism about operators.