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7.3 Determinism

Hat die Zeit nicht Zeit? (Does time not have time?)


Beyond Good and Evil

Loosely related to unitarity, but more general, is the concept of determinism. This is usually weakened in quantum mechanics anyway since in general one makes only probabilistic statements. Nevertheless, the wave function is determined at all times by its initial values, which is sometimes seen as the appropriate substitute for deterministic behavior. In loop quantum cosmology the situation again changes slightly since, as discussed in Section 5.18, the wave function may not be determined by the evolution equation everywhere, i.e., not at points of classical singularities, and instead acquire new conditions on its initial values. This could be seen as a form of indeterministic behavior, even though the values of a wave function at classical singularities would not have any effect on the behavior for non-degenerate configurations.2 (If they had such an effect, the evolution would be singular.) In this situation one deals with determinism in a background independent context, which requires a new view. In fact, rather than interpreting the freedom of choosing values at classical singularities as indeterministic behavior, it seems more appropriate to see this as an example for deterministic behavior in a background independent theory. The internal time label m first appears as a kinematical object through the eigenvalues of the triad operator (46View Equation). It then plays a role in the constraint equation (49View Equation) when formulated in the triad representation. Choosing internal time is just made for convenience, and it is the constraint equation that must be used to see if this choice makes sense in order to formulate evolution. This is indeed the case at non-zero m where we obtain a difference operator in the evolution parameter. At zero m, however, the operator changes and does not allow us to determine the wave function there from previous values. Now, we can interpret this simply as a consequence of the constraint equation rejecting the internal time value m = 0. The background independent evolution selects the values of internal time it needs to propagate a wave function uniquely. As it turns out, m = 0 is not always necessary for this and thus simply decouples. In hindsight, one could already have split off |0> from the kinematical Hilbert space, thereby removing the classical singularity by hand. Since we did not do this, it is the evolution equation that tells us that this is happening anyway. Recall, however, that this is only one possible scenario obtained from a non-symmetric constraint. For the evolution (50View Equation) following from the symmetric constraint, no decoupling happens and m = 0 is just like any other internal time value.

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