Dies alles dauerte eine lange Zeit, oder eine kurze Zeit: denn, recht gesprochen, gibt es für dergleichen Dinge auf Erden keine Zeit.
(All this took a long time, or a short time: for, strictly speaking, for such things no time on earth exists.)
Thus Spoke Zarathustra
Often, time is intuitively viewed as coordinate time, i.e., one direction of spacetime. However, this does not have invariant physical meaning in general relativity, and conceptually an internal time is more appropriate. Evolution is then measured in a relational manner of some degrees of freedom with respect to others [40, 252, 154]. In quantum cosmology, as we have seen, this concept is even more general, since internal time keeps making sense at the quantum level, even around singularities, where classical spacetime dissolves.
The wave function thus extends to a new branch beyond the classical singularity, i.e., to a classically disconnected region. Intuitively this leads to a picture of a collapsing universe preceding the Big Bang, but one has to keep in mind that this is the picture obtained from internal time, where other time concepts are not available. In such a situation it is not clear, intuitive pictures notwithstanding, how this transition would be perceived by observers, were they able to withstand the extreme conditions. It can be said reliably that the wave function is defined on both sides, “before” and “after”, and every computation of physical predictions, e.g., using observables, that we can do at “our” side, can also be done at the other side. In this sense, quantum gravity is free of singularities and provides a transition between the two branches. The more complex question is what this means for the evolution in a literal sense of our usual concept of time (see also ).
Effective equations displaying bounces in internal or coordinate-time evolution indicate that indeed classical singularities are replaced by a bouncing behavior. However, this does not occur completely generally and does not say anything about the orientation reversal, which is characteristic for the quantum transition. In fact, effective equations describe the motion of semiclassical wave packets, which become less reliable at very small volume. And even if the effective bounce happens far away from the classical singularity, will there in general be a part of the wave function splitting off and traversing to the other orientation, as can be seen in the example of Figure 12.
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It is not clear in general that a wave function penetrating a classical singularity enters a new classical regime, even if the volume becomes large again. For instance, there can be oscillations on small scales, i.e., violations of pre-classicality, picked up by the wave function when it travels through the classical singularity. As discussed in Section 5.18, the question of what conditions to require on a wave function to require for a classical regime is still open, but even if one can confidently say that there is such a new classical region does the question arise if time continues during the transition through the pure quantum regime? At least in the special model of a free massless scalar in isotropic cosmology, the answer to both questions is affirmative, based on the availability of a physical inner product and quantum observables in this model . More realistic models remain to be studied, which also must include parity-violating matter Hamiltonians, in which the difference equation would not be invariant under orientation reversal.
Also related to this context is the question of unitary evolution. Even if one uses a self-adjoint constraint operator, unitary evolution is not guaranteed. First, the constraint splits into a time-generator part containing derivatives or difference operators with respect to internal time and a source part containing, for instance, the matter Hamiltonian. It is then not guaranteed that the time generator will lead to unitary evolution. Secondly, it is not obvious in which inner product one should measure unitarity, since the constraint is formulated in the kinematical Hilbert space, but the physical inner product is relevant for its solutions. This shows that the usual expectation of unitary evolution, commonly motivated by the preservation of probability or the normalization of a wave function in an absolute time parameter, is not reliable in quantum cosmology. It must be replaced by suitable conditions on relational probabilities computed from physical wave functions.
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