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7.2 The role of time

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 space-time. 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 [33170106]. In quantum cosmology, as we have seen, this concept is even more general since internal time keeps making sense at the quantum level also around singularities where the classical space-time 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 at both sides, “before” and “after”, and every computation of physical predictions, e.g., using observables, 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 complicated question is what this means for evolution in a literal sense of our usual concept of time (see also [200]).

Effective equations displaying bounces in 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 becomes 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 10Watch/download Movie.

Watch/download Movie

Figure 10: Movie showing the coordinate time evolution [72Jump To The Next Citation Point] of a wave packet starting at the bottom and moving toward the classical singularity (vertical dotted line) for different values of an ambiguity parameter. Some part of the wave packet bounces back (and deforms) according to the effective classical solution (dashed), but other parts penetrate to negative m. The farther away from a = 0 the effective bounce happens, depending on the ambiguity parameter, the smaller the part penetrating to negative m is. The coordinate time evolution represents a physical state obtained after integrating over t [72Jump To The Next Citation Point].
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 5.17, the question of what conditions 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 [24Jump To The Next Citation Point].

Also related to this context is the question of unitary evolution. Even if one uses a selfadjoint 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 what inner product to 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 preservation of probability or 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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