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5.18 Dynamical initial conditions

I am Aton when I am alone in the Nun, but I am Re when he appears, in the moment when he starts to govern what he has created.

Book of the Dead

The traditional subject of quantum cosmology is the imposition of initial conditions for the wave function of a universe in order to guarantee its uniqueness. In the Wheeler-DeWitt framework this is done at the singularity a = 0, sometimes combined with final conditions in the classical regime. One usually uses intuitive pictures as guidance, akin to Lemaitre’s primitive atom whose decay is supposed to have created the world, Tryon’s and Vilenkin’s tunneling event from nothing, or the closure of space-time into a Euclidean domain by Hartle and Hawking.

In the latter approaches, which have been formulated as initial conditions for solutions of the Wheeler-DeWitt equation [204121], the singularity is still present at a = 0, but re-interpreted as a meaningful physical event through the conditions. In particular, the wave function is still supported at the classical singularity, i.e., y(0) /= 0, in contrast to DeWitt’s original idea of requiring y(0) = 0 as a means to argue for the absence of singularities in quantum gravity [104]. DeWitt’s initial condition is in fact, though most appealing conceptually, not feasible in general since it does not lead to a well-posed initial value formulation in more complicated models: the only solution would then vanish identically. Zeh tried to circumvent this problem, for instance by proposing an ad hoc Planck potential, which is noticeable only at the Planck scale and makes the initial problem well-defined [89]. However, the problem remains that in general there is no satisfying origin of initial values.

In all these ideas, the usual picture in physics has been taken that there are dynamical laws describing the general behavior of a physical system, and independently initial or boundary conditions to select a particular situation. This is reasonable since usually one can prepare a system, corresponding to choosing initial and boundary values, and then study its behavior as determined by the dynamical laws. For cosmology, however, this is not appropriate since there is no way to prepare the universe.

At this point, there is a new possibility opened up by loop quantum cosmology where the dynamical law and initial conditions can be part of the same entity [407849]. This is a specialty of difference equations whose order can change locally, in contrast to differential equations. Mathematically, such a difference equation would be called singular since its leading order coefficients can become zero. However, physically we have already seen that the behavior is non-singular since the evolution does not break down.

The difference equation follows from the constraint equation, which is the primary object in canonical quantum gravity. As discussed before, it is usually of high order in classical regimes, where the number of solutions can be restricted, e.g., by pre-classicality. But this, at most, brings us to the number of solutions that we have for the Wheeler-DeWitt equation such that one needs additional conditions as in this approach. The new aspect now is that this can follow from the constraint equation itself: since the order of the difference equation can become smaller at the classical singularity, there are less solutions than expected from the semiclassical behavior. In the simplest models, this is just enough to result in a unique solution up to norm, as appropriate for a wave function describing a universe. In those cases, the dynamical initial conditions are comparable to DeWitt’s initial condition, albeit in a manner that is well-posed even in some cases where DeWitt’s condition is not [64].

In general, the issue is not clear but should be seen as a new option presented by the discrete formulation of loop quantum cosmology. Since there can be many conditions to be imposed on wave functions in different regimes, one has to see in each model whether or not suitable non-zero solutions remain at all. In fact, some first investigations indicate that different requirements taken together can be very restrictive [86], which seems to relate well with the non-separability of the kinematical Hilbert space [98]. So far, only homogeneous models have been investigated in detail, but the mechanism of decoupling is known not to be realized in an identical manner in inhomogeneous models.

Inhomogeneous models can add qualitatively new ingredients also to the issue of initial conditions due to the fact that there are many coupled difference equations. There can then be consistency conditions for solutions to the combined system which can strongly restrict the number of independent solutions. This may be welcome, e.g., in spherical symmetry where a single physical parameter remains, but the restriction can easily become too strong even below the classically expected one. Since the consistency between difference equations is related to the anomaly issue, there may be an important role played by quantum anomalies. While classically anomalies should be absent, the quantum situation can be different since it also takes the behavior at the classical singularity into account and is supposed to describe the whole universe. Anomalies can then be precisely what one needs in order to have a unique wave function of a universe even in inhomogeneous cases where initially there is much more freedom. This does not mean that anomalies are simply ignored or taken lightly since it is difficult to arrange having the right balance between many solutions and no non-zero solution at all. However, quantum cosmology suggests that it is worthwhile to have a less restricted, unconventional view on the anomaly issue.


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