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6.5 Applications

Effective equations in the precise sense have already provided several applications to physical as well as conceptual questions of wider interest in quantum gravity.

6.5.1 Bounces

From the linear Hamiltonian Hˆ = − 12i( ˆJ − Jˆ†) of Section 6.3 one obtains equations of motion

-d- 1- † -d- 1- -d- † d φ⟨ˆp⟩ = − 2(⟨Jˆ⟩ + ⟨Jˆ⟩) , dφ ⟨ ˆJ⟩ = − 2(⟨ˆp⟩ + ℏ) = dφ ⟨ ˆJ ⟩. (78 )
(Again, we can replace p by v in Equation (77View Equation) and thus work with any refinement model in the same way.) They decouple from fluctuations or higher moments thanks to the solvability of the model. Solutions for the partially complex variables are subject to the reality condition
ˆ 2 1 2 pp JJ¯ 1- 2 |⟨J⟩| − (⟨ˆp⟩ + 2ℏ) = G − G + 4 ℏ , (79 )
which follows from the identity ˆˆ† 2 J J = ˆp after taking expectation values and reformulating it in terms of expectation values p and J and their fluctuations. The general solution satisfying the reality condition (and assuming a state being semiclassical at least once) is [69Jump To The Next Citation Point]
⟨ˆp⟩(φ) = ⟨ ˆH ⟩cosh(φ − δ ) − ℏ , ⟨ ˆJ⟩(φ) = − ⟨ ˆH ⟩(sinh (φ − δ) − i), (80 )
where δ is a constant of integration. The solution for p obviously bounces, confirming the numerical results of [26Jump To The Next Citation Point].

For such a solvable model one can determine many more properties in detail, such as the behavior of fluctuations through the bounce and the role of coherent states. But the solvability of the model also reveals its very special nature, and, as well, that of all related numerical results, which are available so far. Initially, it came as a surprise that those numerical investigations were showing a nearly smooth bounce, with hardly any quantum effects deforming the wave packet or even leading to significant spreading. As quantum systems go, this is certainly not the expected behavior. Relating those models to a solvable one, where, as in the harmonic oscillator, moments only couple weakly or not at all to expectation values, demonstrates that the results are, after all, consistent with general expectations.

On the other hand, these models by themselves cannot be taken as an indication of the general behavior. If a matter potential is included, or anisotropies or inhomogeneities are taken care of, there will be quantum backreaction, as in typical quantum systems. States spread and deform during evolution, and it is no longer guaranteed that a state starting out semiclassically at large volume will remain so when it approaches the classical singularity. While the underlying difference equation is non-singular independent of quantum backreaction, the geometrical picture of non-singular behavior may well deviate strongly from a smooth bounce when the state is no longer semiclassical on small scales. (Note that phenomenological equations including higher powers of the connection, which are sometimes taken as indications for bounces, are themselves also subject to severe additional corrections in models with quantum backreaction.)

Moreover, even in these models one makes use of the large size of matter in the form of a large ˆ ⟨H ⟩. For smaller values even semiclassical states can enter the small-volume regime much more deeply, which then would require other corrections, such as those from effective densities. Assuming a large ⟨Hˆ⟩ for the dynamics of local geometrical variables would no longer be justified in an inhomogeneous context, in addition to the then unavoidable quantum backreaction. Currently, robust demonstrations of bounces in loop quantum cosmology only refer to cases where quantum backreaction can safely be ignored, which are hardly realistic ones. Work on extending those results and on understanding the more general picture of non-singular quantum evolution is currently in progress; see also [110] for a discussion of the generality of present bounce results.

6.5.2 Before the Big Bang

The model of Section 6.3 for loop quantum cosmology allows precise results about the behavior of dynamical coherent states when they evolve through the point of the classical singularity. Not only can expectation values for volume and curvature describing the bounce be computed, but also fluctuations and higher moments of the state. Thus, one can see how a state evolves and whether the regime around the bounce has any implications. One can also analyze the full range of parameters determining such states in general, and thus address questions about how general certain properties, such as the approach to semiclassicality, are.

Of particular interest in the context of coherent states are fluctuations, which for the harmonic oscillator would remain constant. The system relevant for cosmology, however, is different and here fluctuations cannot be constant. Nevertheless, dynamical coherent states for the Wheeler–DeWitt model or the loop quantized model demonstrate that the ratios (Δp )∕p and (Δc )∕c remain constant for any part of the universe before or after the bounce. This implies that fluctuations can be huge because the solvable model has an unbounded p, but fluctuations relative to p stay small if they are small in a semiclassical initial state.

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Figure 10: Internal time evolution for expectation values and spread of two bouncing states, one, which is unsqueezed (solid lines), and one, which is being squeezed (dashed).

This is not true, however, if we consider the transition through the bounce. The bounce connects a contracting and an expanding phase, each of which is described well by Wheeler–DeWitt evolution. In these phases, fluctuations relative to expectation values remain nearly constant. During the transition through the bounce, however, the magnitude of fluctuations can change dramatically by factors, which do not need to be near 1 because fluctuations before and after the bounce are determined by independent free parameters of dynamical coherent states, as illustrated in Figure 10View Image. The ratio of these parameters is related to the squeezing of the state of the universe [69Jump To The Next Citation Point]. For unsqueezed states, fluctuations before and after the bounce are symmetric, but this is an additional assumption for which no observational basis exists. While the uncertainty relation (74View Equation) restricts squeezing, and thus the asymmetry, for given fluctuations through a bound on the covariance, a controlled application would require tight control on fluctuations of isotropic variables of the universe.

If one were to ask whether the state before the Big Bang was as classical as the state after the Big Bang, this question could not be answered based solely on observational information available after the Big Bang; fluctuations before the bounce simply have such a weak influence that they could not be discerned from observations afterwards [72Jump To The Next Citation Point].3 Moreover, the ratio of fluctuations before and after the bounce depends very sensitively on state parameters such that a state with symmetric fluctuations requires extreme fine tuning [74Jump To The Next Citation Point].

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Figure 11: Ratio of fluctuations before (Δ −) and after (Δ+) the bounce in the form |1 − Δ ∕Δ | − + as a function of a state parameter A in relation to the Hamiltonian H = ⟨ ˆH ⟩ for H = 1000 in Planck units. For larger H, the curve becomes even steeper, showing how precisely one has to tune the state parameter A in order to have a near symmetric state with |1 − Δ − ∕Δ+ | = 0. See [74Jump To The Next Citation Point] for further details.

6.5.3 Physical inner product

As mentioned in Section 6.3, after solving effective equations one has to impose reality conditions to ensure that expectation values and fluctuations of observables are real. This is the same condition one requires to determine the physical inner product, which at the representation level can be very complicated. For solutions to effective equations, on the other hand, reality conditions are as straightforward to implement as in the classical case. In this way, physical inner product issues are under much better control in the effective treatment. This has been demonstrated not only for the exactly effective solvable system of a free scalar in a flat isotropic universe, but also in the presence of a perturbative potential [87Jump To The Next Citation Point], where quantum backreaction occurs. This issue is also important for the effective treatment of constrained systems.

6.5.4 Anomaly issue

In addition to the physical inner product, potential anomalies are one major issue in quantum gravity. Here also, though direct calculations for operators are hard to perform in general, effective constraints provide a much more practical route. One can first derive effective equations for a given set of constraint operators, not worrying about anomalies. Inconsistencies can only arise when one tries to solve the resulting effective constraints, so before doing so one must analyze the anomaly issue at the effective level. This is feasible because the effective constraints are of the classical type, although amended by quantum corrections. Calculations thus require only the use of Poisson relations rather than commutators of operators. Moreover, after effective constraints have been computed one can often incorporate systematic approximation schemes such as perturbation theory, and then make sure that anomalies are absent order by order. This is a further simplification, which has been used in several cases of quantum cosmological perturbation theory. As one of the results, the possibility of anomaly freedom in perturbative loop quantizations in the presence of non-trivial quantum corrections was demonstrated [929091Jump To The Next Citation Point], and is studied for non-perturbative spherical symmetry in [102]. There are then standard techniques to compute evolution equations for gauge invariant observables from the anomaly-free effective constraints, which can immediately be employed in cosmological phenomenology.

These conclusions are also conceptually important for the general framework: the loop quantization does not necessarily remove covariance, as it is sometimes said. Non-trivial quantum corrections of the characteristic forms of loop quantum gravity are allowed while respecting covariance in effective equations. This issue is related to the question of local Lorentz invariance, although it has not been fully evaluated yet. Naive corrections in Hamiltonians or equations of motion could imply superluminal propagation, e.g., of gravitational waves. This would certainly violate causality, but superluminal motion disappears when anomaly freedom is properly implemented [91]. And yet, non-trivial quantum corrections due to the loop quantization remain.

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