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5.6 Semiclassical limit and correction terms

When replacing c2 by holonomies we have amended the constraint as a function on the classical phase space by higher powers of the connection. This is necessary since otherwise the function cannot be quantized, but is different from the quantization of densities because now the replacements are not equivalent to the original constraint classically (although the corrections are small in small curvature regimes, where the classical theory can be trusted). In addition, the limit lim δ −2sin2δc δ→0, which would give the classical result, does not exist at the operator level.

This situation is different from the full theory, which is again related to the presence of a partial background [16]. There, the parameter length of edges used to construct appropriate loops is irrelevant and thus can shrink to zero. In the model, however, changing the edge length with respect to the background does change the operator and the limit does not exist. Intuitively, this can be understood as follows: The full constraint operator (15View Equation) is a vertex sum obtained after introducing a discretization of space used to choose loops αIJ. This classical regularization sums over all tetrahedra in the discretization, whose number diverges in the limit where the discretization size shrinks to zero. In the quantization, however, almost all these contributions vanish, since a tetrahedron must contain a vertex of a state in order to contribute non-trivially. The result is independent of the discretization size, once it is fine enough, and the limit can thus be taken trivially.

In a homogeneous model, on the other hand, contributions from different tetrahedra of the triangulation must be identical, owing to homogeneity. The coordinate size of tetrahedra drops out of the construction in the full background-independent quantization, as emphasized in Section 3.6, which is part of the reason for the discretization independence. In a homogeneous configuration the number of contributions thus increases in the limit, but their size does not change. This results in an ill-defined limit, as we have already seen within the model itself.

Therefore the difference between models and the full theory is only a consequence of symmetry and not of different approaches. This will also become clear later in inhomogeneous models, where one obtains a mixture of the two situations; see Section 5.11. Moreover, in the full theory one has a situation similar to symmetric models, if one does not only look at the operator limit when the regularization is removed, but also checks the classical limit on semiclassical states. In homogeneous models, the expression in terms of holonomies implies corrections to the classical constraint when curvature becomes larger. This is in analogy to other quantum field theories, where effective actions generally have higher curvature terms. In the full theory, those correction terms can be seen when one computes expectation values of the Hamiltonian constraint in semiclassical states peaked at classical configurations for the connection and triad. When this classical configuration becomes small in volume or large in curvature, correction terms to the classical constraint arise. In this case, the semiclassical state provides the background with respect to which these corrections appear. In a homogeneous model, the symmetry already provides a partial background such that correction terms can be noticed already for the constraint operator itself.

5.6.1 WKB approximation

There are various procedures for making contact between the difference equation and classical constraints. The most straightforward way is to expand the difference operators in a Taylor series, assuming that the wave function is sufficiently smooth. On large scales, this indeed results in the Wheeler–DeWitt equation as a continuum limit in a particular ordering [52]. From then on, one can use the WKB approximation or Wigner functions as usually. (Wigner functions can be defined directly on the Hilbert space of loop quantum cosmology making use of the Bohr compactification [165].

That this is possible may be surprising because, as just discussed, the continuum limit as δ → 0 does not exist for the constraint operator. And indeed, the limit of the constraint equation, i.e., the operator applied to a wave function, does not exist in general. Even for a wave function, the limit as δ → 0 does not exist in general, since some solutions are sensitive to the discreteness and do not have a continuum limit at all. When performing the Taylor expansion we already assumed certain properties of the wave function such as that the continuum limit does exist. This then reduces the number of independent wave functions to that present in the Wheeler–DeWitt framework, subject to the Wheeler–DeWitt equation. That this is possible demonstrates that the constraint, in terms of holonomies, does not have problems with the classical limit.

The Wheeler–DeWitt equation results at leading order, and in addition, higher-order terms arise in an expansion of difference operators in terms of δ or γ. Similarly, after the WKB or other semiclassical approximation, there are correction terms to the classical constraint in terms of γ as well as ℏ [146241].

This procedure is intuitive, but it is not suitable for inhomogeneous models where the Wheeler–DeWitt representation becomes ill defined. One can evade this by performing the continuum and semiclassical limit together. This again leads to corrections in terms of γ as well as ℏ, which are mainly of the following form [36]: matter Hamiltonians receive corrections through the modified density d(p), and there are similar terms in the gravitational part containing ∘ --- |p|. These are purely from triad coefficients, and similarly, connection components lead to higher-order corrections as well as additional contributions summarized in a quantum geometry potential. A possible interpretation of this potential in analogy to the Casimir effect has been put forward in [185]. A related procedure to extract semiclassical properties from the difference operator, based on the Bohmian interpretation of quantum mechanics, has been discussed in [273276].

5.6.2 Effective formulation

In general, one not only expects higher-order corrections for a gravitational action but also higher derivative terms. The situation is then qualitatively different since not only correction terms arise to a given equation, but also new degrees of freedom coming from higher derivatives being independent of lower ones. In a WKB approximation, this could be introduced by parameterizing the amplitude of the wave function in a suitable way, but it has not been fully worked out yet [144]. Quantum degrees of freedom arise because a quantum state is described by a wave function ψ (q), which, compared to a classical canonical pair (q, p), has infinitely many degrees of freedom. The classical canonical pair can be related to expectation values (⟨ψ |ˆq|ψ⟩,⟨ψ |pˆ|ψ ⟩), while quantum degrees of freedom appear as higher moments, parameterizable as

( ) Ga,n = ⟨ψ | (ˆq − ⟨ˆq⟩)n−a(ˆp − ⟨ˆp⟩)a |ψ⟩, (56 ) Weyl
where totally symmetric ordering is understood and 2 ≤ n ∈ ℕ, 0 ≤ a ≤ n. The infinite dimensional space of the expectation values, together with all higher moments, is symplectic and provides a geometrical formulation of quantum mechanics [20118030]; instead of using linear operators on a Hilbert space, one can formulate quantum mechanics on this infinite-dimensional phase space. It is directly obtained from the Hilbert space, where the inner product defines a metric, as well as a symplectic form, on its linear vector space (which in this way even becomes Kähler).

We thus obtain a quantum phase space with infinitely many degrees of freedom, together with a flow defined by the Schrödinger equation. Operators become functions on this phase space through expectation values. The projection π: ℋ → ℝ2, ψ ↦→ (⟨ψ|ˆq|ψ ⟩,⟨ψ|ˆp|ψ⟩) defines the quantum phase space as a fiber bundle over the classical phase space with infinite-dimensional fibers. Sections of this bundle can be defined by embedding the classical phase space into the quantum phase space by means of suitable semiclassical states. For a harmonic oscillator such embeddings can be defined precisely by dynamical coherent states, which are preserved by the quantum evolution. This means that the quantum flow is tangential to the embedding of the classical phase space such that it agrees with the classical flow. Moreover, the section can be chosen to be horizontal with respect to a connection, whose horizontal subspaces are by definition symplectically orthogonal to the fibers. This is possible because the harmonic oscillator allows coherent states which do not spread at all during evolution: quantum variables remain constant and thus there is no evolution along the fibers of the quantum phase space.

In more general systems, however, quantum variables do change: states spread and are deformed in other ways in a rather complicated manner, which can also affect the expectation values. This is exactly the phenomenon that gives rise to quantum corrections to classical equations, which one can capture in effective descriptions. In fact, the dynamics of quantum variables provides a means for the systematic derivation of effective equations that are analogous to effective actions but can be computed in a purely canonical way [105Jump To The Next Citation Point282Jump To The Next Citation Point106]. They can thus be applied to canonical quantum gravity; more details and applications are provided in Section 6. In this way, one can derive a suitable, non-horizontal section of the quantum phase space. In some simple cases, part of the effective dynamics can also be studied by finding a suitable section by inspection of the equations of motion, without explicitly deriving the behavior of quantum fluctuations [30917].

Gravity, as a constrained system, also requires one to deal with constraints. One computes the expectation value of the Hamiltonian constraint, i.e., first goes to the effective picture and then solves equations of motion. Otherwise, there would simply be no effective equations left after the constraints have been solved by physical states used in expectation values. This provides the relation between fundamental constraint operators giving rise to difference equations for physical states and effective equations. The terms used in the phenomenological equations, such as those of Section 4, are justified in this way, but one has to keep in mind that a complete analysis of most of the models discussed there remains to be done. There will be additional correction terms due to the backreaction of quantum variables on expectation values, which are more complicated to derive and have rarely been included in phenomenological studies. The status of precise effective equations is described in Section 6.


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