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 (15) is a vertex sum obtained after introducing a
discretization of space used to choose loops
. 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.
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
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
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 or
. Similarly, after the WKB or other semiclassical
approximation, there are correction terms to the classical constraint in terms of
as well as
[146, 241].
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 as well as
, which are mainly of the following
form [36]: matter Hamiltonians receive corrections through the modified density
, and there are
similar terms in the gravitational part containing
. 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 [273, 276].
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 , which, compared to a classical canonical pair
, has
infinitely many degrees of freedom. The classical canonical pair can be related to expectation values
, while quantum degrees of freedom appear as higher moments, parameterizable as
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 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 [105, 282
, 106]. 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 [309, 17].
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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