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5.5 Semiclassical limit and correction terms

When replacing c2 by holonomies we have modified the constraint as a function on the classical phase space. 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. Also the limit limd -->0 d-2 sin2 dc, which would give the classical result, does not exist at the operator level.

This situation is different from the full theory, again related to the presence of a partial background [15]. 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 a IJ. 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.

The difference between models and the full theory is thus only a consequence of the symmetry and not of different approaches. This will also become clear later in inhomogeneous models where one obtains a mixture between the two situations. 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.

5.5.1 WKB approximation

There are different procedures to make 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 [44]. From then on, one can use the WKB approximation or Wigner functions as usually.

That this is possible may be surprising because as just discussed the continuum limit d --> 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 d --> 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 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 d or g. Similarly, after the WKB or other semiclassical approximation there are correction terms to the classical constraint in terms of g as well as h [99].

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 g as well as h, which are mainly of the following form [29]: Matter Hamiltonians receive corrections through the modified density d(p), and there are similar terms in the gravitational part containing V~ --- |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 [128]. A related procedure to extract semiclassical properties from the difference operator, based on the Bohmian interpretation of quantum mechanics, has been discussed in [185].

5.5.2 Effective formulation

In general, one does not only expect higher order corrections for a gravitational action but also higher derivative terms. The situation is then qualitatively different since not only correction terms to a given equation arise, 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 worked out yet. An alternative approach makes use of a geometrical formulation of quantum mechanics [26], which not only provides a geometrical picture of the classical limit but also a clear-cut procedure for computing effective Hamiltonians in analogy to effective actions [73Jump To The Next Citation Point].

Instead of using linear operators on a Hilbert space, one can formulate quantum mechanics on an infinite-dimensional phase space. This space 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). This formulation brings quantum mechanics conceptually much closer to classical physics, which also facilitates a comparison in a semiclassical analysis.

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. Coordinates can be chosen by suitable parameterizations of a general wave function, in particular using the expectation values q = <^q> and p = <^p> together with uncertainties and higher moments. The projection p: H --> R2, y '--> (<y|^q|y>,<y|^p|y>) 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 this embedding can be done by 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. The harmonic oscillator thus does not receive quantum corrections as is well known from effective actions for free field theories. Other systems, however, behave in a more complicated manner where in general states spread. This means that additional coordinates of the quantum phase space are dynamical and may become excited. If this is the case, the quantum flow differs from the classical flow and an effective Hamiltonian arises with correction terms that can be computed systematically. This effective Hamiltonian is given by the expectation value <H^> in approximate coherent states [16206188]. In these calculations, one can include higher degrees of freedom along the fibers, which, through the effective equations of motion, can be related to higher derivatives or higher curvature in the case of gravity.

For a constrained system, such as gravity, one has to compute the expectation value of the Hamiltonian constraint, i.e., first go to the classical picture and then solve equations of motion. Otherwise, there would simply be no effective equations left after the constraints would already have been solved. This is the same procedure as in standard effective actions, which one can also formulate in a constrained manner if one chooses to parameterize time. Indeed, also for non-constrained systems agreement between the geometrical way to derive effective equations and standard path integral methods has been shown for perturbations around a harmonic oscillator [73Jump To The Next Citation Point].


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