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3.5 Composite operators

From the basic operators we can construct more complicated ones which, with growing degrees of complexity, will be more and more ambiguous for, example from factor ordering choices. Quite simple expressions exist for the area and volume operator [2602122], which are constructed solely from fluxes. Thus, they are less ambiguous, since no factor-ordering issues with holonomies arise. This is true because the area of a surface and volume of a region can be written classically as functionals of the densitized triad alone, ∫ ∘ ---------- AS = S Eai naEbinbd2y and ∫ ∘ -------- VR = R |det Eai |d3x. At the quantum level this implies that, like fluxes, area and volume also have discrete spectra, showing that spatial quantum geometry is discrete. (For discrete approaches to quantum gravity in general see [218].) All area eigenvalues are known explicitly, but this is not possible, even in principle, for the volume operator. Nevertheless, some closed formulas and numerical techniques exist [217150149118].

The length of a curve, on the other hand, requires the co-triad, which is an inverse of the densitized triad and is more problematic. Since fluxes have discrete spectra containing zero, they do not have densely defined inverse operators. As we will describe below, it is possible to quantize those expressions, but it requires one to use holonomies. Thus, we encounter here more ambiguities from factor ordering. Still, one can show that length operators also have discrete spectra [290].

Inverse-densitized triad components also arise when we try to quantize matter Hamiltonians, such as

( ) ∫ 3 1 p2φ + Eai Ebi∂aφ ∂bφ ∘ |------| H φ = d x( ------∘-|------|----+ |det Ecj|V (φ)) , (12 ) 2 |detEcj|
for a scalar field φ with momentum pφ and potential V(φ ) (not to be confused with volume). The inverse determinant again cannot be quantized directly by using, e.g., an inverse of the volume operator, which does not exist. This seems, at first, to be a severe problem, not unlike the situation in quantum field theory on a background where matter Hamiltonians are divergent. Yet it turns out that quantum geometry allows one to quantize these expressions in a well-defined manner [291Jump To The Next Citation Point].

To do this, we notice that the Poisson bracket of the volume with connection components,

∫ ------- b c {Ai , ∘ |det E |d3x } = 2πγG εijkε ∘-EjE-k--, (13 ) a abc |detE |
amounts to an inverse of densitized triad components and allows a well-defined quantization: we can express the connection component through holonomies, use the volume operator and turn the Poisson bracket into a commutator. Since all operators involved have a dense intersection of their domains of definition, the resulting operator is densely defined and provides a quantization of inverse powers of the densitized triad.

This also shows that connection components or holonomies are required in this process and, thus, ambiguities can arise, even if initially one starts with an expression such as ∘ ------- |detE |−1, which only depends on the triad. There are also many different ways to rewrite expressions as above, which all are equivalent classically but result in different quantizations. In classical regimes this would not be relevant, but can have sizeable effects at small scales. In fact, this particular aspect, which as a general mechanism is a direct consequence of the background-independent quantization with its discrete fluxes, implies characteristic modifications of the classical expressions on small scales. We will discuss this and more detailed examples in the cosmological context in Section 4.


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