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 requires one to use holonomies. Thus, here we encounter more ambiguities from factor ordering. Still, one can show that also length operators have discrete spectra [192].
Inverse densitized triad components also arise when we try to quantize matter Hamiltonians such as
for a scalar fieldTo do this, we notice that the Poisson bracket of the volume with connection components,
amounts to an inverse of densitized triad components and does allow 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 amounts to 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 , 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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