One could conventionally split the spacetime metric into two terms: one to be considered a background, which gives a metric structure to spacetime; the other to be treated as a fluctuating quantum field. This, indeed, is the procedure on which old perturbative quantum gravity, perturbative strings, as well as current non-perturbative string theories (M-theory), are based. In following this path, one assumes, for instance, that the causal structure of spacetime is determined by the underlying background metric alone, and not by the full metric. Contrary to this, in loop quantum gravity we assume that the identification between the gravitational field and the metric-causal structure of spacetime holds, and must be taken into account, in the quantum regime as well. Thus, no split of the metric is made, and there is no background metric on spacetime.
We can still describe spacetime as a (differentiable) manifold (a space without metric structure), over which quantum fields are defined. A classical metric structure will then be defined by expectation values of the gravitational field operator. Thus, the problem of quantum gravity is the problem of understanding what is a quantum field theory on a manifold, as opposed to quantum field theory on a metric space. This is what gives quantum gravity its distinctive flavor, so different from ordinary quantum field theory. In all versions of ordinary quantum field theory, the metric of spacetime plays an essential role in the construction of the basic theoretical tools (creation and annihilation operators, canonical commutation relations, gaussian measures, propagators ...); these tools cannot be used in quantum field over a manifold.
Technically, the difficulty due to the absence of a background
metric is circumvented in loop quantum gravity by defining the
quantum theory as a representation of a Poisson algebra of
classical observables which can be defined without using a
background metric. The idea that the quantum algebra at the basis
of quantum gravity is not the canonical commutation relation
algebra, but the Poisson algebra of a different set of
observables, has long been advocated by Chris Isham [118], whose ideas have been very influential in the birth of loop
quantum gravity.
The algebra on which loop gravity is the loop algebra [184
]. The particular choice of
this
algebra is not harmless, as I discuss below.
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Loop Quantum Gravity
Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |