5.2 One additional assumption5 Main Ideas and Physical 5 Main Ideas and Physical

5.1 Quantum field theory on a differentiable manifold

The main idea beyond loop quantum gravity is to take general relativity seriously. We have learned with general relativity that the spacetime metric and the gravitational field are the same physical entity. Thus, a quantum theory of the gravitational field is a quantum theory of the spacetime metric as well. It follows that quantum gravity cannot be formulated as a quantum field theory over a metric manifold, because there is no (classical) metric manifold whatsoever in a regime in which gravity (and therefore the metric) is a quantum variable.

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. Popup Footnote The algebra on which loop gravity is the loop algebra [184Jump To The Next Citation Point In The Article]. The particular choice of this algebra is not harmless, as I discuss below.

5.2 One additional assumption5 Main Ideas and Physical 5 Main Ideas and Physical

image Loop Quantum Gravity
Carlo Rovelli
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