5.3 Physical meaning of diffeomorphism invariance, and its implementation in the quantum theory 

Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields. (Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything .) General relativity, on the contrary, is invariant under such transformations. More precisely, every general relativistic theory has this property. Thus, diffeomorphism invariance is not a feature of just the gravitational field: It is a feature of physics, once the existence of relativistic gravity is taken into account. Thus, one can say that the gravitational field is not particularly ``special'' in this regard, but that diff-invariance is a property of the physical world that can be disregarded only in the approximation in which the dynamics of gravity is neglected. What is this property? What is the physical meaning of diffeomorphism invariance?

Diffeomorphism invariance is the technical implementation of a physical idea, due to Einstein. The idea is a deep modification of the pre-general-relativistic (pre-GR) notions of space and time. In pre-GR physics, we assume that physical objects can be localized in space and time with respect to a fixed non-dynamical background structure. Operationally, this background spacetime can be defined by means of physical reference-system objects, but these objects are considered as dynamically decoupled from the physical system that one studies. This conceptual structure fails in a relativistic gravitational regime. In general relativistic physics, the physical objects are localized in space and time only with respect to each other. Therefore if we ``displace'' all dynamical objects in spacetime at once, we are not generating a different state, but an equivalent mathematical description of the same physical state. Hence, diffeomorphism invariance.

Accordingly, a physical state in GR is not ``located'' somewhere [180, 169, 177] (unless an appropriate gauge fixing is made). Pictorially, GR is not physics over a stage, it is the dynamical theory of (or including) the stage itself.

Loop quantum gravity is an attempt to implement this subtle relational notion of spacetime localization in quantum field theory. In particular, the basic quantum field theoretical excitations cannot be localized somewhere (localized with respect to what?) as, say, photons are. They are quantum excitations of the ``stage'' itself, not excitations over a stage. Intuitively, one can understand from this discussion how knot theory plays a role in the theory. First, we define quantum states that correspond to loop-like excitations of the gravitational field, but then, when factoring away diffeomorphism invariance, the location of the loop becomes irrelevant. The only remaining information contained in the loop is then its knotting (a knot is a loop up to its location). Thus, diffeomorphism invariant physical states are labeled by knots. A knot represents an elementary quantum excitation of space. It is not here or there, since it is the space with respect to which here and there can be defined. A knot state is an elementary quantum of space.

In this manner, loop quantum gravity ties the new notion of space and time introduced by general relativity with quantum mechanics. As I will illustrate later on, the existence of such elementary quanta of space is then made concrete by the quantization of the spectra of geometrical quantities.

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