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Of course, “quantization” is far from being a straightforward algorithm, particularly for a nonlinear field theory. Rather, it is a poorly-understood inverse problem: to find a quantum theory with the given classical limit. Various choices are made in constructing the quantum theory. I discuss these choices below.
The main idea behind loop quantum gravity is that one take general relativity seriously. General relativity is the discovery that the spacetime metric and the gravitational field are the same physical entity. A quantum theory of the gravitational field is therefore also a quantum theory of the spacetime metric. 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 can conventionally split the spacetime metric into two terms, consider one of the the two terms a background that gives a metric structure to spacetime and treat the other as the quantum field. This is the procedure on which perturbative quantum gravity, perturbative strings, as well as several current nonperturbative string theories, are based. In this framework one assumes that the causal structure of spacetime is determined by the underlying background metric alone, and not by the full metric. Contrary to this, loop quantum gravity assumes that the identification between the gravitational field and the metric-causal structure of spacetime holds, and must be taken into account even in the quantum regime. No split of the metric is made, and there is no background metric on spacetime.
One can still describe spacetime as a (differentiable) manifold (a space without metric structure), over which quantum fields live. A classical metric structure will then be defined only 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 basic theoretical tools (creation and annihilation operators, canonical commutation relations, Gaussian measures, propagators …); these tools cannot be used in quantum fields 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 the 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 [154
], whose ideas have been very influential in the birth of loop quantum
gravity.6
The algebra on which loop gravity is based, is the loop algebra [265]. Why this algebra?
In choosing the loop algebra as the basis for the quantization, we are essentially assuming that Wilson loop operators are well defined in the Hilbert space of the theory; in other words, that certain states concentrated on one-dimensional structures (loops and graphs) have finite norm. This is a subtle nontrivial assumption. It is the key assumption that characterizes loop gravity, and is the one that looks most suspicious to scientists that have the habit of conventional background-dependent quantum field theory. If the approach turned out to be wrong, it will likely be because this assumption is wrong. Where does this assumption comes from and why is it dependable?
It comes from an old line of thinking in theoretical physics, according to which the natural variables describing gauge theories are loop-like. This idea has been variously defended by Wilson, Polyakov, Mandelstam, and many others, and, in a sense, can be traced to the very origin of gauge theory, in the intuition of Faraday. According to Faraday, the degrees of freedom of the electromagnetic field are best understood as lines in space: Faraday lines. Can we describe a quantum field theory in terms of its “Faraday lines”?
Consider first this question in a simplified context: on a lattice. The answer is then yes. In a lattice formulation of Yang–Mills theory, the physical Hilbert space of the theory is spanned by well-defined quantum states that are supported by loops on the lattice. These states can be written as traces of the holonomy operator around the loop. They are eigenstates of the electric-field operator, and they precisely represent quantized excitations of a single Faraday line. They are the exact analog of the loop quantum gravity spin-network states.
The attempt to take the continuum limit of this picture, however, fails in Yang–Mills theory. The reason is that when the lattice spacing converges to zero, the “physical width” of the individual loop states shrinks to zero, and the loop states become ill-defined infinite-norm states with one-dimensional support.
However, remarkably, this does not happen in a diffeomorphism-invariant theory. This is because, in the absence of a metric background, there is no sense in “shrinking down” the states. In fact, the size of the state is determined by the metric, which is determined by the gravitational field, which, in turn, is determined by the state itself. An explicit computation shows that refining the lattice space has no effect on the size of the loop states themselves: it only reflects a physically irrelevant change of coordinates.
Thus, in a diffeomorphism invariant theory, we can take the formal continuum limit of the lattice loop states. Once we factor away the gauge transformations defined by the diffeomorphisms, what remains are the abstract physical loop states, which are not localized in a space, but rather that define themselves by the physical excitations of the geometry, as will become clear in the following section.
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. One can say that the gravitational field is not particularly “special” in this regard: rather, diff-invariance is a property of the physical world that can be disregarded only in the approximation in which the dynamics of gravity are 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 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. These objects are considered as dynamically decoupled from the physical system that one studies. This conceptual structure works well in pre-GR physics, but it fails in a relativistic gravitational regime. In general relativistic physics, the physical objects are localized in space and time only with respect to one another. 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 [256, 246, 244, 259] (unless an appropriate gauge fixing is made). Pictorially, GR is not physics over a stage, it is the dynamical theory of everything, including the stage itself.
Loop quantum gravity is an implementation of this relational notion of spacetime localization in
quantum field theory. In particular, the basic quantum field theoretical excitations are not
excitations over a space, but rather excitations of the “stage” itself. In greater detail, we define
quantum states that correspond to loop-like and, more generally, graph-like excitations of the
gravitational field on a differential manifold (spin networks); but then, when factoring away
diffeomorphism invariance, the location of the states becomes irrelevant. The only remaining
information contained in the graph is then its abstract graph structure and its knotting. Thus,
diffeomorphism-invariant physical states are labeled by s-knots: equivalence classes of graphs under
diffeomorphisms. An s-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. An
s-knot state is an elementary quantum of space. See Figure 1
, and the relative discussion in
Section 7.
In this manner, loop quantum gravity binds the new notion of space and time introduced by general relativity with quantum mechanics. As I illustrate later on, the existence of such elementary quanta of space is a consequence of the quantization of the spectra of geometrical quantities.
Quantum gravity is an open problem that has been investigated for over seventy years now. When one contemplates two deep problems, one is tempted to believe that they are related. In the history of physics, there are surprising examples of two deep problems solved by one stroke (the unification of electricity and magnetism and the nature of light, for instance); but there are also many examples in which a great hope to solve more than one problem at once was disappointed (finding the theory of strong interactions and getting rid of quantum-field-theory infinities, for instance). Quantum gravity has been asked, at one time or another, to address almost every deep open problem in theoretical physics (and beyond). Here is a list of problems that have been connected to quantum gravity in the past, but about which loop quantum gravity has little to say:
Loop quantum gravity is a standard quantum (field) theory. Pick your favorite interpretation of quantum
mechanics, and use it for interpreting the quantum aspects of the theory. I will refer to two such
interpretations below. When discussing the quantization of area and volume, I will use the
relation between eigenvalues and outcomes of measurements performed with classical physical
apparatuses; when discussing evolution, I will refer to the histories interpretation. The peculiar way of
describing time evolution in a general relativistic theory may require some appropriate variants of
standard interpretations, such as Hartle’s generalized quantum mechanics [140], or a suitable
generalization of canonical quantum theory [261, 243, 245, 242]. But loop quantum gravity has no help
to offer the scientists who have speculated that quantum gravity will solve the measurement
problem. For a different point of view, see [278]. My own ideas on the interpretation of quantum
mechanics are in [255] and [261]. On the other hand, I think that solving the problem of the
interpretation of quantum mechanics might require relational ideas connected with the relational
nature of spacetime revealed by general relativity. These issues are discussed in detail in my
book [261].
The expression “Quantum cosmology” is used with several different meanings. First, it is used to designate the quantum theory of the cosmological gravitational degrees of freedom of our universe. The application of loop gravity to this problem is substantial. Second, it is used to designate the theory of the entire universe as a quantum system without external observer [139], with or without gravity. The two meanings are unrelated, but confusion is common. Quantum gravity is the theory of one dynamical entity: the quantum gravitational field (or the spacetime metric), just one field among the many degrees of freedom of the universe. Precisely as for the theory of the quantum electromagnetic field, we can always assume that we have a classical observer with classical measuring apparatuses measuring gravitational phenomena, and therefore study quantum gravity under the assumption that there is an observer, which is not part of the quantum system studied.
A common criticism of loop quantum gravity is that it does not unify all interactions. But the idea that quantum gravity can be understood only in conjunction with other fields is an interesting hypothesis, certainly not an established truth.
As far as I see, nothing in loop quantum gravity suggests that one could compute masses from quantum gravity.
A sound quantum theory of gravity is needed to understand the physics of the Big Bang. The converse is probably not true: we should be able to understand the small-scale structure of spacetime, even if we do not understand the origin of the Universe.
Roger Penrose has argued for some time that it should be possible to trace the time asymmetry in the observable Universe to quantum gravity.
Penrose has also speculated that quantum gravity is responsible for the wave function collapse, and, indirectly, governs the physics of the mind [226].
A problem that has been repeatedly tied to quantum gravity, and which loop quantum gravity is able to address, is the problem of the ultraviolet infinities in quantum field theory. The very peculiar nonperturbative short-scale structure of loop quantum gravity introduces a physical cutoff. Since physical spacetime itself comes in quanta in the theory, there is literally no space in the theory for the very high momentum integrations that originate from the ultraviolet divergences.
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