Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case.
It is an attempt to develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation, general relativity. As a theory, LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length, approximately 10−35 meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure.
The areas of research, which involve about 30 research groups worldwide, share the basic physical assumptions and the mathematical description of quantum space. Research has evolved in two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity, called spin foam theory. The most well-developed theory that has been advanced as a direct result of loop quantum gravity is called loop quantum cosmology (LQC). LQC advances the study of the early universe, incorporating the concept of the Big Bang into the broader theory of the Big Bounce, which envisions the Big Bang as the beginning of a period of expansion, that follows a period of contraction, which has been described as the Big Crunch.
History
In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics, specifically Yang–Mills theory. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler–DeWitt equation, admitted solutions labelled by loops when rewritten in the new Ashtekar variables. Carlo Rovelli and Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Jorge Pullin and Jerzy Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, and the theory should be formulated in terms of intersecting loops, or graphs.
In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum. That is, geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins.
The canonical version of the dynamics was established by Thomas Thiemann, who defined an anomaly-free Hamiltonian operator and showed the existence of a mathematically consistent background-independent theory. The covariant, or "spin foam", version of the dynamics was developed jointly over several decades by research groups in France, Canada, UK, Poland, and Germany. It was completed in 2008, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, which is consistent with observed acceleration in the expansion of the Universe.
Background independence LQG is formally background independent, meaning the equations of LQG are not embedded in, or dependent on, space and time (except for its invariant topology). Instead, they are expected to give rise to space and time at distances which are 10 times the Planck length. The issue of background independence in LQG still has some unresolved subtleties. For example, some derivations require a fixed choice of the topology, while any consistent quantum theory of gravity should include topology change as a dynamical process.
Spacetime as a "container" over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world. This is known as the relationalist interpretation of spacetime. In LQG this aspect of general relativity is taken seriously and this symmetry is preserved by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffeomorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called "problem of time" in general relativity. A generally accepted calculational framework to account for this constraint has yet to be found. A plausible candidate for the quantum Hamiltonian constraint is the operator introduced by Thiemann.
Constraints and their Poisson bracket algebra
Dirac observables The constraints define a constraint surface in the original phase space. The gauge motions of the constraints apply to all phase space but have the feature that they leave the constraint surface where it is, and thus the orbit of a point in the hypersurface under gauge transformations will be an orbit entirely within it. Dirac observables are defined as phase space functions, , that Poisson commute with all the constraints when the constraint equations are imposed,
that is, they are quantities defined on the constraint surface that are invariant under the gauge transformations of the theory.
Then, solving only the constraint and determining the Dirac observables with respect to it leads us back to the Arnowitt–Deser–Misner (ADM) phase space with constraints . The dynamics of general relativity is generated by the constraints, it can be shown that six Einstein equations describing time evolution (really a gauge transformation) can be obtained by calculating the Poisson brackets of the three-metric and its conjugate momentum with a linear combination of the spatial diffeomorphism and Hamiltonian constraint. The vanishing of the constraints, giving the physical phase space, are the four other Einstein equations.
Quantization of the constraints – the equations of quantum general relativity Pre-history and Ashtekar new variables
Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to quantum operators because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads (a triad is simply three orthogonal vector fields labeled by and the densitized triad is defined by ) to encode information about the spatial metric,
(where is the flat space metric, and the above equation expresses that , when written in terms of the basis , is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space rotation with respect to the internal indices . The canonically conjugate variable is related to the extrinsic curvature by . But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,
that behaves as a complex connection where is related to the so-called spin connection via . Here is called the chiral spin connection. It defines a covariant derivative . It turns out that is the conjugate momentum of , and together these form Ashtekar's new variables.
The expressions for the constraints in Ashtekar variables; Gauss's theorem, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:
respectively, where is the field strength tensor of the connection and where is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the gauge invariance here expressed by Gauss's theorem. Note that these constraints are polynomial in the fundamental variables, unlike the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article Self-dual Palatini action for a derivation of Ashtekar's formalism).
With Ashtekar's new variables, given the configuration variable , it is natural to consider wavefunctions . This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable and wavefunctions . The configuration variable gets promoted to a quantum operator via:
(analogous to ) and the triads are (functional) derivatives,
(analogous to ). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained Yang–Mills Hilbert space). Note that different ordering of the 's and 's when replacing the 's with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read
There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with . There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.
Quantum constraints as the equations of quantum general relativity The classical result of the Poisson bracket of the smeared Gauss' law with the connections is
The quantum Gauss' law reads
If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of by an infinitesimal (in the sense of the parameter small) gauge transformation,
and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions have to be gauge invariant functions of the connection. The same idea is true for the other constraints.
Therefore, the two step process in the classical theory of solving the constraints (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the 'evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions of the quantum equations . This is because it solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because is the quantum generator of gauge transformations (gauge invariant functions are constant along the gauge orbits and thus characterize them). Recall that, at the classical level, solving the admissibility conditions and evolution equations was equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in canonical quantum gravity.
Introduction of the loop representation
It was in particular the inability to have good control over the space of solutions to Gauss's law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider the loop representation in gauge theories and quantum gravity.