Among the available quantum gravity proposals, string theory, loop quantum gravity, non-commutative geometry, group field theory, causal sets, asymptotic safety, causal dynamical triangulation, emergent gravity are the best motivated models. As an introductory summary to this dossier of Comptes Rendus Physique , I explain how those different theories can be tested or constrained by cosmological observations. Journal page Archives Articles in press. Article Article Outline. Access to the text HTML.
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Loop quantum gravity LQG is a theory of quantum gravity attempting to merge quantum mechanics and general relativity , including the incorporation of the matter of the standard model into the framework established for the pure quantum gravity case.
LQG competes with string theory as a candidate for quantum gravity. According to Albert Einstein , gravity is not a force — it is a property of spacetime itself.
So far, all attempts to treat gravity as another quantum force equal in importance to electromagnetism and the nuclear forces have failed, and loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a force. To do this, in LQG theory space and time are quantized analogously to the way quantities like energy and momentum are quantized in quantum mechanics.
The theory gives a physical picture of spacetime where space and time are granular and discrete directly because of quantization just like photons in the quantum theory of electromagnetism and the discrete energy levels of atoms. An implication of a quantized space is that a minimum distance exists.
LQG postulates that the structure of space is composed of finite loops woven into an extremely fine fabric or network.
These networks of loops are called spin networks. Consequently, not just matter, but space itself, prefers an atomic structure. The vast areas of research involve about 30 research groups worldwide. 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 one could talk of as the Big Crunch.
In , Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics. Carlo Rovelli and Lee 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 , 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 put on firm ground by Thomas Thiemann , who defined an anomaly-free Hamiltonian operator, showing the existence of a mathematically consistent background-independent theory.
The covariant or spin foam version of the dynamics developed during several decades, and crystallized in , from the joint work of research groups in France, Canada, UK, Poland, and Germany, 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.
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations.
The essential idea is that coordinates are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.
A more significant requirement is the principle of general relativity that states that the laws of physics take the same form in all reference systems. This is a generalization of the principle of special relativity which states that the laws of physics take the same form in all inertial frames.
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.
These are the defining symmetry transformations of General Relativity since the theory is formulated only in terms of a differentiable manifold. In general relativity, general covariance is intimately related to "diffeomorphism invariance".
This symmetry is one of the defining features of the theory. However, it is a common misunderstanding that "diffeomorphism invariance" refers to the invariance of the physical predictions of a theory under arbitrary coordinate transformations ; this is untrue and in fact every physical theory is invariant under coordinate transformations this way.
Diffeomorphisms , as mathematicians define them, correspond to something much more radical; intuitively a way they can be envisaged is as simultaneously dragging all the physical fields including the gravitational field over the bare differentiable manifold while staying in the same coordinate system. Diffeomorphisms are the true symmetry transformations of general relativity, and come about from the assertion that the formulation of the theory is based on a bare differentiable manifold, but not on any prior geometry — the theory is background-independent this is a profound shift, as all physical theories before general relativity had as part of their formulation a prior geometry.
What is preserved under such transformations are the coincidences between the values the gravitational field takes at such and such a "place" and the values the matter fields take there. From these relationships one can form a notion of matter being located with respect to the gravitational field, or vice versa.
This is what Einstein discovered: that physical entities are located with respect to one another only and not with respect to the spacetime manifold. As Carlo Rovelli puts it: "No more fields on spacetime: just fields on fields".
This is known as the relationalist interpretation of space-time. The realization by Einstein that general relativity should be interpreted this way is the origin of his remark "Beyond my wildest expectations".
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.
LQG is formally background independent. 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 large compared to 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.
General relativity is an example of a constrained system. In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept.
A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations,. With the use of Poisson brackets, the Hamilton's equations can be rewritten as,. In a similar way the Poisson bracket between a constraint and the phase space variables generates a flow along an orbit in the unconstrained phase space generated by the constraint.
There are three types of constraints in Ashtekar's reformulation of classical general relativity:. See Ashtekar variables. These smeared constraints defined with respect to a suitable space of smearing functions give an equivalent description to the original constraints. The dynamics of such a theory are thus very different from that of ordinary Yang—Mills theory. The constraints are certain functions of these phase space variables.
Of particular importance is the Poisson bracket algebra formed between the smeared constraints themselves as it completely determines the theory. In terms of the above smeared constraints, the constraint algebra amongst the Gauss' law reads,. So the Poisson bracket of those two is equivalent to a single Gauss' law evaluated on the commutator of the smearings. The Poisson bracket amongst spatial diffeomorphisms constraints reads.
The reason for this is that the smearing functions are not functions of the canonical variables and so the spatial diffeomorphism does not generate diffeomorphims on them. They do however generate diffeomorphims on everything else. This is equivalent to leaving everything else fixed while shifting the smearing.
The action of the spatial diffeomorphism on the Gauss law is. The Gauss law has vanishing Poisson bracket with the Hamiltonian constraint. The spatial diffeomorphism constraint with a Hamiltonian gives a Hamiltonian with its smearing shifted,. That is, it is a sum over infinitesimal spatial diffeomorphisms constraints where the coefficients of proportionality are not constants but have non-trivial phase space dependence.
The above Poisson bracket algebra for General relativity does not form a true Lie algebra because there are structure functions rather than structure constants for the Poisson bracket between two Hamiltonians. This leads to difficulties. 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.
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. 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. Formulating general relativity with triads instead of metrics was not new.
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,. The expressions for the constraints in Ashtekar variables; the Gauss's law, the spatial diffeomorphism constraint and the densitized Hamiltonian constraint then read:.
Note that these constraints are polynomial in the fundamental variables, unlike as with 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. This is the connection representation. The configuration variable gets promoted to a quantum operator via:. Formally they read. There are still problems in properly defining all these equations and solving them. 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. The same idea is true for the other constraints.
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. LQG includes the concept of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop; it is denoted. Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence.
Holonomies can also be associated with an edge; under a Gauss Law these transform as.
Loop quantum gravity