Loop quantum gravity (LQG) is a theory of quantum gravity, attempting to merge quantum mechanics and general relativity, while incorporating the standard model particles. It takes seriously the key insight from general relativity that space-time is a dynamic entity, not a fixed framework. It competes with string theory, another candidate for a theory of quantum gravity. However, unlike string theory, LQG is not a candidate for a theory of everything the goal of which is to explain all of particle physics, unifying gravity with the other forces at the same time. In contrast to LQG, string theory (for the most part) is background-dependent (built on a fixed framework) and doesn’t account for the dynamic nature of space-time at the heart of relativity.
From the point of view of Einstein's theory, all attempts to treat gravity as another quantum force equal in importance to electromagnetism and the nuclear forces have failed. According to Einstein, gravity is not a force – it is a property of spacetime itself. Loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation.
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. A minimum distance exists.
Space's structure prefers an extremely fine fabric or network woven of finite loops. 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 metres, and smaller scales do not exist. Consequently, not just matter, but space itself, prefers an atomic structure.
The vast areas of research developed in several directions that involve about 30 research groups worldwide. They all share the basic physical assumptions and the mathematical description of quantum space. Research follows two directions: the more traditional canonical loop quantum gravity, and the newer covariant loop quantum gravity, called spin foam theory.
Physical consequences of the theory proceed in several directions. The most well-developed applies to cosmology, called loop quantum cosmology (LQC), the study of the early universe and the physics of the Big Bang. Its greatest consequence sees the evolution of the universe continuing beyond the Big Bang called the Big Bounce.
In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics. 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 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 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 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 2008, 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. The finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, and this is consistent with observed acceleration in the expansion of the Universe.
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 take 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 the true meaning of the saying "The stage disappears and becomes one of the actors"; space-time 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 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. 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.
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,
where the Poisson bracket is given by
for arbitrary phase space functions and . With the use of Poisson brackets, the Hamilton's equations can be rewritten as,
These equations describe a "flow" or orbit in phase space generated by the Hamiltonian . Given any phase space function , yields
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:
The Gauss constraints
This represents an infinite number of constraints one for each value of . These come about from re-expressing General relativity as an Yang–Mills type gauge theory (Yang–Mills is a generalization of Maxwell's theory where the gauge field transforms as a vector under Gauss transformations, that is, the Gauge field is of the form where is an internal index. See Ashtekar variables). These infinite number of Gauss gauge constraints can be “smeared” by test fields with internal indices, ,
which must vanish for any such function. These smeared constraints defined with respect to a suitable space of smearing functions give an equivalent description to the original constraints.
Ashtekar's formulation may be thought of as ordinary Yang–Mills theory together with the following special constraints, resulting from diffeomorphism invariance, and a Hamiltonian that vanishes. The dynamics of such a theory are thus very different from that of ordinary Yang–Mills theory.
The spatial diffeomorphism constraints
can be smeared by the so-called shift functions to give an equivalent set of smeared spatial diffeomorphism constraints,
These generate spatial diffeomorphisms along orbits defined by the shift function .
can be smeared by the so-called lapse functions to give an equivalent set of smeared Hamiltonian constraints,
These generate time diffeomorphisms along orbits defined by the lapse function .
In Ashtekar formulation the gauge field is the configuration variable (the configuration variable being analogous to in ordinary mechanics) and its conjugate momentum is the (densitized) triad (electrical field) . The constraints are certain functions of these phase space variables.
An important aspect of the action of the constraints on arbitrary phase space functions is the Lie derivative, , which is basically a derivative operation that infinitesimally "shifts" functions along some orbit with tangent vector .
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,
where . 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
and its effect is to "shift the smearing". 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
again, it shifts the test field . 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,
Finally, the poisson bracket of two Hamiltonians is a spatial diffeomorphism,
where is some phase space function. 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.
A (Poisson bracket) Lie algebra, with constraints , is of the form
where are constants (the so-called structure constants). 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. 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 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.
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; the Gauss's law, 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 the Gauss law. 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).
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.
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 obviously 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.
It was in particular the inability to have good control over the space of solutions to the Gauss' law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider a new representation – the loop representation in gauge theories and quantum gravity.
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
For a closed loop and assuming , yields
The trace of an holonomy around a closed loop is written
and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit form of the Holonomy is
where is the curve along which the holonomy is evaluated, and is a parameter along the curve, denotes path ordering meaning factors for smaller values of appear to the left, and are matrices that satisfy the algebra
The Pauli matrices satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises matrices with , and where none of these can be thought to 'decompose' into two or more examples of lower dimension. They are called different irreducible representations of the algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer according to the irreducible representation used.
The use of Wilson loops explicitly solves the Gauss gauge constraint. Loop representation is required handle the spatial diffeomorphism constraint. With Wilson loops as a basis, any Gauss gauge invariant function expands as,
This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see Position and momentum space). The QM representation has a basis of states labelled by a number and expands as
and works with the coefficients of the expansion
The inverse loop transform is defined by
This defines the loop representation. Given an operator in the connection representation,
one should define the corresponding operator on in the loop representation via,
where is defined by the usual inverse loop transform,
A transformation formula giving the action of the operator on in terms of the action of the operator on is then obtained by equating the R.H.S. of with the R.H.S. of with substituted into , namely
where means the operator but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). The action of this operator on the Wilson loop is evaluated as a calculation in the connection representation and the result is rearranged purely as a manipulation in terms of loops (with regard to the action on the Wilson loop, the chosen transformed operator is the one with the opposite factor ordering compared to the one used for its action on wavefunctions ). This gives the physical meaning of the operator . For example, if corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field of where it is while performing a spatial diffeomorphism on instead. Therefore, the meaning of is a spatial diffeomorphism on , the argument of .
In the loop representation, the spatial diffeomorphism constraint is solved by considering functions of loops that are invariant under spatial diffeomorphisms of the loop . That is, knot invariants are used. This opens up an unexpected connection between knot theory and quantum gravity.
Any collection of non-intersecting Wilson loops satisfy Ashtekar's quantum Hamiltonian constraint. Using a particular ordering of terms and replacing by a derivative, the action of the quantum Hamiltonian constraint on a Wilson loop is
When a derivative is taken it brings down the tangent vector, , of the loop, . So,
However, as is anti-symmetric in the indices and this vanishes (this assumes that is not discontinuous anywhere and so the tangent vector is unique).
With regard to loop representation, the wavefunctions vanish when the loop has discontinuities and are knot invariants. Such functions solve the Gauss law, the spatial diffeomorphism constraint and (formally) the Hamiltonian constraint. This yields an infinite set of exact (if only formal) solutions to all the equations of quantum general relativity! This generated a lot of interest in the approach and eventually led to LQG.
The easiest geometric quantity is the area. Let us choose coordinates so that the surface is characterized by . The area of small parallelogram of the surface is the product of length of each side times where is the angle between the sides. Say one edge is given by the vector and the other by then,
In the space spanned by and there is an infinitesimal parallelogram described by and . Using (where the indices and run from 1 to 2), yields the area of the surface given by
where and is the determinant of the metric induced on . The latter can be rewritten where the indices go from 1 to 2. This can be further rewritten as
The standard formula for an inverse matrix is
There is a similarity between this and the expression for . But in Ashtekar variables, . Therefore,
According to the rules of canonical quantization the triads should be promoted to quantum operators,
The area can be promoted to a well defined quantum operator despite the fact that it contains a product of two functional derivatives and a square-root. Putting (-th representation),
This quantity is important in the final formula for the area spectrum. The result is
where the sum is over all edges of the Wilson loop that pierce the surface .
The formula for the volume of a region is given by
The quantization of the volume proceeds the same way as with the area. Each time the derivative is taken, it brings down the tangent vector , and when the volume operator acts on non-intersecting Wilson loops the result vanishes. Quantum states with non-zero volume must therefore involve intersections. Given that the anti-symmetric summation is taken over in the formula for the volume, it needs intersections with at least three non-coplanar lines. At least four-valent vertices are needed for the volume operator to be non-vanishing.
Assuming the real representation where the gauge group is , Wilson loops are an over complete basis as there are identities relating different Wilson loops. These occur because Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities. Given any two matrices and ,
This implies that given two loops and that intersect,
where by we mean the loop traversed in the opposite direction and means the loop obtained by going around the loop and then along . See figure below. Given that the matrices are unitary one has that . Also given the cyclic property of the matrix traces (i.e. ) one has that . These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-compleness entirely) and actually constitute a basis for all gauge invariant functions.
As mentioned above the holonomy tells one how to propagate test spin half particles. A spin network state assigns an amplitude to a set of spin half particles tracing out a path in space, merging and splitting. These are described by spin networks : the edges are labelled by spins together with 'intertwiners' at the vertices which are prescription for how to sum over different ways the spins are rerouted. The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.
Let us go into more detail about the technical difficulties associated with using Ashtekar's variables:
With Ashtekar's variables one uses a complex connection and so the relevant gauge group is actually and not . As is non-compact it creates serious problems for the rigorous construction of the necessary mathematical machinery. The group , on the other hand, is compact and the needed constructions have been developed.
As mentioned above, because Ashtekar's variables are complex the resulting general relativity is complex. To recover the real theory, one has to impose what are known as the "reality conditions." These require that the densitized triad be real and that the real part of the Ashtekar connection equals the compatible spin connection (the compatibility condition being ) determined by the densitized triad. The expression for compatible connection is rather complicated and as such non-polynomial formula enters through the back door.
also appears as a factor, i.e.
It is impossible, on general grounds, to construct a UV-finite, diffeomorphism non-violating operator corresponding to . The reason is that the rescaled Hamiltonian constraint is a scalar density of weight two while it can be shown that only scalar densities of weight one have a chance to result in a well defined operator. Thus, one is forced to work with the original unrescaled, density one-valued, Hamiltonian constraint. However, this is non-polynomial and the whole virtue of the complex variables is questioned. In fact, all the solutions constructed for Ashtekar's Hamiltonian constraint only vanished for finite regularization (physics), however, this violates spatial diffeomorphism invariance.
Without the implementation and solution of the Hamiltonian constraint no progress can be made and no reliable predictions are possible.
To overcome the first problem one works with the configuration variable
where is real (as pointed out by Barbero, who introduced real variables some time after Ashtekar's variables). The Guass law and the spatial diffeomorphism constraints are the same. In real Ashtekar variables the Hamiltonian is
The complicated relationship between and the desitized triads causes serious problems upon quantization. It is with the choice that the second more complicated term is made to vanish. However, as mentioned above reappears in the reality conditions. There is still the problem of the factor.
Thiemann was able to make it work for real . First he could simplify the troublesome by using the identity
where is the volume. Combining this identity with the simple identity
Contracting both sides with gives
The smeared Euclidean Hamiltonian constraint functional can then be written ( is the lapse function)
The , and can be promoted to well defined operators in the loop representation and the Poisson bracket is replaced by a commutator upon quantization; this takes care of the first term. It turns out that a similar trick can be used to treat the second term. One introduces the quantity
and notes that
The reason the quantity is easier to work with at the time of quantization is that it can be written as
where we have used that the integrated densitized trace of the extrinsic curvature, , is the "time derivative of the volume".
In the long history of canonical quantum gravity formulating the Hamiltonian constraint as a quantum operator (Wheeler–DeWitt equation) in a mathematically rigorous manner has been a formidable problem. It was in the loop representation that a mathematically well defined Hamiltonian constraint was finally formulated in 1996. We leave more details of its construction to the article Hamiltonian constraint of LQG. This together with the quantum versions of the Gauss law and spatial diffeomorphism constrains written in the loop representation are the central equations of LQG (modern canonical quantum General relativity).
Finding the states that are annihilated by these constraints (the physical states), and finding the corresponding physical inner product, and observables is the main goal of the technical side of LQG.
A very important aspect of the Hamiltonian operator is that it only acts at vertices (a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning). More precisely, its action is non-zero on at least vertices of valence three and greater and results in a linear combination of new spin networks where the original graph has been modified by the addition of lines at each vertex together and a change in the labels of the adjacent links of the vertex.
We solve, at least approximately, all the quantum constraint equations and for the physical inner product to make physical predictions.
Before we move on to the constraints of LQG, lets us consider certain cases. We start with a kinematic Hilbert space as so is equipped with an inner product—the kinematic inner product .
i) Say we have constraints whose zero eigenvalues lie in their discrete spectrum. Solutions of the first constraint, , correspond to a subspace of the kinematic Hilbert space, . There will be a projection operator mapping onto . The kinematic inner product structure is easily employed to provide the inner product structure after solving this first constraint; the new inner product is simply
They are based on the same inner product and are states normalizable with respect to it.
ii) The zero point is not contained in the point spectrum of all the , there is then no non-trivial solution to the system of quantum constraint equations for all .
For example, the zero eigenvalue of the operator
on lies in the continuous spectrum but the formal "eigenstate" is not normalizable in the kinematic inner product,
and so does not belong to the kinematic Hilbert space . In these cases we take a dense subset of (intuitively this means either any point in is either in or arbitrarily close to a point in ) with very good convergence properties and consider its dual space (intuitively these map elements of onto finite complex numbers in a linear manner), then (as contains distributional functions). The constraint operator is then implemented on this larger dual space, which contains distributional functions, under the adjoint action on the operator. One looks for solutions on this larger space. This comes at the price that the solutions must be given a new Hilbert space inner product with respect to which they are normalizable (see article on rigged Hilbert space). In this case we have a generalized projection operator on the new space of states. We cannot use the above formula for the new inner product as it diverges, instead the new inner product is given by the simply modification of the above,
The generalized projector is known as a rigging map.
Implementation and solution the quantum constraints of LQG.
Let us move to LQG, additional complications will arise from that one cannot define an operator for the quantum spatial diffeomorphism constraint as the infinitesimal generator of finite diffeomorphism transformations and the fact the constraint algebra is not a Lie algebra due to the bracket between two Hamiltonian constraints.
Implementation and solution the Gauss constraint:
One does not actually need to promote the Gauss constraint to an operator since we can work directly with Gauss-gauge-invariant functions (that is, one solves the constraint classically and quantizes only the phase space reduced with respect to the Gauss constraint). The Gauss law is solved by the use of spin network states. They provide a basis for the Kinematic Hilbert space .
Implementation of the quantum spatial diffeomorphism constraint:
It turns out that one cannot define an operator for the quantum spatial diffeomorphism constraint as the infinitesimal generator of finite diffeomorphism transformations, represented on . The representation of finite diffeomorphisms is a family of unitary operators acting on a spin-network state by
for any spatial diffeomorphism on . To understand why one cannot define an operator for the quantum spatial diffeomorphism constraint consider what is called a 1-parameter subgroup in the group of spatial diffeomorphisms, this is then represented as a 1-parameter unitary group on . However, is not weakly continuous since the subspace belongs to and the subspace belongs to are orthogonal to each other no matter how small the parameter is. So one always has
even in the limit when goes to zero. Therefore, the infinitesimal generator of does not exist.
Solution of the spatial diffeomorphism constraint.
The spatial diffeomorphism constraint has been solved. The induced inner product on (we do not pursue the details) has a very simple description in terms of spin network states; given two spin networks and , with associated spin network states and , the inner product is 1 if and are related to each other by a spatial diffeomorphism and zero otherwise.
We have provided a description of the implemented and complete solution of the kinematic constraints, the Gauss and spatial diffeomorphisms constraints which will be the same for any background-independent gauge field theory. The feature that distinguishes such different theories is the Hamiltonian constraint which is the only one that depends on the Lagrangian of the classical theory.
Problem arising from the Hamiltonian constraint.
Details of the implementation the quantum Hamiltonian constraint and solutions are treated in a different article Hamiltonian constraint of LQG. However, in this article we introduce an approximation scheme for the formal solution of the Hamiltonian constraint operator given in the section below on spinfoams. Here we just mention issues that arises with the Hamiltonian constraint.
The Hamiltonian constraint maps diffeomorphism invariant states onto non-diffeomorphism invariant states as so does not preserve the diffeomorphism Hilbert space . This is an unavoidable consequence of the operator algebra, in particular the commutator:
as can be seen by applying this to ,
and using to obtain
and so is not in .
This means that one cannot just solve the spatial diffeomorphism constraint and then the Hamiltonian constraint. This problem can be circumvented by the introduction of the master constraint, with its trivial operator algebra, one is then able in principle to construct the physical inner product from .
In loop quantum gravity (LQG), a spin network represents a "quantum state" of the gravitational field on a 3-dimensional hypersurface. The set of all possible spin networks (or, more accurately, "s-knots" – that is, equivalence classes of spin networks under diffeomorphisms) is countable; it constitutes a basis of LQG Hilbert space.
In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration) description of quantum gravity. It is closely related to loop quantum gravity.
The Hamiltonian constraint generates 'time' evolution. Solving the Hamiltonian constraint should tell us how quantum states evolve in 'time' from an initial spin network state to a final spin network state. One approach to solving the Hamiltonian constraint starts with what is called the Dirac delta function. This is a rather singular function of the real line, denoted , that is zero everywhere except at but whose integral is finite and nonzero. It can be represented as a Fourier integral,
One can employ the idea of the delta function to impose the condition that the Hamiltonian constraint should vanish.
is non-zero only when for all in . Using this we can 'project' out solutions to the Hamiltonian constraint. With analogy to the Fourier integral given above, this (generalized) projector can formally be written as
This is formally spatially diffeomorphism-invariant. As such it can be applied at the spatially diffeomorphism-invariant level. Using this the physical inner product is formally given by
where are the initial spin network and is the final spin network.
The exponential can be expanded
and each time a Hamiltonian operator acts it does so by adding a new edge at the vertex. The summation over different sequences of actions of can be visualized as a summation over different histories of 'interaction vertices' in the 'time' evolution sending the initial spin network to the final spin network. This then naturally gives rise to the two-complex (a combinatorial set of faces that join along edges, which in turn join on vertices) underlying the spin foam description; we evolve forward an initial spin network sweeping out a surface, the action of the Hamiltonian constraint operator is to produce a new planar surface starting at the vertex. We are able to use the action of the Hamiltonian constraint on the vertex of a spin network state to associate an amplitude to each "interaction" (in analogy to Feynman diagrams). See figure below. This opens up a way of trying to directly link canonical LQG to a path integral description. Now just as a spin networks describe quantum space, each configuration contributing to these path integrals, or sums over history, describe 'quantum space-time'. Because of their resemblance to soap foams and the way they are labeled John Baez gave these 'quantum space-times' the name 'spin foams'.
There are however severe difficulties with this particular approach, for example the Hamiltonian operator is not self-adjoint, in fact it is not even a normal operator (i.e. the operator does not commute with its adjoint) and so the spectral theorem cannot be used to define the exponential in general. The most serious problem is that the 's are not mutually commuting, it can then be shown the formal quantity cannot even define a (generalized) projector. The master constraint (see below) does not suffer from these problems and as such offers a way of connecting the canonical theory to the path integral formulation.
It turns out there are alternative routes to formulating the path integral, however their connection to the Hamiltonian formalism is less clear. One way is to start with the BF theory. This is a simpler theory than general relativity, it has no local degrees of freedom and as such depends only on topological aspects of the fields. BF theory is what is known as a topological field theory. Surprisingly, it turns out that general relativity can be obtained from BF theory by imposing a constraint, BF theory involves a field and if one chooses the field to be the (anti-symmetric) product of two tetrads
(tetrads are like triads but in four spacetime dimensions), one recovers general relativity. The condition that the field be given by the product of two tetrads is called the simplicity constraint. The spin foam dynamics of the topological field theory is well understood. Given the spin foam 'interaction' amplitudes for this simple theory, one then tries to implement the simplicity conditions to obtain a path integral for general relativity. The non-trivial task of constructing a spin foam model is then reduced to the question of how this simplicity constraint should be imposed in the quantum theory. The first attempt at this was the famous Barrett–Crane model. However this model was shown to be problematic, for example there did not seem to be enough degrees of freedom to ensure the correct classical limit. It has been argued that the simplicity constraint was imposed too strongly at the quantum level and should only be imposed in the sense of expectation values just as with the Lorenz gauge condition in the Gupta–Bleuler formalism of quantum electrodynamics. New models have now been put forward, sometimes motivated by imposing the simplicity conditions in a weaker sense.
Another difficulty here is that spin foams are defined on a discretization of spacetime. While this presents no problems for a topological field theory as it has no local degrees of freedom, it presents problems for GR. This is known as the problem triangularization dependence.
Just as imposing the classical simplicity constraint recovers general relativity from BF theory, one expects an appropriate quantum simplicity constraint will recover quantum gravity from quantum BF theory.
An attempt to make contact between EPRL-FK spin foam and the canonical formulation of LQG has been made.
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.
There are two basic requirements in establishing the semiclassical limit of any quantum theory:
i) reproduction of the Poisson brackets (of the diffeomorphism constraints in the case of general relativity). This is extremely important because, as noted above, the Poisson bracket algebra formed between the (smeared) constraints themselves completely determines the classical theory. This is analogous to establishing Ehrenfest's theorem;
ii) the specification of a complete set of classical observables whose corresponding operators, when acted on by appropriate semiclassical states, reproduce the same classical variables with small quantum corrections (a subtle point is that states that are semiclassical for one class of observables may not be semiclassical for a different class of observables).
This may be easily done, for example, in ordinary quantum mechanics for a particle but in general relativity this becomes a highly non-trivial problem as we will see below.
Any candidate theory of quantum gravity must be able to reproduce Einstein's theory of general relativity as a classical limit of a quantum theory. This is not guaranteed because of a feature of quantum field theories which is that they have different sectors, these are analogous to the different phases that come about in the thermodynamical limit of statistical systems. Just as different phases are physically different, so are different sectors of a quantum field theory. It may turn out that LQG belongs to an unphysical sector – one in which one does not recover general relativity in the semiclassical limit (in fact there might not be any physical sector at all).
Moreover, the physical Hilbert space must contain enough semiclassical states to guarantee that the quantum theory one obtains can return to the classical theory when . In order to guarantee this one must avoid quantum anomalies at all cost, because if we do not there will be restrictions on the physical Hilbert space that have no counterpart in the classical theory, implying that the quantum theory has fewer degrees of freedom than the classical theory.
Theorems establishing the uniqueness of the loop representation as defined by Ashtekar et al. (i.e. a certain concrete realization of a Hilbert space and associated operators reproducing the correct loop algebra – the realization that everybody was using) have been given by two groups (Lewandowski, Okolow, Sahlmann and Thiemann; and Christian Fleischhack). Before this result was established it was not known whether there could be other examples of Hilbert spaces with operators invoking the same loop algebra, other realizations, not equivalent to the one that had been used so far. These uniqueness theorems imply no others exist and so if LQG does not have the correct semiclassical limit then this would mean the end of the loop representation of quantum gravity altogether.
There are difficulties in trying to establish LQG gives Einstein's theory of general relativity in the semiclassical limit. There are a number of particular difficulties in establishing the semiclassical limit:
Difficulties in trying to examine the semiclassical limit of the theory should not be confused with it having the wrong semiclassical limit.
Concerning issue number 2 above, one can consider so-called weave states. Ordinary measurements of geometric quantities are macroscopic, and planckian discreteness is smoothed out. The fabric of a T-shirt is analogous: At a distance it is a smooth curved two-dimensional surface, but on closer inspection we see that it is actually composed of thousands of one-dimensional linked threads. The image of space given in LQG is similar. Consider a very large spin network formed by a very large number of nodes and links, each of Planck scale. Probed at a macroscopic scale, it appears as a three-dimensional continuous metric geometry.
As far as is currently known, problem 4 of having semiclassical machinery for non-graph changing operators is at the moment still out of reach.
To make contact with familiar low energy physics it is mandatory to have to develop approximation schemes both for the physical inner product and for Dirac observables.
The spin foam models that have been intensively studied can be viewed as avenues toward approximation schemes for the physical inner product.
Markopoulou, et al. adopted the idea of noiseless subsystems in an attempt to solve the problem of the low energy limit in background independent quantum gravity theories The idea has even led to the intriguing possibility of matter of the standard model being identified with emergent degrees of freedom from some versions of LQG (see section below: LQG and related research programs).
As Wightman emphasized in the 1950s, in Minkowski QFTs the point functions
completely determine the theory. In particular, one can calculate the scattering amplitudes from these quantities. As explained below in the section on the Background independent scattering amplitudes, in the background-independent context, the point functions refer to a state and in gravity that state can naturally encode information about a specific geometry which can then appear in the expressions of these quantities. To leading order, LQG calculations have been shown to agree in an appropriate sense with the point functions calculated in the effective low energy quantum general relativity.
Thiemann's master constraint should not be confused with the master equation which has to do with random processes. The Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Hamiltonian constraint equations
( being a continuous index) in terms of a single master constraint,
which involves the square of the constraints in question. Note that were infinitely many whereas the master constraint is only one. It is clear that if vanishes then so do the infinitely many 's. Conversely, if all the 's vanish then so does , therefore they are equivalent. The master constraint involves an appropriate averaging over all space and so is invariant under spatial diffeomorphisms (it is invariant under spatial "shifts" as it is a summation over all such spatial "shifts" of a quantity that transforms as a scalar). Hence its Poisson bracket with the (smeared) spatial diffeomorphism constraint, , is simple:
(it is invariant as well). Also, obviously as any quantity Poisson commutes with itself, and the master constraint being a single constraint, it satisfies
We also have the usual algebra between spatial diffeomorphisms. This represents a dramatic simplification of the Poisson bracket structure, and raises new hope in understanding the dynamics and establishing the semiclassical limit.
An initial objection to the use of the master constraint was that on first sight it did not seem to encode information about the observables; because the Master constraint is quadratic in the constraint, when one computes its Poisson bracket with any quantity, the result is proportional to the constraint, therefore it always vanishes when the constraints are imposed and as such does not select out particular phase space functions. However, it was realized that the condition
is equivalent to being a Dirac observable. So the master constraint does capture information about the observables. Because of its significance this is known as the master equation.
That the master constraint Poisson algebra is an honest Lie algebra opens up the possibility of using a certain method, known as group averaging, in order to construct solutions of the infinite number of Hamiltonian constraints, a physical inner product thereon and Dirac observables via what is known as refined algebraic quantization RAQ.
Define the quantum master constraint (regularisation issues aside) as
for all implies . Conversely, if then
What is done first is, we are able to compute the matrix elements of the would-be operator , that is, we compute the quadratic form . It turns out that as is a graph changing, diffeomorphism invariant quadratic form it cannot exist on the kinematic Hilbert space , and must be defined on . Since the master constraint operator is densely defined on , then is a positive and symmetric operator in . Therefore, the quadratic form associated with is closable. The closure of is the quadratic form of a unique self-adjoint operator , called the Friedrichs extension of . We relabel as for simplicity.
Note that the presence of an inner product, viz Eq 4, means there are no superfluous solutions i.e. there are no such that
but for which .
It is also possible to construct a quadratic form for what is called the extended master constraint (discussed below) on which also involves the weighted integral of the square of the spatial diffeomorphism constraint (this is possible because is not graph changing).
The spectrum of the master constraint may not contain zero due to normal or factor ordering effects which are finite but similar in nature to the infinite vacuum energies of background-dependent quantum field theories. In this case it turns out to be physically correct to replace with provided that the "normal ordering constant" vanishes in the classical limit, that is,
so that is a valid quantisation of .
The constraints in their primitive form are rather singular, this was the reason for integrating them over test functions to obtain smeared constraints. However, it would appear that the equation for the master constraint, given above, is even more singular involving the product of two primitive constraints (although integrated over space). Squaring the constraint is dangerous as it could lead to worsened ultraviolet behaviour of the corresponding operator and hence the master constraint programme must be approached with due care.
In doing so the master constraint programme has been satisfactorily tested in a number of model systems with non-trivial constraint algebras, free and interacting field theories. The master constraint for LQG was established as a genuine positive self-adjoint operator and the physical Hilbert space of LQG was shown to be non-empty, an obvious consistency test LQG must pass to be a viable theory of quantum General relativity.
It turns out that the master constraint is easily generalized to incorporate the other constraints. It is then referred to as the extended master constraint, denoted . We can define the extended master constraint which imposes both the Hamiltonian constraint and spatial diffeomorphism constraint as a single operator,
Setting this single constraint to zero is equivalent to and for all in . This constraint implements the spatial diffeomorphism and Hamiltonian constraint at the same time on the Kinematic Hilbert space. The physical inner product is then defined as
(as ). A spin foam representation of this expression is obtained by splitting the -parameter in discrete steps and writing
The spin foam description then follows from the application of on a spin network resulting in a linear combination of new spin networks whose graph and labels have been modified. Obviously an approximation is made by truncating the value of to some finite integer. An advantage of the extended master constraint is that we are working at the kinematic level and so far it is only here we have access semiclassical coherent states. Moreover, one can find none graph changing versions of this master constraint operator, which are the only type of operators appropriate for these coherent states.
The master constraint programme has evolved into a fully combinatorial treatment of gravity known as Algebraic Quantum Gravity (AQG). The non-graph changing master constraint operator is adapted in the framework of algebraic quantum gravity. While AQG is inspired by LQG, it differs drastically from it because in AQG there is fundamentally no topology or differential structure – it is background independent in a more generalized sense and could possibly have something to say about topology change. In this new formulation of quantum gravity AQG semiclassical states always control the fluctuations of all present degrees of freedom. This makes the AQG semiclassical analysis superior over that of LQG, and progress has been made in establishing it has the correct semiclassical limit and providing contact with familiar low energy physics.
The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity. It may take real or imaginary values.
Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. The no hair conjecture of general relativity states that a black hole is characterized only by its mass, its charge, and its angular momentum; hence, it has no entropy. It appears, then, that one can violate the second law of thermodynamics by dropping an object with nonzero entropy into a black hole. Work by Stephen Hawking and Jacob Bekenstein showed that one can preserve the second law of thermodynamics by assigning to each black hole a black-hole entropy
where is the area of the hole's event horizon, is the Boltzmann constant, and is the Planck length. The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.
An oversight in the application of the no-hair theorem is the assumption that the relevant degrees of freedom accounting for the entropy of the black hole must be classical in nature; what if they were purely quantum mechanical instead and had non-zero entropy? Actually, this is what is realized in the LQG derivation of black hole entropy, and can be seen as a consequence of its background-independence – the classical black hole spacetime comes about from the semiclassical limit of the quantum state of the gravitational field, but there are many quantum states that have the same semiclassical limit. Specifically, in LQG it is possible to associate a quantum geometrical interpretation to the microstates: These are the quantum geometries of the horizon which are consistent with the area, , of the black hole and the topology of the horizon (i.e. spherical). LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon. These calculations have been generalized to rotating black holes.
It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes.
A recent success of the theory in this direction is the computation of the entropy of all non singular black holes directly from theory and independent of Immirzi parameter. The result is the expected formula , where is the entropy and the area of the black hole, derived by Bekenstein and Hawking on heuristic grounds. This is the only known derivation of this formula from a fundamental theory, for the case of generic non singular black holes. Older attempts at this calculation had difficulties. The problem was that although Loop quantum gravity predicted that the entropy of a black hole is proportional to the area of the event horizon, the result depended on a crucial free parameter in the theory, the above-mentioned Immirzi parameter. However, there is no known computation of the Immirzi parameter, so it had to be fixed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy.
A detailed study of the quantum geometry of a black hole horizon has been made using loop quantum gravity. Loop-quantization reproduces the result for black hole entropy originally discovered by Bekenstein and Hawking. Further, it led to the computation of quantum gravity corrections to the entropy and radiation of black holes.
Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking spectrum that would be observable were X-rays from Hawking radiation of evaporating primordial black holes to be observed. The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of Hawking radiation spectrum.
In 2014 Carlo Rovelli and Francesca Vidotto proposed that there is a Planck star inside black holes. Based on LQG, the theory states that as stars are collapsing into black holes, the energy density reaches the planck energy density, causing a repulsive force that creates a star. Furthermore, the existence of such a star would resolve the black hole firewall and black hole information paradox.
The popular and technical literature makes extensive references to LQG-related topic of loop quantum cosmology. LQC was mainly developed by Martin Bojowald, it was popularized Loop quantum cosmology in Scientific American for predicting a Big Bounce prior to the Big Bang. Loop quantum cosmology (LQC) is a symmetry-reduced model of classical general relativity quantized using methods that mimic those of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches.
Achievements of LQC have been the resolution of the big bang singularity, the prediction of a Big Bounce, and a natural mechanism for inflation (cosmology).
LQC models share features of LQG and so is a useful toy model. However, the results obtained are subject to the usual restriction that a truncated classical theory, then quantized, might not display the true behaviour of the full theory due to artificial suppression of degrees of freedom that might have large quantum fluctuations in the full theory. It has been argued that singularity avoidance in LQC are by mechanisms only available in these restrictive models and that singularity avoidance in the full theory can still be obtained but by a more subtle feature of LQG.
Quantum gravity effects are notoriously difficult to measure because the Planck length is so incredibly small. However recently physicists have started to consider the possibility of measuring quantum gravity effects mostly from astrophysical observations and gravitational wave detectors. The energy of those fluctuations at scales this small cause space-perturbations which are visible at higher scales.
Loop quantum gravity is formulated in a background-independent language. No spacetime is assumed a priori, but rather it is built up by the states of theory themselves – however scattering amplitudes are derived from -point functions (Correlation function (quantum field theory)) and these, formulated in conventional quantum field theory, are functions of points of a background space-time. The relation between the background-independent formalism and the conventional formalism of quantum field theory on a given spacetime is far from obvious, and it is far from obvious how to recover low-energy quantities from the full background-independent theory. One would like to derive the -point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.
A strategy for addressing this problem has been suggested; the idea is to study the boundary amplitude, namely a path integral over a finite space-time region, seen as a function of the boundary value of the field. In conventional quantum field theory, this boundary amplitude is well–defined and codes the physical information of the theory; it does so in quantum gravity as well, but in a fully background–independent manner. A generally covariant definition of -point functions can then be based on the idea that the distance between physical points –arguments of the -point function is determined by the state of the gravitational field on the boundary of the spacetime region considered.
Progress has been made in calculating background independent scattering amplitudes this way with the use of spin foams. This is a way to extract physical information from the theory. Claims to have reproduced the correct behaviour for graviton scattering amplitudes and to have recovered classical gravity have been made. "We have calculated Newton's law starting from a world with no space and no time." – Carlo Rovelli.
Some quantum theories of gravity posit a spin-2 quantum field that is quantized, giving rise to gravitons. In string theory one generally starts with quantized excitations on top of a classically fixed background. This theory is thus described as background dependent. Particles like photons as well as changes in the spacetime geometry (gravitons) are both described as excitations on the string worldsheet. The background dependence of string theory can have important physical consequences, such as determining the number of quark generations. In contrast, loop quantum gravity, like general relativity, is manifestly background independent, eliminating the background required in string theory. Loop quantum gravity, like string theory, also aims to overcome the nonrenormalizable divergences of quantum field theories.
LQG never introduces a background and excitations living on this background, so LQG does not use gravitons as building blocks. Instead one expects that one may recover a kind of semiclassical limit or weak field limit where something like "gravitons" will show up again. In contrast, gravitons play a key role in string theory where they are among the first (massless) level of excitations of a superstring.
LQG differs from string theory in that it is formulated in 3 and 4 dimensions and without supersymmetry or Kaluza-Klein extra dimensions, while the latter requires both to be true. There is no experimental evidence to date that confirms string theory's predictions of supersymmetry and Kaluza–Klein extra dimensions. In a 2003 paper "A Dialog on Quantum Gravity", Carlo Rovelli regards the fact LQG is formulated in 4 dimensions and without supersymmetry as a strength of the theory as it represents the most parsimonious explanation, consistent with current experimental results, over its rival string/M-theory. Proponents of string theory will often point to the fact that, among other things, it demonstrably reproduces the established theories of general relativity and quantum field theory in the appropriate limits, which loop quantum gravity has struggled to do. In that sense string theory's connection to established physics may be considered more reliable and less speculative, at the mathematical level. Loop quantum gravity has nothing to say about the matter (fermions) in the universe.
Since LQG has been formulated in 4 dimensions (with and without supersymmetry), and M-theory requires supersymmetry and 11 dimensions, a direct comparison between the two has not been possible. It is possible to extend mainstream LQG formalism to higher-dimensional supergravity, general relativity with supersymmetry and Kaluza–Klein extra dimensions should experimental evidence establish their existence. It would therefore be desirable to have higher-dimensional Supergravity loop quantizations at one's disposal in order to compare these approaches. In fact a series of recent papers have been published attempting just this. Most recently, Thiemann (and alumni) have made progress toward calculating black hole entropy for supergravity in higher dimensions. It will be interesting to compare these results to the corresponding super string calculations.
Several research groups have attempted to combine LQG with other research programs: Johannes Aastrup, Jesper M. Grimstrup et al. research combines noncommutative geometry with canonical quantum gravity and Ashtekar variables, Laurent Freidel, Simone Speziale, et al., spinors and twistor theory with loop quantum gravity, and Lee Smolin et al. with Verlinde entropic gravity and loop gravity. Stephon Alexander, Antonino Marciano and Lee Smolin have attempted to explain the origins of weak force chirality in terms of Ashketar's variables, which describe gravity as chiral, and LQG with Yang–Mills theory fields in four dimensions. Sundance Bilson-Thompson, Hackett et al., has attempted to introduce standard model via LQG's degrees of freedom as an emergent property (by employing the idea noiseless subsystems a useful notion introduced in more general situation for constrained systems by Fotini Markopoulou-Kalamara et al.)
Furthermore, LQG has drawn philosophical comparisons with causal dynamical triangulation and asymptotically safe gravity, and the spinfoam with group field theory and AdS/CFT correspondence. Smolin and Wen have suggested combining LQG with string-net liquid, tensors, and Smolin and Fotini Markopoulou-Kalamara quantum graphity. There is the consistent discretizations approach. Also, Pullin and Gambini provide a framework to connect the path integral and canonical approaches to quantum gravity. They may help reconcile the spin foam and canonical loop representation approaches. Recent research by Chris Duston and Matilde Marcolli introduces topology change via topspin networks.
Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail.
Many of these problems apply to LQG, including:
The theory of LQG is one possible solution to the problem of quantum gravity, as is string theory. There are substantial differences however. For example, string theory also addresses unification, the understanding of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries. Contrary to this, LQG is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction. On the other hand, the consequences of LQG are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.
Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections). Specifically, the dynamics of the theory are encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian. Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see one-loop Feynman diagram in Feynman diagram).
While there has been a recent proposal relating to observation of naked singularities, and doubly special relativity as a part of a program called loop quantum cosmology, there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity). Because of the above-mentioned lack of a semiclassical limit, LQG has not yet even reproduced the predictions made by general relativity.
An alternative criticism is that general relativity may be an effective field theory, and therefore quantization ignores the fundamental degrees of freedom.
Abhay Vasant Ashtekar (born 5 July 1949) is an Indian theoretical physicist. He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University. As the creator of Ashtekar variables, he is one of the founders of loop quantum gravity and its subfield loop quantum cosmology. He has also written a number of descriptions of loop quantum gravity that are accessible to non-physicists. In 1999, Ashtekar and his colleagues were able to calculate the entropy for a black hole, matching a legendary 1974 prediction by Hawking. Oxford mathematical physicist Roger Penrose has described Ashtekar's approach to quantum gravity as "The most important of all the attempts at 'quantizing' general relativity." Ashtekar was elected as Member to National Academy of Sciences in May 2016.Barrett–Crane model
The Barrett–Crane model is a model in quantum gravity, first published in 1998, which was defined using the Plebanski action.
The field in the action is supposed to be a -valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term
in the action has the same symmetries as it does to provide the Einstein–Hilbert action. But the form of
is not unique and can be posed by the different forms:
where is the tetrad and is the antisymmetric symbol of the -valued 2-form fields.
The Plebanski action can be constrained to produce the BF model which is a theory of no local degrees of freedom. John W. Barrett and Louis Crane modeled the analogous constraint on the summation over spin foam.
The Barrett–Crane model on spin foam quantizes the Plebanski action, but its path integral amplitude corresponds to the degenerate field and not the specific definition
which formally satisfies the Einstein's field equation of general relativity. However, if analysed with the tools of loop quantum gravity the Barrett–Crane model gives an incorrect long-distance limit , and so the model is not identical to loop quantum gravity.Carlo Rovelli
Carlo Rovelli (born 3 May 1956) is an Italian theoretical physicist and writer who has worked in Italy, the United States, and since 2000 in France. He works mainly in the field of quantum gravity, and is a founder of loop quantum gravity theory. He has also worked in the history and philosophy of science. He collaborates with several Italian newspapers, including the cultural supplements of the Corriere della Sera, Il Sole 24 Ore, and La Repubblica.
His popular-science book Seven Brief Lessons on Physics has been translated into 41 languages and has sold over a million copies worldwide. In 2019 he was included by Foreign Policy magazine in a list of 100 most influential global thinkers.Causal dynamical triangulation
Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent.
This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.
The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work. A brief article appeared in the February 2007 issue of Scientific American, which gives an overview of the theory, explained why some physicists are excited about it, and put it in historical perspective. The same publication gives CDT, and its primary authors, a feature article in its July 2008 issue.Eugenio Bianchi
Eugenio Bianchi is an Italian theoretical physicist and assistant professor at the Pennsylvania State University who works on loop quantum gravity and black hole thermodynamics. He has derived the Bekenstein-Hawking formula S=A/4 for the entropy of non-extremal black holes from loop quantum gravity, for all values of the Immirzi parameter.Group field theory
Group field theory (GFT) is a quantum field theory in which the base manifold is taken to be a Lie group. It is closely related to background independent quantum gravity approaches such as loop quantum gravity, the spin foam formalism and causal dynamical triangulation. It can be shown that its perturbative expansion can be interpreted as spin foams and simplicial pseudo-manifolds (depending on the representation of the fields). Thus, its partition function defines a non-perturbative sum over all simplicial topologies and geometries, giving a path integral formulation of quantum spacetime.History of loop quantum gravity
This article is a historical introduction to the subject. For the main encyclopedia article, see Loop quantum gravity.The history of loop quantum gravity spans more than three decades of intense research.Immirzi parameter
The Immirzi parameter (also known as the Barbero–Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity.Lee Smolin
Lee Smolin (; born June 6, 1955) is an American theoretical physicist, a faculty member at the Perimeter Institute for Theoretical Physics, an adjunct professor of physics at the University of Waterloo and a member of the graduate faculty of the philosophy department at the University of Toronto. Smolin's 2006 book The Trouble with Physics criticized string theory as a viable scientific theory. He has made contributions to quantum gravity theory, in particular the approach known as loop quantum gravity. He advocates that the two primary approaches to quantum gravity, loop quantum gravity and string theory, can be reconciled as different aspects of the same underlying theory. His research interests also include cosmology, elementary particle theory, the foundations of quantum mechanics, and theoretical biology.List of loop quantum gravity researchers
This is a list researchers in the physics field of loop quantum gravity that are present in Wikipedia.
Abhay Ashtekar, Pennsylvania State University, USA
John Baez, University of California, Riverside, USA
Aurélien Barrau, Université J. Fourier, Grenoble, France
John W. Barrett, University of Nottingham, UK
Eugenio Bianchi, Pennsylvania State University, USA
Martin Bojowald, Pennsylvania State University, USA
Alejandro Corichi, National Autonomous University of Mexico, Mexico
Bianca Dittrich, Perimeter Institute for Theoretical Physics, Canada
Laurent Freidel, Perimeter Institute for Theoretical Physics, Canada
Rodolfo Gambini, University of the Republic, Uruguay
Jorge Pullin, Louisiana State University, USA
Carlo Rovelli, Centre de Physique Théorique, Centre National de la Recherche Scientifique (CNRS), Marseille, France
Lee Smolin, Perimeter Institute for Theoretical Physics, CanadaList of quantum gravity researchers
This is a list of (some of) the researchers in quantum gravity.
Jan Ambjørn: Expert on dynamical triangulations who helped develop the causal dynamical triangulations approach to quantum gravity.
Giovanni Amelino-Camelia: Physicist who developed the idea of doubly special relativity, and founded Quantum-Gravity phenomenology.
Abhay Ashtekar: Inventor of the Ashtekar variables, one of the founders of loop quantum gravity.
John Baez: Mathematical physicist who introduced the notion of spin foam in loop quantum gravity (a term originally introduced by Wheeler).
Julian Barbour: Philosopher and author of The End of Time, Absolute or Relative Motion?: The Discovery of Dynamics.
John W. Barrett: Mathematical physicist who helped develop the Barrett–Crane model of quantum gravity.
Martin Bojowald: Physicist who developed the application of loop quantum gravity to cosmology.
Steve Carlip: Expert on 3-dimensional quantum gravity.
Louis Crane: Mathematician who helped develop the Barrett–Crane model of quantum gravity.
Bryce DeWitt: Formulated the Wheeler–DeWitt equation for the wavefunction of the Universe with John Archibald Wheeler.
Bianca Dittrich: Mathematical physicist known for her contributions to loop quantum gravity and spin foam models, currently working on coarse-graining of spin foams.
Fay Dowker: Physicist working on causal sets as well as the interpretation of quantum mechanics.
David Finkelstein: Physicist who has contributed much quantum relativity and the logical foundations of QR.
Rodolfo Gambini: Physicist who helped introduce loop quantum gravity; coauthor of Loops, Knots, Gauge Theories and Quantum Gravity.
Gary Gibbons: Physicist who has done important work on black holes.
Brian Greene: Physicist who is considered one of the world's foremost string theorists.
James Hartle: Physicist who helped develop the Hartle-Hawking wavefunction for the universe.
Stephen Hawking: Leading physicist, expert on black holes and discoverer of Hawking radiation who helped develop the Hartle-Hawking wavefunction for the universe.
Laurent Freidel: Mathematical physicist known for his contributions to loop quantum gravity and spin foam models, in particular the Freidel-Krasnov model.
Christopher Isham: Physicist who focuses on conceptual problems in quantum gravity.
Ted Jacobson: Physicist who helped develop loop quantum gravity.
Michio Kaku: Physicist one of the foremost leading String theorist and also known for the Popular Science.
Renate Loll: Physicist who worked on loop quantum gravity and more recently helped develop the causal dynamical triangulations approach to quantum gravity.
Fotini Markopoulou-Kalamara: Physicist who works on loop quantum gravity and spin network models that take causality into account.
Leonardo Modesto: Physicist who works on Nonlocal Quantum Gravity a unitary and finite theory of quantum gravity in the quantum field theory framework.
Roger Penrose: Mathematical physicist who invented spin networks and twistor theory.
Jorge Pullin: Physicist who helped develop loop quantum gravity, co-author of Loops, Knots, Gauge Theories and Quantum Gravity.
Carlo Rovelli: One of the founders and major contributors to loop quantum gravity.
Lee Smolin: One of the founders and major contributors to loop quantum gravity.
Rafael Sorkin: Physicist, primary proponent of the causal set approach to quantum gravity.
Andrew Strominger: Physicist who works on string theory.
Leonard Susskind: Leading physicist, who is considered to be one of the three fathers of string theory.
Frank J. Tipler: Mathematical physicist who incorporates quantum gravity into his ideas of a Judeo-Christian God.
Bill Unruh: Canadian physicist engaged in the study of semiclassical gravity and responsible for the discovery of the so-called Unruh effect.
Cumrun Vafa: Leading physicist, developer of F-theory, known for Vafa-Witten theorem and Gopakumar-Vafa conjecture.
Robert Wald: Leading physicist in the field of quantum field theory in curved spacetime.
Anzhong Wang: Physicist, major contributor to Horava-Lifshitz gravity; String theory and applications to cosmology.
Paul S. Wesson: Physicist, cosmologist and writer, known as founder of the "Space-time Consortium" and his work on Kaluza-Klein theory.
John Archibald Wheeler: Pioneer in the field of quantum gravity due to his development, with Bryce DeWitt, of the Wheeler–DeWitt equation.
Edward Witten: Leading mathematical physicist, does research in string theory and M-theory.Loop quantum cosmology
Loop quantum cosmology (LQC) is a finite, symmetry-reduced model of loop quantum gravity (LQG) that predicts a "quantum bridge" between contracting and expanding cosmological branches.
The distinguishing feature of LQC is the prominent role played by the quantum geometry effects of loop quantum gravity (LQG). In particular, quantum geometry creates a brand new repulsive force which is totally negligible at low space-time curvature but rises very rapidly in the Planck regime, overwhelming the classical gravitational attraction and thereby resolving singularities of general relativity. Once singularities are resolved, the conceptual paradigm of cosmology changes and one has to revisit many of the standard issues—e.g., the "horizon problem"—from a new perspective.
Since LQG is based on a specific quantum theory of Riemannian geometry, geometric observables display a fundamental discreteness that play a key role in quantum dynamics: While predictions of LQC are very close to those of quantum geometrodynamics (QGD) away from the Planck regime, there is a dramatic difference once densities and curvatures enter the Planck scale. In LQC the Big Bang is replaced by a quantum bounce.
Study of LQC has led to many successes, including the emergence of a possible mechanism for cosmic inflation, resolution of gravitational singularities, as well as the development of effective semi-classical Hamiltonians.
This subfield originated in 1999 by Martin Bojowald, and further developed in particular by Abhay Ashtekar and Jerzy Lewandowski, as well as Tomasz Pawłowski and Parampreet Singh, et al. In late 2012 LQC represents a very active field in physics, with about three hundred papers on the subject published in the literature. There has also recently been work by Carlo Rovelli, et al. on relating LQC to the spinfoam-based spinfoam cosmology.
However, the results obtained in LQC are subject to the usual restriction that a truncated classical theory, then quantized, might not display the true behaviour of the full theory due to artificial suppression of degrees of freedom that might have large quantum fluctuations in the full theory. It has been argued that singularity avoidance in LQC are by mechanisms only available in these restrictive models and that singularity avoidance in the full theory can still be obtained but by a more subtle feature of LQG.Due to the quantum geometry, the Big Bang is replaced by a big bounce without any assumptions on the matter content or any fine tuning. An important feature of loop quantum cosmology is the effective space-time description of the underlying quantum evolution. The effective dynamics approach has been extensively used in loop quantum cosmology to describe physics at the Planck scale and the very early universe. Rigorous numerical simulations have confirmed the validity of the effective dynamics, which provides an excellent approximation to the full loop quantum dynamics. It has been shown that only when the states have very large quantum fluctuations at late times, which means that they do not lead to macroscopic universes as described by general relativity, that the effective dynamics has departures from the quantum dynamics near bounce and the subsequent evolution. In such a case, the effective dynamics overestimates the density at the bounce, but still captures the qualitative aspects extremely well.Lorentz invariance in loop quantum gravity
Lorentz invariance is a measure of universal features in hypothetical loop quantum gravity universes. The various hypothetical multiverse loop quantum gravity universe design models could have various Lorentz invariance results.
Because loop quantum gravity models universes, space gravity theories are contenders to build and answer unification theory; the Lorentz invariance helps grade the spread of universal features throughout a proposed multiverse in time.Planck star
In loop quantum gravity theory, a Planck star is a hypothetical astronomical object that is created when the energy density of a collapsing star reaches the Planck energy density. Under these conditions, assuming gravity and spacetime are quantized, there arises a repulsive 'force' derived from Heisenberg's uncertainty principle. The accumulation of mass-energy inside the Planck star cannot collapse beyond this limit because it violates the uncertainty principle for spacetime itself.The key feature of this theoretical object is that this repulsion arises from the energy density, not the Planck length, and starts taking effect far earlier than might be expected. This repulsive 'force' is strong enough to stop the collapse of the star well before a singularity is formed, and indeed, well before the Planck scale for distance. Since a Planck star is calculated to be considerably larger than the Planck scale, this means there is adequate room for all the information captured inside of a black hole to be encoded in the star, thus avoiding information loss.While it might be expected that such a repulsion would act very quickly to reverse the collapse of a star, it turns out that the relativistic effects of the extreme gravity such an object generates slow down time for the Planck star to a similarly extreme degree. Seen from outside the star's Schwartzschild radius, the rebound from a Planck star takes approximately fourteen billion years, such that even primordial black holes are only now starting to rebound from an outside perspective.
Furthermore, the emission of Hawking radiation can be calculated to correspond to the timescale of gravitational effects on time, such that the event horizon that 'forms' a black hole evaporates as the rebound proceeds.The existence of Planck stars was first proposed by Carlo Rovelli and Francesca Vidotto, who theorized in 2014 that Planck stars form inside black holes as a solution to the black hole firewall and black hole information paradox. Confirmation of emissions from rebounding black holes could possibly provide evidence for loop quantum gravity. Recent work demonstrates that Planck stars may exist inside of black holes as part of a cycle between black hole to white hole.Quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored, such as near compact astrophysical objects where the effects of gravity are strong.
The current understanding of gravity is based on Albert Einstein's general theory of relativity, which is formulated within the framework of classical physics. On the other hand, the other three fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory, radically different formalisms for describing physical phenomena. It is sometimes argued that a quantum mechanical description of gravity is necessary on the grounds that one cannot consistently couple a classical system to a quantum one.While a quantum theory of gravity may be needed to reconcile general relativity with the principles of quantum mechanics, difficulties arise when applying the usual prescriptions of quantum field theory to the force of gravity via graviton bosons. The problem is that the theory one gets in this way is not renormalizable (it predicts infinite values for some observable properties such as the mass of particles) and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity. Although some quantum gravity theories, such as string theory, try to unify gravity with the other fundamental forces, others, such as loop quantum gravity, make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces.
Strictly speaking, the aim of quantum gravity is only to describe the quantum behavior of the gravitational field and should not be confused with the objective of unifying all fundamental interactions into a single mathematical framework. A quantum field theory of gravity that is unified with a grand unified theory is sometimes referred to as a theory of everything (TOE). While any substantial improvement into the present understanding of gravity would aid further work towards unification, the study of quantum gravity is a field in its own right with various branches having different approaches to unification.
One of the difficulties of formulating a quantum gravity theory is that quantum gravitational effects only appear at length scales near the Planck scale, around 10−35 meter, a scale far smaller, and equivalently far larger in energy, than those currently accessible by high energy particle accelerators. Therefore physicists lack experimental data which could distinguish between the competing theories which have been proposed and thus thought experiment approaches are suggested as a testing tool for these theories.S-knot
In loop quantum gravity, an s-knot is an equivalence class of spin networks under diffeomorphisms. In this formalism, s-knots represent the quantum states of the gravitational field.Spin foam
In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. Also, see loop quantum gravity.Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups. The diagrammatic notation often simplifies calculation because simple diagrams may be used to represent complicated functions.
Roger Penrose is credited with the invention of spin networks in 1971, although similar diagrammatic techniques existed before his time. Spin networks have been applied to the theory of quantum gravity by Carlo Rovelli, Lee Smolin, Jorge Pullin, Rodolfo Gambini and others.
Spin networks can also be used to construct a particular functional on the space of connections which is invariant under local gauge transformations.Sundance Bilson-Thompson
Sundance Osland Bilson-Thompson is an Australian theoretical particle physicist. He has developed the idea that certain preon models may be represented topologically, rather than by treating preons as pointlike particles. His ideas have attracted interest in the field of loop quantum gravity, as they may represent a way of incorporating the Standard Model into loop quantum gravity. This would make loop quantum gravity a candidate theory of everything.
|Quantum field theory|
in curved spacetime