The starting point of the construction of quantum theory is classical general relativity, formulated in terms
of the Sen–Ashtekar–Barbero connection [271, 16, 61]. Detailed introductions to the (complex) Ashtekar
formalism can be found in the book [18
] and in the conference proceedings [103]. The real version of the
theory is presently the most widely used.
Classical general relativity can be formulated in phase-space form as follows [18, 61]. Fix a
three-dimensional manifold (compact and without boundaries) and consider a smooth real
connection
and a vector density
, transforming in the vector representation of
on
. We use
for spatial indices and
for internal indices.
The internal indices can be viewed as labeling a basis in the Lie algebra of
or the
three axis of a local triad. We indicate coordinates on
as
. The relation between these
fields and conventional metric gravitational variables is as follows:
is the (densitized)
inverse triad, related to the three-dimensional metric
of constant-time surfaces by
In Equation (2),
is a constant, denoted the Immirzi parameter, that can be chosen arbitrarily (it
will enter the Hamiltonian constraint) [152, 151, 150]. Different choices for
yield different
versions of the formalism, all equivalent in the classical domain. If we choose
to be equal to
the imaginary unit,
, then
is the standard Ashtekar connection, which can be
shown to be the projection of the self-dual part of the four-dimensional spin connection on
the constant-time surface. If we choose
, we obtain the real Barbero connection. The
Hamiltonian constraint of Lorentzian general relativity has a particularly simple form in the
formalism; while the Hamiltonian constraint of Euclidean general relativity has a
simple form when expressed in terms of the
real connection. Other choices of
are
viable as well. Different choices of
are genuinely physical physically? nonequivalent in the
quantum theory, since they yield “geometrical quanta” of different magnitude [270
]. It has
been argued that there is a unique choice of
yielding the correct
coefficient in the
Bekenstein–Hawking formula [170
, 171
, 253
, 22
, 254
, 92
], but the matter is still under discussion; see for
instance [160
].
The spinorial version of the Ashtekar variables is given in terms of the Pauli matrices , or
the
generators
, by
The theory is invariant under local gauge transformations, three-dimensional diffeomorphisms
of the manifold on which the fields are defined, as well as under (coordinate) time translations generated by
the Hamiltonian constraint. The full dynamical content of general relativity is captured by the three
constraints that generate these gauge invariances.
The Lorentzian Hamiltonian constraint does not have a simple polynomial form if we use the real
connection (2). For a while, this fact was considered an obstacle to defining the quantum Hamiltonian
constraint; therefore, the complex version of the connection was mostly used. However, Thiemann has
succeeded in constructing a Lorentzian quantum-Hamiltonian constraint [285
, 289
, 291
] in
spite of the non-polynomiality of the classical expression. This is why the real connection is
now widely used. This choice has the advantage of eliminating the old “reality conditions”
problem, namely the problem of implementing nontrivial reality conditions in the quantum
theory.
Alternative versions of the classical formalism used as a starting point for the quantization have been
explored in the literature. Of particular interest is the approach followed by Alexandrov, who has
argued for a formalism where the full local symmetry of the tetrad formalism is
manifestly maintained [4
, 5
, 6
]. One of the advantages of this approach is that it sheds light
on the relationship with covariant spin-foam formalism (see below). Its main difficulty is to
fully keep track of the complicated second-class constraints and the resulting nontrivial Dirac
algebra.
Certain classical quantities play a very important role in the quantum theory. These are: traces of the
holonomy of the connection, which are labeled by loops on the 3-manifold; and surface integrals of the triad.
Given a loop in
define:
Consider a “Schrödinger-like” representation formed by quantum states that are functionals of
the connection. On these states, the two quantities
and
act naturally: the first as a
multiplicative operator, the second as the functional derivative operator
The class of functionals that we will use is formed by (the closure in the Hilbert-space norm of the linear
span of) functionals of a particular class, denoted “cylindrical states”. These are defined as follows.
Pick a graph , say with
links, denoted
, immersed in the manifold
. Let
be the parallel transport operator of the connection
along
.
is
an element of
. Pick a function
on
. The graph
and the function
determine a functional of the connection as follows
The main property of is that it carries a natural unitary representation of the diffeomorphism group
and of the group of the local
transformations, obtained transforming the argument of the
functionals. In fact, the essential property of the scalar product (11
) is that it is invariant under both these
transformations. The operators
and
are well-defined self-adjoint operators in this Hilbert
space.
A number of observations are in order.
Using Dirac notation, we write
in the same manner in which one may write
A subspace of
is formed by states invariant under
gauge transformations. We now
define an orthonormal basis in
. This basis represents a very important tool for using
the theory. It was introduced in [269] and developed in [47, 48
]; it is denoted ’spin network
basis’.
First, given a loop in
, there is a normalized state
in
, which is obtained by taking
and
. Namely
Next, consider a graph . A “coloring” of
is given by the following.
Indicate a colored graph by , or simply
, and denote it a “spin
network”. (It was Penrose who first had the intuition that this mathematics could be relevant for
describing the quantum properties of the geometry, and who gave the first version of spin-network
theory [223, 224].)
Given a spin network , we can construct a state
as follows. Take the propagator of the
connection along each link of the graph in the representation associated to that link, and then, at each
node, contract the matrices of the representation with the invariant tensor. We obtain a state
,
which we also write as
The spin network states provide a very convenient basis for the quantum theory, with a direct physical
interpretation. This follows from the fact that the spin network states are eigenstates of area and volume
operators, therefore they are states in which the three-dimensional geometry is well defined. See [261] for
details.
Consider the relations between the loop states
and the states
The next step in the construction of the theory is to factor away diffeomorphism invariance. This is a key
step for two reasons. First of all, is a “huge” nonseparable space. It is far “too large” for a quantum
field theory. However, most of this redundancy is gauge, and disappears when one solves the
diffeomorphism constraint, defining the diff-invariant Hilbert space
. This is the reason for
which the loop representation, as defined here, is only of value in diffeomorphism invariant
theories.
The second reason is that turns out to have a natural basis labeled by knots. More precisely by
“s-knots”. An s-knot s is an equivalence class of spin networks S under diffeomorphisms. An s-knot is
characterized by its “abstract” graph (defined only by the adjacency relations between links and
nodes), by the coloring, and by its knotting and linking properties, as in knot theory. Thus, the
physical quantum states of the gravitational field turn out to be essentially classified by knot
theory.
There are various equivalent ways of obtaining from
. One can use regularization techniques
for defining the quantum operator corresponding to the classical diffeomorphism constraint in
terms of elementary loop operators, and then find the kernel of such operator. Equivalently,
one can factor
by the natural action of the diffeomorphism group that it carries. Namely
There are several rigorous ways for defining the quotient of a Hilbert space by the unitary action
of a group. See in particular the construction in [34], which follows the ideas of Marolf and
Higuchi [194, 196, 197, 145].
In the quantum gravity literature, a big deal has been made of the problem that a scalar product is not
defined on the space of solutions of a constraint , defined on a Hilbert space
. This, however, is a
false problem. It is true that if zero is in the continuum spectrum of
, then the corresponding
eigenstates are generalized states and the
scalar product is not defined between them. But the
generalized eigenspaces of
, including the kernel, nevertheless inherit a scalar product from
. This
can be seen in a variety of equivalent ways. For instance, it can be seen from the following theorem. If
is self-adjoint, then there exists a measure
on its spectrum and a family of Hilbert spaces
such that
When factoring away the diffeomorphisms in the quantum-theory finite-dimensional moduli spaces
associated with high valence, nodes appear [137]. Because of these, the resulting Hilbert space is still
nonseparable. These moduli parameters, however, have no physical significance and do not play any role in
the quantum theory. They can be discarded by judicious choice of the functional space in which the fields
are defined [110, 299, 300], or in other ways [294].
The mathematical foundations of loop quantum gravity have been developed to the level of rigor of mathematical physics. This has introduced some heavy mathematical tools, sometimes unfamiliar to the average physicist, at the price of widening the language gap between scientists who study quantum gravity and other parts of the community. There is good reason for seeking a mathematical-physics level of precision in quantum gravity. In the development of conventional quantum field theory mathematical rigor could be low because extremely accurate empirical verifications assured physicists that “the theory may be mathematically meaningless, but it is nevertheless physically correct, and therefore the theory must make sense, even if we do not understand well how.” In quantum gravity this indirect experimental reassurance is lacking and the claim that the theory is well founded can be based only on a solid mathematical control. Given the unlikelihood of finding direct experimental corroboration, the research can only aim for the moment at the goal of finding a consistent theory, with correct limits in the regimes that we control experimentally. High mathematical rigor is the only assurance of the consistency of the theory. Quantum field theory on manifolds is an unfamiliar terrain in which the experience accumulated in conventional quantum field theory is often useless and sometimes misleading.
One may object that a rigorous definition of quantum gravity is a vain hope, given that we do not even have a rigorous definition of QED, presumably a much simpler theory. The objection is particularly valid from the point of view of a physicist who views gravity “just as any other field theory; like the ones we already understand”. But the (serious) difficulties of QED and of other conventional field theories are ultraviolet. The physical hope supporting the quantum gravity research program is that the ultraviolet structure of a diffeomorphism-invariant quantum field theory is profoundly different from the one of conventional theories. Indeed, recall that in a very precise sense there is no short distance limit in the theory; the theory naturally cuts itself off at the Planck scale, due to the very quantum discreteness of spacetime. Thus, the hope that quantum gravity could be defined rigorously may be optimistic, but it is not ill founded.
After these comments, let me briefly mention some of the structures that have been explored in .
First of all, the spin-network states satisfy the Kauffman axioms of the tangle theoretical version of
recoupling theory [162
] (in the “classical” case
) at all the points (in 3D space) where
they meet. For instance, consider a 4-valent node of four links colored
. The color
of the node is determined by expanding the 4-valent node into a trivalent tree; in this case,
we have a single internal link. The expansion can be done in different ways (by pairing links
differently). These are related to each other by the recoupling theorem of pg. 60 in Ref. [162
]
Since spin network states satisfy recoupling theory, they form a Temperley–Lieb algebra [162]. The
scalar product (11) in
is also given by the Temperley–Lieb trace of the spin networks, or,
equivalently by the Kauffman brackets, or, equivalently, by the chromatic evaluation of the spin
network.
Next, admits a rigorous representation as an
space, namely a space of square-integrable
functions. To obtain this representation, however, we have to extend the notion of connection, to a notion of
“distributional connection”. The space of the distributional connections is the closure of the space of
smooth connection in a certain topology. Thus, distributional connections can be seen as limits of sequences
of connections, in the same manner in which distributions can be seen as limits of sequences of
functions. Usual distributions are defined as elements of the topological dual of certain spaces
of functions. Here, there is no natural linear structure in the space of the connections, but
there is a natural duality between connections and curves in
: a smooth connection
assigns a group element
to every segment
. The group elements satisfy certain
properties. For instance if
is the composition of the two segments
and
, then
.
A generalized connection is defined as a map that assigns an element of
, which we denote
as
or
, to each (oriented) curve
in
, satisfying the following requirements: i)
; and ii)
, where
is obtained from
by reversing
its orientation,
denotes the composition of the two curves (obtained by connecting the end of
with the beginning of
) and
is the composition in
. The space of such
generalized connections is denoted
. The cylindrical functions
, defined in Section 6.3 as
functions on the space of smooth connections, extend immediately to generalized connections
Furthermore, can be seen as the projective limit of the projective family of the Hilbert
spaces
, associated to each graph
immersed in
.
is defined as the space
, where
is the number of links in
. The cylindrical function
is
naturally associated to the function
in
, and the projective structure is given by the natural
map (10
) [34, 198].
Finally, Ashtekar and Isham [27] have recovered the representation of the loop algebra by using
C*-algebra representation theory: the space , where
is the group of local
transformations (which acts in the obvious way on generalized connections), is precisely the
Gelfand spectrum of the Abelian part of the loop algebra. One can show that this is a suitable
norm closure of the space of smooth
connections over physical space, modulo gauge
transformations.
Thus, a number of powerful mathematical tools are at hand for dealing with nonperturbative quantum
gravity. These include Penrose’s spin network theory, representation theory, Kauffman tangle
theoretical recoupling theory, Temperley–Lieb algebras, Gelfand’s
algebra, spectral-representation
theory, infinite-dimensional measure theory and differential geometry over infinite-dimensional
spaces.
The definition of the theory is completed by giving the Hamiltonian constraint. A number of approaches to
the definition of a Hamiltonian constraint have been attempted in the past, with various degrees of success.
Thiemann has succeeded in providing a regularization of the Hamiltonian constraint that yields
a well-defined, finite operator. Thiemann’s construction [285, 289, 291] is based on several
clever ideas. I will not describe it here. Rather, I will sketch below in a simple manner the final
form of the constraint (for the Lapse = 1 case), following [252]. For a complete treatment,
see [294].
I begin with the Euclidean Hamiltonian constraint. We have
Here The coefficients , which are finite, can be expressed explicitly (but in a rather laborious
way) in terms of products of linear combinations of Wigner 6
symbols of
. The
Lorentzian Hamiltonian constraint is given by a similar expression, but quadratic in the
operators.
The operator defined above is obtained by introducing a regularized expression for the classical
Hamiltonian constraint, written in terms of elementary loop observables, turning these observables into
the corresponding operators and taking the limit. The construction works rather magically,
relying on the fact [267] that certain operator limits turn out to be finite on
diff-invariant states, thanks to the fact that, for
and
sufficiently small,
and
are diffeomorphic equivalent. Thus, here diff invariance plays again the crucial role in the
theory.
During the last years, Thomas Thiemann has introduced the idea of replacing the full set of quantum constraints with a single (“master”) constraint [293]. The development of this formulation of the dynamics is in progress.
Alternatively, the dynamics of the spin-network states can be defined via the spin-foam formalism. This is a covariant, rather than canonical, language, which can be used to define the dynamics of loop quantum gravity, in the same sense in which giving the covariant vertex amplitude defines the dynamics of the photon and electron states.
In three dimensions, the spin-foam formalism gives the well-known Ponzano–Regge model. Here the
vertex amplitude turns out to be given by an Wigner 6
symbol. The relation between this
model and loop quantum gravity has been pointed out long ago [248], and the complete equivalence has
been proven by Alejandro Perez and Karim Noui [210].
In four dimensions, an important role has been been played by the Barrett–Crane model, a
much-studied spin-foam theory constructed [97, 221] from the definition of the vertex as an
Wigner 10
symbol, given by Louis Crane and John Barrett in [66]. There are several difficulties in using
the Barrett–Crane model for defining the dynamics of 4D loop quantum gravity [53, 52]. First, the
Barrett–Crane model is formulated keeping local Lorentz invariance manifest, while loop quantum gravity is
not. This problem has been investigated by Sergei Alexandrov, see [4
, 5
, 7
] and references therein.
Second, the Barrett–Crane model appears to have fewer degrees of freedom than general relativity
on a spacelike surface, because it fixes the values of the intertwiners. More importantly, the
low-energy limit of the propagator defined by the Barrett–Crane model does not seem to be
correct [2
, 3
]. An important recent development, however, has been the introduction of a new vertex
amplitude, defined by the square of the
Wigner 15
symbol, which may correct all
these problems [106]. This has given rise to a rapid development [105
, 181, 182, 227] and
formulation of a class of spin-foam models that may provide a viable definition of the LQG
dynamics [104
, 114, 6], and are currently under intense investigation. Some preliminary results appear to
be encouraging [188].
What characterizes the spin-foam formalism is the fact that it can be derived in a surprising variety of
different ways, which all converge to essentially the same structure. The formalism can be obtained (i) from
loop gravity, (ii) as a quantization of a discretization of general relativity on a simplicial triangulation, (iii)
as a generalization of matrix models to higher dimensions, (iv) from a quantization of the elementary
geometry of tetrahedra and 4-simplices, and (v) by quantizing general relativity from its formulation as a
constrained BF theory, by imposing constraints on quantum topological theory. Each of these
derivations sheds some light on the formalism. For detailed introductions to the spin-foam formalism
see [50, 229, 230, 212, 64, 177]. A recent derivation as the quantization of a discretization of
general relativity is in [105, 104], which can also be seen as an independent derivation of the
loop-gravity canonical formalism itself. The PhD thesis of Daniele Oriti [213] is also a very good
introduction. Here I give only a simple heuristic description of the way spin foams appear from loop
gravity.
In his PhD thesis, Feynman introduced a path-integral formulation of quantum mechanics, deriving it from the canonical formalism. He considered a perturbation expansion for the matrix elements of the evolution operator
One can search for a covariant formulation of loop quantum gravity following the same idea [257, 236]. The matrix elements of the operator This is a consequence of the fact that the Hamiltonian operators acts only on nodes. The histories of
s-knots (abstract spin networks), evolving under such an action, are branched surfaces, which carry spins on
the faces (swept by the links of the spin network) and intertwiners on the edges (swept by the nodes of the
spin network). These have been called “spin foams” by John Baez [48], who has studied the general
structure of theories defined in this manner [49, 50].
Thus, the time evolution of a spin network, which generates spacetime, is given by a spin foam. The spin foam describes the evolving gravitational degrees of freedom. The formulation is “topological” in the sense that one must sum over topologically-nonequivalent surfaces only, and the contribution of each surface depends on its topology only. This contribution is given by the product of the amplitude of the elementary “vertices”, namely points where the edges branch.
The physical transition amplitude between two s-knot states and
turns out to be given by
summing over all (branching, colored) surfaces
that are bounded by the two s-knots
and
The contribution of each vertex is given by the matrix elements of the Hamiltonian constraint
operator between the two s-knots obtained by slicing
immediately below and immediately above the
vertex. They turn out to depend only on the colors of the surface components immediately
adjacent the vertex
. The sum turns out to be finite and explicitly computable order by
order.
As in the usual Feynman diagrams, the vertices describe the elementary interactions of the theory. Here,
in particular, one sees that the complicated structure of the Thiemann Hamiltonian, which makes a node
split into three nodes, corresponds to a geometrically very simple vertex. Figure 3 is a picture of the
elementary vertex. Notice that it represents nothing but the spacetime evolution of the elementary action of
the Hamiltonian constraint, given in Figure 2
. An example of a surface in the sum is given in
Figure 4
.
The resulting form of the covariant formulation of quantum gravity as a sum over spin-foam amplitudes is quite different from that of background-dependent quantum field theory.
Spin foams have a very nice geometric interpretation as the dual to a simplicial decomposition of spacetime. Vertices corresponding to four-volumes, edges to three-volumes, etc. This last point is a very elegant feature, and has an immediately intuitive explanation in terms of loop quantum gravity operators. The area operator counts lines in a spin network, corresponding to faces in a spin foam; the volume operator counts vertices in a spin network, corresponding to lines in a spin foam.
Finiteness of some spin-foam models at all orders of perturbation theory has been proven [228, 94, 93].
Very strictly related to the spin-foam language is an intriguing formalism that has been developing in recent years: group field theory. This has emerged from dual formulations of topological theories [76], and can be seen as a higher-dimensional version of the duality between matrix models and fluctuating geometries in 2D. Group field theories are standard quantum field theories defined over a group manifold, characterized by a peculiar nonlocal interaction term. They have a remarkable property: the Feynman expansion generates a sum over Feynman graphs that have a direct interpretation as a spin-foam model. In other words, the “discrete geometries” summed over can be seen as Feynman graphs of the group field theory.
Intuitively, the individual quanta of the group field theory can be seen as the quanta of space predicted by loop quantum gravity (see Section 7), and their Feynman histories make up spacetime.
The advantage of this formulation of quantum gravity is that it precisely fixes the sum over spin foams, and that it allows a number of theoretical tools from standard quantum field theory to be imported directly into the background independent formalism. In this sense, this approach has similarities with the philosophy of the Maldacena duality in string theory: a nonperturbative theory is dual to a more quantum field theory. But here there is no conjecture involved: the duality between certain spin-foam models and certain group field theories is a theorem.
The application of group field theory to quantum gravity, begun in the first attempts to frame the Barrett–Crane vertex into a complete theory [97]. It was later shown that any spin-foam model can be written as a group field theory [237, 238]. The subject of group field theory has greatly developed recently, and I refer to the reviews [113, 218, 216] for an update and complete references.
The coupling of fermions to the theory [206, 207, 54, 287] works easily. All the important results of the pure GR case survive in the GR + fermions theory. Not surprisingly, fermions can be described as open ends of “open-spin networks”.
The extension of the theory to the Maxwell field [169, 126] and Yang–Mills [290] also works smoothly.
Remarkably, the Yang–Mills term in the quantum Hamiltonian constraint can be defined in a rigorous
manner, extending the pure gravity methods, and ultraviolet divergences do not appear, strongly supporting
the expectation that the natural cutoff introduced by quantum gravity might cure the ultraviolet difficulties
of conventional quantum field theory. For an up-to-date account and complete references, see Thiemann’s
book [294].
The coupling of matter in the spin-foam and group field theory formalism is not yet clear, in spite of
considerable work in this direction. In the spin-foam formalism, an interesting line of investigation has
studied the coupling of particles in 3D, treated as topological defects [118, 119, 209, 117, 116, 220, 108].
Extensions of this formalism to 4D are considered in [59, 56]. On the coupling of gauge fields to spin foams,
see [219, 201, 281].
Matter couplings in the group field theory formalism have been studied especially in 3D. See [120, 215, 167, 109].
What I have describe above is the conservative and most widely used formulation of loop quantum gravity. A number of variants have appeared over the years. I sketch here some of those that are currently under investigation.
The common theme of these variants is to take seriously spacetime discreteness, which in loop quantum gravity is derived as a result of conservative quantization of continuum classical general relativity, and to use it, instead, as a starting point for the formulation of the fundamental theory.
The common point of view is then that the fundamental quantum theory will be discrete. There is no continuum limit in the sense in which lattice QCD has (presumably) a continuum quantum field-theory limit. Rather, states can approximate classical continuum GR.
A point of view on the dynamics somewhat intermediate between the canonical one and
the spin-foam one has been developed over the years by Fotini Markopoulou and Lee
Smolin [191, 192, 190]. See Smolin’s introduction [277] and complete references therein. The
idea is that, after having understood that quantum space can be described by a basis of
spin-network states, and that evolution happens discretely at the nodes, then the problem of
determining the dynamics is reduced to the study of the possible elementary “moves”, such as
the one that takes the graph in the left-hand side of Figure 1
into the one in the right-hand
side, and their amplitudes. Ideally, a space of background-independent theories is given by the
space of these possible moves and their amplitudes.
Markopoulou, Smolin and their collaborators have developed this point of view in a number of intriguing directions. See for instance [193]. I refer to Smolin’s review for an overview, and I mention here only one recent idea that I have found particularly intriguing. Together with T. Konopka, they have introduced in [165] a model where the degrees of freedom live on a complete graph (all nodes connected to one another) and the physics is invariant under the permutations of all the points. This opens up the possibility of the model having a “low-energy” phase in which physics on a low-dimensional lattice emerges, the permutation symmetry is broken to the translation group of that lattice, and a “high-temperature”, as well as a disordered, phase, where the permutation symmetry is respected and the average distance between degrees of freedom is small. This may serve as a paradigm for the emergence of classical geometry in background-independent models of spacetime. The appeal of the idea is its possible application in a cosmological scenario, in relation to the horizon problem. The horizon problem is generally presented as the puzzle raised by the fact that different parts of the universe appear to have emerged thermalized from an initial phase of the life of the universe in which the causal structure determined by classical general relativity has them causally disconnected. It seems to me natural to suspect that the problem is in the application of classical ideas to the very early universe, where, instead, quantum gravity dominates and there is no classical causal structure in place. The approach of Konopka, Markopoulou and Smolin gives a means to model a possibility of avoiding the horizon problem with a transition from the high-temperature phase, in which the points of the universe are all in direct causal connection, to the low-temperature phase, in which the classical causal structure gets established.
A different approach to the introduction of causality at the microscopic level in the spin-foam formalism is due to Daniele Oriti and Etera Livine [178, 179]; see also [214]. This includes the construction of new causal models, as well as the extraction of causal amplitudes from existing models.
Thomas Thiemann and Kristina Giesel have introduced a top down approach [136, 135] called
Algebraic Quantum Gravity (AQG). The quantum kinematics of AQG is determined by an
abstract algebra generated by a countable set of elementary operators labelled by a given
algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint
operator. AQG is inspired by loop quantum gravity; the difference is the absence of any
fundamental topology or differential structure in the setting up of the theory. This is quite
appealing, since in loop quantum gravity one has the strong impression that the topology and
the differential structure used in the setting up of the theory are residual of the classical limit,
namely of the “inverse-problem” way in which the theory is deduced from its classical limit,
and should play no real role in the fundamental theory. In AQG, the information about the
topology and differential structure of the spacetime manifold, as well as about the background
metric to be approximated, comes from the quantum state itself, when this state is a coherent
state that approximates a classical state.
Motivated by the same problems as those considered by Giesel and Thiemann in [134],
Campiglia, Di Bartolo, Gambini and Pullin have studied a lattice formulation of the
fundamental theory, denoted “uniform discretization” [82]. Though starting from different
points, the end prescription of this approach and AQG can, in some cases, be quite close in
practical implementation.
The uniform discretization is an evolution of the “consistent discretization” [131] approach,
studied by Rodolfo Gambini and Jorge Pullin. The key idea is that, when one uses connection
variables and holonomies, a natural regularization arena for theories is the lattice. The
introduction of a lattice in a theory like general relativity is a highly nontrivial affair. It
destroys at the most basic level the fundamental symmetry of the theory, the invariance under
diffeomorphisms. The consequences of this can be far reaching. If one just discretizes the
equations of general relativity, the resulting set of discrete equations is not even consistent and
the equations cannot be solved simultaneously (this is well known, for instance, in numerical
relativity, where “free evolution” schemes violate the constraints). The uniform (and the
earlier consistent) discretization approaches attempt to take these discrete theories seriously. In
particular, to recognize that they do not have the same symmetries as continuum GR, but have,
at best, “approximate symmetries”. Quantities that were constraints in the continuum theory
become evolution equations, and quantities that were Lagrange multipliers become dynamical
variables determined by evolution. In the simplest view, called consistent discretization, these
theories were analyzed directly as they were constructed by the discretization procedure [101].
In the “uniform discretization” [82] one constructs theories in which one controls precisely via
the initial data “how much they depart from the continuum” in the sense of how large the
constraints are.
The resulting theories are straightforward to quantize. Since the constraints are not identically zero, all the conceptual problems of having to enforce the constraints go away. One can construct relational descriptions of reality [124] and probe properties like how much does the use of real clocks and rulers destroy unitarity and entanglement in quantum theories [123, 125].
This approach has been explored in several finite-dimensional models [82], and is being studied in midi-superspace implementations, where it could yield a viable numerical quantum gravity approach to spherical gravitational collapse [83].
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