Loop quantum gravity is a theory that
attempts to describe the quantum
properties of the universe and gravity.
It is also a theory of quantum spacetime
because, according to general
relativity, gravity is a manifestation
of the geometry of spacetime. LQG is an
attempt to merge quantum mechanics and
general relativity. The main output of
the theory is a physical picture of
space where space is granular. The
granularity is a direct consequence of
the quantization. It has the same nature
as the granularity of the photons in the
quantum theory of electromagnetism and
the discrete levels of the energy of the
atoms. Here, it is space itself that is
discrete. In other words, there is a
minimum distance possible to travel
through it.
More precisely, space can be viewed as
an extremely fine fabric or network
"woven" of finite loops. These networks
of loops are called spin networks. The
evolution of a spin network over time is
called a spin foam. The predicted size
of this structure is the Planck length,
which is approximately 10−35 meters.
According to the theory, there is no
meaning to distance at scales smaller
than the Planck scale. Therefore, LQG
predicts that not just matter, but space
itself, has an atomic structure.
Today LQG is a vast area of research,
developing in several directions, which
involves about 30 research groups
worldwide. They all share the basic
physical assumptions and the
mathematical description of quantum
space. The full development of the
theory is being pursued in two
directions: the more traditional
canonical loop quantum gravity, and the
newer covariant loop quantum gravity,
more commonly called spin foam theory.
Research into the physical consequences
of the theory is proceeding in several
directions. Among these, the most
well-developed is the application of LQG
to cosmology, called loop quantum
cosmology. LQC applies LQG ideas to the
study of the early universe and the
physics of the Big Bang. Its most
spectacular consequence is that the
evolution of the universe can be
continued beyond the Big Bang. The Big
Bang appears thus to be replaced by a
sort of cosmic Big Bounce.
History
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, and 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 spinfoam 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,
lead 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.
General covariance and background
independence
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 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. 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 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.
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.
Constraints and their Poisson bracket
algebra
= The constraints of classical canonical
general relativity=
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 , we have
Let us consider constrained systems, of
which General relativity is an example.
In a similar way the Poisson bracket
between a constraint and the phase space
variables generates a flow along an
orbit in phase space generated by the
constraint. There are three types of
constraints in Ashtekar's reformulation
of classical general relativity:
Gauss gauge constraints
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. These infinite number
of Gauss gauge constraints can be
smeared with test fields with internal
indices, ,
which we demand 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.
In fact 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.
Spatial diffeomorphisms constraints
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 .
Hamiltonian constraints
The Hamiltonian
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 and its
conjugate momentum is the triad . The
constraints are certain functions of
these phase space variables.
We consider the action of the
constraints on arbitrary phase space
functions. An important notion here is
the Lie derivative, , which is basically
a derivative operation that
infinitesimally "shifts" functions along
some orbit with tangent vector .
= The Poisson bracket algebra=
Of particular importance is the Poisson
bracket algebra formed between the
constraints themselves as it completely
determines the theory. In terms of the
above smeared constraints the constraint
algebra amongst the Gauss' law reads,
where . And so we see that the Poisson
bracket of two Gauss' law is equivalent
to a single Gauss' law evaluated on the
commutator of the smearings. The Poisson
bracket amongst spatial diffeomorphisms
constraints reads
and we see that 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 Lie algebra, with constraints , is of
the form
where  are constants. The above Poisson
bracket algebra for General relativity
does not form a true Lie algebra as we
have structure functions rather than
structure constants for the Poisson
bracket between two Hamiltonians. This
leads to difficulties.
= Dirac observables=
The constraints define a constraint
surface in the original phase space. The
gauge motions of the constraints apply
to all phase space but have the feature
that they leave the constraint surface
where it is, and thus the orbit of a
point in the hypersurface under gauge
transformations will be an orbit
entirely within it. Dirac observables
are defined as phase space functions, ,
that Poisson commute with all the
constraints when the constraint
equations are imposed,
that is, they are quantities defined on
the constraint surface that are
invariant under the gauge
transformations of the theory.
Then, solving only the constraint  and
determining the Dirac observables with
respect to it leads us back to the 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 can be
obtained by calculating the Poisson
brackets of the three-metric and its
conjugate momentum with a linear
combination of the spatial
diffeomorphism and Hamiltonian
constraint. The vanishing of the
constraints, giving the physical phase
space, are the four other Einstein
equations.
Quantization of the constraints – the
equations of quantum general relativity
= Pre-history and Ashtekar new
variables=
Many of the technical problems in
canonical quantum gravity revolve around
the constraints. Canonical general
relativity was originally formulated in
terms of metric variables, but there
seemed to be insurmountable mathematical
difficulties in promoting the
constraints to quantum operators because
of their highly non-linear dependence on
the canonical variables. The equations
were much simplified with the
introduction of Ashtekars new variables.
Ashtekar variables describe canonical
general relativity in terms of a new
pair canonical variables closer to that
of gauge theories. The first step
consists of using densitized triads  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). 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 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..
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
derivatives,
(analogous to ). In passing over to the
quantum theory the constraints become
operators on a kinematic Hilbert space.
Note that different ordering of the 's
and 's when replacing the 's with
derivatives give rise to different
operators - the choice made is called
the factor ordering and should be chosen
via physical reasoning. Formally they
read
There are still problems in properly
defining all these equations and solving
them. For example the Hamiltonian
constraint Ashtekar worked with was the
densitized version instead of the
original Hamiltonian, that is, he worked
with . There were serious difficulties
in promoting this quantity to a quantum
operator. Moreover, although Ashtekar
variables had the virtue of simplifying
the Hamiltonian, they are complex. When
one quantizes the theory, it is
difficult to ensure that one recovers
real general relativity as opposed to
complex general relativity.
= Quantum constraints as the equations
of quantum general relativity=
We now move on to demonstrate an
important aspect of the quantum
constraints. We consider Gauss' law
only. First we state the classical
result that 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 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  and looking for the gauge
orbits 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. Recall that, at the
classical level, solving the
admissibility conditions and evolution
equations was equivalent to solving all
of Einstein's field equations, this
underlines the central role of the
quantum constraint equations in
canonical quantum gravity.
= Introduction of the loop
representation=
It was in particular the inability to
have good control over the space of
solutions to the Gauss' law and spacial
diffeomorphism constraints that led
Rovelli and Smolin to consider a new
representation - the loop representation
in gauge theories and quantum gravity.
We need the notion of a holonomy. A
holonomy is a measure of how much the
initial and final values of a spinor or
vector differ after parallel transport
around a closed loop; it is denoted
Knowledge of the holonomies is
equivalent to knowledge of the
connection, up to gauge equivalence.
Holonomies can also be associated with
an edge; under a Gauss Law these
transform as
For a closed loop  if we take the trace
of this, that is, putting  and summing
we obtain
or
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. To
handle the spatial diffeomorphism
constraint we need to go over to the
loop representation. As Wilson loops
form a basis we can formally expand any
Gauss gauge invariant function as,
This is called the loop transform. We
can see the analogy with going to the
momentum representation in quantum
mechanics(see Position and momentum
space). There one has a basis of states 
labelled by a number  and one expands
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
or
where by  we mean the operator  but with
the reverse factor ordering. We evaluate
the action of this operator on the
Wilson loop as a calculation in the
connection representation and
rearranging the result as a manipulation
purely in terms of loops. 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 we can then
solve the spatial diffeomorphism
constraint by considering functions of
loops  that are invariant under spatial
diffeomorphisms of the loop . That is,
we construct what mathematicians call
knot invariants. This opened up an
unexpected connection between knot
theory and quantum gravity.
What about the Hamiltonian constraint?
Let us go back to the connection
representation. Any collection of
non-intersecting Wilson loops satisfy
Ashtekar's quantum Hamiltonian
constraint. This can be seen from the
following. With 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 we have something like
However, as  is anti-symmetric in the
indices  and  this vanishes. Now let us
go back to the loop representation.
We consider wavefunctions  that vanish
if the loop has discontinuities and that
are knot invariants. Such functions
solve the Gauss law, the spatial
diffeomorphism constraint and the
Hamiltonian constraint. Thus we have
identified an infinite set of exact
solutions to all the equations of
quantum general relativity! This
generated a lot of interest in the
approach and eventually led to LQG.
= Geometric operators, the need for
intersecting Wilson loops and spin
network states=
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,
From this we get the area of the surface
to be given by
where  and is the determinant of the
metric induced on . This can be
rewritten as
The standard formula for an inverse
matrix is
Note the similarity between this and the
expression for . But in Ashtekar
variables we have . Therefore
According to the rules of canonical
quantization we should promote the
triads  to quantum operators,
It turns out that the area  can be
promoted to a well defined quantum
operator despite the fact that we are
dealing with product of two functional
derivatives and worse we have a
square-root to contend with as well.
Putting , we talk of being in the -th
representation. We note that . This
quantity is important in the final
formula for the area spectrum. We simply
state the result below,
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. As we
take the derivative, and each time we do
so we bring down the tangent vector ,
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 we would
need at least intersections with three
non-coplanar lines. Actually it turns
out that one needs at least four-valent
vertices for the volume operator to be
non-vanishing.
We now consider Wilson loops with
intersections. We assume the real
representation where the gauge group is
. Wilson loops are an over complete
basis as there are identities relating
different Wilson loops. These come about
from the fact that Wilson loops are
based on matrices and these matrices
satisfy identities. Given any two 
matrices  and  it is easy to check that,
This implies that given two loops  and 
that intersect, we will have,
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 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 and
actually constitute a basis for all
gauge invariant functions.
As mentioned above the holonomy tells
you 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.
= Real variables, modern analysis and
LQG=
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 as actually  and not . As 
is non-compact it creates serious
problems for the rigorous construction
of the necessary mathematical machinery.
The group  is on the other hand is
compact and the relevant constructions
needed have been developed.
As mentioned above, because Ashtekar's
variables are complex it results in
complex general relativity. 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 determined by
the desitized triad. The expression for
compatible connection  is rather
complicated and as such non-polynomial
formula enters through the back door.
Before we state the next difficulty we
should give a definition; a tensor
density of weight  transforms like an
ordinary tensor, except that in
additional the th power of the Jacobian,
appears as a factor, i.e.
It turns out that 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, 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. 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. Also we still
have 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. 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
We are then able to write
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 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.
Finding the states that are annihilated
by these constraints, 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. 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.
= Solving the quantum constraints=
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  with very good
convergence properties and consider its
dual space , then . 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. 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.
Let us move to LQG, additional
complications will arise from the fact
the constraint algebra is not a Lie
algebra due to the bracket between two
Hamiltonian constraints.
The Gauss law is solved by the use of
spin network states. They provide a
basis for the Kinematic Hilbert space .
The spatial diffeomorphism constraint
has been solved. The induced inner
product on  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.
The Hamiltonian constraint maps
diffeomorphism invariant states onto
non-diffeomorphism invaiant 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 you can't just solve the
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 .
Spin foams
In loop quantum gravity, a spin network
represents a "quantum state" of the
gravitational field on a 3-dimensional
hypersurface. The set of all possible
spin networks 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
description of quantum gravity. It is
closely related to loop quantum gravity.
= Spin foam derived from the Hamiltonian
constraint operator=
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. It is obvious that
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 projector can formally
be written as
Interestingly, 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 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". 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 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
projector. The master constraint does
not suffer from these problems and as
such offers a way of connecting the
canonical theory to the path integral
formulation.
= Spin foams from BF theory=
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 to 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 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.
= Modern formulation of spin foams=
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.
Much progress has been made with regard
to this issue by Engle, Pereira, and
Rovelli and Freidal and Krasnov in
defining spin foam interaction
amplitudes with much better behaviour.
An attempt to make contact between
EPRL-FK spin foam and the canonical
formulation of LQG has been made.
= Spin foam derived from the master
constraint operator=
See below.
The semi-classical limit
= What is the semiclassical limit?=
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 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.
The principle was formulated by Niels
Bohr in 1920, though he had previously
made use of it as early as 1913 in
developing his model of the atom.
There are two basic requirements in
establishing the semi-classical limit of
any quantum theory:
i) reproduction of the Poisson brackets.
This is extremely important because, as
noted above, the Poisson bracket algebra
formed between the 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 semi-classical states
reproduce the same classical variables
with small quantum corrections.
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.
= Why might LQG not have general
relativity as its semiclassical limit?=
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 you do
not recover general relativity in the
semi classical limit.
Theorems establishing the uniqueness of
the loop representation as defined by
Ashtekar et al. have been given by two
groups. 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.
= Difficulties checking the
semiclassical limit of LQG=
There are difficulties in trying to
establish LQG gives Einstein's theory of
general relativity in the semi classical
limit. There are a number of particular
difficulties in establishing the
semi-classical limit
There is no operator corresponding to
infinitesimal spacial diffeomorphisms.
Instead it must be approximated by
finite spatial diffeomorphisms and so
the Poisson bracket structure of the
classical theory is not exactly
reproduced. This problem can be
circumvented with the introduction of
the so-called master constraint
There is the problem of reconciling the
discrete combinatorial nature of the
quantum states with the continuous
nature of the fields of the classical
theory.
There are serious difficulties arising
from the structure of the Poisson
brackets involving the spatial
diffeomorphism and Hamiltonian
constraints. In particular, the algebra
of Hamiltonian constraints does not
close, it is proportional to a sum over
infinitesimal spatial diffeomorphisms
where the coefficients of
proportionality are not constants but
have non-trivial phase space dependence
- as such it does not form a Lie
algebra. However, the situation is much
improved by the introduction of the
master constraint.
The semi-classical machinery developed
so far is only appropriate to
non-graph-changing operators, however,
Thiemann's Hamiltonian constraint is a
graph-changing operator - the new graph
it generates has degrees of freedom upon
which the coherent state does not depend
and so their quantum fluctuations are
not suppressed. There is also the
restriction, so far, that these coherent
states are only defined at the Kinematic
level, and now one has to lift them to
the level of  and . It can be shown that
Thiemann's Hamiltonian constraint is
required to be graph changing in order
to resolve problem 3 in some sense. The
master constraint algebra however is
trivial and so the requirement that it
be graph changing can be lifted and
indeed non-graph changing master
constraint operators have been defined.
Formulating observables for classical
general relativity is a formidable
problem by itself because of its
non-linear nature and space-time
diffeomorphism invariance. In fact a
systematic approximation scheme to
calculate observables has only been
recently developed.
Difficulties in trying to examine the
semi classical limit of the theory
should not be confused with it having
the wrong semi classical limit.
= Progress in demonstrating LQG has the
correct semiclassical limit=
Much details here to be written up...
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 a 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. But probed at a
macroscopic scale, it appears as a
three-dimensional continuous metric
geometry.
As far as the editor knows problem 4 of
having semi-classical machinery for
non-graph changing operators is as 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 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.
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.
Improved dynamics and the master
constraint
= The master constraint=
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 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. Hence its
Poisson bracket with the spacial
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 semi-classical 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 Mater constraint is
quadratic in the constraint, when you
compute 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
= The quantum master constraint=
Define the quantum master constraint as
Obviously,
for all  implies . Conversely, if  then
implies
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 . The fact that the
master constraint operator  is densely
defined on , it is obvious that  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..
It is also possible to construct a
quadratic form  for what is called the
extended master constraint on  which
also involves the weighted integral of
the square of the spatial diffeomorphism
constraint.
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 .
= Testing the master constraint=
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. Squaring
the constraint is dangerous as it could
lead to worsened ultraviolent 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.
= Applications of the master constraint=
The master constraint has been employed
in attempts to approximate the physical
inner product and define more rigorous
path integrals.
The Consistent Discretizations approach
to LQG, is an application of the master
constraint program to construct the
physical Hilbert space of the canonical
theory.
= Spin foam from the master constraint=
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 semi-classical
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.
= Algebraic quantum gravity=
The master constraint programme has
evolved into a fully combinatorial
treatment of gravity known as Algebraic
Quantum Gravity. 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. See Thiemann's book for
details.
Physical applications of LQG
= Black hole entropy=
The 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 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 semi-classical 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. 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 the correct relation between
energy and area, 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.
= Loop quantum cosmology=
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 is a
symmetry-reduced model of classical
general relativity quantized using
methods that mimic those of loop quantum
gravity 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.
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.
= Loop quantum gravity phenomenology=
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.
= Background independent scattering
amplitudes=
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) 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.
Gravitons, string theory, supersymmetry,
extra dimensions in LQG
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 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 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. Peter Woit in Not
Even Wrong and Lee Smolin in The Trouble
with Physics regard string/M-theory to
be in conflict with current known
experimental results.
Since LQG has been formulated in 4
dimensions, 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 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.
As of April 2013 LHC has failed to find
evidence of supersymmetry or
Kaluza–Klein extra dimensions, which has
encouraged LQG researchers. Shaposhnikov
in his paper "Is there a new physics
between electroweak and Planck scales?"
has proposed the neutrino minimal
standard model, which claims the most
parsimonious theory is a standard model
extended with neutrinos, plus gravity,
and that extra dimensions, GUT physics,
and supersymmetry, string/M-theory
physics are unrealized in nature, and
that any theory of quantum gravity must
be four dimensional, like loop quantum
gravity.
LQG and related research programs
Several research groups have attempted
to combine LQG with other research
programs: Johannes Aastrup, Jesper M.
Grimstrup et al. research combines
noncommutative geometry with loop
quantum gravity, 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 LQG has also 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. In addition to
what has already mentioned above, 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.
Problems and comparisons with
alternative approaches
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.
Can quantum mechanics and general
relativity be realized as a fully
consistent theory? Is spacetime
fundamentally continuous or discrete?
Would a consistent theory involve a
force mediated by a hypothetical
graviton, or be a product of a discrete
structure of spacetime itself? Are there
deviations from the predictions of
general relativity at very small or very
large scales or in other extreme
circumstances that flow from a quantum
gravity theory?
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. Specifically, the
dynamics of the theory is 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.
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. 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.
See also
Notes
References
Topical Reviews
Rovelli, Carlo. "Zakopane lectures on
loop gravity". arXiv:1102.3660. 
Rovelli, Carlo. "Loop Quantum Gravity".
Living Reviews in Relativity 1.
Retrieved 2008-03-13. 
Thiemann, Thomas. "Lectures on Loop
Quantum Gravity". Lectures Notes in
Physics. Lecture Notes in Physics 631:
41–135. arXiv:gr-qc/0210094.
Bibcode:2003LNP...631...41T.
doi:10.1007/978-3-540-45230-0_3. ISBN
978-3-540-40810-9. 
Ashtekar, Abhay; Lewandowski, Jerzy.
"Background Independent Quantum Gravity:
A Status Report". Classical and Quantum
Gravity 21: R53–R152.
arXiv:gr-qc/0404018.
Bibcode:2004CQGra..21R..53A.
doi:10.108821R01. 
Carlo Rovelli and Marcus Gaul, Loop
Quantum Gravity and the Meaning of
Diffeomorphism Invariance, e-print
available as gr-qc/9910079.
Lee Smolin, The case for background
independence, e-print available as
hep-th/0507235.
Alejandro Corichi, Loop Quantum
Geometry: A primer, e-print available as
[1].
Alejandro Perez, Introduction to loop
quantum gravity and spin foams, e-print
available as [2].
Hermann Nicolai and Kasper Peeters Loop
and spin foam quantum gravity: A Brief
guide for beginners., e-print available
as [3].
Popular books:
Lee Smolin, Three Roads to Quantum
Gravity
Carlo Rovelli, Che cos'è il tempo? Che
cos'è lo spazio?, Di Renzo Editore,
Roma, 2004. French translation: Qu'est
ce que le temps? Qu'est ce que
l'espace?, Bernard Gilson ed, Brussel,
2006. English translation: What is Time?
What is space?, Di Renzo Editore, Roma,
2006.
Julian Barbour, The End of Time: The
Next Revolution in Our Understanding of
the Universe
Musser, George. "The Complete Idiot's
Guide to String Theory". The Physics
Teacher 47: 368.
Bibcode:2009PhTea..47Q.128H.
doi:10.1119/1.3072469. ISBN
978-1-59257-702-6.  – Focuses on string
theory but has an extended discussion of
loop gravity as well.
Magazine articles:
Lee Smolin, "Atoms of Space and Time",
Scientific American, January 2004
Martin Bojowald, "Following the Bouncing
Universe", Scientific American, October
2008
Easier introductory, expository or
critical works:
Abhay Ashtekar, Gravity and the quantum,
e-print available as gr-qc/0410054
John C. Baez and Javier Perez de
Muniain, Gauge Fields, Knots and Quantum
Gravity, World Scientific
Carlo Rovelli, A Dialog on Quantum
Gravity, e-print available as
hep-th/0310077
Rodolfo Gambini and Jorge Pullin, A
First Course in Loop Quantum Gravity,
Oxford
Carlo Rovelli and Francesca Vidotto,
Covariant Loop Quantum Gravity,
Cambridge; draft available online
More advanced introductory/expository
works:
Carlo Rovelli, Quantum Gravity,
Cambridge University Press; draft
available online
Thomas Thiemann, Introduction to modern
canonical quantum general relativity,
e-print available as gr-qc/0110034
Thomas Thiemann, Introduction to Modern
Canonical Quantum General Relativity,
Cambridge University Press
Abhay Ashtekar, New Perspectives in
Canonical Gravity, Bibliopolis.
Abhay Ashtekar, Lectures on
Non-Perturbative Canonical Gravity,
World Scientific
Rodolfo Gambini and Jorge Pullin, Loops,
Knots, Gauge Theories and Quantum
Gravity, Cambridge University Press
Hermann Nicolai, Kasper Peeters, Marija
Zamaklar, Loop quantum gravity: an
outside view, e-print available as
hep-th/0501114
H. Nicolai and K. Peeters, Loop and Spin
Foam Quantum Gravity: A Brief Guide for
Beginners, e-print available as
hep-th/0601129
T. Thiemann The LQG – String: Loop
Quantum Gravity Quantization of String
Theory
Conference proceedings:
John C. Baez, Knots and Quantum Gravity
Fundamental research papers:
Ashtekar, Abhay. "New variables for
classical and quantum gravity". Physical
Review Letters 57: 2244–2247.
Bibcode:1986PhRvL..57.2244A.
doi:10.1103/PhysRevLett.57.2244. PMID
10033673 
Ashtekar, Abhay. "New Hamiltonian
formulation of general relativity".
Physical Review D 36: 1587–1602.
Bibcode:1987PhRvD..36.1587A.
doi:10.1103/PhysRevD.36.1587 
Roger Penrose, Angular momentum: an
approach to combinatorial space-time in
Quantum Theory and Beyond, ed. Ted
Bastin, Cambridge University Press, 1971
Rovelli, Carlo; Smolin, Lee. "Knot
theory and quantum gravity". Physical
Review Letters 61: 1155–1158.
Bibcode:1988PhRvL..61.1155R.
doi:10.1103/PhysRevLett.61.1155. 
Rovelli, Carlo; Smolin, Lee. "Loop space
representation of quantum general
relativity". Nuclear Physics B331:
80–152. 
Carlo Rovelli and Lee Smolin,
Discreteness of area and volume in
quantum gravity, Nucl. Phys., B442
593-622, e-print available as
gr-qc/9411005
Kuchař, Karel. "Canonical Quantization
of Gravity". In Israel, Werner.
Relativity, Astrophysics and Cosmology.
D. Reidel. pp. 237–288. ISBN
90-277-0369-8. 
Thiemann, Thomas. "Loop Quantum Gravity:
An Inside View". Approaches to
Fundamental Physics. Lecture Notes in
Physics 721: 185–263.
arXiv:hep-th/0608210.
Bibcode:2007LNP...721..185T.
doi:10.1007/978-3-540-71117-9_10. ISBN
978-3-540-71115-5. 
External links
"Loop Quantum Gravity" by Carlo Rovelli
Physics World, November 2003
Quantum Foam and Loop Quantum Gravity
Abhay Ashtekar: Semi-Popular Articles .
Some excellent popular articles suitable
for beginners about space, time, GR, and
LQG.
Loop Quantum Gravity: Lee Smolin.
Loop Quantum Gravity on arxiv.org
A list of LQG references catered to
fresh graduates
Loop Quantum Gravity Lectures Online by
Lee Smolin
Spin networks, spin foams and loop
quantum gravity
Wired magazine, News: Moving Beyond
String Theory
April 2006 Scientific American Special
Issue, A Matter of Time, has Lee Smolin
LQG Article Atoms of Space and Time
September 2006, The Economist, article
Looping the loop
Gamma-ray Large Area Space Telescope:
http:glast.gsfc.nasa.gov/
Zeno meets modern science. Article from
Acta Physica Polonica B by Z.K.
Silagadze.
Did pre-big bang universe leave its mark
on the sky? - According to a model based
on "loop quantum gravity" theory, a
parent universe that existed before ours
may have left an imprint
