One of the key problems
in our era is the problem of quantum gravity- whether we can realize quantum
mechanics and general relativity (GR) as a single consistent theory.

One logical way of
getting a quantum theory of gravity would be to obtain a perturbation expansion
of canonical GR, and quantizing it using the rules in quantum field theory. But
we see that the terms of this perturbation expansion are divergent; in fact
there are infinitely many divergences 1, making it necessary to have infinitely
many counter-terms to be of any physical significance. This is the infamous non-renormalizability of gravity
and there are many intuitive arguments for the same 2. For example, Newton’s
gravitational constant multiplied by energy is used as a coupling constant in
GR. At high energies, coupling constant is also high which results in an
infinite number of divergent Feynman diagrams.

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To proceed from here, we
could assume GR is not a fundamental theory but a low energy limit of a more
general underlying theory, which leads to the approach of string theory. Or we
could assume perturbation expansion in Newton’s constant is not well defined,
but that GR can still be quantized non- perturbatively. This leads to the
approach of loop quantum gravity. LQG begins with GR and tries to integrate
quantum features, while string theory, begins with quantum field theory and tries
to add GR to its framework.

Non- perturbative
approach leads to a background independent theory. There are arguments that the
correct quantum theory of gravity must be background independent 3. Of
course, if a theory of quantum gravity pre-supposes GR to be valid, it (also)
must be background independent.

Formalism

Quantizing GRT without
using a perturbation expansion yields a plethora of difficulties. For
non-perturbative quantization of GR we assume the Einstein Hilbert action from
which GR is derived is exact and not a low-energy limit of a more general,
underlying theory. There are three constraint equations- the Hamiltonian, Gauss
and diffeomorphism that come from this approach.

 To describe GR in a canonical way; we separate
space and time with a method called 3+1 decomposition from which we get
Lagrangian density and the Hamiltonian density. The constraint equations follow
from the Hamiltonian density. This was done for the first time by Arnowitt,
Deser and Misner in 1962 4. This is the ADM formulation of General
Relativity. From here one can quantize gravity by calculating the poisson
bracket

 

And replacing the variables
by operators

The constraint equation
shows that Hamiltonian density

 at all times. The Schrödinger equation reduces
to

. From this we get the Wheeler DeWitt
equation

 

 

Where {displaystyle {hat {H}}(x)}

 is the Hamiltonian constraint in
quantized general relativity and

 stands for the wave function of the universe.
The equations turn out to highly singular and no known solutions to these
constraint equations exist so far. To overcome this, Abbhay Ashtekar,
introduced a new set of variables (Ashtekar variables) which turn the
constraint equations into simple polynomials.

The initial promise
that the Ashtekar variables would simplify the constraint equations was
lessened due to the necessity of introducing the Barbero-Immirzi parameter in
the new variables. When we choose the Barbero-Immirzi parameter to be complex,
it gives polynomial constraint equations. But this choice leads to a complex
phase space of GR. To obtain physical solutions reality conditions must be
imposed. In the classical case this is not a problem, but after quantizing the
theory it becomes a major problem to find such reality conditions. Thus this
complex form is abandoned.

Also, the numerical
value of the Barbero Immirzi parameter is currently fixed by matching the semi-classical black hole entropy, as calculated
by Stephen Hawking, and the counting of
microstates in loop quantum gravity. There are, however, no physical reasons
for this value.

In the complex form
when the theory is quantized, the metric is no longer a simple operator. This
is an indication that theories like LQG, which uses the Ashtekar variables to
quantize GR will have difficulties finding semi-classical states. Also the
change of the metric to ashtekar variables still does not yield any results.
Thus we need another change of variables- the loop representation.

It was found that
certain functionals, loops, annihilate the Hamiltonian constraint. They depend
only on the Ashtekar variables through the trace of the holonomy, a measure of
the change of the direction of a vector when its parallel transported over a
closed circle (a loop) i.e. they are SO(3)-invariant or rotational invariant.

Space can be viewed as networks
of loops called spin networks5. A spin network, as formulated by
Penrose 6 is a kind of diagram in which each line segment represents
the world line of a “unit”. Three line
segments join at each vertex. A vertex may be interpreted as an event in which
either a single unit splits into two or two units collide and join into a
single unit. 

More technically, a
spin network is a directed graph whose edges are
associated with irreducible representations of
a compact Lie group and
whose vertices are associated
with intertwiners of the edge representations
adjacent to it. A spin network, immersed into a manifold, can be used to define
a functional on the space
of connections on this
manifold. In fact a loop is a closed spin network (For example, certain
linear combinations of Wilson loops are called spin network states).

  The evolution of a spin network over
time is called a spin foam which is about the size of the Planck
length. 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. 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 LQG space and time
are quantized, it gives a
physical picture of spacetime where space and time are “granular”,
analogous to photons in quantum electrodynamics or discrete values of angular
momentum and energy in quantum mechanics. For example, quantization of areas: the operator
of the area A of a two-dimensional surface ? should have a
discrete spectrum. Every spin network is
an eigenstate of
each such operator, and the area eigenvalue equals

Where the sum goes over all intersections i of ? with
the spin network and

 is the Planck length

  is
the Immirzi parameter and

 = 0, 1/2, 1, 3/2,… is the spin associated
with the link i of the spin network. The two-dimensional area is
therefore “concentrated” in the intersections with the spin network.

According to this formula, the lowest possible non-zero eigenvalue
of the area operator corresponds to a link that carries spin 1/2
representation. Assuming an Immirzi
parameter on the order of 1, this gives the smallest possible measurable
area of ~10?66 cm2.

Criticism
for LQG

·       
LQG
might not get general relativity in the semi classical limit

·       
Possible
Lorentz violation

·       
Testability-
though this remains a problem for all existing theories of quantum gravity

 

Advantages of LQG

·       
It
is a background independent approach

·       
It
doesn’t invoke extra mathematical structures other than that dealt by quantum
field theory and general relativity

·       
LQG
naturally gives the discreteness of space time 

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