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.

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