Loop Quantum Gravity
To describe the world around us different kinds of physical theories are used.
The everyday world we are used to is described by mechanics that uses concepts
like mass, velocity, force and energy. How the position of an object changes over
time is described by Newton’s equations. Electrical phenomena are described by
classical electrodynamics that uses electric and magnetic fields that has strength
and direction in every point of space and time. The evolution of the fields are
described by Maxwell’s equations. These theories are called classical. The state of the system
is described by a set of values from a continuous range and they evolve deterministically as
described by differential equations. The energy of an object or the strength
of a field can be as small as you want and in principle you can measure everything with
arbitrary precision.
To describe the world at very small scales these theories are no good. The discrete energy
levels of an atom and the spectral lines associated to each substance need new theories.
The answer is quantum mechanics ruled by the Schrödinger equation. Quantum mechanics
introduce randomness in a fundamental way and certain quantities like angular momentum have a minimal value.
Discreteness in the allowed range of values is a central part of quantum mechanics and there is a
new constant of nature. The Planck constant ħ=1.055∙10^{-34} Nm is responsible for the size of
the discrete chunks that replaces the previous idea of a continuous range of values for everything
that can be measured.
When quantum mechanics evolved in the beginning of the 20^{th} century another theory challenged
the old theories. It was Einstein’s special and general theory of relativity. The special
theory of relativity introduced a maximum relative speed between objects, the speed of light
in vacuum, c=299792458 m/s. As a consequence of this speed limit, space and time had to be united
into a new concept, spacetime. Observers in different frames of reference could no longer agree on certain measures,
distances in space and separations in time were frame-dependent. What could be agreed upon
was the distance in spacetime. Space and time were no longer separate they had merged in a
4-dimensional spacetime. When quantum mechanics is combined with special relativity you get a
quantum field theory. Of the four fundamental forces in nature three have successfully been
described by quantum field theories. The electric, weak and strong forces are now part of the
standard model of particle physics.
The remaining force, gravity, is described by general relativity. This is what you get from special
relativity when you want each frame of reference to be treated on an equal footing. Gravity creeps
in when you realize that the effect of gravity is the same as being in an accelerated frame of
reference. Gravity too is associated with a fundamental constant, the gravitational constant
G=6.673∙10^{-11} Nm^{2}/kg^{2}. General relativity is still a classical theory since the
metric field that describes spacetime is classical to its nature, no discrete chunks and no
randomness. A quantum mechanical description of gravity would break up the spacetime continuum at
distances smaller than the Planck length l_{p}=√(ħG/c^{3})=1.62∙10^{-35} m
and at times smaller than the Planck time t_{p}=√(ħG/c^{5})=5.39∙10^{-44} s.
To describe what happens when a lot of mass is concentrated in a very small region you need a theory
that unites general relativity and quantum mechanics. To really understand a black hole and
the first stages in the evolution of the universe you need a quantum theory of gravity.
The most ambitious project to unify gravity and quantum theory is superstring theory that combines
all four forces in a quantum-field-theoretical framework. Instead of treating elementary particles
as pointlike they are described by strings or loops. One disadvantage of the theory is that the strings
still move in a continuous spacetime background, the very thing that you wanted to replace with
something more discrete and random. The theory is said to be background dependent.
Loop Quantum Gravity (LQG) is another theory that unifies general relativity and quantum mechanics.
It does not incorporate the other forces in a unified theory but it is background independent.
The spacetime in its most basic form is random and discrete on the Planck scale. The loop part
of LQG comes from spacetime being built from small “spacetime-atoms” in the form of loops where
all spacetime curvature is concentrated. How the loops interconnect decides the number of dimensions
and when many loops form a web there will emerge a classical curved spacetime.
The evolution of the network (spin foam) is described by methods from QFT,
Feynman path integrals. I will not go into any detail on this but refer the interested reader
to the listed articles and books. The first reference is popular enough for readers with a basic knowledge
of physics.
Loop Quantum Gravity, Physics World, November 2003
Covariant Loop Quantum Gravity by Carlo Rovelli and Francesca Vidotto
Quantum Gravity by Carlo Rovelli, Cambridge monographs on mathematical physics
In the autumn of 2013 I took a course in cosmology and astroparticle physics. Part of the course
was to hold a short lecture on a chosen topic. I chose Loop Quantum Cosmology, which basically
is LQG applied to cosmology and the universe at the very beginning. The powerpoint presentation
can be seen if you choose it from the menu above. I was also inspired to illustrate Loop Quantum
Gravity with some artwork. My interconnected rings shown below illustrate how you can build
three-dimensional flat space and two-dimensional curved space just by interconnecting “atomic loops
of curvature” in a certain way. The first picture shows the material and tools I used.
The rings do not incorporate the time aspect of LQG but it does give the
general idea of building curved or flat space from discrete units without any background space.
There is no physical meaning in the surrounding space. It is only how the rings are interconnected
that decides the dimension and the gravitational field.