QUANTUM GRAVITY
Conflicts and Inconsistencies
QM requires relationships and interactions between particles to take place on
some independent background lattice of space and time, this is known as ‘background
dependence’. For example any two particles can be described by two sets of three
parameters, the set of three parameters being special coordinates (say x, y, and z) that
unambiguously determine their position or point in space (but not simultaneously with
momentum), this is a Newtonian perspective in regards to a coordinate system. In GR
however the only meaning of a point or position in space is a given network of
relationships between interacting gravitational field lines with matter interacting with
the gravitational field lines to change the network of relationships. Furthermore GR
has no part of GR restricts knowing simultaneously both the precise position in spacetime and momentum of a particle.
LOOP QUANTUM GRAVITY
Loop Quantum Gravity (LQG) is a theory that attempts to resolve some of the
inconsistencies that arise with the unification of GR and QM such as the background
dependence of QM. Its basic function to date is a development of a quantised
description of space-time or gravity. Mathematically it is a framework for a nonperturbative quantisation of diffeomorphism-invariant gauge theories.
The theory was developed in collaboration with many physicists and mathematicians
but the first presentations of the theory were by Lee Smolin and Carlo Rovelli where
Smolin had been tackling the problem for many years.1
The theory implies that any state of space-time is characterised by the interaction of
loops or knots. These states are known as spin states (not to be confused spin in QM).
Spin states can be further described by spin networks by a method developed by
Roger Penrose,1 and give the possible quantum states for the geometry of space time.
fig 1.1 an example of a spin state (only some numbers are shown)
A spin networks can be represented graphically, as above, the lines represent elements
of area where the integers represent spins, and the nodes (dots) represent elements of
volume. Below are representations of spin networks evolving with time, it must be
emphasised that the evolution is quantised the networks do not slowly or continuously
morph into a new state but rather, on the plank time scale are one state, than the other,
this is why LQG is a quantized description of space time.
fig 1.2 spin networks quantized evolution in time
Spin states are solutions of the Wheeler-DeWitt equation, which plays an analogous
role to the Schrödinger equation in QM. The Wheeler-DeWitt equation is given by2
 d2
9 2
 2 
4G 2 k 2
 da
 2 8G
4 
a  3k    vac a  S  0

where S is a spin state. The equation is given in the minisuperspace approximation for
arbitrary k and with matter or radiation (ρ term) included.
1
L. Smolin, Three Roads To Quantum Gravity, Weidenfeld & Nicolson, Great Briain, 2000, 134.
J. W. Norberry, From Newton’s Laws To Wheeler-DeWitt Equation,
http://arxiv.org/PS_cache/physics/pdf/9806/9806004.pdf, 2005.
2
Thus the spin states are analogous to the role wavefunctions play in QM, except that
spin states describe states of space-time and not particles. Furthermore spin states are
eigenfunctions of area and volume operators, where the eigenvalues of the area and
volume operators are quantized values of area and volume respectively e.g.
2
Aˆ  S  nhl S
where we have the area operator acting over some surface ∑ on a spin state S giving
an area eigenvalue which is some multiple of the plank length squared.
The main physical implications of LQG are that time, area and volume like energy
and electromagnetic radiation in QM are quantised, thus space-time is quantised. The
fundamental quantity of area and volume are the plank length squared and the plank
length cubed respectively.
The plank length hl is given by3
hl  Gc 3
The plank length is extremely small compared with our everyday experience ~10-35m
and plank time unit is ~10-43 seconds.
At present it is unclear whether on macroscopic scales space-time appears to be
continuous which would be some support for the theory. The theory has not been
shown to predict the appearance of gravitons, however it has been suggested that
properties of gravitons may be inherent in the present theory.
Support for LQG as a correct description of space-time are that it leads to derivations
of the black hole entropy formula and also predicts Hawking radiation which are not
features of string theory (ST).
LQG is not a theory particles and their interactions like ST but rather is more simply a
theory of the structure of space-time. It has been proposed that the space-time
structural aspects of LQG and the fundamental particle interaction aspects ST could
be combined to formulate a theory of everything (TOE).
NOTES TO THE GROUP:
Limitations of the theory…etc need to be explained yet because so far there is no
reference to particles and their interactions only the structure of space time, this is
where string theory comes in. Smolin suggests that some aspects of string theory and
some aspects of quantum gravity will both be needed to explain a theory of
everything, but this is beyond the scope of this our project we are focussing on
quantum gravity.
3
R. Penrose, The Emperor’s New Mind, Oxford University Press, New York, 1990, 451.
Underlined words I am hoping can be used as links to the other areas of the web page
that are explained in more detail in your sections.
Also I have references to the information I sourced, this is probably really important
so I sent both the html and word files since references did not show up in the html.
If there are any suggestions I would really like and appreciate them don’t be scared to
be critical as I am unsure what balance I should make with the math and references to
other sections of the site, it will need to look coherent.
I have added a little logo as a fun little thing and put it in a few places to see what it
looks like, what do you guys think? I suggest it could be a theme throughout the page
in order to help give a gelled appearance.