quantum setting. Following WWII, rapid discoveries of a whole
zoo of new elementary particles led theorists to postulate the
existence of two more forces of nature - in addition to the already
known electromagnetic and gravitational forces - known as the
“weak” and “strong” forces. Theoretical work by giants of 20th
century physics, such as Feynman, Gell-Mann and Gerard ’t Hooft
among many others, led to a unified description of the weak force
and electromagnetism in a framework known as the “electroweak”
theory. This process culminated in the last quarter of the 20th
century with the establishment of the Standard Model of particle
physics which provides a unified - albeit, in some ways flawed
- description of the weak, strong and electromagnetic forces
as excitations of the quantum mechanical “vacuum”. Gravity,
however, remains outside the grasp of such unified frameworks.
Consequently we are left in the awkward situation where the most
complete formulation of physical law must be written in the form:
Standard Model + Gravity.
Superpositions of Geometry
The challenge in unifying gravity with quantum mechanics can be
understood in the following way. The central feature of quantum
mechanics is the principle of superposition; that the wavefunctions which describe two different particles or systems can
overlap. Consequently two systems described by two different
wavefunctions
and
can instead be treated as a
single composite system with wavefunction
Wavefunctions are just that - functions - and must therefore be
defined on some set. In quantum mechanics the set is taken to be
the co-ordinates (x,t) of the spacetime our system is embedded in.
So we can always write down what the wavefunction of a particle
passing through two slits at the same time must look like. Similarly
we can write down the superposition of two particles in terms of
their momenta, rather than their position, by working in the (p,t)
basis instead of the (x,t) basis with the momenta p being related
to the position x by the usual fourier transform:
Regardless of whether we work in the position basis or the
momentum basis, there is an implicit assumption at work in this
prescription - the assumption of a flat background geometry for
which we can assign a set of co-ordinates (x,y,z,t) to each point of
the spacetime.
Now, General Relativity teaches us that physics should be independent of the particular co-ordinates used to describe a system.
Moreover, any theory which is consistent with GR must also be
well-defined both on curved space as on flat space. It turns out
that while we can perform quantum mechanical calculations to
our heart’s content with wavefunctions defined on flat space, the
case of wavefunctions living on curved space becomes tricky.
The complications associated with spacetime curvature can be
dealt with by resorting to sufficiently sophisticated mathematical
methods. The resulting framework is known as Quantum Field
Theory on Curved Spacetime (QFT-CS) – not a terribly memorable phrase. It was by using the methods of QFT-CS that Stephen
Hawking obtained his historic result that a blac k hole must emit
thermal radiation at a rate inversely proportional to its mass.
However, even QFT-CS does not qualify as a theory of “quantum
gravity”, as explained next.
Quantum Mechanics is about assigning various attributes to a
system and then constructing states of the system corresponding
to each attribute. These states can be superimposed and the result-
38
The Shoreline
ing system will manifest all non-intuitive phenomena associated
with quantum behaviour such as interference and entanglement.
As mentioned previously, gravity in the modern conception arises
from the non-trivial geometry of a region of spacetime induced
by some distribution of matter. Some of the geometric attributes
that one can assign to a given region of spacetime are length, area,
volume, angles etc. A theory of quantum gravity should be able
to tell us how to write down the wavefunction, not defined on a
given region of spacetime, but a wavefunction of a given region
of spacetime, allowing us to construct states which correspond to
superpositions of different geometries. However, as mentioned
previously traditional quantum mechanics tells us only how to
write down the wavefunction on a given geometry rather than of a
given geometry. The traditional language of quantum mechanics is
thus insufficient to describe quantum states of geometry.
For the same reasons QFT-CS is also not a theory of “quantum
gravity”. There the curved spacetime merely serves as an arena
on which quantum states can be defined, but there is no notion
of states of the geometry itself, rather than of the matter which
moves about on that geometry.
At present there are several approaches towards tackling the open
question of writing quantum states of geometry. These include
String Theory, Loop Quantum Gravity and Causal Dynamical
Triangulations among others. Most laypersons with an interest in
science have heard of String Theory, simply because it is the oldest of these approaches and thus also the most widely taught and
practiced. LQG was born about a decade after String Theory and
has only recently reached a level of maturity and acceptability as a
valid physical theory. It would take us far afield to go into details
- even at a non-technical level - of these approaches and their
similarities and differences. I will try to briefly summarize the two
approaches and the basic idea behind each one.
New Paradigms
The idea behind String Theory is that instead of a description of
fundamental particles as point-like objects we should switch to
a picture where the basic entities are extended one-dimensional
objects called strings. These strings move and interact in some
background spacetime. Requirements of physical and theoretical
consistency restrict the number of dimensions of the spacetime in
which strings can live to 26, 11 and 10 depending on the particular
characteristics - fermionic, bosonic, open and closed - we choose
to endow the strings with. The excitations of a string happen to
include a part which can be identified with gravitons - which are
excitations of the background geometry the string is propagating
in. Though gravitons are often thought of as the quanta of the
gravitational field, in the same way that photons are quanta of the
electromagnetic field, this belief is only partially correct.
As mentioned in previous sections, the gravitational field is
characterized by geometric attributes such as lengths, areas and
volumes. Therefore, quanta of the gravitational field should
correspond to quantized lengths, areas and volumes, in the same
way that a quantum of the electromagnetic field corresponds to
a quantized amount of energy given by Planck’s relationship between the energy of a photon and its frequency, E=hv. However,
the graviton picture does not predict any such relations - such as
the area of a given region of spacetime and the frequency of a
gravitational wave which passes through that region - between
any fundamental geometric quantities and so cannot be said to
provide a picture of quantum geometry. Moreover, gravitons are
perturbations of the background spacetime which is, by default,