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General relativity is a geometric theory of gravitation as a consequence of spacetime’s curvature and the uneven distribution of mass. It is applicable to very large scales, such as to: planets, stars, galaxies and clusters of galaxies.
Quantum mechanics, on the other hand, describes the other 3 fundamental forces of nature: electromagnetism, the strong and weak nuclear forces, on the scale of the atom and of subatomic particles, using principles of probability.
The problem at hand for quantum gravity theorists
Both of these theories (general relativity and quantum mechanics) are tremendously accurate in their own domain, however, when one attempts to harmonize them, by say, throwing a graviton into quantum field theory, to see how gravity functions at the quantum scale, theoretical problems arise. The theory that results is not renormalizable and infinities arise, such as the curvature of spacetime becoming infinite. This means that the theory is not able to make any kind of meaningful predictions whatsoever. This is the necessity for a theory of quantum gravity, to describe both the very small and the very massive, such as at a gravitational singularity, like at a black hole or at the instant of the Big Bang.
John Wheeler proposed that if we could zoom into spacetime at the Planck length it would no longer be smooth, however, it would be more like a foam.
If we could zoom into spacetime at the scale of quantum gravity, or, the Planck length of 10^-33 cm, to observe the so-called "quantum foam", we would observe that the very foundation of the fabric of spacetime is not the smoothness of spacetime, that we observe macroscopically. However, spacetime, can fluctuate, like a foam. This would be due to the Heisenberg uncertainty principle's application to general relativity. However, the nature of spacetime at the Planck scale is uncertain, until we have a full theory of quantum gravity.
Nonrenormalizability:
The difficulty with harmonizing general relativity and quantum mechanics comes from a phenomenon known as nonrenormalizability. A renormalizable theory, such as quantum electrodynamics, is a good theory. A nonrenormalizable theory, like most of our theoretical attempts at quantum gravity, are no theory at all. What is the problem? Well, photons respond to charge, while gravitons don’t respond to charge, however, to mass and energy. Photons don’t respond to themselves, however, they do respond to electric charge. However, gravitons, since they respond to energy, do, respond to themselves. To understand gravity quantum mechanically, we must take into account wave particle duality. Also, a graviton has a gravitational field, corresponding to other gravitons. The closer you are to the graviton, the stronger the gravitational field. The energy and momentum of these offspring gravitons (virtual gravitons) have tremendous momentum and energy. Gravitons respond to momentum. Each of the gravitons emitted by each of these gravitons has it's own field of gravitons! We can't keep track of all of these gravitons.
Gravitons produce so many virtual gravitons that the methods of renormalization cannot keep track of them. Renormalization is the mathematical method to keep track of all of these virtual particles. This is why we say that general relativity is nonrenormalizable.
Graviton:
The graviton is the hypothetical force mediating particle for gravitation. Indeed, it has never been observed. The properties of the graviton are, for the most part, two-fold:
The graviton must be massless, since the range of the gravitational force is infinite.
The graviton must be a tensor boson of spin-2, since, the source of gravitation is the stress-energy tensor.
String theory, is currently, our leading candidate as a theory of quantum gravity. The idea, that every particle and interaction of the Standard Model, can be modeled as a 1-dimensional vibrating string, is a proposed solution to the conflict between general relativity and quantum mechanics. Each mode of the string's vibration, will determine the properties of it's corresponding subatomic particle, that would be observed at larger length scales.
String theory, began as a theory of the strong nuclear force. We now know that the strong force is the consequence of gluons, acting to bind our quarks into protons and neutrons, and our protons and neutrons into the nucleus of our atoms. However, this understanding did not exist, until a theory was put forth in 1973 called quantum chromodynamics. One year later, in 1974, Tamiaki Yoneya, and independently, John H. Schwarz and Joel Scherk, proposed that the presence of the spin-2 massless particle (potentially the graviton) in string theory, was indicative of the fact that it is not a theory of the strong nuclear force, however, a theory of quantum gravity.
A 2D projection of a Calabi-yau manifold, a way to compactify the extra dimensions in string theory.
Tamiaki Yoneya
John H. Schwarz
Joel Scherk
The idea is that graviton would arise from one of the vibrational states of a 1-dimensional closed string in string theory.
For further reading on theories of quantum gravity, see:
Loop quantum gravity
Causal dynamical triangulation
Causal sets
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