To explore fluctuations greater than the Planck mass, or equivalently to measure distances shorter than the Planck length Rp, neither general relativity nor quantum field theory can be used alone. We can also get a sense of the energy of the fluctuation and the time limit by going backto the uncertainty relation.
Again, to describe fundamental particles with an energy of the scale of the Planck energy or time intervals less than the Planck time requires a theory of quantum gravity. We can take this time limit as the approximate limit in exploring the initial conditions of the big bang. What this argument suggests is that we can push the separate theories of general relativity and quantum field theory (quantum mechanics united with special relativity) back to a time of about 10-42 s or so. Prior to this time physics is governed by the unknown theory of quantum gravity. If a theory of quantum gravity is developed, the hope is that it will describe the initial conditions of the universe and answer all questions about its development. Hence, such a theory would be enormously powerful. Over the past 50 years it has become clear that such a theory is not going to be easily developed.
What are the difficulties in uniting these two powerful theories? There are several different ways to point out the conflicts. First, it is clear that general relativity needs to have some modification on the extremely small, high energy scale. At the center of black holes and the beginning of the universe, the theory calls for a singularity. This singularity is a point of infinite spacetime curvature and energy density. Such singularities are mathematically unacceptable. However in low curvature, low energy, regions the general theory is an accurate theory. Quantum field theory (the unification of quantum mechanics and special relativity) on the other hand is an accurate theory on short distance, moderately high energy, scales. On the large scale, low energy scale, quantum mechanics transitions to classical mechanics. A transition which is not entirely well understood. There is much work today on this transition regime between quantum mechanics and classical physics.Another problem in bringing together these two theories is the question of what exactly is being quantized. To discuss quantization, first consider classical electromagnetic theory and its quantized form quantum electrodynamics. The process of quantizing the electromagnetic theory replaces the notion of an electromagnetic wave with particles (quanta) which mediate the electric and magnetic forces. The photon is the quanta of EM radiation. Classically it is electromagnetic waves (or electric and magnetic fields) which mediate the forces. This is what is being quantized.(The process is more complicated then simply replacing waves with particles but there is nospace to discuss the details). For general relativity what is to be quantized? Recall that the Einstein equation gives the metric solution for a particular distribution of energy. The metric is the ‘field’ which mediates the gravitational force (we are drawing an analogy to electricity and magnetism, again, there is no gravitational force but the curvature of spacetime). So the quantity to quantize is the metric itself. Or, put another way, spacetime itself must be quantized! This is surely a strange requirement. What does it mean to replace spacetime with quanta (gravitons) which mediate the gravitational force? What do these particles propagate through, since there is no longer a continuum of spacetime? Tying this together with quantum mechanics which employs time and space as parameters to describe the wave functions of the quanta – here being space and time itself. It is somewhat self-referential and causes difficulties in even beginning to construct a theory. Quantum theory relies on a spacetime background, however here we are doing away with such a concept and replacing it with discrete particles.
