When computing scattering amplitudes in Quantum Field Theory, one normally works "perturbatively". This requires that each amplitude (which is associated with a well-defined set of field interactions) be characterized by a power of a constant much less than one. The constant chosen is usually the "coupling constant", which determines the strength of the interactions, and such a "perturbative expansion" implies that the computation is only valid for relatively weak interactions. For amplitudes associated with more complicated interactions (and hence higher powers of the coupling constant and therefore smaller contributions to the whole), there are field interactions which involve the exchange of quanta of arbitrary momentum. In order to allow for all possible momenta and still maintain Lorentz invariance, the amplitude must be integrated over all values of the momentum exchanged. Arbitrarily large values of momentum correspond to arbitrarily small distances, due to the conjugate nature of momentum and coordinate. Hence the local nature of quantum field theory requires some sort of "averaging" over finite distances which will eliminate the problems associated with dividing by distances which may be zero!
This averaging procedure is called "renormalization" and involves scaling of the coupling constant: at larger momentum (or energy), the value of the coupling constant is different from that at small momentum. For the coupling associated with electromagnetism, for example, the smaller the region investigated, the higher the coupling constant, and vise versa. This is similar to the dependence of measurement on frame of reference. By assuming that the coupling at one length scale is a function only of the coupling at another length scale and the ratio of the scales, one can determine the behavior of the system at all scales. It is interesting to note that perturbative Quantum Electrodynamics is "asymptotic": at sufficiently small orders of magnitude, the amplitudes begin to grow large again, due to the renormalization of the coupling.
One outstanding example of differing physical behavior at different scales is quark "confinement". At very small distances, quarks are "asymptotically free": they are essentially non-interacting. However, as the distance between them grows, the force of attraction grows as well. This phenomenon can be "explained" by the assumption that the vacuum is a perfect color dielectric. By requiring the color dielectric constant of the vacuum to be zero, the vacuum becomes color "antiscreening": induced color charge (with the vacuum analogous to polar molecules) is of the same sign as the source. Since the dielectric constant in the region of a single quark is one, a domain wall of induced color charge forms around it. It requires an infinite amount of work to shrink the domain to zero volume, which corresponds to the quark having infinite mass. This situation is only overcome when the total color inside the region is neutral. Hence quarks are confined to regions containing color singlets.
It is possible to determine the natural scale of physical variables by examining the fundamental constants of nature. These constants include the Planck Constant, which determines the quantum of action, the speed of light, which determines the maximum velocity in any reference frame, and Newton's Constant, which is the gravitational coupling. By constructing a constant with dimensions of length from these, called the Planck Length, we obtain an idea of the distance scale at which Quantum Gravity is relevant. The Planck Scale is many orders of magnitude smaller than is currently experimentally accessible. It is interesting to note that at distance scales slightly larger than the Planck Scale, the coupling constants of the Standard Model of Particle Physics converge to a single value.
Because the "laws" of physics can be very different depending on scale, the notion of a "fundamental" theory is less useful than one might expect. Instead, each theory should be viewed within the context of the scale to which it applies and has been experimentally tested. In this light, Quantum Field Theory and General Relativity need only be married because of the equivalence of gravitational and inertial mass; inertial mass is a quantum number associated with the Poincare Group, while gravitational mass is a source of spacetime curvature.
Quantum Gravity Concept Map Index:
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©1997, Kenneth R. Koehler. All Rights Reserved.