Eric D. Carlson
Gravity, alone of the four forces of nature, has not been incorporated into the Standard Model of particle physics. There is simply no obvious way to quantize the theory, and make sure that all standard model particles are included. Einstein's General Theory of Relativity (G.R.), explains how space and time will respond to the classical stress-energy-momentum tensor.
The conventional approach to applying general relativity to physically relevant situations is to treat space-time as locally flat, work out any relevant non-gravity properties of the matter (such as the equation of state) including quantum effects, and then to solve the G.R. equations. This might seem the best you can do. In the presence of strong gravitational fields, or high curvature, it should be expected that this approach yields, at best, a crude approximation.
The goal of Semi-Classical Gravity is to go one step beyond this. Although it is impossible (at present) to predict what the quantum effect of strong curvature on space-time might be, it is possible to determine the effect of strong curvature on various matter fields. Furthermore, if the number of matter fields is large compared to the number of new fields coming from quantum gravity, it is to be expected that the quantum effects will be dominated by the effects of the matter fields. If we treat space-time as fixed, then we can determine how quantum fluctuations in matter fields can cause changes in the local stress-energy-momentum tensor. Quantum mechanically, these will have unpredictable random fluctuations. As an approximation, we can take a quantum mechanical average of these fluctuations. This averaging is the essence of semi-classical gravity. Matter fields are treated as quantum mechanical in a curved space-time background. The equations of G.R. are solved classically, with the classical stress-energy-momentum tensor replaced with the quantum mechanical average value.
For simplicity, semi-classical gravity is normally calculated using non-interacting scalar fields. However, the only real fundamental scalar field, the Higgs, has at best an uncertain detection. The vast majority of matter fields are spin-half fields, or fermions. In conjunction with Dr. Paul Anderson and Peter Groves, we are currently attempting to develop a practical computational formalism for finding the stress-energy-momentum tensor due to fermion fields. We are working initially with static, spherically symmetric spacetimes and working out a WKB-type analytic approximation for the stress-energy-momentum tensor.
At first, we are working only with massless fermions, but we plan to shortly generalize to massive fermions. Once the WKB approximation has been worked out, we can numerically solve the differential equations to find more exact solutions.
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