Let be a globally hyperbolic four-dimensional spacetime manifold with metric , and consider a real scalar quantum field of mass propagating on that manifold; we just assume a scalar field for simplicity. The classical action for this matter field is given by the functional
The field may be quantized in the manifold using the standard canonical quantization formalism [25, 100, 285]. The field operator in the Heisenberg representation is an operator-valued distribution solution of the Klein–Gordon equation, the field equation derived from Equation (1),
The classical stress-energy tensor is obtained by functional derivation of this action in the usual way, , leading to
The next step is to define a stress-energy tensor operator . Naively one would replace the classical field in the above functional by the quantum operator , but this procedure involves taking the product of two distributions at the same spacetime point. This is ill-defined and we need a regularization procedure. There are several regularization methods which one may use; one is the point-splitting or point-separation regularization method [67, 68], in which one introduces a point in a neighborhood of the point and then uses as the regulator the vector tangent at the point of the geodesic joining and ; this method is discussed for instance in [242, 243, 245] and in Section 5. Another well known method is dimensional regularization in which one works in arbitrary dimensions, where is not necessarily an integer, and then uses as the regulator the parameter ; this method is implicitly used in this section. The regularized stress-energy operator using the Weyl ordering prescription, i.e. symmetrical ordering, can be written as for details). The field states can be chosen in such a way that for any pair of physically acceptable states (i.e., Hadamard states in the sense of ), and , the matrix element , defined as the limit when the regulator takes the physical value is finite and satisfies Wald’s axioms [100, 282]. These counterterms can be extracted from the singular part of a Schwinger–DeWitt series [100, 67, 68, 31]. The choice of these counterterms is not unique, but this ambiguity can be absorbed into the renormalized coupling constants which appear in the equations of motion for the gravitational field.
The semiclassical Einstein equation for the metric can then be written as
A solution of semiclassical gravity consists of a spacetime (), a quantum field operator which satisfies the evolution equation (2), and a physically acceptable state for this field, such that Equation (7) is satisfied when the expectation value of the renormalized stress-energy operator is evaluated in this state.
For a free quantum field this theory is robust in the sense that it is self-consistent and fairly well understood. As long as the gravitational field is assumed to be described by a classical metric, the above semiclassical Einstein equations seems to be the only plausible dynamical equation for this metric: The metric couples to matter fields via the stress-energy tensor, and for a given quantum state the only physically observable c-number stress-energy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantum gravity theory, the scope and limits of the theory are not so well understood. It is assumed that the semiclassical theory should break down at Planck scales, which is when simple order of magnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Planck size region, as determined by the Heisenberg uncertainty principle, is comparable to the gravitational energy of that fluctuation.
The theory is expected to break down when the fluctuations of the stress-energy operator are large . A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford  (see also work via zeta-function methods [241, 69]). This proposal was questioned by Phillips and Hu [163, 242, 243] because it does not contain a scale at which the theory is probed or how accurately the theory can be resolved. They suggested the use of a smearing scale or point-separation distance for integrating over the bi-tensor quantities, equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [93, 95]. A different criterion is recently suggested by Anderson et al. [10, 9] based on linear response theory. A partial summary of this issue can be found in our Erice Lectures .
© Max Planck Society and the author(s)