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
, 286
]. 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 [243
, 244
, 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
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 [92]. A criterion based on
the ratio of the fluctuations to the mean was proposed by Kuo and
Ford [194] (see also work via
zeta-function methods [242
, 69
]). This proposal was
questioned by Phillips and Hu [163
, 243
, 244
] 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. [9
, 10
] based on linear
response theory. A partial summary of this issue can be found in
our Erice Lectures [168
].