For quantum systems in curved space we are still
interested in the gravitational action, Equation (28), possibly
supplemented by a matter action such as that of Equation (36
). As before,
quantization is performed by splitting the metric into a classical
background
and a quantum fluctuation
according to
. A
similar expansion may be required for the matter fields
, if these acquire vacuum expectation values
.
The main difference from previous sections is
that is not assumed to be the Minkowski metric. Typically
we imagine the spacetime may be foliated by a set of observers, as
in Equation (38
),
Many of the general issues which arise in this
problem are similar to those which arise outside of gravitational
physics when quantum fields are considered in the presence of
background classical scalar or electromagnetic fields. In
particular, there is a qualitative difference between the cases
where the background fields are time-independent or time-dependent.
The purpose of this section is to argue that we expect an effective
field theory to work provided that the background fields vary
sufficiently slowly with respect to time, an argument which in its
relativistic context is called the ‘nice slice’
criterion [131].
A similar statement holds for background
gravitational fields, with a conserved Hamiltonian following from
the existence of a time-like Killing vector field for the background metric
4
.
For matter fields
is defined in terms of
and the stress tensor
by the
integral
If the timelike Killing vector is hypersurface orthogonal, then the spacetime is
called static, and it is possible to adapt our coordinates so that
and so the metric of Equation (43
) specializes to
, with
and
independent of
. In this case we call the observers corresponding to
these coordinates ‘Killing’ observers. In order for these observers
to describe physics everywhere, it is implicit that the timelike
Killing vector
be globally defined throughout the
spacetime of interest.
If is defined and is bounded from below,
its lowest eigenstate defines a stable vacuum and allows the
creation of a Fock space of fluctuations about this vacuum. As was
true for non-gravitational background fields, in such a case we
might again expect to be able to define an effective theory, using
to distinguish ‘low-energy’ from ‘high-energy’
fluctuations about any given vacuum.
It is often true that is not unique
because there is more than one globally-defined timelike Killing
vector in a given spacetime. For instance, this occurs in flat
space where different inertial observers can be rotated,
translated, or boosted relative to one another. In this case the
Fock space of states to which each of the observers is led are
typically all unitarily equivalent to one another, and so each
observer has an equivalent description of the physics of the
system.
An important complication to this picture arises when the timelike Killing vector field cannot be globally defined throughout all of the spacetime. In this case horizons exist, which divide the regions having timelike Killing vectors from those which do not. Often the Killing vector of interest becomes null at the boundaries of these regions. Examples of this type include the accelerated ‘Rindler’ observers of flat space, as well as the static observers outside of a black hole. In this case it is impossible to foliate the entire spacetime using static slices which are adapted to the Killing observer, and the above construction must be reconsidered.
Putting the case of horizons aside for the
moment, we expect that a sensible low-energy/high-energy split
should be possible if the background spacetime is everywhere
static, and if the conserved energy is bounded from
below.
Once the system is partitioned in this way into low-energy and high-energy state, one can ask whether a purely low-energy description of time evolution is possible using only a low-energy, local effective Lagrangian. The main danger which arises with time-dependent backgrounds is that the time evolution of the system need not keep low-energy states at low energies, or high-energy states at high energies. There are several ways in which this might happen:
What emerges from this is that an effective field
theory can make sense despite the presence of time-dependent
backgrounds, provided one can focus on the evolution of low-energy
states () without worrying about losing
probability into high-energy states (
). This is
usually ensured if the background time evolution is sufficiently
adiabatic.
A similar story should also hold for background
spacetimes which are not globally static, but for which a
globally-defined timelike hypersurface-orthogonal vector field exists. For such a spacetime the observers for whom
is tangent to world-lines can define a foliation of
spacetime, as in Equation (43
), but with the various
metric components not being
-independent. In this case the
quantity defined by Equation (44
) need not be
conserved,
, for these observers. A low-energy
effective theory should nonetheless be possible, provided
is bounded from below and is sufficiently slowly
varying (in the senses described above). If such a foliation of
spacetime exists, following [131
] we call it a ‘nice
slice’.
Given an effective field theory, the next question is to analyze systematically how small energy ratios arise within perturbation theory. Since the key power-counting arguments of the previous sections were given in momentum space, a natural question is to ask how much of the previous discussion need apply to quantum fluctuations about more general curved spaces. In particular, does the argument that shows how large quantum effects are at arbitrary-loop order apply more generally to quantum field theory in curved space?
An estimate of higher-loop contributions performed in position space is required in order to properly apply the previous arguments to more general settings. Such a calculation is possible because the crucial part of the earlier estimate relied on an estimate for the high-energy dependence of a generic Feynman graph. This estimate was possible on dimensional grounds given the high-energy behaviour of the relevant vertices and propagators. The analogous computation in position space is also possible, where it instead relies on the operator product expansion [157, 41]. In position space one’s interest is in how effective amplitudes behave in the short-distance regime, rather than the limit of high energy. But the short-distance limit of propagators and vertices are equally well-known, and resemble the short-distance limit which is obtained on flat space. Consequently general statements can be made about the contributions to the low-energy effective theory.
Physically, the equivalence of the short-distance
position-space and high-energy momentum-space estimates is expected
because the high-energy contributions arise due to the propagation
of modes having very small wavelength . Provided this
wavelength is very small compared with the local radius of
curvature
particle propagation should behave just
as if it had taken place in flat space. One expects the most
singular behaviour to be just as for flat space, with curvature
effects appearing in subdominant corrections as powers of
.
Unfortunately, although the result is not in serious doubt, such a general position-space estimates for gravitational physics on curved space has not yet been done explicitly at arbitrary orders of perturbation theory. Partial results are known, however, including general calculations of the leading one-loop ultraviolet divergences in curved space [76, 36, 37, 116].
Among the most interesting applications of
effective field theory ideas to curved space is the study of
quantum effects near black holes and in the early universe. In
particular, for massive black holes () one expects
semi-classical arguments to be valid since the curvature at the
horizon is small and the interesting phenomena (like Hawking
radiation) rely only on the existence of the horizon rather than on
any properties of the spacetime near the internal curvature
singularity [142]. Although
the power-counting near the horizon has not been done in the same
detail as it has been for the asymptotic regions, semi-classical
effective-field-theory arguments at the horizon are expected to be
valid. Similar statements are also expected to be true for
calculations of particle production in inflationary universes.
An objection has been raised to the validity of
effective field theory arguments in both the black hole [140, 94, 95
] and
inflationary [96
, 115
, 23
] contexts. For both
of these cases the potential difficulty arises if one compares the
energy of the modes as measured by different observers situated
throughout the spacetime. For instance, a mode which emerges far
from a black hole at late times with an energy (as seen by static
and freely-falling observers) close to the Hawking temperature
starts off having extremely high energies as seen by freely-falling
observers very close to, but outside of, the black hole’s event
horizon just as it forms. The energy measured at infinity is much
smaller because the state experiences an extremely large redshift
as it climbs out of the black hole’s gravitational well. The
corresponding situation in inflation is the phenomenon in which
modes get enormously redshifted (all the way from microscopic to
cosmological scales) as the universe expands.
It has been argued that these effects prevent a consistent low-energy effective theory from being built in these situations, because very high-energy states are continuously turning up at later times at low energies. If so, this would seem to imply that a reliable calculation of phenomena like Hawking radiation (or inflationary fluctuations of the CMB) necessarily require an understanding of very high energy physics. Since we do not know what this very high energy physics is, this is another way of saying that these predictions are theoretically unreliable, since uncontrolled theoretical errors potentially contribute with the same size as the predicted effect.
The remainder of this section argues that although the concerns raised are legitimate, they are special cases of the general conditions mentioned earlier which govern the applicability of effective field theory ideas to time-dependent backgrounds also in non-gravitational settings. As such one expects to find robustness against adiabatic physics at high energies, and sensitivity to non-adiabatic effects. (This is borne out by the explicit calculations to date.) Given a concrete theory of what the high-energy physics is, one can then ask into which category it falls, and so better quantify the theoretical error.
The fact that freely-falling observers measure different energies for outgoing particles, depending on their distance from the horizon, underlines that there is a certain amount of frame dependence in any effective-theory description, even in flat space. This is so because energy is used as the criterion for deciding which states fall into the effective theory and which do not, yet any nominally low-energy particle has a large energy as seen by a sufficiently boosted observer. In practice this is not a problem, because the validity of the effective theory description only requires the existence of low-energy observers, not that all observers be at low energy. What is important is that the physically relevant energies for the process of interest - for instance, the centre-of-mass energies in a scattering event - are small in order for this process to be describable using a low-energy theory. Once this is true, invariant quantities like cross sections take a simple low-energy form when expressed in terms of physical kinematic variables, regardless of the energies which the particles involved have as measured by observers who are highly boosted compared to the centre-of-mass frame.
Flat-space experience therefore suggests that
there need not be a problem associated with escaping modes having
large energies as seen by freely-falling observers. This only
indicates that the use of some observers near the horizon may be
problematic. So long as the physics involved does not rely
crucially on these observers, it may in any case allow an
effective-theory description. This is essentially the point of view
put forward in [94, 95] and [131]6
,
where it is argued that the robustness of the Hawking radiation to
high-energy physics is most simply understood if one is careful to
foliate the spacetime using slices which are chosen to be ‘nice
slices’ (in the sense described above), which cut through the
horizon in such a way as to encounter only small curvatures and
adiabatic time variation. Since such slices exist, a low-energy
theory may be set up in terms of the slowly-varying which these slices define. A great many calculations
using nice slices have been done, including for example [44, 47, 126, 118, 119].
Of course, calculations need not explicitly use the nice slices in order to profit from their existence. In the same way that dimensional regularization can be more useful in practice for calculations in effective field theories, despite its inclusion of arbitrarily high energy modes, the sensitivity of Hawking radiation to high energies can be investigated using a convenient covariant regularization. This is because if nice slices exist, covariant calculations must reproduce the insensitivity to high energies which they guarantee. This is borne out by explicit calculations of the sensitivity of the Hawking radiation to high energies [87] using a simple covariant regularization.
The most important manner in which high-energy
states can influence the Hawking radiation has been identified from
non-covariant studies, such as those which model the high-energy
physics as non-Lorentz invariant dispersion relations for otherwise
free particles [141, 24, 42].
(See [96] for a review, with
references, of these calculations.) These identify the second
pertinent issue mentioned above: the descent of higher-energy
states into the low-energy theory. In these calculations
high-energy modes cross into the low-energy theory because of their
redshift as they climb out of the gravitational potential well of
the black hole. The usual expression for the Hawking radiation
follows provided that these modes enter the low-energy theory near
the black hole horizon in their adiabatic ground state (a result
which can also be seen in covariant approaches, where it can be
shown that the Hawking radiation depends only on the form of the
singularity of the propagator near the light cone [72]). If these
modes do not start off in their ground state, then they potentially
cause observable changes to the Hawking radiation.
The condition that high-energy modes enter the low-energy theory in their ground state is reminiscent of the same condition which was encountered in previous sections as a general pre-condition for the validity of a low-energy effective description when there are time-dependent backgrounds (including, for example, the descent of Landau levels in a decreasing magnetic field). In non-gravitational contexts it is automatically satisfied if the background evolution is adiabatic, and this can also be expected to be true in the gravitational case. Of course, this expectation cannot be checked explicitly unless the theory for the relevant high-energy physics is specified, but it is borne out by all of the existing calculations. To the extent that high-energy modes do not arise in their adiabatic vacua, their effects might be observable in the Hawking radiation as well as in possibly many other observables which would otherwise be expected to be insensitive to high-energy physics.
Clearly this is good news, since it tells us that
we can believe that generic quantum effects do not ruin the
classical calculations using general relativity, which tell us that
black holes exist. Nor do they ruin the semiclassical calculations
which lead to effects like the Hawking radiation [88, 89] in the
vicinity of black holes - provided that the black hole mass is much
larger than (which we shall see is required if
quantum effects are to remains small at the event horizon). On the
other hand, it means that we cannot predict the final stages of
black hole evaporation, since these inevitably lead to small black
hole masses, where the semiclassical approximation breaks down.
It is beyond the scope of the this article to summarize all of the intricacies associated with quantum field theory in de Sitter space, so we focus only on the parallels with the black hole situation. The bottom line for cosmology is similar to what was found for the Hawking radiation:
Again the final picture which emerges is encouraging. The criteria for validity of effective field theories appear to be the same for gravity as they are in non-gravitational situations. In particular, for a very broad class of high-energy physics effective field theory arguments apply, and so theoretical predictions for the fluctuations in the CMB are robust in the sense that they are insensitive to most of the details of this physics. But some kinds of high-energy effects can produce observable phenomena, and these should be searched for.