Effective field theory in curved space

3.3 Effective field theory in curved space

All of the explicit calculations of the previous sections are performed for weak gravitational fields, which are well described as perturbations about flat space. This has the great virtue of being sufficiently simple to make explicit calculations possible, including the widespread use of momentum-space techniques. Much less is known in detail about effective field theory in more general curved spaces, although its validity is implicitly assumed by the many extant calculation of quantum effects in curved space [19 ], including the famous calculation of Hawking radiation [87

, 88

] by black holes. This section provides a brief sketch of some effective-field theory issues which arise in curved-space applications.

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 $ˆgμν$ and a quantum fluctuation $hμν$ according to $gμν = ˆgμν + hμν$ . A similar expansion may be required for the matter fields $ˆ φ = φ + δφ$ , if these acquire vacuum expectation values $ˆφ$ .

The main difference from previous sections is that $ˆgμν$ 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 ),

and we imagine that the slices of constant $t$ are Cauchy surfaces, in the sense that the future evolution of the fields is uniquely specified by initial data on a constant- $t$ slice. The validity of this initial-value problem may also require boundary conditions for the fluctuations $hμν$ such as that they vanish at spatial infinity.

3.3.1 When should effective Lagrangians work?

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 [130 ].

3.3.1.1 Time-independent backgrounds

In flat space, background fields which are time-translation invariant allow the construction of a conserved energy $H$ , which evolves forward the fields from one $t$ slice to another. If $H$ is bounded below³ , then a stable ground state for the quantum fields (the ‘vacuum’) typically exists, about which small fluctuations can be explored perturbatively. Since energy can be defined and is conserved, it can be used as a criterion for defining an effective theory which distinguishes between states which are ‘low-energy’ or ‘high-energy’ as measured using $H$ . Once this is possible, a low-energy effective theory may be defined, and we expect the general uncertainty-principle arguments given earlier to ensure that it is local in time.

A similar statement holds for background gravitational fields, with a conserved Hamiltonian following from the existence of a time-like Killing vector field $μ K$ for the background metric $ˆgμν$ ⁴ . For matter fields $H$ is defined in terms of $K μ$ and the stress tensor $T μν$ by the integral

Here the integral is over any spacelike slice whose measure is $dΣμ$ . Given appropriate boundary conditions this definition can also be extended to the fluctuations $hμν$ of the gravitational field [10 , 11 , 12 , 6 , 16 , 8 , 7 , 13 , 14 , 15 , 1 ].

If the timelike Killing vector $μ K$ is hypersurface orthogonal, then the spacetime is called static, and it is possible to adapt our coordinates so that $μ K ∂ μ = ∂t$ and so the metric of Equation (43 ) specializes to $Ni = 0$ , with $φ$ and $γij$ independent of $t$ . 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 $K μ$ be globally defined throughout the spacetime of interest.

If $H$ 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 $H$ to distinguish ‘low-energy’ from ‘high-energy’ fluctuations about any given vacuum.

It is often true that $μ K$ 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 $H$ is bounded from below.

3.3.1.2 Time-dependent backgrounds

In non-gravitational problems, time-dependence of the background fields need not completely destroy the utility of a Hamiltonian or of a ground state, provided that this time dependence is sufficiently weak as to be treated adiabatically. In this case the lowest eigenstate of $H (t)$ for any given time defines both an approximate ground state and an energy in terms of which low-energy and high-energy states can be defined (such as by using an appropriate eigenvalue $Λ (t)$ of $H (t)$ ).

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:

The time dependence of the background can be rapid, for instance having Fourier components which are larger than the dividing line $Λ (t)$ between low-energy and high-energy states. If so, the background evolution is not sufficiently adiabatic to prevent the direct production of ‘heavy’ particles, and an effective field theory need not exist. The criterion for this not to happen would be that the energies of any produced particles be smaller than the dividing eigenvalue $Λ(t)$ , which typically requires that $Λ (t)$ be much smaller than the background time dependence, for instance if $Λ ≪ Λ˙∕Λ$ .
Even for slowly-varying backgrounds there can be other dangers, such as the danger of level-crossing. For instance the dividing eigenvalue $Λ (t)$ may slowly evolve to smaller values and so eventually invalidate an expansion in inverse powers of $Λ(t)$ . In this case states which are prepared in the low-energy regime may evolve out of it, leading to a breakdown in the low-energy approximation. For instance, this might happen if $Λ (t)$ were chosen to be the mass of an initially heavy particle, which is integrated out to obtain the effective theory. If evolution of the background fields allows this mass to evolve to become too low, then eventually it becomes a bad approximation to have integrated it out.
A related potential problem can arise if states whose energy is initially larger than $Λ(t)$ enter the low-energy theory by evolving into states having energy lower than $Λ (t)$ ⁵ . This usually is only a problem for the effective-theory formulation if the states which enter in this way are not in their ground state when they do so, since then new physical excitations would arise at low energies. Conversely, they are not a problem for the effective field theory description so long as they enter in their ground states, as can usually be ensured if the background time evolution is adiabatic.

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 ( $E < Λ (t)$ ) without worrying about losing probability into high-energy states ( $E > Λ(t)$ ). 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 $μ V$ exists. For such a spacetime the observers for whom $V μ$ is tangent to world-lines can define a foliation of spacetime, as in Equation (43 ), but with the various metric components not being $t$ -independent. In this case the quantity defined by Equation (44 ) need not be conserved, $H = H (t)$ , for these observers. A low-energy effective theory should nonetheless be possible, provided $H (t)$ is bounded from below and is sufficiently slowly varying (in the senses described above). If such a foliation of spacetime exists, following [130 ] we call it a ‘nice slice’.

3.3.2 General power counting

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 [156 , 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 $λ$ . Provided this wavelength is very small compared with the local radius of curvature $rc$ 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 $λ ∕rc$ .

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 [75 , 36 , 37 , 115 ].

3.3.3 Horizons and large redshifts

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 ( $M ≫ Mp$ ) 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 [141 ]. 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 [139 , 94 , 95 ] and inflationary [93 , 114 , 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.

3.3.3.1 Black holes

For black holes the potential high-energy difficulty can be traced to the fact that the energy of an freely-falling object as seen by static observers becomes arbitrarily large as one approaches the event horizon. This is why the escaping modes have such high energies as seen by static observers near the horizon. This raises two separate issues for effective field theories, which are worth separating: the issue of the relevance of static observers measuring very large energies for freely-falling objects, and the issue of high-energy states descending into the low-energy theory as time progresses. Each of these is now discussed in turn.

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 [130 ]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 $H (t)$ which these slices define. A great many calculations using nice slices have been done, including for example [44 , 47 , 125 , 117 , 118 ].

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 [86 ] 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 [140 , 24 , 42 ]. (See [93 ] 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 [71 ]). 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 [87 , 88 ] in the vicinity of black holes – provided that the black hole mass is much larger than $Mp$ (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.

3.3.3.2 Inflation

Many of the issues concerning the validity of effective field theories which arise for the Hawking radiation also arise within inflationary cosmology, and have generated considerable discussion due to the recent advent of precise measurements of CMB temperature fluctuations. By analogy with the black hole case, it has been proposed [93 , 114 , 23 ] that very-high-energy physics may not decouple from inflationary predictions due to the exponential expansion of space. If so, there might be detectable imprints on the observed temperature fluctuations in the cosmic microwave background [96 ]. Conversely, if high-energy effects do contaminate inflationary predictions for CMB fluctuations at an observable level, then inflationary predictions themselves must be recognized as containing an uncontrollable theoretical uncertainty. If so, their successful description of the observations cannot be deemed to be credible evidence of the existence of an earlier inflationary phase. There is clearly much at stake.

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:

Observable effects can be obtained from physics associated with energies $E$ much higher than the inflationary expansion rate $H$ , if the states associated with the heavy physics are chosen not to be in their adiabatic vacuum⁷ . Potentially observable effects have been obtained by explicit calculations which incorporate non-adiabatic physics of various types. For many of these the high-energy physics is modelled as a free particle having a non-Lorentz-invariant dispersion relation (for a review with references, see [22 ]). However large Lorentz-violating interactions need not be required since similar effects are also obtained using more conventional inflationary field theories, for which background scalar fields are allowed to roll non-adiabatically [31 ].
If the states associated with high-energy physics are prepared in their adiabatic vacua then an effective field theory description applies. In this case most kinds of heavy physics decouple, and the vast majority of effects it can produce for the microwave background can be argued to be smaller than $2 2 𝒪 (H ∕M )$ [98 , 97 ]. Even in this case, however, larger contributions can be obtained using ordinary inflationary field-theory examples, where low-energy effects can instead be suppressed by powers of $m ∕M$ rather than $H ∕M$ , where $m$ is a light scale which need not be as small as $H$ [30 ].

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.