3.4 Non-Gaussian fixed point and asymptotic safety
As explained before the function
plays the role of a generating functional for an infinite set of
essential couplings. As such it is subject to a flow equation
where
is the renormalization scale and
is the ‘running’ coupling function. The flow
equation can be obtained by the usual procedure starting from the fact that the left-hand-side of
Equation (3.58) is
-independent. One finds [154
]
where we suppress the
dependence of
. Inserting Equation (3.69) and setting
in the result
gives
Here
is again the conventional (numerical) beta function of the
nonlinear sigma-model without coupling to gravity, computed in the minimal subtraction scheme.
can thus be regarded as a “gravitationally dressed” version of
. The term is borrowed from [120]
where a similar phenomenon was found in a different context and to lowest order. In contrast to
Equation (3.85) the effect of quantum gravity on the running of the couplings in these Liouville-type
theories cannot be represented as a simple “dressing relation” beyond lowest order [168]. The flow equation
resulting from Equation (3.85) will be studied in more detail below. We anticipate however
that the appropriate boundary conditions are such that the solution
is stationary
(constant in
) for
. This guarantees that the renormalization flow is exclusively
driven by the counterterms, as it should. We add some comments on the structure of the beta
function (3.85).
An initially puzzling feature of
is that it comes out as a total
-derivative.
Restoring the interpretation of
as a field on the 2D base space, however, it has a natural
interpretation: An immediate consequence of Equations (3.83, 3.85) is that contour integrals of the form
are
-independent for any closed contour
in the base space, as can be seen by differentiating
Equation (3.86) with respect to
. They are thus invariants of the flow and can be used to discriminate
the inequivalent quantum theories parameterized by the
. With the initial condition
the
-independence of Equation (3.86) is equivalent to
. On the
other hand the (classical and quantum) equations of motion for
with respect to the
-modified action
are just
. Combining both we find that the significance of
being a total
-derivative is that this feature preserves the equations of motion for
under the
-evolution of
:
This provides an important consistency check as Equation (3.87) is also required by the non-renormalization of
the
Noether current in Equation (3.78).
Another intriguing property of Equation (3.85) can be seen from the second line in Equation (3.84).
The first two terms correspond to the beta function of a
sigma-model without coupling to 2D
gravity (i.e. with nondynamical
and
). The last term is crucial for all the subsequent properties of
the flow (3.83). Comparing with Equation (3.75) one sees that it describes a backreaction of the scale
dependent area radius
on the coupling flow, which is mediated by the quantum dynamics of the other
fields. We shall return to this point below.
As usual fixed points of the flow correspond to zeros of the generalized beta function. The flow has two
fixed points, a degenerate Gaussian one corresponding to
formally, and a non-Gaussian fixed point
which is of main interest here. We postpone the discussion of the Gaussian fixed point and focus
on the non-Gaussian one here. The defining relation for such a nontrivial fixed point amounts by
Equation (3.85) to the differential equation
where
is a constant, which in principle could be
-dependent. The Equation (3.88)
is to be interpreted with Equation (3.67) inserted and expanded in powers of
. This can
be solved recursively for
,
, etc. We denote the solutions by
. The unique
solution adhering to the above boundary condition corresponds to a
-independent
, for
which we write
, where
is the first beta function coefficient. The solution then reads
This is a nontrivial fixed point, in the sense that gravity remains self-interacting and coupled to
matter.
The renormalized action (at or away from the fixed point) has no direct significance in the quantum
theory. It is however instructive to note that
in Equation (3.56) can for any
not be
written in the form
for any metric with two commuting Killing vectors. In accordance
with the general renormalization group picture one can of course write
.
Interestingly, in the 2-Killing vector reduction one can write the renormalized Lagrangian
as a sum of
two terms which are reductions of
. Since the parameter
in
can be changed at will by
redefining
, we may assume that
, for some
. Then
modulo total derivatives, where in the first term
is the metric with line element (3.8) and
is a
metric with line element
The extra term
involves only the
area radius
and the conformal factor
. It vanishes iff
is proportional to
.
From Equation (3.89) one sees that
always; by construction in the minimal
subtraction scheme, but since the coefficients
,
are universal, the same holds in any other
scheme.
The origin of this feature is the seeming violation of scale invariance on the level of the renormalized
action. Recall from after Equation (3.62) that
, so that for
the action
is no longer scale invariant. However this is precisely the property which allows one to cancel the (otherwise)
anomalous term in the trace of the would-be energy momentum tensor, as discussed before, rendering
the system conformally invariant at the fixed point. Due to the lack of naive scale invariance
on the level of the renormalized action the dynamics of quantum gravidynamics is different
from that of quantum general relativity, in the sector considered, even at the fixed point. The
moral presumably generalizes: The form of the (bare and/or renormalized) action may have to
differ from the Einstein–Hilbert action in order to incorporate the physics properties aimed
at.
In the present context the most important feature of the fixed point
is its ultraviolet
stability: For
all linearized perturbations around the fixed point function (3.89) are
driven back to the fixed point. Since the fixed point (3.89) has the form of a power series in the
loop counting parameter
, the proper concept of a “linearized perturbation” has the form
where the
are functions of
and
. Note that the perturbation involves
infinitely many functions of two variables. The boundary condition mentioned before, which guarantees that
the full
flow is driven by the counterterms, only amounts to the requirement that all the
vanish for
uniformly in
. Inserting the ansatz (3.92) into the flow equation
and linearizing in
gives a recursive system of inhomogeneous integro-differential equations for
the
,
Here
, while the
,
, are complicated integro-differential operators acting (linearly) on
the
(see [155
] for the explicit expressions). The lowest order equation (3.93) is homogeneous
and its solution is given by
where
is an arbitrary smooth function of one variable satisfying
for
. This
function can essentially be identified with the initial datum at some renormalization time
, as
. Evidently
for
, if
. This condition is indeed
satisfied by all the systems (3.1, 3.56) considered, precisely because the coset space
is noncompact.
Interestingly one has the simple formula [155
]
It follows from the value of the quadratic Casimir in the appropriate representation and is consistent
with [47]. Since
always, Equation (3.94) shows that the lowest order perturbation
will always
die out for
, for arbitrary smooth initial data prescribed at
. It can be shown that
this continues to hold for all higher order
irrespective of the signs of the coefficients
,
.
Result (UV stability):
Given smooth initial
with
,
, the solution of the linearized flow
equations (3.93) is unique and satisfies
where the convergence is uniform in
.
The situation is illustrated in Figure 1. The proof of this result is somewhat technical and can be found
in [155
].
Often the stability properties of fixed points are not discussed by solving the linear flow equations
directly, but by studying the spectral properties of the linearized perturbation operator (the “stability
matrix” in Equation (A.10)). Since the generalized couplings here are functions, the linearized perturbation
operator
is a formal integral operator,
with a distributional kernel
, which can be computed from the explicit
formula (3.85). For example
where
is the step function. Writing
,
,
, the
spectral problem
decomposes into a sequence of integro-differential equations for the
, where the right-hand-side is determined by the solution of the lower order
equations. Only the
equation is a spectral problem proper,
,
.
The relevant and irrelevant perturbations have spectra with negative and positive real parts,
respectively. Remarkably all (nontrivial) eigenfunctions of
are normalizable; the spectrum
is “purely discrete” and consists of the entire halfplane
. Indeed, the
general solution to
is
, with
. The first
term merely corresponds to a change of normalization of
and we may set
,
without loss of generality. The second term corresponds to Equation (3.94)
with
. This clearly confirms the above result from a different perspective. For
the parameters
are not spectral values for
. Moreover, since the kernels
are
distributions it is not quite clear which precise functional analytic setting one should choose for
the full spectral problem
. This is why above we adopted the direct strategy and
determined the solutions of the linearized flow equations. Their asymptotic decay shows the
ultraviolet stability of the fixed point unambiguously and independent of functional analytical
subtleties.
We can put this result into the context the general discussion in Section 2 and arrive at the following
conclusion:
Conclusion:
With respect to the non-Gaussian fixed point
all couplings in the generating functional
are asymptotically safe. All symmetry reduced gravity systems satisfy the Criteria (PTC1) and (PTC2) to
all loop orders of sigma-model perturbation theory. As explained in Section 3.2 from the viewpoint of the
graviton loop expansion the distinction between a perturbative and a non-perturbative treatment is blurred
here.
It is instructive to compare these properties to that of the Gaussian fixed point. The Gaussian fixed
point of the flow (3.83) is best understood in analogy to the Gaussian fixed-point of a conventional
nonlinear sigma-model. For a
nonlinear sigma-model with Lagrangian
(with
satisfying Equation (3.12)) the beta function
has only the trivial
zero
. As
the renormalized Lagrangian blows up, but in an expansion around
one can see that for
the interaction terms vanish. In this sense the fixed point
is Gaussian. This holds irrespective of the sign of
, which however determines the stability
properties of the flow. The stability ‘matrix’ vanishes so that the linearized stability analysis is empty. By
direct inspection of the differential equation one sees that the unstable manifold of
is one-dimensional
for
(typical for
compact) and empty for
(typical for
noncompact). Indeed,
, and if one insists on
for positivity-of-energy reasons, the flow will
be attracted to
for
iff
. In particular for
these models
are, based on the Criterion (PTC2) of Section 2, not expected to have a genuine continuum
limit.
The Gaussian fixed point of the symmetry reduced gravity theories can be analyzed similarly. In terms
of
the flow equation (3.83) reads
Clearly
is a fixed point (function) and in a similar sense as before it can be interpreted as a
Gaussian fixed point. In contrast to the non-Gaussian fixed point the linearized stability analysis is now
empty (just as it is for the
sigma-model flow). One thus has to cope with the nonlinear flow
equation (3.99) at least to quadratic order. This is cumbersome but the qualitative feature of interest here
can readily be understood: With respect to the Gaussian fixed point
not all couplings
contained in the generating functional
are asymptotically safe. That is, there exists initial
data
(with
, for
) for which the
asymptotics
does not vanish identically in
. To see this, it suffices to note that the right-hand-side of
Equation (3.99) to quadratic order reads
, with
. Since
always, initial data
for which
is a strictly increasing function of
will give
rise to solutions having the tendency to be driven towards larger values (pointwise in
) as
increases. Conversely, only initial data
for which
is strictly decreasing in
can be expected to give rise to a solution
which vanishes identically in
as
.
A rigorous theorem describing the stable and the unstable manifold is presumably hard to
come by, but for our purposes it is enough to know that there exist initial data which do not
give rise to solutions decaying to
for
. For example
or
,
, have this property. The upshot is that the Gaussian fixed
point of the symmetry truncated gravity theories is not UV stable, the Condition (PTC2) is
not satisfied, and one can presumably not use it for the construction of a genuine continuum
limit.
At this point it may be instructive to contrast the quantum properties of the dimensionally reduced
gravity theories with those of the same noncompact
sigma-model without coupling to gravity (which
effectively amounts to setting
constant in Equation (3.16)). The qualitative differences are summarized
in Table 2.
The comparison highlights why the above conclusion is surprising and significant. While the noncompact
sigma-models are renormalizable with just one relevant coupling (denoted by
in the table), at
least in the known constructions they do not have a fixed point at which they are conformally invariant.
Their gravitational counterparts require infinitely many relevant couplings for their UV renormalization.
This infinite coupling flow has a nontrivial UV fixed point at which the theory is conformally invariant.
Most importantly the stability properties of the renormalization flow are reversed (compared
to the flow of
) for all of the infinitely many relevant couplings. As there appears to be
no structural reason for this surprising reversal in the reduced theory itself, we regard it as
strong evidence for the existence of an UV stable fixed point for the full renormalization group
dynamics.
In the table we anticipated that at the fixed point the trace anomaly of the would-be energy momentum
tensor vanishes for the symmetry truncated gravity theories. This allows one to construct quantum
counterparts of the constraints
and
as well-defined composite operators. In detail this comes
about as follows. Taking the trace in Equation (3.81) gives
, again
modulo the equations of motion operator. The first term has a nonzero trace anomaly given by
Here
is the field vector in Equation (3.76); furthermore
, where
is a functional of
which receives contributions only at three and higher loop orders (see Section B.3
and [154
]). The improvement term is determined by the function
in Equation (3.81),
which in turn is largely determined by
and thus cannot be freely chosen as a function of
.
It is therefore a very nontrivial match that (i) upon insertion of the fixed point
the function
becomes stationary (
-independent), (ii) the equation
turns out to determine
completely, and (iii) the so-determined function has
the property that
precisely cancels the second term in Equation (3.100) evaluated
for
. Thus the trace anomaly vanishes precisely at the fixed point of the coupling flow
As should be clear from the derivation this is a nontrivial property of the system, rather than one which is
used to define the renormalized target space metric
. The latter is already determined by the
warped product structure (3.59, 3.60) and the renormalization result (3.57). At the fixed point
the
first term in Equation (3.100) vanishes, but the second then is completely determined. On the other
hand by Equation (3.81) one is not free to choose the improvement potential as a function of
-independent of
, so the cancellation is not built in. One can also verify that the only
solution for
such that
is
with the above
,
such that Equation (3.101) holds. That is, scale invariance implies here conformal invariance.
Since the target space metric (3.60) has indefinite signature this does not follow from general
grounds [176, 194].
Due to Equation (3.101) we can now define the quantum constraints by
The linear combinations
are thus expected to generate two commuting copies of a Virasoro
algebra with formal central charge
. This central charge is only formal because it refers
to a state space with indefinite norm (see [127, 128] for an anomaly-free implementation of the Virasoro
constraints in essentially noninteracting systems). In the case at hand the proper positivity
requirement will be determined by the quantum observables commuting with the constraints. Their
construction and the exploration of the physical state space is a major desideratum. In summary,
the systems should at the fixed point be described by one whose physical states can be set
into correspondence to the above quantum observables. The infrared problem has not been
investigated so far, but based on results in the polarized subsector [153] one might expect it to be
benign.
Despite the fact that the system is conformally invariant at the fixed point there are still scale
dependent running parameters. At first sight this seems paradoxical. However, already the example of the
massless continuum limit in a 3D scalar field theory exhibits this behavior. The remaining
scale parameter is related to the direction of instability within the critical manifold, pointing
from the Gaussian to the Fisher–Wilson fixed point, in the direction of coarse graining. The
systems considered here provide an intriguing other example of this phenomenon. The critical
manifold can be identified with the subset of parameter values where the system is scale (and here
conformally) invariant. This fixes
, but the inessential parameters contained in the
renormalized fields are left unconstrained. This allows one to introduce a running parameter as
follows. One evaluates the running coupling function
at the ‘comoving’ field
and sets
This quantity carries a two-fold
-dependence, one via the running coupling
and one
because now the argument at which the function is evaluated is likewise
-dependent. Since
is a field
on the base manifold, the quantity
depends parametrically on the value of
– and hence
on
. Combining Equation (3.83) with Equation (3.75) one finds the following flow equation:
These are the usual flow equations for the single coupling
sigma-model without coupling to gravity!
In other words the ‘gravitationally dressed’ functional flow for
has been ‘undressed’ by reference to the
scale dependent ‘rod field’
(the term is adapted from H. Weyl’s “Maßstabsfeld”). The
Equations (3.104) are not by themselves useful for renormalization purposes – which requires determination
of the flow of
with respect to a fixed set of field coordinates. Moreover in the technical sense
is an “inessential” parameter. The fact that the scale dependence of
is governed by the beta function
means that for increasing
it will be driven away from the fixed point
. The condition
can be traded for the specification of the Gaussian fixed point
.
Thus the parameter flow
may be viewed as a coupling flow emanating (in the direction of
increasing
) from the Gaussian fixed point. At the non-Gaussian fixed point, on the other
hand,
governs the scale dependence of the ‘rod field’
via
This follows from Equation (3.75) and the relation
. Here we indicated the
functional dependence of
on
, which at the non-Gaussian fixed point gives a dependence of
on
. Since
, one sees from Equation (3.105) that
is pointwise for all
a decreasing function of
, at least locally in
. In addition Equation (3.87) implies
Here
are functions of one variable which by Equation (3.105) are locally decreasing in
and
are lightcone coordinates. Since the theory is conformally invariant at the fixed point
(of the couplings) one can change coordinates
to bring scaling operators into a standard
form. The upshot is, as anticipated in Section 3.2, that the rod field
describes the local
scale changes dynamically induced by quantum gravity and defines a resolution scale for the
geometries.
For orientation we summarize here our results on the renormalization of the symmetry reduced
Quantum Einstein Gravity theories (3.1):
- The systems inherit the lack of standard perturbative renormalizability from the full theory. A
cut-off independent quantum theory can be achieved at the expense of introducing infinitely
many couplings combined into a generating function
of one variable.
- The argument of this function is the ‘area radius’ field
associated with the two Killing
vectors. The field
is (nonlinearly) renormalized but no extra renormalizations are needed
to define arbitrary powers thereof.
- A universal formula for the beta functional for
and hence for the infinitely many couplings
contained in it can be given. The flow possesses a Gaussian as well as a non-Gaussian fixed
point. With respect to the non-Gaussian fixed point all couplings in
are asymptotically
safe.
- At the fixed point the trace anomaly vanishes and the quantum constraints (well-defined as
composite operators)
,
can in principle be imposed. The linear combinations
are expected to generate commuting copies of a centrally extended conformal
algebra acting on an indefinite metric Hilbert space.
- Despite the conformal invariance at the fixed point there is a scale dependent local parameter,
whose scale dependence is governed by the beta function of the
sigma-model without
coupling to gravity.
So far we considered the renormalization of the symmetry reduced theories in its own right, leaving the
embedding into the full Quantum Gravidynamics open. The proposed relation to qualitative aspects of the
Quantum Gravidynamics in the extreme UV has already been mentioned. Here we offer some tentative
remarks on the embedding otherwise. The constructions presented in this section can be extended to
dimensions in the spirit of an
-expansion. At the same time this mimics quantum aspects of the
1-Killing vector reduction. One finds that the qualitative features of the renormalization flow –
non-Gaussian fixed point and asymptotic safety – are still present [156
]. A cosmological ‘constant’ term can
likewise be included and displays a similar pattern as outlined at the end of Section 3.2. The advantage
of this setting is that the UV cutoff can strictly be removed, which is hard to achieve with a
nonperturbative technique. The extension of these results from a quasi-perturbative analysis to a
nonperturbative one, ideally via controlled approximations, is an important open problem. The same holds
for the analysis of the 1-Killing vector reduction, which holds the potential for cosmological
applications. These truncations can be viewed as complementary to the ‘hierarchical’ truncations used
in Section 4: A manifest truncation is initially imposed on the functional integral, but the
infinite coupling renormalization flow can then be studied in great detail, often without further
approximations.