The framework closest in spirit to the present one are discretized approaches to the gravitational functional
integral, where a continuum limit in the statistical field theory sense is aimed at. See [138] for a general
review and [8] for the dynamical triangulations approach. Arguably the most promising variant of the latter
is the causal dynamical triangulations approach by Ambjørn, Jurkiewicz, and Loll [5]. In this setting the
formal four dimensional quantum gravity functional integral is replaced by a sum over discrete
geometries,
. The geometries
are piecewise Minkowskian and selected such
that they admit a Wick rotation to piecewise Euclidean geometries. The edge lengths in the
spatial and the temporal directions are
and
, where
sets the
discretization scale and
is an adjustable parameter. The flip
defines a Wick
rotation under which the weights in the partition function become real:
.
For
the usual expressions for the action used [8
] in equilateral Euclidean dynamical
triangulations are recovered, but the sum is only over those Euclidean triangulations
which lie in
the image of the above Wick rotation. The weight factor
is the inverse of the order
of the automorphism group of the triangulation, i.e.
for almost all of them. With these
specifications the goal is to construct a continuum limit by sending
and the number
of
simplices to infinity, while adjusting the two bare parameters (corresponding to Newton’s constant
and a cosmological constant) in
as well as the overall scale of
. Very likely, in
order for such a continuum limit to exist and to be insensitive against modifications of the
discretized setting, a renormalization group fixed point in the coupling flow is needed. Assuming that
the system indeed has a fixed point, this fixed point would by construction have a nontrivial
unstable manifold, and ideally both couplings would be asymptotically safe, thereby realizing the
strong asymptotic safety scenario (using the terminology of Section 1.3). Consistent with this
picture and the previously described dimensional reduction phenomenon for asymptotically safe
functional measures (see also Section 2.4), the microscopic geometries appear to be effectively
two-dimensional [6, 7].
Despite these similarities there are (for the time being) also important differences. First the discretized
action depends on two parameters only and it is hoped that a renormalized trajectory can be found by
tuning only these two parameters. Since in dynamical triangulations there is no naive (classical) continuum
limit, one cannot directly compare the discretized action used with a microscopic action in the
previous sense. Conceptually one can assign a microscopic action to the two parametric measure
defined by the causal dynamical triangulations by requiring that combined with the regularized
kinematical continuum measure (see Section 2.3.3) it reproduces the same correlation functions in
the continuum limit. The microscopic action defined that way would presumably be different
from the Einstein–Hilbert action, but it would still contain only two tunable parameters. In
other words the hope is that the particular non-naive discretization procedure gets all but two
coordinates of the unstable manifold automatically right. A second difference concerns the role of
averages of the metric. The transfer matrix used in [8] is presumed to have a unique ground
state for both finite and infinite triangulations. Expectation values in a reconstructed Hilbert
space will refer to this ground state and hence be unique for a given operator. A microscopic
metric operator does not exist in a dynamical triangulations approach but if one were to define
coarse grained variants, their expectation value would have to be unique. In contrast the field
theoretical formulations based on a background effective action allow for a large class of averaged
metrics.
The term loop quantum gravity is by now used for a number of interrelated formulations (see [199] for a
guide). For definiteness we confine our comparative remarks to the original canonical formulation using loop
(holonomy) variables.
Here a reformulation of general relativity in terms of Ashtekar variables is taken as a starting
point, where schematically
and
are defined on a three-dimensional time slice and are conjugate to
each other,
, with respect to the canonical symplectic structure (see [199
, 15]). From
one can form holonomies (line integrals along loops) and from
one can form fluxes
(integrals over two-dimensional hypersurfaces) without using more than the manifold structure.
The Poisson bracket
is converted into a Poisson algebra for the holonomy and
the flux variables. Two basic assumptions then govern the transition to the quantum theory:
First the Poisson bracket
is replaced by a commutator
and is
subsequently converted into an algebraic structure among the holonomy and flux variables.
Second, representations of this algebra are sought on a state space built from multiple products of
holonomies associated with a graph (spin network states). The inner product on this space
is sensitive only to the coincidence or non-coincidence of the graphs labeling the states (not
to their embedding into the three-manifold). Based on a Gelfand triple associated with this
kinematical state space one then aims at the incorporation of dynamics via a (weak) solution of the
Hamiltonian constraint of general relativity (or a ‘squared’ variant thereof). To this end one
has to transplant the constraint into holonomy and flux variables so that it can act on the
above state space. This step is technically difficult and the results obtained do not allow one to
address the off-shell closure of the constraint algebra, an essential requirement emphasized
in [151
].
As far as comparison with Quantum Gravidynamics is concerned, important differences occur both on a
kinematical and on a dynamical level, even if a variant of Gravidynamics formulated in terms of the
Ashtekar variables was used [156
]. Step one in the above quantization procedure keeps the
right-hand-side of the commutator
free from dynamical information. In any field theoretical
framework, on the other hand, one would expect the right-hand-side to be modified: minimally (if
and
are multiplicatively renormalized) by multiplication with a (divergent) wave function renormalization
constant, or (if
and
are nonlinearly renormalized) by having
replaced with a more general,
possibly field dependent, distribution. Stipulation of unmodified canonical commutation relations might
put severe constraints on the allowed interactions, as it does in quantum field theories with a
sufficiently soft ultraviolet behavior. (We have in mind here “triviality” results, where e.g. for scalar
quantum field theories in dimensions
a finite wave function renormalization constant
goes hand in hand with the absence of interaction [24, 77
]). A second marked difference to
Quantum Gravidynamics is that in Loop Quantum Gravity there appears to be no room for the
distinction between fine grained (‘rough’) and coarse grained (‘smooth’) geometries. The inner
product used in the second of the above steps sees only whether the graphs of two spin network
states coincide or not, but is insensitive to the ‘roughness’ of the geometry encoded initially in
the
pair. This information appears to be lost [151
]. In a field theory the geometries
would be sampled according to some underlying measure and the typical configurations are very
rough (non-differentiable). As long as the above ‘holonomy inner product’ on such sampled
geometries is well defined and depends only on the coincidence or non-coincidence of the graphs the
information about the measure according to which the sampling is done appears to be lost.
Every measure will look the same. This property seems to match the existence of a preferred
diffeomorphism invariant measure [14
] (on a space generated by the holonomies) which is uniquely
determined by some natural requirements. The typical
configurations are also of distributional
type [14, 140]. This uniqueness translates into the uniqueness of the associated representation of the
holonomy-flux algebra (which rephrases the content of the original
algebra). In a field
theory based on the
variables, on the other hand, there would be a cone of regularized
measures which incorporate dynamical information and on which the renormalization group
acts.
Another difference concerns the interplay between the dynamics and the canonical commutation
relations. In a field theory the moral from Haag’s theorem is that “the choice of the representation of the
canonical commutation relations is a dynamical problem” [99]. Further the inability to pick the ‘good’
representation beforehand is one way to look at the origin of the divergencies in a canonically quantized
relativistic field theory. (To a certain extent the implications of Haag’s theorem can be avoided by
considering scattering states and spatially cutoff interactions; in a quantum gravity context, however, it is
unclear what this would amount to.) In contrast, in the above holonomy setting a preferred representation
of the holonomy-flux algebra is uniquely determined by a set of natural requirements which do not refer to
the dynamics. The dynamics formulated in terms of the Hamiltonian constraint thus must be
automatically well-defined on the above kinematical arena (see [151, 173
] for a discussion of the
ambiguities in such constructions). In a field theoretical framework, on the other hand, the
constraints would be defined as composite operators in a way that explicitly requires dynamical
information (fed in through the renormalized action). So the constraints and the space on which
they act are dynamically correlated. In loop quantum gravity, in contrast, both aspects are
decoupled.
Finally, the microscopic action for asymptotically safe Quantum Gravidynamics is very likely different from the Einstein–Hilbert action and thus not of second order. This changes the perspective on a canonical formulation considerably.
String theory provides a possible context for the unification of known and unknown forces including quantum gravity [97, 177, 112]. As far as quantum gravity is concerned the point of departure is the presupposition that the renormalization problem for the quantized gravitational field is both insoluble and irrelevant. Presently a clearly defined dynamical principle that could serve as a substitute seems to be available only for so-called perturbative first quantized string theory, to which we therefore confine the following comparative comments.
In this setting certain two-dimensional (supersymmetric) conformal field theories are believed to capture (some of) the ‘ultimate degrees of freedom of Nature’. The attribute ‘perturbative’ mostly refers to the fact that a functional integral over the two-surfaces on which the theories are defined is meant to be performed, too, but in a genus expansion this gives rise to a divergent and not Borel summable series. (In a non-perturbative formulation aimed at degrees of freedom corresponding to other extended objects are meant to occur and to cure this problem.) For the relation to gravity it is mainly the bosonic part of the conformal field theories which is relevant, so we take the 2D fermions to be implicitly present in the following without displaying them.
A loose relation to a gravitational functional integral then can be set up as follows. Schematically, the
so-called low energy effective action arises by functional integration over the fields of a Riemannian
sigma-model
target pace with metric
. Thus
is a functional integral that depends parametrically on the metric
which is viewed as a set of
generalized couplings. The functional integral one (morally speaking) would like to make sense of in the
present context is however
Since arguments presented after Equation (1.5) suggest a kind of ‘dimensional reduction’ to
,
one might be tempted to see this as a vindication of string theory from the present viewpoint. However
string theory’s very departure was the presupposition that no fixed point exists for the gravitational
functional integral. Moreover in string theory the sigma-model fields relate the worldsheet to a
(
-dimensional) target manifold with a prescribed metric (or pairs thereof related by
T-duality). The at least perturbatively known dynamics of the sigma-model fields does not appear to
simulate the functional integral over metrics (see Equation (1.16
)). The additional functional
integral over Euclidean worldsheet geometries is problematic in itself and leaves unanswered the
question how and why it successfully captures or replaces the ultaviolet aspects of the original
functional integral, other than by definition. In the context of the asymptotic safety scenario, on
the other hand, the presumed reduction to effectively two-dimensional propagating degrees of
freedom is a consequence of the renormalization group dynamics, which in this case acts like a
‘holographic map’. This holographic map is of course not explicitly known, nor is it off-hand
likely that it can be described by some effective string theory. A more immediate difference is
that Quantum Gravidynamics does not require the introduction of hitherto unseen degrees of
freedom.
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