## A Reminder on Kadanoff–Wilson Renormalization

The modern view of renormalization has been shaped by Kadanoff and Wilson. See [114] and [234231232233] for first hand accounts and a guide to the original articles. In the present context the relevance of a Kadanoff–Wilson view on renormalization is two-fold: First it allows one to formulate the notion of renormalizability without reference to perturbation theory, and second it allows one to treat at least in principle renormalizable and non-renormalizable theories on the same footing. For convenience we briefly summarize the main principles of the Kadanoff–Wilson approach to renormalization here:

Kadanoff–Wilson view on renormalization – Main principles:

1. A theory is not defined in terms of a given action, but in terms of a field content and the Steps 26 below.
2. The functional integral is performed in piecemeal, integrating out fast modes, retaining slow modes, while keeping the values of observables fixed. This “coarse graining” process results in a flow in the space of actions which depends on the chosen coarse graining operation.
3. Starting from a retroactively justified initial action ideally all interaction monomials generated by the flow are included in a typical action; in any case many more than just the power-counting renormalizable ones. Then one classifies the coefficients of the monomials into essential (couplings) and inessential (field redefinitions).
4. A fixed point (FP) in the flow of couplings is searched for. The position of the FP depends on the chosen coarse graining operation, but the rates of approach to it typically do not (“universality”).
5. The flow itself decides which monomials are relevant in the vicinity of a FP and hence defines the dynamics. The scaling dimensions with respect to a non-Gaussian FP may be different from (corrected) power-counting dimensions referring to the Gaussian FP.
6. The dimension of the unstable manifold and hence the “degree” of renormalizability depends on the FP!

We first add some general remarks and then elaborate on the Points 16.

The more familar perturbative notion of renormalizability is neither sufficient (e.g.  theory in ) nor necessary (e.g. Gross–Neveu model in ) for renormalizability in the above sense.

As summarized here, these principles describe the construction of a so-called massive continuum limit of a statistical field theory initially formulated on a lattice, say. A brief reminder: In a lattice field theory there is typically a dynamically generated scale, the correlation length , which allows one to convert lattice distances into a physical length scale, such that say, lattice spacings equal . The lattice points are then traded for dimensionful distances . Taking the lattice spacing to zero amounts to sending to infinity while keeping fixed. If the correlation functions of some lattice fields are rescaled accordingly (including a ‘wave function’ renormalization factor) and the limit exists, this defines a massive continuum limit of the lattice theory.

Let us now elaborate on the various points. The comments are of a generic nature, whenever a formula is needed to make the point, we consider the case of a scalar field theory on a -dimensional Euclidean lattice with lattice spacing and . Then denotes the scalar field multiplet at point and is its Fourier transform. We freely combine results and viewpoints from the following reviews [1022177146111].

1. This will become clear from Point 5 below.
2. The rationale for the piecemeal performance of the functional integral is that in statistical mechanics language a critical problem is decomposed into a sequence of subcritical ones. Here a critical problem is one where fluctuations of the dynamical variables over vastly different length scales have to be taken into account; for a subcritical problem the opposite is true. In more detail, let be a function of the fields whose functional average is meant to be a macroscopic observable, but whose statistical average is sensitive to fluctuations of the microscopic fields on very different length scales. The replacement by a sequence of subcritical problems is done by specifying a “blocking” kernel , , such that
1. has support mostly on configurations with .
2. .

Then

with
Taking defines the coarse grained action functional , after which Equation (A.2) can be used to define the coarse grained observables . Property 1 entails that only field configurations with a similar ‘degree of roughness’ have to be considered in evaluating the functional integral in Equation (A.2). It should thus be much more amenable to (numerical or analytical) approximation techniques than the original functional integral (A.1).

Once Equation (A.2) has been evaluated one can iterate the procedure. The formulas (A.1, A.2) remain valid with the basic kernel replaced by its -fold convolution product, for which we write . For most choices a kernel will not be reproducing, i.e.  will not (despite the suggestive notation) coincide with the original kernel , just with modified parameters. Technically it is thus easier to specify the iterated kernel directly, which is of course still normalized. The -fold iterated kernel will have support mostly on configurations with , if , and is the fraction of the momentum modes over which the functional integral has been performed after iterations. In the above terminology the critical problem (A.1) has been replaced by the sequence of subcritical problems (A.2). In each iteration, referred to as a coarse graining step defined by the kernel , only a small fraction of the degrees of freedom is integrated out. The action at the cutoff scale is called the microsocopic (or bare) action, the reached after integrating out the ‘fast’ modes in the range is called the coarse grained action at scale , and similarly for the fields . Note that the action as a functional is defined for all field configurations, though for the evaluation of Equation (A.1) only is needed.

Throughout we shall follow the sloppy field theory convention that the coarse graining operates on the action. Of course what really gets updated is the functional measure

In the (lattice) regularized theory the decomposition of the measure into a flat reference measure and a Boltzmann factor parameterized by the action is unproblematic. The flow in the measures can thus be traded for a flow in the actions (as long as the Jacobian is taken into account that comes from the reference measure upon a change of field variables ). The Wilsonian “space of actions” refers to a cone of positive measures (A.3) which is preserved under the coarse graining operation considered.

Using Equation (A.2) for the kernel , one readily gets the flow equation

For a given coarse graining kernel and a given initial action this flow equation in principle determines the flow of actions . Of course the usefulness of such a flow equation will largely depend on a good choice for the kernel. In particular the kernel should be ‘almost diagonal’ in the field configurations , so that the multiple integrals are replaced by something simpler, typically a single remaining momentum integral. Specific action-dependent choices for such kernels lead to the various Wilson type (Wegner–Houghton, Polchinski) flow equations for , which have been employed in the literature (see [130] for the kernel giving Polchinski’s version and [19221130] for the relation to field redefinitions). Flow equations of this type are known as (exact) functional renormalization group equations (FRGEs) of Wilsonian type.

For the sake of contradistinction let us already mention here another type of flow equations, which is formulated in terms of the generating functional for the vertex (1-PI) functions and which uses a mode suppression scheme rather than a coarse graining procedure, namely the effective average action. To set up a functional renormalization flow one does not specify the coarse graining flow by iteration of a 1-step kernel, but rather starts from a functional integral of the form

It differs from the standard one (formally) defining the generating functional for the connected correlation functions only by the presence of the factor, where can be thought of as a -dependent modification of the bare action. The factor later is chosen such that it suppresses the momentum modes with , while the modes with are integrated out unsuppressed. The response to a variation in the extra scale allows one to write down flow equations for
Here is the usual effective action, which computed for is just the Legendre transform of in Equation (A.5). This flow equation contains to a certain extent the same information as the original functional integral. This framework will be described in more detail in Appendix C.
3. In general the functional form of the action will change drastically in each coarse graining step . A way to keep track of the change is to consider all (or sufficiently many) interaction monomials , , compatible with the symmetries of the theory. As an organizing principle one can take the number of derivatives (derivative expansion) or the power of the field (see Equation (A.7)). The only constraints on the possible terms come from symmetry requirements, here e.g. Euclidean invariance or evenness under , as well as locality requirements for the putative fixed point action. ‘Sufficiently many’ means that the action can be replaced with a good degree of accuracy (as far the evaluation of observables is concerned) by its expansion in . In this sense, we shall refer to the set loosely as a ‘basis’ in the space of interaction monomials. Importantly it will typically include many more monomials than just the power-counting renormalizable ones. Different bases may be related nonlinearly and nonlocally. For example often an explicit parameterization through running vertex functions is used, which upon separation of the dependence, , is formally equivalent to a parameterization in terms of monomials :
The upshot is that the flow in the action functionals is replaced by a flow in the parameters. The parameters may be dimensionful, in which case a dependence on both and (not just on the ratio) enters. For later reference we also indicated the dependence on the initial values at scale . Our conventions will be such that carries mass dimension and mass dimension . The kinetic term and its coefficient are taken to be dimensionless which fixes to have mass dimension in Euclidean dimensions.

The parameters can be classified into essential and inessential ones. A parameter combination is called inessential if the response of the bare Lagrangian to a change in it can be absorbed by a field reparameterization. Explicitly the existence of an inessential parameter combination is signaled by the fact that there exists (locally) a vector field on coupling space such that

for some possibly nonlocal functional of the fields . The right-hand-side is the response of the measure (A.3) with respect to a field reparameterization . The concept of field reparameterizations is familiar from power counting renormalizable field theories where linear field parameterizations give rise to wave function renormalizations, e.g.  for . In the Kadanoff–Wilson setting the bare Lagrangian contains arbitrary interaction monomials and as a consequence also nonlinear and nonlocal field reparameterizations are allowed (see [19221] for further discussion). The first term in Equation (A.8) comes from expanding , with , the second term from expanding the Jacobian . For the mode suppressed generating functional (A.5) one obtains a field reparameterization (pre-) Ward identity ([237], see Section 5.4.3)
where in all terms are evaluated at the same point.

The number of commuting linearly independent vector fields with the property (A.8) is a characteristic of the Lagrangian, and as in [227] we shall assume that one can choose adapted coordinates such that , . The remaining parameters , , are called essential parameters or coupling constants. By definition they are such that is a diffeomorphism. For convenience we also assume that they have been made dimensionless (with respect to mass dimensions) by a redefinition , if is the mass dimension of . The dimensionless couplings then depend only on the ratio and we write , , . Note that this affects the flow direction: Decreasing (increasing number of coarse graining steps) corresponds to increasing . The variable offset is useful because by making it large it formally allows one to ask “where a coarse graining trajectory comes from”. For large the offset can roughly be identified with the (logarithm of the) renormalization point used in perturbative quantum field theory. Under the above conditions the parameter flow of the will typically decompose into an autonomous flow equation for the couplings and a non-autonomous flow equation for the inessential parameters. That is decomposes into

The beta functions (which carry no explicit dependence as the are dimensionless) define a vector field ; its integral curves are the renormalization group trajectories. The flow equations (A.9) of course depend on the chosen coarse graining operation.
4. A fixed point is a zero of the beta functions, . The position of thus also depends on the chosen coarse graining operation. A fixed point is called a UV fixed point for a given trajectory if this trajectory emanates from the fixed point, i.e. . Formally this can be described by viewing the flow equation for the couplings as one in the offset parameter , viz. . The condition for a UV fixed point then translates into . A fixed point is intrinsic to a given coarse graining flow to the same extent that defines a (coordinate independent) vector field.

Certain inessential parameters are still allowed to ‘run’ at the fixed point of the couplings. As a consequence the action with is not unique, rather the fixed point in the couplings corresponds to a submanifold of fixed point actions . More precisely the class of field redefinitions which commute with the given coarse graining operation will give rise to marginal perturbations (see below) of the fixed point and typically the vector space spanned by these marginal perturbations coincides with the tangent space of at the fixed point [77]. In this case is unique modulo reparameterization terms (like the ones on the right-hand-side of Equation (A.3)) and we shall refer to it as ‘the’ fixed point action.

Most statistical field theories have at least one fixed point, the so-called Gaussian fixed point. This means there exists a choice of field variables for which the fixed point action is quadratic in the fields, i.e. only the terms in Equation (A.4) are nonzero. In a local field theory the fixed point action will typically be local, here , but more generally one could allow for nonlocal ones, here e.g. with smooth.

Given a fixed point and a coarse graining operation one can (under suitable regularity conditions) decompose the space of actions (the cone of measures) into a stable manifold and an unstable manifold. All actions in the stable manifold are driven into the fixed point. The set of points reached on a trajectory emanating from the fixed point is called the unstable manifold; any individual emanating trajectory is called a renormalized trajectory. The stable manifold is typically infinite-dimensional; this corresponds to the infinitely many interaction monomials that die out under the successive coarse graining. The dimension of the unstable manifold is of crucial importance because it determines the “degree of renormalizability”.

So far the entire discussion was for a fixed coarse graining operation, , say. All concepts (fixed point, stable and unstable manifold, etc.) referred to a given . If one now changes , the location of the fixed point will change in the given coordinate system provided by the essential couplings. The set of points reached belongs to the critical manifold [77]. One aspect of universality is that the rates of approach to the fixed point are typically independent of the choice of . More generally all quantities defined through a scaling limit are expected to be independent of the choice of (within a certain class). Limitations may arise as follows. One parametric families of coarse graining operations may have ‘bifurcation points’ where the dimension of equals the number of independent marginal perturbations for and is smaller for . One expects the emergence of new fixed points (or periodic cycles) at such bifurcation points. The physics interpretation of may be the (analytically continued) dimension of the system or the range of the interaction.

5. A lower bound on the dimension of the unstable manifold is obtained from a linearized analysis. The tangent spaces to the stable and unstable manifold at the fixed point are called the spaces of relevant and irrelevant perturbations, respectively. In terms of the couplings this amounts to the familiar criterion in terms of the right eigenvectors and eigenvalues of the stability matrix
The solution of the linearized flow equation in Equation (A.5) reads , for constants . One should add that is often degenerate and that it is not necessarily symmetric. The eigenvectors are thus not assured to span the tangent space at the fixed point and the eigenvalues are not assured to be real. Whatever eigenvectors has, however, they are of interest. The space of irrelevant perturbations is spanned by the eigenvectors with ; the linearized coupling perturbations then decay exponentially in . Similarly the space of relevant perturbations is spanned by the eigenvectors with ; for them grows exponentially in . The borderline case are marginal perturbations spanned by the eigenvectors with . For them a linearized analysis is insufficient and a refined analysis is needed to decide whether is driven towards (respectively away from) the fixed point for increasing , in which case is sometimes said to be marginally irrelevant (respectively marginally relevant).

The significance of the stability matrix can be illustrated at the Gaussian fixed point. If the fixed point action is not just quadratic in the fields but also local, in Equation (A.4), say, the eigenvalues and eigenvectors of the stability matrix reproduce the structure based on mass (or power counting) scaling dimensions. If one adopts a parameterization where the local Gaussian fixed point is described by , the matrix (A.10) has a set of right eigenvectors whose eigenvalues are , where are the mass dimensions of the dimensionful couplings . In the setting of Equation (A.1, A.2) this amounts to the following. Consider a coarse graining transformation where momenta in the range are integrated out. To every monomial with mass dimension there corresponds an eigenoperator whose highest dimensional element is and the corresponding eigenvalue is . Here one can see directly that the monomials which are irrelevant with respect to a local Gaussian fixed point (those which ‘die out’ under successive coarse graining operations) are the ones with mass dimension . For example with the conventions set after Equation (A.4) an term has mass dimension in Euclidean dimensions.

On the other hand the amount of information that can be extracted from the stability matrix is often limited by the fact that it is degenerate. For illustration let us consider some simple examples. It is convenient to consider the flow as a function of the off-set so that is the appropriate flow equation. Let us assume that (for reasons of positivity of energy, say) the couplings are required to be non-negative. In the case of a single coupling, then has the fixed point , but with the upper sign the unstable manifold is one-dimensional, while with the lower sign the unstable manifold is empty. Indeed the solution is in the first case positive for all and approaches the fixed point for , while with the lower sign the fixed point cannot be reached with positive values of the coupling. Both are ‘paradigmatic’ situations mimicking the perturbative behavior of a Yang–Mills coupling and a coupling, respectively. Note that in both cases the stability matrix (A.10) vanishes identically, so the attempt to gain insight into the unstable manifold via the linearized analysis already fails in this trivial example. A multi-dimensional generalization is , where the unstable manifold of the fixed point consists only of the halfline , (of co-dimension ).

An important case when a linearized analysis is insufficient is when the number of independent marginal perturbations is larger than the dimension of the tanget space to (see the remark at the end of Comment 4). One may then be able to enlarge the stable or the unstable manifold (or both) by submanifolds of points which are driven towards or away from it with ‘vanishing speed’.

Returning to the general discussion, a schematic pattern of a coarse graining flow in the vicinity of a fixed point is shown in Figure 6. Individual flow lines starting outside the stable manifold will in general first approach the fixed point, without touching down, and then shoot away from it. In order to (almost) touch down at the fixed point the initial values have to be carefully fine tuned. With ideal fine tuning the trajectory then splits into two parts. One part that moves within the stable manifold into the fixed point and another part that emerges from it. The latter has the fixed point couplings as its initial values, , and is called a renormalized trajectory; with repect to it the fixed point is an ultaviolet one. Its physics significance is that the actions associated with points on the renormalized trajectory are perfect, in the sense that the effect of the cutoff on observables is completely erased, even when the couplings are not close to their fixed point values. More realistic than the construction of perfect actions is that of improved actions, designed such that for given values of the couplings the cutoff effects are systematically diminished (see [102] for further discussion). In order to identify a renormalized trajectory the initial values of the relevant couplings have to be fine tuned. A statistical field theory for which this amounts to a manifestly finite-dimensional problem is called strictly renormalizable in the ultraviolet, otherwise we suggest to call it weakly renormalizable.

UV strict (weak) renormalizability:

1. There exists an UV fixed point, i.e.  and .
2. Its unstable manifold is finite (infinite)-dimensional.

Importantly this is a nonperturbative definition of renormalizability. The averages defined by backtracing a perfect action along the renormalized trajectory into the fixed point have all desirable properties: They are independent of the cutoff scale and independent of all the irrelevant couplings, in the sense that they become computable functions of the relevant couplings. Whenever the unstable manifold is well defined as a geometric object, its dimension and structure is of course a coordinate independent notion. In practice the existence of (patches of) the unstable manifold is only established retroactively once a good ‘basis’ of interaction monomials has been found. Then a large class of other choices will be on an equal footing giving the concept a geometric flavor. The condition of renormalizability as discussed so far only singles out a subclass of regularized (discretized) statistical field theories. We shall outline below why and how this subclass can be used to construct continuum quantum field systems with certain robustness properties. Before turning to this we complete our list of comments on the Kadanoff–Wilson renormalization.

6. This is a direct consequence of Point 4. The ‘degree’ of renormalizability is the dimension of the unstable manifold, which depends on the fixed point considered.

In the framework of statistical lattice field theories the importance of the construction principle 16 lies in the fact that it allows one to construct a scaling limit without introducing uncontrolled approximations. Usually this employs the concept of a correlation length and of the critical manifold as the locus of all points in the space of measures with infinite correlation length. In a gravitational context it is not obvious how to adapt this notion adequately, which is why we tried to avoid direct use of it, see the discussion at the end of Point 4. In statistical field theories a scaling limit is usually constructed in terms of multipoint functions of the basic fields. The lattice spacing is sent to zero while simultaneously the couplings are moved back into the fixed point along a renormalized trajectory. As mentioned before we are interested here in massive scaling limits, meaning that among the systems on the renormalized trajectory used only the one ‘at’ the fixed point is scale invariant. In the case of quantum gravity the multipoint functions of the basic fields might not be physical quantities, so the appropriate requirement is that a scaling limit exists for generic physical quantities (see Section 1.1).

Initially all concepts in the construction 16 refer to a choice of coarse graining operation. One aspect of universality is that all statistical field theories based on fixed points related by a change of coarse graining operation have the same scaling limit. One can thus refer to an equivalence class of scaling limits as a continuum limit. The construction then entails that whenever a continuum limit exists physical quantities become independent of the UV cutoff and of the choice of the coarse graining operation (within a certain class).

So far we did not presuppose unitarity/positivity of the resulting quantum field system. For a Euclidean statistical field theory, however, reflection positivity of the class of actions used provides an easily verifyable sufficient condition for the existence of a positive definite inner product on the physical state space of the theory. This is why relativistic unitary continuum quantum field theories (QFTs) can be constructed along the above lines in a way that does not require uncontrolled approximations from the outset. Of course the rigorous construction of a relativistic unitary QFT along these lines remains an extremely challenging problem and has only been achieved in a few cases. However numerical techniques often allow one to verify the renormalizability properties to good accuracy. The resulting QFT then is renormalizable in the above sense “for all practical purposes”; an example are Yang-Mills theories. As a consequence the extracted continuum physics will have the desired universal properties for all practical purposes likewise.

This concludes our ‘birds eye’ summary of basic renormalization group concepts. The concrete implementation in a given field theory quickly becomes fairly technical. A key problem is to obtain mathematical control over the scale dependencies and these are often extremely hard to come by (see e.g. [195]). Outside a dedicated group of specialists one usually resorts to uncontrolled approximations. In this review we do so likewise. First, because of the early stage in which the investigation of the ‘gravitational renormalization group’ is, and second, because also the controlled approximations – where they exist – often draw from experience gained from uncontrolled approximations.