Holographic Wilson's RG

In an earlier paper (arXiv:1706.03371) a holographic form of the Exact Renormalization Group (ERG) evolution operator for a (perturbed) free scalar field (CFT) in D dimensions was formulated. It was shown to be equivalent, after a change of variables, to a free scalar field action in AdS_{D+1} space time. We attempt to extend this result to a theory where the scalar field has an anomalous dimension. Instead of the ERG evolution operator, we examine the generating functional with an infrared cutoff, and derive the prescription of alternative quantization by using the change of variables introduced in the previous paper. The anomalous dimension is thus related in the usual way to the mass of the bulk scalar field. Computation of higher point functions remains difficult in this theory, but should be tractable in the large N version.


I. INTRODUCTION
The idea of holography has been with us for some time since the publication of the first papers [1,2]. It became a mathematically precise idea with the discovery of the AdS/CFT correspondence [3][4][5][6] where an ordinary conformal field theory (N=4 Super Yang Mills) in D flat dimensions is conjectured to be dual to a gravity theory (IIB Superstrings) in AdS D+1 . By now much evidence has been collected for the correctness of this conjecture. This correspondence has a natural interpretation in string theory where there is a world sheet duality that relates open and closed strings. Nevertheless it is worth exploring to what extent string theory is required for a holographic AdS description of a CFT. String theory may be required for a UV completion on the gravity side. But the duality itself may be more general if we are only interested in an effective field theory description. It is certainly known that in some limits of the parameter space gravity is sufficient for the correspondence to be correct.
Another intriguing aspect of this correspondence is the possibility of interpreting the extra radial dimension as the renormalization scale of the boundary theory. This gives rise to the idea of "holographic" RG, in which radial evolution in the bulk gravity theory is identified with RG evolution of the boundary theory [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. It is natural to ask whether this identification can be made more precise, i.e., whether it is possible to derive the holographic RG equation from the RG equation of the boundary field theory. In an earlier paper [21] this was answered in the affirmative for the simple case of a free massless scalar field theory.
It was shown first that the evolution operator for Wilson's Exact Renormalization Group (ERG) [22][23][24], in the simpler Polchinski form [25], could be written as a functional integral of a D+1 dimensional field theory. (See [26][27][28][29] for reviews on ERG.) A change of field variables then transformed this operator into the action for a free scalar field theory in AdS D+1 . A contact was thus made with the standard AdS/CFT methods for the calculation of twopoint correlators. One important point is that the bulk field took the value of the boundary field at the boundary (rather than the source) so this is more naturally understood as the alternative quantization procedure introduced in [16].
In the present paper we change the course of approach a little by considering the generating functional of correlation functions with an infrared cutoff [30][31][32] instead of the ERG evolution operator. The generating functional is closely related to a Wilson action, and it reduces to the ordinary generating functional in the limit of the vanishing infrared cutoff.
We follow section 5.2 of [21] by introducing an elementary scalar field of scale dimension between (D − 2)/2 and D/2 to represent a composite field. We then construct a quadratic Wilson action that gives the expected two-point function with an anomalous dimension.
Normally anomalous dimensions arise due to interactions. However it is very hard to write down fixed point Wilson  Having constructed a fixed point Wilson action, we construct a corresponding generating functional with an infrared cutoff following a recipe well known in the ERG literature. Using the same change of variables that we introduced in 2.3 of [21], we rewrite the generating functional to reproduce the prescription of the alternative quantization [16] (reviewed nicely in Appendix of [15]) for computing the two-point function.
This paper is organized as follows. In Sec. II we overview the ERG formalism to introduce a quadratic Wilson action that gives a two-point function with an anomalous dimension. We then introduce a corresponding generating functional with an infrared cutoff in Sec. III to apply the change of variables of [21]. By a judicious adjustment of the change of variables, we can derive the prescription of alternative quantization. We discuss our result and method in Sec. IV. In Sec. V we give some background regarding anomalous dimension in ERG and its connection with the change of variables used in [21]. Sec. VI contains a preliminary discussion of a situation where one might obtain a nontrivial (i.e. cubic and higher order ) bulk action starting from a generalized ERG equation. We conclude the paper in Sec. VII.

A. ERG formalism
For the convenience of the reader, we would like to collect relevant background material from the exact renormalization group formalism (ERG).
Let S Λ [φ] be a Wilson action of a generic scalar field theory. To preserve physics independent of Λ, we impose the ERG differential equation where The cutoff function K Λ (p) has three properties: 3. it approaches 0 rapidly for p 2 ≫ Λ 2 .
An example is K Λ (p) = exp (−p 2 /Λ 2 ). We denote the correlation functions by The ERG differential equation (1) implies that the correlation functions defined below are independent of Λ: where The modification of the two-point function by k Λ (p)/p 2 does not affect the physics in the infrared as long as k Λ (p) vanishes as p 2 at p 2 = 0. (In other words, k Λ (p)/p 2 does not correspond to the propagation of a free massless particle.) We then define the generating functional W[J ] of the connected correlation functions by This can be written as a functional integral in the presence of a source term: where We now define a field and We then obtain Since we obtain Hence, we obtain [30,31,33] We can think of W Λ [J] as the generating functional with an IR cutoff Λ. [30,31] Its Λ-dependence can be obtained from the ERG differential equation (1) as which can be solved as Since R Λ 1 (p) − R Λ 2 (p) is non-vanishing mainly for Λ 2 < p < Λ 1 , the above equation implies by integrating fluctuations of momenta between Λ 2 and Λ 1 . In the limit Λ → 0+, all the momentum modes are integrated to give (14).

B. Quadratic Wilson action with an anomalous dimension
We now consider a simple quadratic Wilson action with an anomalous dimension: where η = 2γ (0 < γ < 1) is a positive anomalous dimension, and µ is an arbitrary reference momentum cale. This action reproduces a two-point function with an anomalous dimension: The corresponding generating functional with an IR cutoff Λ is given by where 1/ p 2 (p/µ) −η + R Λ (p) is the high-momentum propagator, or the two-point function with an infrared cutoff. Using (13), we get Please observe that for small p ≪ Λ the action is approximately given by which is a non-analytic (non-local) action. We have two comments: 1. For an elementary field φ, we expect the action is analytic at zero momentum, and any non-analyticity comes from interactions. As φ, we have composite fields in mind.
2. For example, in the massless free theory, the composite field 1 2 φ 2 has scale dimension D − 2 so that the anomalous dimension, compared with the free elementary field, is 0 < γ < 1 implies 2 < D < 4. This is an example of our φ.
We will discuss the first point further in Sec. IV.
C. ERG formalism with explicit dependence on the anomalous dimension Then, (15) implies thatW Λ [J] satisfies the alternate ERG equation with an explicit dependence on η: whereR To derive (23) from (15), we have used The integral formula (16) implies the corresponding integral formula: (13) can now be written as and we obtain, from (14), So far, we have not assumed that W Λ [J] is quadratic and given by (19). Let us now assume it so thatW This has no more dependence on the reference momentum µ. For the quadraticW Λ [J], (23) reduces to gives As the original ERG equation (15) does not single out a particular η in (19), the ERG equation (23) with an explicit dependence on η has a more general quadratic solution given byW where ∆η is an arbitrary shift of the anomalous dimension. Given (23), what makes (29) stand out? It is the relation to an RG fixed point. Let us elaborate on this a little. We assume that the cutoff functionR Λ (p) has a particular cutoff dependence: We define a dimensionless fieldJ with the dimensionless momentump. We then definē where We find thatW [J] is a fixed point action, satisfying the equation for scale invariance At the same time it also satisfies the equation for special conformal invariance [34] pJ In Appendix C we show how to derive (35) either from (37) or from (38).

III. DERIVATION OF THE ALTERNATIVE QUANTIZATION
To simplify our notation, we omit the tilde fromW altogether, and consider Let Λ 1 = 1 ǫ be a large cutoff, and Λ 2 = 1 z 0 be a small cutoff, compared with a reference momentum scale µ. Our goal is to derive the prescription of the alternative quantization of AdS/CFT from the integral formula (26): Since (39) is quadratic, we can expand and rewrite the above as Rewriting the integration variable as J ′ (p) to normalize the JJ ′ term, we obtain where we have defined With G z (p), we can write Using (27) lim we obtain the infrared limit This is obviously independent of ǫ. (This is in fact thanks to the scale invariance (37).) We wish to rewrite (42) in the AdS form by using the change of variables introduced in [21]. We first rewrite by introducing a field y(z, p) in D + 1 dimensions as where the boundary values of the field y(z, p) are fixed by Then, following 2.3 of [21], we change field variables from y(z, p) to Y (z, p) defined by where the positive function f (z, p) is defined by (Note that the mass dimensions of y(z, p), where we have used Y (z 0 , p) = 0.
We now choose f (z, p) to satisfy where m 2 is a squared mass parameter. This gives where the action is defined for a massive scalar field Y in the D + 1-dimensional AdS space with radius of curvature 1 µ . Let us solve (53), which amounts to The general solution is given by where and I ν , K ν are the modified Bessel functions. We will shortly determine the functions of momenta A(p), B(p) (both with the mass dimension 1) and the value of ν (equivalently m 2 µ 2 ). Now, (51) and (53) imply that G z (p)/f (z, p) satisfies the same differential equation as (Note the mass dimension of C, D is −1.) Moreover, (51) gives Substituting (57) and (59) into the above, and using the Wronskian we obtain We can determine the coefficient functions A(p) to D(p), and the constant ν as follows.
From (57) and (59), we obtain Let us consider the limit z → 0+. We must find since this corresponds to the integration of no momentum mode. Hence, we obtain This gives Next, consider the limit z → +∞. From lim z→∞ G z (p) = 1 we obtain D(p) B(p) = 1 Combining the three equations (62, 66, 68), we obtain where we have taken c to be a constant for simplicity.
To determine c, we must examine f (z, p). From (57), we obtain Demanding that the change of variables from y(z, p) to Y (z, p) = y(z, p)/f (z, p) be analytic at p 2 = 0, we must first choose Then, since we must choose so that is analytic at p 2 = 0.
Note that the resulting high-momentum propagator G z (p) = 1 is analytic at p 2 = 0 as long as z is finite. Only as z → +∞, we find non-analyticity: To summarize so far, for the choice of the high-momentum propagator (76), we can rewrite (42) as where the AdS action is given by (55). By construction, (78) is independent of ǫ, but it depends on z 0 .
In the limit z 0 → +∞, we obtain the generating functional: Since this is independent of ǫ, we can also take the limit ǫ → 0+. Using we obtain, for ǫ → 0+, where In the literature it may be more common to write αJ ′ (p) as J ′ (p) and 1 α J(p) as J(p) so that We then obtain This reproduces the prescription of the alternative quantization of the AdS/CFT correspondence [16] (reviewed nicely in Appendix of [15]) for computing the two-point function.
In [21] it was pointed out that when the ERG equation is mapped to AdS space there remains a boundary term depending on the function f (p). We see this in (85) also. But note that it is analytic in p and therefore does not affect the all important non-analytic piece.

IV. DISCUSSION
We have managed to derive the AdS/CFT correspondence from ERG, but our derivation is not without faults. We discuss three issues here.
A. Non-analyticity of the Wilson action at zero momentum As we have already pointed out in Sec. II, our Wilson action (17) is not analytic at p 2 = 0 (hence non-local) due to the anomalous dimension η. This is partially because we are treating a composite field as an elementary field φ. Even in the free massless theory in D dimensions, the composite field φ 2 has scale dimension D − 2 so that its anomalous dimension is (D − 2)/2 compared with the canonical scale dimension (D − 2)/2 of φ.
Another reason for the non-analyticity is that we are not taking interactions into account.
Consider a Wilson action whose quadratic part is given by where the cutoff dependence of the squared mass m and it reproduces the same two-point function in the limit Λ → 0+, if we assume B. Non-analyticity of the cutoff function K In the ERG formulation, the choice of a cutoff function K(p/Λ) is totally arbitrary as long as it satisfies K(0) = 1, and it decreases rapidly for p > Λ. But we usually assume K(p/Λ) to be analytic at p 2 = 0: Now, in Sec. III we have chosen a particular cutoff function so that This is obtained from (76). This implies (Note that this has the form (33).) We then obtain, from (8), is analytic at p 2 = 0 since 2ν = 2 − η. Therefore, we find This is not analytic. Since no physics depends on the choice of K(p/Λ), one could argue, perhaps, that using a cutoff function non-analytic at p 2 = 0 is acceptable.

C. Absence of interactions
We have already discussed the importance of introducing interactions to restore the analyticity or locality of the Wilson action. From the ERG perspectives, it is natural to introduce interactions only at the boundary z = ǫ of the AdS space. Whether or not interactions are induced in the bulk of the AdS space is left for a future study. We make some preliminary remarks on this in Section VI. As pointed out originally by Wilson and others, (see for instance, [22] or Bell and Wilson, [35,36]), there are two important aspects in an RG -one is the coarse graining and the other is a rescaling of the field variables. In [35,36], they exemplify this with the following simple transformation, T (in their notation): The initial field variable is σ and the final field variable if S. The subscript on the field variable has changed from q/2 to q. q is dimensionless and this corresponds to a change of the cutoff from Λ → Λ/2. This is the coarse graining. The factor b is the rescaling. After n steps the rescaling becomes b n . The factor a can be changed by a scaling of both σ and S, and has no effect on the physics.
In a field theory the field variables are integrated over. However, if one wants to make the Hamiltonian mathematically form invariant under the ERG, then a particular rescaling parameter (b in the above example) has to be chosen -the precise value depends on the details of the interacting theory. This is a choice of field normalization. A natural choice is to make sure that the kinetic term has a fixed normalization.
If this normalization is implemented then the fixed point Hamiltonian is mathematically identical after the transformation. This is convenient in an actual calculation because requiring that the Hamiltonian be mathematically identical after an RG transformation leads to a well defined mathematical "fixed point equatio". [38] Let us now go to the connection between b and the anomalous dimension. t . If we denote by x i the initial variable (σ in the above example) and x f the final variable (S in the above example) then the transformation (96) can be written in the form of a time dependent rescaling as (the notation used in [21] where momentum labels are suppressed) The conclusion we reach is that to determine the anomalous dimension of the evolution This also suggests that if we do a field redefinition involving a time dependent rescaling, of the form x(t) = y(t)e µt , the scaling dimension is changed by an amount µ. Thus an ERG with anomalous dimension can be related to one without anomalous dimension, by such a field redefinition. If one wants a mathematical fixed point the anomalous dimension has to be chosen correctly. The precise value will depend on the interactions.
This then answers the question raised at the beginning of this section: In [21] the starting point was Polchinski's ERG without anomalous dimension. The integrating kernel is of the With the change of variables x = f y we obtain We have seen that f ≈ e −( D−2+η) 2 t . Clearly the ERG equation obeyed in the new variables will have an anomalous dimension parameter. This explains the appearance of this parameter in the AdS equation in [21]. In Sec. II C we have already seen how (23) is related to (15) by change of variables.
In this section we have shown the role of field redefinitions (or "wave function renormalization" in perturbative calculations) in introducing anomalous dimension in an ERG equation. This is important for locating the mathematical fixed point of the equation. We have also seen that the dimension can be read off from the integral formula. Some other examples of these are given in Appendix A.

VI. NONTRIVIAL FIXED POINT ACTION
In this section we consider a nontrivial fixed point action. To begin with we use the usual Polchinski ERG formalism. The kinetic term is 1 2 x 2 G −1 and the interacting part is S 0 .
Let the perturbation be S 1 so that the full action is Then in our earlier notation, the "wave functions" are given by Polchinski's equation is What is special is that S 0 by itself satisfies Polchinski's equation -eventually it will be taken to be a fixed point solution. Thus we have the following two equations: and Subtracting (101) from (102) we get Since S 0 is a solution of (101), its form is (in principle) known as a function of time. In the case that S 0 is chosen to be a fixed point solution, its time dependence can be specified very easily: expressed in terms of rescaled and dimensionless variables it has no time dependence.
This is equivalent to saying that the dimensionless couplings are constant in RG-time, t, i.e., they have vanishing beta functions. One can work backwards and determine the exact t-dependence in terms of the original variables. (103) can be used to define a modified Hamiltonian evolution equation for the wave function ψ ′′ = e −S 1 (x,t) : Note that the term involving S 0 is like a gauge field coupling -in fact it is "pure gauge".
The Action functional corresponding to this Hamiltonian is derived in Appendix B using canonical methods. While it involves more algebra, it can be applied even when S 0 is not a solution of the ERG equation.
In the end the result can be summarized very simply: Start with the usual RG evolution Take S 0 (x(t f )), t f ) into the RHS to get Introduce the evolution of S 0 into the functional integral by writing it as an integral of a total derivative: to get Thus the evolution of the perturbation involves an evolution operator that has a non linear term, but is in fact a total derivative.
The total derivative can also be rewritten in other forms using the ERG for S 0 . Thus for instance (using field theory notation) it can be rewritten as (see (B)): This paper discusses only perturbations involving an elementary scalar field. Equally interesting are questions involving composite fields. This was also considered in a general way in [21]. Once again a large N expansion is required to do these computations.
The role of dynamical gravity has not been discussed thus far. This needs to be addressed. dependence suppressed is: Another ERG equation with anomalous dimension was written down in [40], [41] which, in simplified notation, is: The integral evolution operator for both these have the same form: Comparing with the form (98) we see that e and (A2): We now put back the momentum dependence and go back to more standard notation: In the Polchinski case from (A4) For Λ 2 < Λ 1 , this is solved by the kernel substituting in (A3): where the evolution operator is given by Similarly (A2) [40], [41], can be obtained by the same substitution.
The integrating kernel remains the same as in (A9), but R is defined by the more complicated relation e −ηt d dt ( In each case we see that the anomalous dimension can be read off from the powers of Λ multiplying the fields in the exponent in (A9).

Appendix B: Action for the Nontrivial Fixed Point Hamiltonian
In this appendix we derive (109) using Hamiltonian methods.
Our starting point is Note that in principle one can postulate this as an ERG equation even if S 0 is not a solution of Polchinki's ERG equation as assumed in Section VI. To that extant the derivation in this Appendix is more general.
We rotate to Minkowski space: it M = t E . Hence, Thus, S E = −iS M . Thus in the above case S 0 = −iS 0M . Let −τ E = G.
Writing in terms of P = −i ∂ ∂x we get finally for the Hamiltonian in Minkowski space-time, This givesẋ = P + A x = P + ∂S M ∂x and one can obtain using L = xṖ − H an action Here A x (x) is to be understood as A x (x(t)) and ∂Ax ∂x = ∂Ax(x(t)) ∂x (t) .
In Euclidean space this becomes Now reintroduce G: dG = dt E dG dt E = dt EĠ : i)