Exact Form of Boundary Operators Dual to Interacting Bulk Scalar ...

arXiv:1702.00038v1 [hep-th] 31 Jan 2017

Prepared for submission to JHEP

February 2, 2017

Exact Form of Boundary Operators Dual to Interacting Bulk Scalar Fields in the AdS/CFT Correspondence

Gabriele La Navea Philip W. Phillipsb a b

Department of Mathematics, University of Illinois, Urbana, Il. 61820 Department of Physics and Institute for Condensed Matter Theory, University of Illinois, 1110 W. Green Street, Urbana, IL 61801

Abstract: Using holographic renormalization coupled with the Caffarelli/Silvestre[1] extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to a bulk scalar field is an anti-local operator, namely the fractional Laplacian. The propagator associated with such operators is of the general powerlaw (fixed by the dimension of the scalar field) type indicative of the absence of particle-like excitations at the Wilson-Fisher fixed point or the phenomenological unparticle construction. Holographic renormalization also allows us to show how radial quantization can be extended to such non-local conformal operators.

Contents 1 Introduction

1

2 Locality in QFT

2

3 Holographic Renormalization and dual fields 3.1 Holographic Renormalization 3.2 Dual Field

3 3 6

4 Radial quantization of conformal (non-local) operators 4.1 The conformal Laplacian 4.2 Conformal operators and radial quantization

7 7 8

5 Final Remarks

9

6 Appendix: Asymptotics

9

1

Introduction

Despite that fact that many of the key questions in quantum gravity are not at all related to local boundary correlators, our best understanding to date of quantum gravity is through the AdS/CFT correspondence. Central to this correspondence is the duality between bulk fields and boundary conformal operators. Although conformal theories can be entirely delineated by their correlation functions, by Wightman’s reconstruction theorem, the n-point functions (along with a vacuum state Ω) reconstruct the quantum field theory in its entirety in that they completely determine the Hilbert space of states H and the operators. Hence, in principle the AdS/CFT dictionary[2–5] should yield the precise form of the boundary operators, rather than just average values as is typical. In this note, we show how it is possible to obtain the exact form of the boundary operators dual to bulk scalar fields using the holographic renormalization scheme[6] even when the bulk theory is interacting, thereby extending our previous work which applied strictly to a free massive bulk scalar[7]. In either the BDHM[2] or GKPW[3, 4] formulations (completely equivalent as long as the spacetime is asymptotically AdS[8]), the claim is that the AdS/CFT dictionary yields only the expectation value of the boundary operator in response to a probe sourced by the bulk field. Formally, the BDHM formulation of the AdS/CFT correspondence entails that hO(x1 ) · · · O(xn )iCF T = lim z −n∆ hφ(x1 , z) · · · φ(xn , z)ibulk . z→0

–1–

(1.1)

By Wightman’s reconstruction theorem, the limit in Eq. (1.1) determines completely the form of the dual operator O. In this note, we use the holographic renormalization technique of Heemskerk and Polchinski[6] to show that if the bulk is an interactive theory of the form,   Z X X √ |∇φi |2 + m2i φ2i + λij φ2i φj  (1.2) Sbulk = dd+1 x −g  i,j

i

then, the boundary operators O behave to leading order as the fractional Laplacian, and they are thus anti-local, thereby extending our previous work[7] which applied strictly to the free case. This operator identity removes the expectation value restriction in the AdS/CFT dictionary. Anti locality of an operator Tˆ in a space V (x) means that for any function f (x), the only solution to f (x) = 0 ( for some x ∈ V ) and Tˆf (x) = 0 is f (x) = 0 everywhere. Fractional Laplacians naturally satisfy this property of anti-locality as can seen from their Fourier representation. The significance of our result is that fractional Laplacians are not composite operators and hence should be viewed as the primary operators of the boundary theory. That the operator dual to a bulk scalar field should be an anti-local operator at the boundary is not unexpected because the boundary correlators[3] all have anomalous dimensions. If we consider the case of the Wilson-Fisher[9] fixed point where the correlation functions scale as G(k, ω) = k 2−η F (

ω ) kz

(1.3)

where z is the dynamical exponent and η, the anomalous dimension enters through the momentum dependence. The same is true for any scale-invariant Green function, unparticles a case in point[10]. In both cases, anomalous powers of the momentum can be obtained from ∂µη , thereby suggesting that the operator content at the fixed point of WF and that underlying the unparticle construction should be thought of in terms of non-local operators. The analogous reasoning applies to the AdS/CFT conjecture as well and it is this realization that unifies all such anomalous scaling as manifestations of boundary physics in a higher dimensional spacetime. More fundamentally, that the fractional Laplacian enters into the AdS/CFT correspondence is also anticipated in light of the Caffaerlli/Silvestre[1] extension theorem which demonstrates that the limiting form of differential operators in a bounded space in the upper half-plane always yields the fractional Laplacian. It is this limiting procedure that is central to the AdS/CFT construction. While our construction, for sake of convenience, is carried out for AdS, it applies more generally[7] to any asymptotic AdS spacetime.

2

Locality in QFT

First a word on locality in QFT. Arguably the most crucial sense in which locality is imposed is the notion of micro-causality, defined by the condition that for local operators O one has

–2–

that [O(x1 ), O(x2 )] = 0

(2.1)

when x1 and x2 cannot be joined by a light-like geodesic. This condition holds experimentally for the Standard Model (as verified at the L.H.C.) down to scales of order 10−20 m. Nothing we say violates micro-causality, as it can be seen by a straightforward application of Eq. (3.9) and the fact that the fields ϕ(x, t) are micro local. The types of non-localities we are interested in here are non-localities of the operators in the sense defined in the previous section. A local operator is defined as a polynomial in fields φ and their (conventional) derivatives, or in mathematical terms they are (symbols of) differential operators in the fields φ. The operators we derive here are (conformal versions of) fractional Laplacians in the fields, which cannot be constructed from any operator product expansion of local differential operators. In this sense, the boundary operators we derive are not composite operators.

3 3.1

Holographic Renormalization and dual fields Holographic Renormalization

AdS/CFT is ultimately a correspondence between partition functions, where the bulk partition function is defined by means of boundary conditions. Such boundary conditions are not given by a standard asymptotic condition of the field in the limit of approaching the conformal boundary, but rather in terms a renormalization procedure which is dictated by, for example, the classical solutions of the Gaussian theory (i.e., in Euclidean signature, solution to −∆φ + m2 φ = 0) which have the asymptotic form d

d

φ = F z 2 −γ + Gz 2 +γ ,

F, G ∈ C ∞ (H),

F = φ0 + O(z 2 ),

G = g0 + O(z 2 ),

(3.1)

√ where γ = 21 d2 + 4m2 . It is manifest from this asymptotic formula that there is no ambiguity as to what condition one should impose at the conformal infinity: d2 + γ is always positive d

and therefore Gz 2 +γ approaches 0 towards the boundary. Clearly the problematic term is d F z 2 −γ when d2 < γ. One thus imposes that fields φ in the bulk be of the form φ = z ∆ φ0 as z → 0 and φ0 is then named the boundary condition. Here ∆ = d/2 + γ. For convenience in formulating the path integral in the partition function, we let β(x) represent the non-normalizable mode with boundary condition, φ(x) = −∆ β and z =  will represent the inner cutoff for the integration of the bulk fields and z = ` the uppermost limit on the integration. With this in mind, we follow the holographic renormalization formalism of Heemskerk and Polchinski [6] which is based on a separation of the partition function into UV and IR parts, Z ˜ `] ΦU V [β, φ; ˜ , `], Zbulk [β] = Dφ˜ ΦIR [φ, (3.2)

–3–

where ˜ `] = ΦIR [φ,

Z

˜ , `] = ΦU V [β, φ;

Z

Dφ |z>` e−S|z>`

(3.3)

Dφ |