A Note on Commutation Relation in Conformal Field Theory

arXiv:2101.04090v1 [hep-th] 11 Jan 2021 edter nMnosisae nta fteclne,b aigt taking by cylinder, co the a of if for limit. instead commutator even distance space, the theory, Minkowski have those on field also theory for We s conformal field the except Lagrangian. non-free of the commutators a commutators from finite for defined time have well-defined equal the not to Thus o are time not operators. in smearing scalar need also free we but that calculations space explicit in from find We cylinder. omttro clrfilsi a in fields scalar of commutator ∗ † nti oe eepiil opt h aumepcainvleo t of value expectation vacuum the compute explicitly we note, this In terasima(at)yukawa.kyoto-u.ac.jp hep1.c.u-tokyo.ac.jp (at) nagano oeo omtto eainin Relation Commutation on Note A b uaaIsiuefrTertclPyis yt Universit Kyoto Physics, Theoretical for Institute Yukawa a nttt fPyis nvriyo oy,Komaba, Tokyo, of University Physics, of Institute ofra il Theory Field Conformal et Nagano Lento euok,Tko1380,Japan 153-8902, Tokyo Meguro-ku, yt 0-52 Japan 606-8502, Kyoto a d Abstract ∗ dmninlcnomlfil hoyo the on theory field conformal -dimensional n ej Terashima Seiji and b † YITP-21-05 aa fields calar y, hc is which esmall he nformal nly he of Contents

1 Introduction and summary 1

2 OPE 3 2.1 OPE in cylinder coordinates ...... 3

3 Commutation relation 5 3.1 Orderingoftheoperators...... 5 3.2 FreeCFT ...... 6 3.2.1 Equaltimecommutator ...... 7 3.2.2 Flat space limit ...... 7 3.3 GeneralCFT ...... 9 3.3.1 Equaltimecommutator ...... 10 3.3.2 Gaussiansmearedlocaloperators ...... 11 3.3.3 Flat space limit ...... 12 3.3.4 ∆ = d/2case ...... 14

A Conventions and notations 15 A.1 Sphericalharmonics...... 15 A.2 Gegenbauer polynomials and addition theorem ...... 16

∆ l B Approximation of d 2n+l 16

1 Introduction and summary

The commutation relation or commutator of the quantum fields is the fun- damental objects in quantum field theory (QFT) in the operator formalism. Indeed, usually the QFT is given by the canonical commutator, which is de- fined at a fixed time slice, of fundamental fields and the Hamiltonian, which describes the time evolution, in the operator formalism. Even for the theory without the canonical commutator, the commutator is important. For the two dimensional conformal field theory (CFT), the Virasoro algebra and the current algebra, which are the equal time commutators, play the important roles. These can be derived from the operator product expansion of fields by the contour integrals using the infinitely many current conservation laws.

1 Higher dimensional (d 3) CFTs are also very important in theoretical physics, such as condensed≥ matter physics and AdS/CFT correspondence [1], and so on. Recently they have been significantly studied by the conformal bootstrap since a seminal work [2]. 3 However, for the d-dimensional CFT (or general fields in 2d CFT), the commutators have not been studied intensively partly because there are not infinitely many conserved current. 4 On the other hand, the commutator of fields in the cylinder R Sd−1 can be derived × from the operator product expansion (OPE) for the higher dimensional CFT recently [15]. In particular, the vacuum expectation value (VEV) of the commutator is determined by the most singular part of the OPE, which is essentially the two point function. In this note, we explicitly compute the VEV of the commutator of (pri- mary) scalar fields in d-dimensional CFT on the cylinder. The commutators are expressed by an infinite summation. We observe a difference between the commutators for a free CFT and a non-free CFT as follows.

• For the commutators of primary scalar fields in a free CFT, we only need to smear operators in space and don’t need to do so in time to have finite values. So the equal time commutators of operators smeared in space are well-defined.

• For the commutators of primary scalar fields in a non-free CFT, we need to smear operators not only in space but also in time to have finite values. So the equal time commutators of smeared operators are ill-defined.

The latter fact is related to the fact that the weight of the K¨all´en-Lehmann- like representation for the CFT is not normalizable. We also have the commu- tator for the CFT on Rd, instead of the cylinder, by taking the small distance limit. Besides we can explicitly perform a summation when ∆ = d/2. We hope our results will be useful for future studies of the CFT, in particular for the AdS/CFT correspondence in the operator formalism [15, 16, 17]. This paper is organized as follows. In Section 2 we review the OPE in general CFTs and explain how to compute the commutator from the OPE. In section 3, we compute the VEV of the commutators of the scalar fields

3See [3, 4] for reviews on this topic. 4Constraints on commutators and their application were investigated by recent works such as [5, 6, 7, 8, 9]. The stress tensor commutators were studied in old works, e.g. [10, 11, 12] and also in recent works [13, 14].

2 in CFTs. First we evaluate them for a free CFT in Section 3.2, and then discuss them for a non-free CFT in Section 3.3.

2 OPE

In this section we review the OPE of primary scalar fields. Let us consider a scalar primary operator with a conformal dimension ∆. In general, the OPE between two operatorsO in Euclidean flat space is given by

µ1...µlp (x ) (x )= C f (x , ∂ ) p (x ) , (2.1) O 1 O 2 OOOp µ1...µlp 12 2 O 2 Op:primaryX

where x12 = x1 x2 and fµ1...µlp (x12, ∂2) is a function which can be deter- mined only by| the− representation| theory of the conformal symmetry. The most singular term in (2.1) is the contribution from identity operator 5,

(x ) (x ) x−2∆ , (2.2) O 1 O 2 ∼ 12 where we normalized (x ) usually. We focus on a contribution from this O 1 term. Note that if we consider OPE in a two point function, only an identity term contribute, (x ) (x ) = x−2∆ , (2.3) hO 1 O 2 i 12 since the one-point functions of any operators except for an identity vanish on conformaly flat manifold.

2.1 OPE in cylinder coordinates First we parametrize a position in Euclidean flat space Rd by xµ = r eµ(Ω), µ µ where r = x and e (Ω) is a unit vector, i.e. e (Ω)eµ(Ω) = 1. We parametrize unit vector| e|µ by angular variables Ω. Then we move to the cylinder coordi- nates via

r = eτ . (2.4)

Operators which live in the cylinder coordinates are denoted by cyl and O they are related to the corresponding operators in flat space by

cyl(τ, Ω) = r∆ (r, Ω) . (2.5) O O 5We mean ”taking most singular part” by . ∼ 3 In the cylinder coordinates, the OPE can be written as cyl(τ , Ω ) cyl(τ , Ω ) r∆r∆x−2∆ . (2.6) O 1 1 O 2 2 ∼ 1 2 12 −2∆ We suppress a superscript “cyl” below. We can expand x12 as follows [15].

∞ s [s/2] 1 r s−2n x−2∆ = < d∆ Y (Ω )Y (Ω ) (2.7) 12 r2∆ r  s s−2n,m 1 s−2n,m 2 > Xs=0 > Xn=0
Xm ∞ ∞ 2n+l 1 r< ∆ l = 2∆ d 2n+l Yl,m(Ω1)Yl,m(Ω2), (2.8) r r>  > Xn=0 Xl=0
Xm where r and r are the larger and smaller ones of x and x , respectively > < | 1| | 2| and d/2 l 2π Γ(∆ + n + l)Γ(∆ + 1 d/2+ n) d∆ = − . (2.9) 2n+l Γ(∆)Γ(∆ + 1 d/2)Γ(n + 1)Γ(n + l + d/2)
− For the normalization of the spherical harmonics, see the appendix A.1. Thus, we have

(τ1, Ω1) (τ2, Ω2) O ∞ O l e−(∆+2n+l)|τ12| d∆ Y (Ω )Y (Ω ) (2.10) ∼ 2n+l l,m 1 l,m 2 n,lX=0
Xm ∞ l −(∆+2n+l)|τ12| ∆ ˜ = e d 2n+l Cl(Ω12), (2.11) n,lX=0
where τ := τ τ , Ω := eµ(Ω )e (Ω ) = cos θ , where θ is the angle 12 1 − 2 12 1 µ 2 12 12 between the two points in Sd−1, and d +2l 2 C˜ (Ω ) := Y (Ω )Y (Ω )= − Cd/2−1(Ω ), (2.12) l 12 l,m 1 l,m 2 d 2 l 12 Xm − α 6 where Cl (x) is the Gegenbauer polynomial [18]. For the normalization of the Gegenbauer polynomials, see Appendix A.2. We have considered the most singular part of the OPE only, however, other parts also can be expanded in the same way. Below, we continue dealing only with the most singular part. For other parts, we can also compute the commutator formally [15] although we need the explicit OPE data to give an explicit result.

d/2 6For d = 2, Y (Ωµ)Y (Ωµ)= 2π (2 cos(lθ ) δ ). m l,m 1 l,m 2 Γ(d/2) 12 − l,0 P 4 3 Commutation relation

3.1 Ordering of the operators In the previous section, the OPE (2.1) is regarded as the expansion in the correlation function, where the fields, which are regarded as the path-integral variables, can commute each other and the ordering is not relevant. Below, we will consider the CFT in the operator formalism and regard the fields in the OPE (2.1) as operators acting on the Hilbert space. More precisely, we will regard an operator E(τ, Ω) which is the field in the Euclidean cylinder O d−1 7 as the operator acting on the Hilbert space of the CFTd on the sphere S . Then, the operator ordering corresponding to the correlation function with the condition τ > > τ 8 is as follows, 1 ··· n (τ , Ω ) (τ , Ω ) (τ , Ω ), (τ > > τ ), (3.1) OE 1 1 OE 2 2 ···OE n n 1 ··· n τH −τH where E(τ, Ω) = e E(0, Ω)e and H is the Hamiltonian (which is the dilatationO operator).O For this ordering, we can apply the OPE of the two operators using the expansion (2.8). We can define the the product of the operators for a complex τ by the analytic continuation by the expansion (2.8) for

(τ , Ω ) (τ , Ω ) (τ , Ω ), (Re(τ ) > > Re(τ )), (3.2) OE 1 1 OE 2 2 ···OE n n 1 ··· n When we want to move to Lorentzian signature CFT, we evolve a Loren- zian time with the ordering given by the small Euclidean time ǫi fixed and, then take ǫ 0 limit: i → (t , Ω ) (t , Ω ) (t , Ω ) OL 1 1 OL 2 2 ···OL n n := lim E(τ1 = ǫ1 + it1, Ω1) E(τn = ǫn + itn, Ωn) (ǫ1 > > ǫn), ǫi→0 O ···O ··· (3.3)

7 Below we denote the operator whose argument is an Euclidean time by E. 8For a quantum mechanics (with a finite degrees of freedom), we do not needO to expand like (2.8) and we can consider any ordering of the operators. In quantum field theories, the operators which is not ordered as (3.1) have a diverging expectation values. However, the local field should be smeared for a finite expectation value and then the smeared operators which is not ordered as (3.1), where smearing region is small compared with the time distances will have finite values. In this sense, the other orderings are possible.

5 9 where we denote operators whose argument is a Lorentzian time as L(t, Ω). Note that the local operators has a diverging expectation value,O thus we need to consider the smearing (or distributions) of them.

3.2 Free CFT First, we consider the free CFT which means ∆ = d 1, as a simplest 2 − example. For this case, we have

free l ∆=d/2−1 l d 2n+l := d 2n+l (3.4)
2πd/2
Γ(d/2)Γ(d/2 1+ n + l)Γ(n) = − (3.5) Γ(d/2) Γ(d/2 1)Γ(0)Γ(n + 1)Γ(n + l + d/2) − d/2 1 2πd/2 = δ − (3.6) n,0 (l + d/2 1) Γ(d/2) − ∆ 2πd/2 = δ . (3.7) n,0 (∆ + l) Γ(d/2)

With this, we can write down the (singular part of) OPE as

∞ 2πd/2∆ e−(∆+l)τ12 (τ , Ω ) (τ , Ω ) C˜ (Ω ), (3.8) OE 1 1 OE 2 2 ∼ Γ(d/2) (∆ + l) l 12 Xl=0 where we assume Re(τ1 τ2) > 0. Using this, the VEV of the commutation re- lation of the two local operators− is computed as 0 [ (t , Ω ), (t , Ω )] 0 = h | OL 1 1 OL 2 2 | i limǫ→0 Aǫ(t12, Ω12) where

A (t , Ω ) := 0 (τ = ǫ + it , Ωµ) (τ = it , Ωµ) ǫ 12 12 h | OE 1 1 1 OE 2 2 2 (τ = it , Ωµ) (τ = ǫ + it , Ωµ) 0 (3.9) − OE 2 2 2 OE 1 − 1 1 | i = 0 (t iǫ, Ωµ) (t , Ωµ) h | OL 1 − 1 OL 2 2 (t , Ωµ) (t + iǫ, Ωµ) 0 (3.10) − OL 2 2 OL 1 1 | i 2πd/2 ∞ e−(∆+l)ǫ sin ((∆ + l)t ) = 2i ∆ 12 C˜ (Ω ), (3.11) − Γ(d/2) (∆ + l) l 12 Xl=0

9In operator formalism, the Hilbert space and operators acting on it are same for Lorentzian and Euclidean signature. Lorentzian means that the operator is evolved by eiHt, instead of eHτ for Euclidean case.

6 and we assume ǫ > 0 and t12 = t1 t2. Note that for the free theory, we know the commutator contains only− the identity operator. This means [ L(t1, Ω1), L(t2, Ω2)] = limǫ→0 Aǫ(t12, Ω12), where the identity operator is notO explicitlyO written. Precisely speaking, we need to take the limit after the smearing of the local operators. Here we just need a space smearing, not a spacetime smearing. We can easily check that this formally satisfies the equations of motion of the free field (which has the conformal mass term ∆2 on Sd−1) as

2 ∂ 2 Sd−1 2 (Ω1) + ∆ [ L(t1, Ω1), L(t2, Ω2)]=0, (3.12) ∂t1 −△  O O where we assume t = t or Ω = Ω . Here, Sd−1 is the Laplacian acting 1 6 2 1 6 2 △ (Ω1) on Ω1 and satisfies

Sd−1 Y (Ω) = l(l + d 2)Y (Ω). (3.13) △ l,m − − l,m 3.2.1 Equal time commutator The equal time commutation relation can be easily computed as [ (t , Ω ), (t , Ω )] = 0 (3.14) OL 1 1 OL 1 2 because of the symmetry. Instead if we consider [ ˙ , ], then we find O O d 0 L(t1, Ω1), L(t1, Ω2) 0 h | dt1 O O  | i d = Aǫ(t12, Ω12) (3.15) dt1 t12=0 ∞ 2πd/2 = 2i ∆ e−(∆+l)ǫ Y (Ω )Y (Ω ) (3.16) Γ(d/2) l,m 1 l,m 2 Xl=0 Xm 2πd/2 2i ∆ δ(Ω Ω ) (ǫ 0), (3.17) → Γ(d/2) 1 − 2 → where the expression in the final line is interpreted as a distribution.

3.2.2 Flat space limit Finally, we will consider the commutator of very close two operators, i.e. the commutator with t 1 and Ω 1 1. In this limit, we expect | 12| ≪ | 12 − | ≪ 7 that the commutator becomes the one for the theory on the Minkowski space which is given by the invariant Delta function. 10 For this, we will use the formula of the Gegenbaur polynomial as the Jacobi polynomial, Γ(2α + n) Γ(α +1/2) Cα(x)= P α−1/2,α−1/2(x), (3.18) n Γ(2α) Γ(α +1/2+ n) n and the asymptotics of the Jacobi polynomials. Near the point x = 1, we have [21] −α α,β −α lim n Pn (cos(z/n))=(z/2) Jα(z), (3.19) n→∞ where Jα(z) is the Bessel function of the first kind. Using this, we can see that the commutator (3.11) in the limit becomes

d/2 ∞ 2π sin ((∆ + l)t12) lim Aǫ(t12, Ω12)= 2i ∆ C˜l(Ω12) (3.20) ǫ→0 − Γ(d/2) (∆ + l) Xl=0 2πd/2 ∞ 1 = 2i ∆ dk sin (k t) C˜k/ε(cos(εr))+ (3.21) − Γ(d/2) Z0 k ··· 2πd/2 ∞ Γ(d 1+ k/ε) Γ(d/2 1/2) = i dk − − − Γ(d/2) Z0 Γ(d 2) Γ(d/2 1/2+ k/ε) − − − − 1 d 3 , d 3 sin (k t) P 2 2 (cos(εr)) + × k k/ε ··· (3.22) 2πd/2 Γ((d 1)/2) ∞ 1 k d−2 kr −(d−3)/2 = i − dk − Γ(d/2) Γ(d 2) Z k  ε   2  − 0 sin (k t) J (kr)+ , (3.23) × (d−3)/2 ··· where the large l contributions are dominant, then we replace l = k/ε, t12 = εt and Ω12 = cos(θ12) = cos(εr) and means the terms suppressed in the small ε limit. ··· On the other hand, the commutator of the free scalar theory with mass µ on d-dimensional Minkowski space is given by d−1 i i 1 −1 −iω(k)t+ikix iω(k)t−ikix [φ(x),φ(0)] = dki(2ω(k)) (e e ) (2π)d−1 Z − Yi=1 (3.24)

10In this paper we call this limit as flat space limit, but this is different from the usual flat space limit as in [19, 20].

8 i 2 11 where ω(k) = kik + µ and we took the standard normalization. We can integrate outp the angular directions of ki with the the following formula for the expansion of the plain wave in Rd−1 by the spherical harmonics [18]:

(d−1)/2 ∞ i 2π eikix = (d 3)!! il jd−1(kr) Y (Ω′ )Y (Ω′), Γ((d 1)/2) − l l,m k l,m − Xl=0 Xm ∞ 2(d−5)/2J (kr) =4π(d−1)/2 il (d−3)/2+l Y (Ω′ )Y (Ω′), (3.25) (kr)(d−3)/2 l,m k l,m Xl=0 Xm

i i ′ ′ i where r = √x xi, k = √k ki, Ω and Ωk are the angular variables for x d and ki, respectively and jl (z) is the hyper spherical Bessel function which is written as Γ(d/2 1)2d/2−2J (z) jd(z) := − d/2−1+l . (3.26) l (d 4)!!zd/2−1 − ′ After the integration over the angular directions Ωk in the momentum space Rd−1, the constant mode of the spherical harmonics remains and we find that

∞ −(d−3)/2 (d−1)/2 1 d−2 kr [φ(x),φ(0)] = i2π dk (k) sin (k t) J(d−3)/2(kr), − Z0 k  2  (3.27) where we took the massless case µ = 0. This coincides with the commutator in the limit, (3.24), up to a numerical factor and (ε)d−2 which is the scaling factor for the two free scalar fields. Note that the normalizations of the (t, Ω) and φ(x) are different. Thus, we confirmed that the commutator reproducesO the usual commutator of the free theory on Minkowski space in the small distance limit.

3.3 General CFT In this section, we will consider the general scalar primary field whose di- mension is above the unitarity bound, i.e. ∆ > d/2 1. The OPE is given − 11Here we introduced the mass of the scalar field for later convenience. We will finally take µ = 0.

9 by

µ µ E(τ1, Ω1 ) E(τ2, Ω2 ) O ∞ O l e−(∆+2n+l)τ12 d∆ C˜ (Ω ), (3.28) ∼ 2n+l l 12 n,lX=0
for Re(τ1) > Re(τ2). Thus, the VEV of the commutation relation of the two local operators is computed as 0 [ L(t1, Ω1), L(t2, Ω2)] 0 = limǫ→0 Aǫ(t12, Ω12) where h | O O | i

A (t , Ω ) := 0 ( (τ = ǫ + it , Ωµ) (τ = it , Ωµ) ǫ 12 12 h | OE 1 1 1 OE 2 2 2 cyl µ cyl µ E (τ2 = it2, Ω2 ) E (τ1 = ǫ + it1, Ω1 )) 0 (3.29) ∞− O O − | i = e−(∆+2n+l)ǫ( 2i) sin ((∆ + 2n + l)t ) − 12 n,lX=0 l d∆ C˜ (Ω ), (3.30) × 2n+l l 12
Note that because of the factor e−(∆+2n+l)ǫ, the summations over n, l will converge if ǫ> 0. 12 In order to take ǫ 0 limit, we need to smear the local → operators first in general. We will explain this issue in Section 3.3.2 .

3.3.1 Equal time commutator Let us consider the equal time commutator. Usually, the equal time com- mutator is defined on a time slice, thus it should be the distribution, like the delta-function, in the space, not in spacetime. Indeed, in a free CFT the equal time commutator can be written by the delta-function with respect to Ω as (3.17). However, we will see that the equal time commutator in a general CFT except for a free one is not defined even after integrating over space. 13 This can be easily seen by considering the commutator of the l =0

12The summation over n,l can be written by the summation over ω =2n + l and l then ∆ ˜ the summation over l is truncated since d ω vanish unless l ω. Besides, Cl(Ω12) and l ≤ d∆ are power functions of ω,l in theω,l
limit, thus this summation converges ω → ∞ when
ǫ> 0. 13This fact is related to another fact that the expectation value of the energy of the local state smeared over the space is divergent for the operator of CFT except the free fields and homomorphic field in 2d case [17].

10 mode, (t) := dΩ (t, Ω), which is maximally smeared over space. The O0 OL equal time commutatorR of this operator and its derivative are given by

∞ ∞ 0 0 [∂ (t), (t)] 0 = 2i (∆+2n) d∆ n2∆−d+1, (3.31) h | tO0 O0 | i − 2n ∼ Xn=0
Xn=0 which diverges for ∆ > d/2 1 which is satisfied for the unitary CFT except for a free one. 14 As we will see− below, if we smear an operator over spacetime instead of smearing over space we have a finite result. 15 Remind that local operators smeared over the space in a certain time slice should have a finite commutator if the equal time commutators of the original operators in this time slice are well-defined. Thus, for CFT except the free CFT, 16 the equal time commutators can not be defined as clearly seen by this divergence of the one for the maximally smeared local operators. This might be surprising because there are non-trivial CFTs which will have the Lagrangians and can be defined by the canonical commutation relations with Hamiltonians, for example, the 4d = 4 supersymmetric gauge theory. However, the gauge invariant operatorsN are composite operators which need the renormalization. Then, such operators will not have a finite equal time commutator. Note that this divergence will due to the high energy behavior of the theory. Thus, the quantum field theory with a non-trivial UV fixed point, i.e. it is defined by the renormalization flow from the fixed point, will has divergent equal time commutators of the distributions for the local fields. On the other hand, the asymptotic free quantum field theory will have a well-defined equal time commutators as described in the usual text book.

3.3.2 Gaussian smeared local operators To take ǫ 0 limit and obtain the commutator in the Lorentzian spacetime, we need to→ smear the local operators over spacetime instead of smearing

14When ∆ = d/2 1 the above equation does not hold since only the n = 0 term can contribute to the summation,− so the equal time commutator does not diverge in this case. This is consistent with the results in Section 3.2. 15Note that this is consistent with a fact that in the axiomatic quantum field theory only correlators smeared over spacetime are considered. 16The energy momentum tensor and the homomorphic currents of 2d CFT also has the finite equal time commutators because the energy of a state is proportional to the absolute value of the (angular) momentum and then there are no summation over n.

11 over space. Here, as an example of an explicit smearing, we introduce the Gaussian smeared (over time direction) local operators as 17

∞ 1 α2 − 2 (t, Ω)δ dαe 2δ L(t + α, Ω). (3.32) O ≡ √2πδ Z−∞ O The commutator of the Gaussian smeared local operators is given by

0 [ (t1, Ω1)δ, (t2, Ω2)δ 0 (3.33) h | O ∞O | i 2 2 2i (α1) +(α2) − 2 = dα1dα2e 2δ − 2πδ2 Z Xn,l −∞ l sin((∆ + 2n + l)t + α α ) d∆ C˜ (Ω ) (3.34) × 12 1 − 2 2n+l l 12 2 2
l = 2i e−δ (∆+2n+l) sin((∆ + 2n + l)t ) d∆ C˜ (Ω ), (3.35) − 12 2n+l l 12 Xn,l
where we took the ǫ 0 limit. Then the summations over n, l converge even → 2 2 after taking ǫ 0 because of the Gasussian factor e−δ (∆+2n+l) . → Note that for t12 = 0 the commutator is zero because of the symme- try (and locality).

3.3.3 Flat space limit If we only consider a small region in the space-time (=the cylinder) , the theory is expected to become the CFT on the Minkowski space. We will see this for the commutators below. Let us consider the following limit where ε 0 and t, r are fixed finite: → t = εt, Ω (= cos θ )=1 (εr)2/2. (3.36) 12 12 12 − By defining 2πd/2 Γ(d/2) d∆ := , (3.37) ∞ Γ(d/2) Γ(∆)Γ(∆ + 1 d/2)
− 17The small Euclidian time ǫ caused the smearing of the local operator, however, it also specifies the ordering of the operators. To compute the commutator, we need another smearing.

12 the commutator in this limit becomes 18

∞ l A (t , Ω )= e−(∆+2n+l)ǫ( 2i) sin ((∆ + 2n + l)t ) d∆ C˜ (Ω ) ǫ 12 12 − 12 2n+l l 12 n,lX=0
(3.38) ∞ = d∆ e−(2n+l)( 2i) sin ((2n + l)t ) ∞ − 12
n,lX=0 ∆−d/2 (n(n + l)) C˜l(Ω12)+ (3.39) ∞× ∞ ··· ∆ i − µ2+k2 ǫ = d dµ dke √ ε sin µ2 + k2 t − ∞ 2ε2 Z Z
0 0 p  µ2 ∆−d/2 C˜ (cos(εr))+ , (3.40) × 4ε2  k/ε ··· where contributions from n, l 1 are dominant in the summation over n, l ≫ and we defined

k = εl, µ2 = ε2((2n + l)2 l2), (3.41) − which are considered as continuous variables. Thus, the commutator is cor- rectly written as the following form, which is similar to the K¨all´en-Lehmann representation:

∞ ∆ i 1 2 ∆−d/2 lim Aǫ(t12, Ω12) d dµ µ ∆(µ; t, r), (3.42) ǫ→0 ≃ − ∞ 2 ε2(∆−d/2+1) Z
0
where ∞ 2 2 ∆(µ; t, r)= dkk sin µ + kk t C˜k/ε(cos(εr)) (3.43) Z0 p  The function ∆(µ; t, r) is proportional to the commutator of the scalar field with mass µ on the Minkowski spacetime [φ(x),φ(y)] in which we identified t = x y and r = (x y )2, as shown in Section 3.2. 0 − 0 i − i p ∞ 2 ∆−d/2 Here, the integration over the weight, 0 dµ (µ ) , is divergent whereas it should be converged for the K¨all´en-LehmannR representation of the commu- tator. This is because the asymptotic fields (in the flat space case) can not

18For more details on the approximation here, see Appendix B.

13 be defined in the non-trivial CFT. It also clearly be related to the fact that the equal time commutator is not well-defined and we need the smearing of the local operators as we showed before. Instead of the smearing, we can introduce the UV cut-off Λ for the in- Λ tegral of µ, like 0 dµ, to make the integral converge. However, this is not appropriate becauseR this divergence does not due to the theory itself. The divergence appears because we consider the “ill-defined” operators, i.e. the local operators. If we consider the well defined operators, which can be (spacetime) smeared local operators, there are no divergence and there are no need of the cut-off to define the commutator, as we have seen.

3.3.4 ∆= d/2 case

∆ l In general, d 2n+l is complicated and it is difficult to perform the sum- mation overn
explicitly. However, we can perform the summation when ∆=d/2 l 2πd/2 19 ∆= d/2 where it does not depend on n and l: d 2n+l = Γ(d/2) . For this case, the commutator is
2πd/2 ∞ A (t , Ω )= e−(∆+2n+l)ǫ( 2i) sin ((∆ + 2n + l)t ) C˜ (Ω ) ǫ 12 12 Γ(d/2) − 12 l 12 n,lX=0 (3.44) ∞ 2πd/2 e(∆+l)(−ǫ+it12) e(∆+l)(−ǫ−it12) = − + C˜l(Ω12). Γ(d/2)  1 e−2ǫ+2it12 1 e−2ǫ−2it12  Xl=0 − − (3.45)

Thus, if e2it12 = 1, 6 2πd/2 1 ∞ lim Aǫ(t12, Ω12)= cos((d/2 1+ l)t12)C˜l(Ω12), (3.46) ǫ→0 Γ(d/2) i sin t12 − Xl=0 where the expression is interpreted as a distribution in space. Note that it becomes large when t is small because of the 1 factor. This reflects the 12 sin t12 fact that the equal time commutator diverges. Note also that this expression does not vanish at t12 = 0 although we are considering the commutator of the same operators. This is consistent because limt12→0 Aǫ(t12, Ω12) = 0 where ǫ was fixed at finite value. 19If the CFT has the holographic dual, this ∆ corresponds to the scalar in the bulk whose mass saturates the Breitenlohner-Freedman bound [22].

14 We find that the commutator of the ∆ = d/2 scalar operators represented in the “angular momentum” eigen modes, (t) := dΩ (t, Ω)Y (Ω), as Olm OL lm R 1 0 [ (t1)lm, (t2)l′m′ 0 = cos((d/2 1+ l)t12)δl,l′ δm.m′ (3.47) h | O O | i i sin t12 − for e2it12 = 1. 6 Acknowledgments

L.N. would like to thank to Yukawa institute for Theoretical Physics at Kyoto University for hospitality. Discussions during the YITP atom-type visiting program were useful to proceed this work. This work was supported by JSPS KAKENHI Grant Number 17K05414.

Note added: As this article was being completed, we received the preprint [23]. In that paper, they discussed some general aspects of commutators of local operators in CFT from OPE.

A Conventions and notations

A.1 Spherical harmonics In this article we use the convention for spherical harmonics as they satisfy

(here) (here) dΩY (Ω)Y ′ ′ (Ω) = δl,l′ δm,m′ . (A.1) Z l,m l ,m On the other hand, another convention is used in [15] where spherical har- monics satisfy

1 (there) (there) dΩY (Ω)Y ′ ′ (Ω) = δl,l′ δm,m′ , (A.2) Area(Sd−1) Z l,m l ,m where 2πd/2 Area(Sd−1)= dΩ= . (A.3) Z Γ(d/2)

15 which reduces 4π in d = 3. The relation between them are given by 1 Y (here)(Ω) = Y (there)(Ω). (A.4) l,m Area(Sd−1) l,m p A.2 Gegenbauer polynomials and addition theorem

α In this paper, the Gegenbauer polynomials Cn [18] is given by

[ 1 s] 2 ( 1)p(2η)s−2p Γ(α + s p) Cα (η)= − − , (A.5) s p!(s 2p)! Γ(α) Xp=0 − which reduces to the Legendre polynomial for α = 1/2. The Gegenbauer polynomials are normalized as

1−2α 2 π2 Γ(n +2α) dx(1 x2)α−1/2 C(α)(x) = . (A.6) Z − n 2   n!(n + α) [Γ(α)] The spherical harmonics satisfy the following relation which is so-called the addition theorem d +2l 2 Y (Ω )Y (Ω )= − C(d/2−1)(Ω ). (A.7) l,m 1 l,m 2 d 2 l 12 Xm −

∆ l B Approximation of d 2n+l Stirling’s formula is given by

2π z z Γ(z) , ( arg z < π ǫ, z ). (B.1) ≃ r z  e | | − | |→∞ Using this formlua and the following equation,

(x + a)x+a eaxa, (x ), (B.2) xx ≃ →∞

16 we have Γ(x + a) (x) := (B.3) a Γ(x) x ex (x + a)x+a (B.4) ≃ rx + a ex+a xx (x + a)x+a e−a (B.5) ≃ xx e−aeaxa = xa, (x ). (B.6) ≃ →∞ Thus when n, l are large we have

d/2 l 2π Γ(d/2)Γ(∆ + n + l)Γ(∆ + 1 d/2+ n) d∆ = − (B.7) 2n+l Γ(d/2) Γ(∆)Γ(∆ + 1 d/2)Γ(n + 1)Γ(n + l + d/2)
− 2πd/2 Γ(d/2)(n + 1) (∆ + n + l) = ∆−d/2 ∆−d/2 (B.8) Γ(d/2) Γ(∆)Γ(n +1+ d/2) 2πd/2 Γ(d/2) ((n + 1)(n + l + d/2))∆−d/2 (B.9) ≃ Γ(d/2) Γ(∆)Γ(∆ + 1 d/2) − 2πd/2 Γ(d/2) (n(n + l))∆−d/2 . (B.10) ≃ Γ(d/2) Γ(∆)Γ(∆ + 1 d/2) − References [1] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133, [hep-th/9711200]. [2] R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031, [0807.0004]. [3] S. Rychkov, EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions, 1, 2016. 10.1007/978-3-319-43626-5. [4] D. Simmons-Duffin, The Conformal Bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 1–74, 2017. 1602.07982. DOI. [5] T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099, [1509.00014]. [6] T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Con- straints, JHEP 10 (2016) 141, [1601.07904].

17 [7] T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP 07 (2017) 066, [1610.05308]. [8] N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049, [1610.09378]. [9] M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, JHEP 06 (2018) 121, [1705.02934]. [10] J. S. Schwinger, NonAbelian gauge fields. Relativistic invariance, Phys. Rev. 127 (1962) 324–330. [11] J. Schwinger, Energy and Momentum Density in Field Theory, Phys. Rev. 130 (1963) 800–805. [12] S. Deser and D. Boulware, Stress-Tensor Commutators and Schwinger Terms, J. Math. Phys. 8 (1967) 1468. [13] K.-W. Huang, Stress-tensor commutators in conformal field theories near the lightcone, Phys. Rev. D 100 (2019) 061701, [1907.00599]. [14] K.-W. Huang, Lightcone Commutator and Stress-Tensor Exchange in d> 2 CFTs, Phys. Rev. D 102 (2020) 021701, [2002.00110]. [15] S. Terashima, Classical Limit of Large N Gauge Theories with Confor- mal Symmetry, JHEP 02 (2020) 021, [1907.05419]. [16] S. Terashima, AdS/CFT Correspondence in Operator Formalism, JHEP 02 (2018) 019, [1710.07298]. [17] S. Terashima, Bulk Locality in AdS/CFT Correspondence, 2005.05962. [18] J. Avery, Hyperspherical Harmonics Applications in Quantum Theory. Kluwer Academic Publishers, Dordrecht, 1989. [19] J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025, [1011.1485]. [20] T. Okuda and J. Penedones, String scattering in flat space and a scal- ing limit of Yang-Mills correlators, Phys. Rev. D 83 (2011) 086001, [1002.2641]. [21] L.-c. Chen and M. Ismail, On Asymptotics of Jacobi Polynomials, SIAM J. Math. Anal. 22 (5) (1991) 1442–1449. [22] P. Breitenlohner and D. Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249. [23] M. Besken, J. de Boer and G. Mathys, On Local and Integrated Stress- Tensor Commutators, 2012.15724.

18