A Note on Commutation Relation in Conformal Field Theory
aa r X i v : . [ h e p - t h ] J a n YITP-21-05
A Note on Commutation Relation inConformal Field Theory
Lento Nagano a ∗ and Seiji Terashima b † a Institute of Physics, University of Tokyo, Komaba,Meguro-ku, Tokyo 153-8902, Japan b Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan
Abstract
In this note, we explicitly compute the vacuum expectation value of thecommutator of scalar fields in a d -dimensional conformal field theory on thecylinder. We find from explicit calculations that we need smearing not onlyin space but also in time to have finite commutators except for those offree scalar operators. Thus the equal time commutators of the scalar fieldsare not well-defined for a non-free conformal field theory, even if which isdefined from the Lagrangian. We also have the commutator for a conformalfield theory on Minkowski space, instead of the cylinder, by taking the smalldistance limit. ∗ nagano (at) hep1.c.u-tokyo.ac.jp † terasima(at)yukawa.kyoto-u.ac.jp ontents d/ A Conventions and notations 15
A.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . 15A.2 Gegenbauer polynomials and addition theorem . . . . . . . . . 16
B Approximation of (cid:0) d ∆ (cid:1) l n + l 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, whichdescribes the time evolution, in the operator formalism. Even for the theorywithout the canonical commutator, the commutator is important. For thetwo dimensional conformal field theory (CFT), the Virasoro algebra and thecurrent algebra, which are the equal time commutators, play the importantroles. These can be derived from the operator product expansion of fields bythe contour integrals using the infinitely many current conservation laws.1igher dimensional ( d ≥
3) CFTs are also very important in theoreticalphysics, such as condensed matter physics and AdS/CFT correspondence [1],and so on. Recently they have been significantly studied by the conformalbootstrap since a seminal work [2]. However, for the d -dimensional CFT (orgeneral fields in 2d CFT), the commutators have not been studied intensivelypartly because there are not infinitely many conserved current. On theother hand, the commutator of fields in the cylinder R × S d − can be derivedfrom the operator product expansion (OPE) for the higher dimensional CFTrecently [15]. In particular, the vacuum expectation value (VEV) of thecommutator is determined by the most singular part of the OPE, which isessentially 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 commutatorsare expressed by an infinite summation. We observe a difference between thecommutators for a free CFT and a non-free CFT as follows. • For the commutators of primary scalar fields in a free CFT, we onlyneed to smear operators in space and don’t need to do so in time tohave finite values. So the equal time commutators of operators smearedin space are well-defined. • For the commutators of primary scalar fields in a non-free CFT, weneed to smear operators not only in space but also in time to havefinite values. So the equal time commutators of smeared operators areill-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 R d , instead of the cylinder, by taking the small distancelimit. Besides we can explicitly perform a summation when ∆ = d/
2. Wehope our results will be useful for future studies of the CFT, in particularfor the AdS/CFT correspondence in the operator formalism [15, 16, 17].This paper is organized as follows. In Section 2 we review the OPE ingeneral 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 See [3, 4] for reviews on this topic. Constraints on commutators and their application were investigated by recent workssuch 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].
2n CFTs. First we evaluate them for a free CFT in Section 3.2, and thendiscuss them for a non-free CFT in Section 3.3.
In this section we review the OPE of primary scalar fields. Let us considera scalar primary operator O with a conformal dimension ∆. In general, theOPE between two operators in Euclidean flat space is given by O ( x ) O ( x ) = X O p :primary C OOO p f µ ...µ lp ( x , ∂ ) O µ ...µ lp p ( x ) , (2.1)where x = | x − x | and f µ ...µ lp ( x , ∂ ) is a function which can be deter-mined only by the representation theory of the conformal symmetry. Themost singular term in (2.1) is the contribution from identity operator , O ( x ) O ( x ) ∼ x − , (2.2)where we normalized O ( x ) usually. We focus on a contribution from thisterm. Note that if we consider OPE in a two point function, only an identityterm contribute, hO ( x ) O ( x ) i = x − , (2.3)since the one-point functions of any operators except for an identity vanishon conformaly flat manifold. First we parametrize a position in Euclidean flat space R d by x µ = r e µ (Ω),where r = | x | and e µ (Ω) is a unit vector, i.e. e µ (Ω) e µ (Ω) = 1. We parametrizeunit vector e µ by angular variables Ω. Then we move to the cylinder coordi-nates via r = e τ . (2.4)Operators which live in the cylinder coordinates are denoted by O cyl andthey are related to the corresponding operators in flat space by O cyl ( τ, Ω) = r ∆ O ( r, Ω) . (2.5) We mean ”taking most singular part” by ∼ .
3n the cylinder coordinates, the OPE can be written as O cyl ( τ , Ω ) O cyl ( τ , Ω ) ∼ r ∆1 r ∆2 x − . (2.6)We suppress a superscript “cyl” below. We can expand x − as follows [15]. x − = 1 r > ∞ X s =0 (cid:18) r < r > (cid:19) s [ s/ X n =0 (cid:0) d ∆ (cid:1) s − ns X m Y s − n,m (Ω ) Y s − n,m (Ω ) (2.7)= 1 r > ∞ X n =0 ∞ X l =0 (cid:18) r < r > (cid:19) n + l (cid:0) d ∆ (cid:1) l n + l X m Y l,m (Ω ) Y l,m (Ω ) , (2.8)where r > and r < are the larger and smaller ones of | x | and | x | , respectivelyand (cid:0) d ∆ (cid:1) l n + l = 2 π d/ Γ(∆ + n + l )Γ(∆ + 1 − d/ n )Γ(∆)Γ(∆ + 1 − d/ n + 1)Γ( n + l + d/ . (2.9)For the normalization of the spherical harmonics, see the appendix A.1.Thus, we have O ( τ , Ω ) O ( τ , Ω ) ∼ ∞ X n,l =0 e − (∆+2 n + l ) | τ | (cid:0) d ∆ (cid:1) l n + l X m Y l,m (Ω ) Y l,m (Ω ) (2.10)= ∞ X n,l =0 e − (∆+2 n + l ) | τ | (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω ) , (2.11)where τ := τ − τ , Ω := e µ (Ω ) e µ (Ω ) = cos θ , where θ is the anglebetween the two points in S d − , and˜ C l (Ω ) := X m Y l,m (Ω ) Y l,m (Ω ) = d + 2 l − d − C d/ − l (Ω ) , (2.12)where C αl ( x ) is the Gegenbauer polynomial [18]. For the normalization ofthe 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 dealingonly with the most singular part. For other parts, we can also compute thecommutator formally [15] although we need the explicit OPE data to give anexplicit result. For d = 2, P m Y l,m (Ω µ ) Y l,m (Ω µ ) = π d/ Γ( d/ (2 cos( lθ ) − δ l, ). Commutation relation
In the previous section, the OPE (2.1) is regarded as the expansion in thecorrelation function, where the fields, which are regarded as the path-integralvariables, can commute each other and the ordering is not relevant. Below,we will consider the CFT in the operator formalism and regard the fields inthe OPE (2.1) as operators acting on the Hilbert space. More precisely, wewill regard an operator O E ( τ, Ω) which is the field in the Euclidean cylinderas the operator acting on the Hilbert space of the CFT d on the sphere S d − . Then, the operator ordering corresponding to the correlation function withthe condition τ > · · · > τ n is as follows, O E ( τ , Ω ) O E ( τ , Ω ) · · · O E ( τ n , Ω n ) , ( τ > · · · > τ n ) , (3.1)where O E ( τ, Ω) = e τH O E (0 , Ω) e − τH and H is the Hamiltonian (which is thedilatation operator). For this ordering, we can apply the OPE of the twooperators using the expansion (2.8).We can define the the product of the operators for a complex τ by theanalytic continuation by the expansion (2.8) for O E ( τ , Ω ) O E ( τ , Ω ) · · · O E ( τ n , Ω n ) , (Re( τ ) > · · · > Re( τ n )) , (3.2)When we want to move to Lorentzian signature CFT, we evolve a Loren-zian time with the ordering given by the small Euclidean time ǫ i fixed and,then take ǫ i → O L ( t , Ω ) O L ( t , Ω ) · · · O L ( t n , Ω n ):= lim ǫ i → O E ( τ = ǫ + i t , Ω ) · · · O E ( τ n = ǫ n + i t n , Ω n ) ( ǫ > · · · > ǫ n ) , (3.3) Below we denote the operator whose argument is an Euclidean time by O E . For a quantum mechanics (with a finite degrees of freedom), we do not need to expandlike (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 smearedoperators which is not ordered as (3.1), where smearing region is small compared with thetime distances will have finite values. In this sense, the other orderings are possible. O L ( t, Ω). Note that the local operators has a diverging expectation value, thus weneed to consider the smearing (or distributions) of them.
First, we consider the free CFT which means ∆ = d −
1, as a simplestexample. For this case, we have (cid:0) d free (cid:1) l n + l := (cid:0) d ∆= d/ − (cid:1) l n + l (3.4)= 2 π d/ Γ( d/
2) Γ( d/ d/ − n + l )Γ( n )Γ( d/ − n + 1)Γ( n + l + d/
2) (3.5)= δ n, d/ − l + d/ −
1) 2 π d/ Γ( d/
2) (3.6)= δ n, ∆(∆ + l ) 2 π d/ Γ( d/ . (3.7)With this, we can write down the (singular part of) OPE as O E ( τ , Ω ) O E ( τ , Ω ) ∼ π d/ ∆Γ( d/ ∞ X l =0 e − (∆+ l ) τ (∆ + l ) ˜ C l (Ω ) , (3.8)where we assume Re( τ − τ ) >
0. Using this, the VEV of the commutation re-lation of the two local operators is computed as h | [ O L ( t , Ω ) , O L ( t , Ω )] | i =lim ǫ → A ǫ ( t , Ω ) where A ǫ ( t , Ω ) := h | O E ( τ = ǫ + i t , Ω µ ) O E ( τ = i t , Ω µ ) − O E ( τ = i t , Ω µ ) O E ( τ = − ǫ + i t , Ω µ ) | i (3.9)= h | O L ( t − i ǫ, Ω µ ) O L ( t , Ω µ ) − O L ( t , Ω µ ) O L ( t + i ǫ, Ω µ ) | i (3.10)= −
2i 2 π d/ Γ( d/ ∞ X l =0 ∆ e − (∆+ l ) ǫ sin ((∆ + l ) t )(∆ + l ) ˜ C l (Ω ) , (3.11) In operator formalism, the Hilbert space and operators acting on it are same forLorentzian and Euclidean signature. Lorentzian means that the operator is evolved by e i Ht , instead of e Hτ for Euclidean case. ǫ > t = t − t . Note that for the free theory,we know the commutator contains only the identity operator. This means[ O L ( t , Ω ) , O L ( t , Ω )] = lim ǫ → A ǫ ( t , Ω ), where the identity operator isnot explicitly written.Precisely speaking, we need to take the limit after the smearing of the localoperators. Here we just need a space smearing, not a spacetime smearing.We can easily check that this formally satisfies the equations of motionof the free field (which has the conformal mass term ∆ on S d − ) as (cid:18) ∂ ∂t − △ S d − (Ω ) + ∆ (cid:19) [ O L ( t , Ω ) , O L ( t , Ω )] = 0 , (3.12)where we assume t = t or Ω = Ω . Here, △ S d − (Ω ) is the Laplacian actingon Ω and satisfies △ S d − Y l,m (Ω) = − l ( l + d − Y l,m (Ω) . (3.13) The equal time commutation relation can be easily computed as[ O L ( t , Ω ) , O L ( t , Ω )] = 0 (3.14)because of the symmetry. Instead if we consider [ ˙ O , O ], then we find h | (cid:20) ddt O L ( t , Ω ) , O L ( t , Ω ) (cid:21) | i = ddt A ǫ ( t , Ω ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 (3.15)= 2i 2 π d/ Γ( d/
2) ∆ ∞ X l =0 e − (∆+ l ) ǫ X m Y l,m (Ω ) Y l,m (Ω ) (3.16) →
2i 2 π d/ Γ( d/
2) ∆ δ (Ω − Ω ) ( ǫ → , (3.17)where the expression in the final line is interpreted as a distribution. Finally, we will consider the commutator of very close two operators, i.e. the commutator with | t | ≪ | Ω − | ≪
1. In this limit, we expect7hat the commutator becomes the one for the theory on the Minkowski spacewhich is given by the invariant Delta function. For this, we will use theformula of the Gegenbaur polynomial as the Jacobi polynomial, C αn ( x ) = Γ(2 α + n )Γ(2 α ) Γ( α + 1 / α + 1 / n ) P α − / ,α − / n ( x ) , (3.18)and the asymptotics of the Jacobi polynomials. Near the point x = 1, wehave [21] lim n →∞ n − α P α,βn (cos( z/n )) = ( z/ − α J α ( z ) , (3.19)where J α ( z ) is the Bessel function of the first kind. Using this, we can seethat the commutator (3.11) in the limit becomeslim ǫ → A ǫ ( t , Ω ) = −
2i 2 π d/ Γ( d/ ∞ X l =0 ∆ sin ((∆ + l ) t )(∆ + l ) ˜ C l (Ω ) (3.20)= −
2i 2 π d/ Γ( d/
2) ∆ Z ∞ dk k sin ( k t ) ˜ C k/ε (cos( εr )) + · · · (3.21)= − i 2 π d/ Γ( d/ Z ∞ dk Γ( d − k/ε )Γ( d −
2) Γ( d/ − / d/ − / k/ε ) × k sin ( k t ) P d − , d − k/ε (cos( εr )) + · · · (3.22)= − i 2 π d/ Γ( d/
2) Γ(( d − / d − Z ∞ dk k (cid:18) kε (cid:19) d − (cid:18) kr (cid:19) − ( d − / × sin ( k t ) J ( d − / ( kr ) + · · · , (3.23)where the large l contributions are dominant, then we replace l = k/ε , t = εt and Ω = cos( θ ) = cos( εr ) and · · · means the terms suppressed in thesmall ε limit.On the other hand, the commutator of the free scalar theory with mass µ on d -dimensional Minkowski space is given by[ φ ( x ) , φ (0)] = 1(2 π ) d − Z d − Y i =1 d k i (2 ω ( k )) − ( e − i ω ( k ) t +i k i x i − e i ω ( k ) t − i k i x i )(3.24) In this paper we call this limit as flat space limit, but this is different from the usualflat space limit as in [19, 20]. ω ( k ) = p k i k i + µ and we took the standard normalization. Wecan integrate out the angular directions of k i with the the following formulafor the expansion of the plain wave in R d − by the spherical harmonics [18]: e i k i x i = 2 π ( d − / Γ(( d − /
2) ( d − ∞ X l =0 i l j d − l ( kr ) X m Y l,m (Ω ′ k ) Y l,m (Ω ′ ) , = 4 π ( d − / ∞ X l =0 i l ( d − / J ( d − / l ( kr )( kr ) ( d − / X m Y l,m (Ω ′ k ) Y l,m (Ω ′ ) , (3.25)where r = √ x i x i , k = √ k i k i , Ω ′ and Ω ′ k are the angular variables for x i and k i , respectively and j dl ( z ) is the hyper spherical Bessel function which iswritten as j dl ( z ) := Γ( d/ − d/ − J d/ − l ( z )( d − z d/ − . (3.26)After the integration over the angular directions Ω ′ k in the momentum space R d − , the constant mode of the spherical harmonics remains and we find that[ φ ( x ) , φ (0)] = − i2 π ( d − / Z ∞ dk k ( k ) d − (cid:18) kr (cid:19) − ( d − / sin ( k t ) J ( d − / ( kr ) , (3.27)where we took the massless case µ = 0. This coincides with the commutatorin the limit, (3.24), up to a numerical factor and ( ε ) d − which is the scalingfactor for the two free scalar fields. Note that the normalizations of the O ( t, Ω) and φ ( x ) are different. Thus, we confirmed that the commutatorreproduces the usual commutator of the free theory on Minkowski space inthe small distance limit. In this section, we will consider the general scalar primary field whose di-mension is above the unitarity bound, i.e. ∆ > d/ −
1. The OPE is given Here we introduced the mass of the scalar field for later convenience. We will finallytake µ = 0. O E ( τ , Ω µ ) O E ( τ , Ω µ ) ∼ ∞ X n,l =0 e − (∆+2 n + l ) τ (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω ) , (3.28)for Re( τ ) > Re( τ ). Thus, the VEV of the commutation relation of the twolocal operators is computed as h | [ O L ( t , Ω ) , O L ( t , Ω )] | i = lim ǫ → A ǫ ( t , Ω )where A ǫ ( t , Ω ) := h | ( O E ( τ = ǫ + i t , Ω µ ) O E ( τ = i t , Ω µ ) − O cyl E ( τ = i t , Ω µ ) O cyl E ( τ = − ǫ + i t , Ω µ )) | i (3.29)= ∞ X n,l =0 e − (∆+2 n + l ) ǫ ( − n + l ) t ) × (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω ) , (3.30)Note that because of the factor e − (∆+2 n + l ) ǫ , the summations over n, l willconverge if ǫ > In order to take ǫ → 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, likethe delta-function, in the space, not in spacetime. Indeed, in a free CFT theequal time commutator can be written by the delta-function with respectto Ω as (3.17). However, we will see that the equal time commutator in ageneral CFT except for a free one is not defined even after integrating overspace. This can be easily seen by considering the commutator of the l = 0 The summation over n, l can be written by the summation over ω = 2 n + l and l thenthe summation over l is truncated since (cid:0) d ∆ (cid:1) ω vanish unless l ≤ ω . Besides, ˜ C l (Ω ) and (cid:0) d ∆ (cid:1) lω are power functions of ω, l in the ω, l → ∞ limit, thus this summation convergeswhen ǫ > This fact is related to another fact that the expectation value of the energy of thelocal state smeared over the space is divergent for the operator of CFT except the freefields and homomorphic field in 2d case [17]. O ( t ) := R d Ω O L ( t, Ω), which is maximally smeared over space. Theequal time commutator of this operator and its derivative are given by h | [ ∂ t O ( t ) , O ( t )] | i = − ∞ X n =0 (∆ + 2 n ) (cid:0) d ∆ (cid:1) n ∼ ∞ X n =0 n − d +1 , (3.31)which diverges for ∆ > d/ − As we will see below, if we smear an operator over spacetimeinstead of smearing over space we have a finite result. Remind that local operators smeared over the space in a certain timeslice should have a finite commutator if the equal time commutators of theoriginal operators in this time slice are well-defined. Thus, for CFT exceptthe free CFT, the equal time commutators can not be defined as clearlyseen by this divergence of the one for the maximally smeared local operators.This might be surprising because there are non-trivial CFTs which will havethe Lagrangians and can be defined by the canonical commutation relationswith Hamiltonians, for example, the 4d N = 4 supersymmetric gauge theory.However, the gauge invariant operators are composite operators which needthe renormalization. Then, such operators will not have a finite equal timecommutator.Note that this divergence will due to the high energy behavior of thetheory. 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 hasdivergent equal time commutators of the distributions for the local fields.On the other hand, the asymptotic free quantum field theory will have awell-defined equal time commutators as described in the usual text book. To take ǫ → When ∆ = d/ − n = 0 term cancontribute to the summation, so the equal time commutator does not diverge in this case.This is consistent with the results in Section 3.2. Note that this is consistent with a fact that in the axiomatic quantum field theoryonly correlators smeared over spacetime are considered. The energy momentum tensor and the homomorphic currents of 2d CFT also has thefinite equal time commutators because the energy of a state is proportional to the absolutevalue of the (angular) momentum and then there are no summation over n . O ( t, Ω) δ ≡ √ πδ Z ∞−∞ dαe − α δ O L ( t + α, Ω) . (3.32)The commutator of the Gaussian smeared local operators is given by h | [ O ( t , Ω ) δ , O ( t , Ω ) δ | i (3.33)= − πδ X n,l Z ∞−∞ dα dα e − ( α α δ × sin((∆ + 2 n + l ) t + α − α ) (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω ) (3.34)= − X n,l e − δ (∆+2 n + l ) sin((∆ + 2 n + l ) t ) (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω ) , (3.35)where we took the ǫ → n, l converge evenafter taking ǫ → e − δ (∆+2 n + l ) .Note that for t = 0 the commutator is zero because of the symme-try (and locality). If we only consider a small region in the space-time (=the cylinder) , thetheory is expected to become the CFT on the Minkowski space. We will seethis for the commutators below. Let us consider the following limit where ε → t, r are fixed finite: t = εt, Ω (= cos θ ) = 1 − ( εr ) / . (3.36)By defining (cid:0) d ∆ (cid:1) ∞ := 2 π d/ Γ( d/
2) Γ( d/ − d/ , (3.37) The small Euclidian time ǫ caused the smearing of the local operator, however, it alsospecifies the ordering of the operators. To compute the commutator, we need anothersmearing. A ǫ ( t , Ω ) = ∞ X n,l =0 e − (∆+2 n + l ) ǫ ( − n + l ) t ) (cid:0) d ∆ (cid:1) l n + l ˜ C l (Ω )(3.38)= (cid:0) d ∆ (cid:1) ∞ ∞ X n,l =0 e − (2 n + l ) ( − n + l ) t ) × ( n ( n + l )) ∆ − d/ ˜ C l (Ω ) + · · · (3.39)= − (cid:0) d ∆ (cid:1) ∞ i2 ε Z ∞ dµ Z ∞ dke − √ µ + k ǫε sin (cid:16)p µ + k t (cid:17) × (cid:18) µ ε (cid:19) ∆ − d/ ˜ C k/ε (cos( εr )) + · · · , (3.40)where contributions from n, l ≫ n, l and we defined k = εl, µ = ε ((2 n + l ) − l ) , (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-Lehmannrepresentation:lim ǫ → A ǫ ( t , Ω ) ≃ − (cid:0) d ∆ (cid:1) ∞ i2 1 ε − d/ Z ∞ dµ (cid:0) µ (cid:1) ∆ − d/ ∆( µ ; t, r ) , (3.42)where ∆( µ ; t, r ) = Z ∞ dkk sin (cid:16)p µ + kk t (cid:17) ˜ C k/ε (cos( εr )) (3.43)The function ∆( µ ; t, r ) is proportional to the commutator of the scalar fieldwith mass µ on the Minkowski spacetime [ φ ( x ) , φ ( y )] in which we identified t = x − y and r = p ( x i − y i ) , as shown in Section 3.2.Here, the integration over the weight, R ∞ dµ ( µ ) ∆ − d/ , is divergent whereasit should be converged for the K¨all´en-Lehmann representation of the commu-tator. This is because the asymptotic fields (in the flat space case) can not For more details on the approximation here, see Appendix B.
13e defined in the non-trivial CFT. It also clearly be related to the fact thatthe equal time commutator is not well-defined and we need the smearing ofthe local operators as we showed before.Instead of the smearing, we can introduce the UV cut-off Λ for the in-tegral of µ , like R Λ dµ , to make the integral converge. However, this is notappropriate because this divergence does not due to the theory itself. Thedivergence appears because we consider the “ill-defined” operators, i.e. thelocal operators. If we consider the well defined operators, which can be(spacetime) smeared local operators, there are no divergence and there areno need of the cut-off to define the commutator, as we have seen. ∆ = d/ case In general, (cid:0) d ∆ (cid:1) l n + l is complicated and it is difficult to perform the sum-mation over n explicitly. However, we can perform the summation when∆ = d/ n and l : (cid:0) d ∆= d/ (cid:1) l n + l = π d/ Γ( d/ . Forthis case, the commutator is A ǫ ( t , Ω ) = 2 π d/ Γ( d/ ∞ X n,l =0 e − (∆+2 n + l ) ǫ ( − n + l ) t ) ˜ C l (Ω )(3.44)= 2 π d/ Γ( d/ ∞ X l =0 (cid:18) − e (∆+ l )( − ǫ +i t ) − e − ǫ +2i t + e (∆+ l )( − ǫ − i t ) − e − ǫ − t (cid:19) ˜ C l (Ω ) . (3.45)Thus, if e t = 1,lim ǫ → A ǫ ( t , Ω ) = 2 π d/ Γ( d/
2) 1i sin t ∞ X l =0 cos(( d/ − l ) t ) ˜ C l (Ω ) , (3.46)where the expression is interpreted as a distribution in space. Note that itbecomes large when t is small because of the t factor. This reflects thefact that the equal time commutator diverges.Note also that this expression does not vanish at t = 0 although weare considering the commutator of the same operators. This is consistentbecause lim t → A ǫ ( t , Ω ) = 0 where ǫ was fixed at finite value. If the CFT has the holographic dual, this ∆ corresponds to the scalar in the bulkwhose mass saturates the Breitenlohner-Freedman bound [22].
14e find that the commutator of the ∆ = d/ O lm ( t ) := R d Ω O L ( t, Ω) Y lm (Ω), as h | [ O ( t ) lm , O ( t ) l ′ m ′ | i = 1i sin t cos(( d/ − l ) t ) δ l,l ′ δ m.m ′ (3.47)for e t = 1. Acknowledgments
L.N. would like to thank to Yukawa institute for Theoretical Physics at KyotoUniversity for hospitality. Discussions during the YITP atom-type visitingprogram were useful to proceed this work. This work was supported by JSPSKAKENHI Grant Number 17K05414.
Note added :As this article was being completed, we received the preprint [23]. In thatpaper, they discussed some general aspects of commutators of local operatorsin CFT from OPE.
A Conventions and notations
A.1 Spherical harmonics
In this article we use the convention for spherical harmonics as they satisfy Z d Ω Y (here) l,m (Ω) Y (here) l ′ ,m ′ (Ω) = δ l,l ′ δ m,m ′ . (A.1)On the other hand, another convention is used in [15] where spherical har-monics satisfy 1Area( S d − ) Z d Ω Y (there) l,m (Ω) Y (there) l ′ ,m ′ (Ω) = δ l,l ′ δ m,m ′ , (A.2)where Area( S d − ) = Z d Ω = 2 π d/ Γ( d/ . (A.3)15hich reduces 4 π in d = 3. The relation between them are given by Y (here) l,m (Ω) = 1 p Area( S d − ) Y (there) l,m (Ω) . (A.4) A.2 Gegenbauer polynomials and addition theorem
In this paper, the Gegenbauer polynomials C αn [18] is given by C αs ( η ) = [ s ] X p =0 ( − p (2 η ) s − p p !( s − p )! Γ( α + s − p )Γ( α ) , (A.5)which reduces to the Legendre polynomial for α = 1 /
2. The Gegenbauerpolynomials are normalized as Z dx (1 − x ) α − / (cid:2) C ( α ) n ( x ) (cid:3) = π − α Γ( n + 2 α ) n !( n + α ) [Γ( α )] . (A.6)The spherical harmonics satisfy the following relation which is so-calledthe addition theorem X m Y l,m (Ω ) Y l,m (Ω ) = d + 2 l − d − C ( d/ − l (Ω ) . (A.7) B Approximation of (cid:0) d ∆ (cid:1) l n + l Stirling’s formula is given byΓ( z ) ≃ r πz (cid:16) ze (cid:17) z , ( | arg z | < π − ǫ, | z | → ∞ ) . (B.1)Using this formlua and the following equation,( x + a ) x + a x x ≃ e a x a , ( x → ∞ ) , (B.2)16e have ( x ) a := Γ( x + a )Γ( x ) (B.3) ≃ r xx + a e x e x + a ( x + a ) x + a x x (B.4) ≃ e − a ( x + a ) x + a x x (B.5) ≃ e − a e a x a = x a , ( x → ∞ ) . (B.6)Thus when n, l are large we have (cid:0) d ∆ (cid:1) l n + l = 2 π d/ Γ( d/
2) Γ( d/ n + l )Γ(∆ + 1 − d/ n )Γ(∆)Γ(∆ + 1 − d/ n + 1)Γ( n + l + d/
2) (B.7)= 2 π d/ Γ( d/
2) Γ( d/ n + 1) ∆ − d/ (∆ + n + l ) ∆ − d/ Γ(∆)Γ( n + 1 + d/
2) (B.8) ≃ π d/ Γ( d/
2) Γ( d/ − d/
2) (( n + 1)( n + l + d/ ∆ − d/ (B.9) ≃ π d/ Γ( d/
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