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A Note on Commutation Relation in Conformal Field Theory

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A Note on Commutation Relation in Conformal Field Theory YITP-21-05 A Note on Commutation Relation in Conformal Field Theory Lento Naganoa ∗ and Seiji Terashimab † aInstitute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan bYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Abstract In this note, we explicitly compute the vacuum expectation value of the commutator of scalar fields in a d-dimensional conformal field theory on the cylinder. We find from explicit calculations that we need smearing not only in space but also in time to have finite commutators except for those of free scalar operators. Thus the equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian. We also have the commutator for a conformal field theory on Minkowski space, instead of the cylinder, by taking the small distance limit. arXiv:2101.04090v1 [hep-th] 11 Jan 2021 ∗nagano (at) hep1.c.u-tokyo.ac.jp †terasima(at)yukawa.kyoto-u.ac.jp 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 .
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