The Lorentz anomaly via operator product expansion

Transcript

1 JOURNAL OF MATHEMATICAL PHYSICS 56, 3 5 The Lorentz anomaly via operator product expansion Stefan Fredenhagen,,a Jens Hoppe,,b and Mariusz Hynek,c Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut Am Mühlenberg, 4476 Golm, Germany Department of Mathematics, Royal Institute of Technology, KTH 44 Stockholm, Sweden Received 8 August 5; accepted 8 September 5; published online 5 October 5 The emergence of a critical dimension is one of the most striking features of string theory. One way to obtain it is by demanding closure of the Lorentz algebra in the light-cone gauge quantisation, as discovered for bosonic strings more than forty years ago. We give a detailed derivation of this classical result based on the operator product expansion on the Lorentzian world-sheet. C 5 AIP Publishing LLC. [ I. INTRODUCTION More than forty years ago, 6 was noted as a critical dimension for the dual-resonance models that preceded string theory. 3 One way of obtaining the critical dimension has been to show that in lightcone gauge quantisation, the longitudinal Lorentz-operators M i, i =,..., d > only commute in D = d + = 6 space-time dimensions. 4 In this computation, the generators M i are normal-ordered infinite sums cubic in the oscillator modes of the quantised string cp. Ref. 5; standard textbooks have curiously refrained from presenting the calculation in detail. Whereas the quantisation of the string is well understood, much less is known for general M-dimensional extended objects; however, in Ref. 6, it was noted that as a consequence of Lorentz invariance, a dynamical symmetry exists which might give a way to algebraically determine the spectrum if one can understand this symmetry in the quantum theory. Classically, these higher-dimensional objects can be described similarly to the string in the light-cone gauge, but the corresponding world-volume theories are not free as in the case of strings. Therefore, one does not have an expansion in terms of harmonic oscillators that would make it possible to quantise the theory directly. On the other hand, one can still use field theory techniques like operator-product expansions OPEs in the computations. These considerations motivated us to rederive the critical dimension of bosonic string theory in the light-cone gauge quantisation by only using the operator product expansion on the Lorentzian world-sheet. As the world-sheet theory is free, the OPEs are simple and completely equivalent to the harmonic oscillator commutators of the corresponding modes so that it is guaranteed that the result is the same as in the oscillator approach. The computation, however, turned out to be surprisingly tedious and subtle it involves a careful treatment of composite and non-local operators. We decided to write it up and present it in this note in the hope that the approach might be useful for higher dimensional extended objects note also Ref. 7, as well as an alternative derivation of the critical dimension see also Ref. 8 where yet another derivation of the Lorentz anomaly was presented. We should add that, similar to the oscillator computation, it seems that one cannot pinpoint any particular step of the computation where the anomaly arises; it rather appears as a result of an interplay of several anomalous terms that arise due to the regularisations needed to define the composite operators. Note however that because we work on the Lorentzian cylinder, there is no need to artificially introduce a normal-ordering constant in the computation; it is already set to the right value by using the most natural definition of composite operators. a stefan.fredenhagen@aei.mpg.de b hoppe@kth.se c mkhynek@kth.se -488/5/56/3/3/$3. 56, 3-5 AIP Publishing LLC to IP: On: Tue, 4 Nov 5 6:45:5

2 3- Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 Let us describe the computation in a language that can also be used for higher-dimensional extended objects see Ref. 9 for some naive heuristic considerations. The string is parameterised by a map from the Lorentzian cylinder time coordinate t and angular variable ϕ to a flat Minkowski space. In the light-cone gauge, the degrees of freedom are carried by the transversal fields x and their conjugate momenta p, as well as by the zero mode ζ of the coordinate ζ = x and by its conjugate variable. Classically, the longitudinal generators of the Lorentz algebra are given by Here, M i π x i H ζ p i dϕ, i =,..., d = D.. is the classical Hamiltonian density corresponding to p and ζϕ = ζ π π ϕ H ϕ π p p + x x. p x dψ + π p x ψdψ + π ϕ π p x dψ.3 is the reconstructed x coordinate of the string that follows from ζ = π p x. The transversal fields x i ϕ, p j ϕ are constrained by π so that the last term in.3 could be dropped, and ζ can be rewritten as ζϕ = ζ + π ϕ p x dϕ =,.4 p x dψ + π p x ψdψ..5 That the M i Poisson-commute provided.4 holds is a particular case of a result of Goldstone, who for arbitrary dimension M of the extended object solved ζ ϕ a = p a x, a =,..., M.6 ρ for ζ in terms of x and p and some Green s function G ρ is a density satisfying ρ d M ϕ = and then showed that classically, for all M, the generators of the inhomogeneous Lorentz group can be consistently realised on the, ζ, xϕ, pϕ phase-space constrained by the consistency of.6 which for M = and ρ = π simply becomes.4. In the string case considered here, Gϕ,ψ = πψ ϕθψ ϕ ψ ϕ + π + π 6,.7 and the reconstructed x coordinate is ζϕ = ζ π ψ Gϕ,ψ pψ x ψ dψ,.8 which reduces to expression.5 given above. The paper is organised as follows. In Section II, we explain the OPE techniques that are needed to get to a quantum definition of the Lorentz operators. We then derive the commutation relations of all basic fields in Section III. Finally, we compute the crucial commutator [M i, M j ] in Section IV. Appendices A C contain some technical parts of the computation. II. OPERATOR PRODUCT EXPANSION AND COMPOSITE OPERATORS The definition of the Lorentz generators involves products of fields which we have to define properly in the quantum theory. In a free theory, this can be done by using an oscillator expansion of to IP: On: Tue, 4 Nov 5 6:45:5

3 3-3 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 the free fields and then define normal-ordered products by moving annihilation operators to the right of creation operators. Alternatively, we can use the OPE of the fields to define composite operators by subtracting the singular part of the OPE. This leads to an equivalent description for free fields, but it can in principle also be used in more general situations where the usual normal ordering prescription in terms of annihilation and creation operators is not possible. In the case at hand, the x i are massless free fields on the two-dimensional cylinder, and their OPE reads no summation over i x i ϕx i ϕ = π log sin ϕ ϕ + regular.. Then, the OPE of the fields x i is given by with the distribution S sing in two variables ϕ and ϕ given by x i ϕx i ϕ = S sing ϕ, ϕ + regular,. S sing ϕ, ϕ = π ϕ ϕ log sin ϕ ϕ = 4π ϕ cos ϕ ϕ P sin ϕ ϕ,.3 where P denotes the principal value. To define the operator x i ϕx i ϕ, we use point-splitting, so we evaluate. for ϕ = ϕ ϵ and determine the singular piece, x i ϕx i ϕ = + regular..4 ϕ=ϕ ϵ πϵ Note that away from ϕ = ϕ, S sing is a regular function, and we can replace ϕ = ϕ ϵ. This singular piece is then subtracted to define the product of x i with itself, x i x i ϕ lim x i ϕx i ϕ ϵ +..5 ϕ=ϕ ϵ πϵ Up to an additive constant, this is equivalent to the normal ordering prescription using oscillators. Similarly, we have and p i ϕp i ϕ = S sing ϕ, ϕ + regular.6 pi p i ϕ lim p i ϕp i ϕ ϵ + ϕ=ϕ ϵ πϵ This then leads to the quantum definition of H, H ϕ = π pi p i ϕ + x i x i ϕ i..7 π = lim ϵ pi ϕp i ϕ + x i ϕx i ϕ + D ϕ=ϕ ϵ πϵ..9 i In the definition of ζ, we also meet the product of p i and x i. Their operator product expansion only has singularities of contact type,.8 p i ϕx i ϕ = i ϕδ ϕ ϕ + regular.. We therefore get a well-defined composite operator just by point-splitting, pi x i ϕ = lim pi ϕx i ϕ,. ϵ ϕ=ϕ ϵ and we can define the quantum version of ζ as ζϕ = ζ π ψ Gϕ,ψ p x ψ dψ.. When we define the Lorentz generators M i, we also encounter the product of H and x i as well as the product of ζ and p i, which we have to regularise to obtain well-defined expressions. to IP: On: Tue, 4 Nov 5 6:45:5

4 3-4 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 Let us start with the product of H and x i. The singularities in the operator product expansion follow via Wick s theorem from the individual contractions of x i and the x j appearing inside H, x i ϕh ϕ = π ϕ=ϕ ϵ πϵ x i ϕ + regular..3 Therefore, we can define the quantum product of x i and H by xi H ϕ = lim ϵ x i ϕh ϕ ϕ=ϕ ϵ + ϵ x i ϕ..4 A little more work is needed to define the product of p i and ζ, because ζ is defined as a non-local expression in the fields. The possible singularities come from the contact singularity between p i and x j inside ζ and from the singularity between p i and the p j inside ζ. The contact singularity is avoided if we consider the symmetrised product p i ζ + ζ p i, and we find pi ϕ p x ψ + p x ψ p i ϕ = 4π x i ψ ψ cos ψ ϕ P ψ ϕ sin + regular..5 The possible singularity in the symmetrised product of ζ and p i is then p i ϕζϕ + ζϕp i ϕ ϕ=ϕ ϵ = π ψ Gϕ,ψ x i 4π ψ ψ cos ψ ϕ P ψ ϕ sin dψ + regular.6 ϕ=ϕ ϵ = π x i ψ ψ ψ Gϕ,ψ ψ cos ψ ϕ P 4π ψ ϕ sin dψ + regular ϕ=ϕ ϵ.7 = ϕ π x i ψ 4π ψ Gϕ,ψ ψ cos ψ ϕ P ψ ϕ sin dψ + regular. ϕ=ϕ ϵ.8 The possible singular part is therefore a total derivative in ϕ, which means that it does not matter in the expression for M i, which involves an integration over ϕ. Similarly, also the singular part of the product of x i and H is a total derivative see.3, which vanishes upon integration. Therefore, the quantum definition of M i using symmetrised products and point-splitting is given by M i = lim i ϵ, δ ϵ,δ.9 xi = lim ϕ + ϵh ϕ + H ϕx i ϕ + ϵ ζϕ + δp i ϕ p i ϕζϕ + δ dϕ. ϵ,δ. This is our starting point for analysing the commutator of M i and M j. III. BASIC COMMUTATION RELATIONS To compute the commutators of the Lorentz algebra generators, we need to determine the commutators of the fields x i ϕ, p j ϕ,h ϕ, ζϕ, which follow from the canonical commutation relations of x i and p j and and ζ. We first list the results and present the derivation subsequently. The commutators are [, ζ ] = i, 3. [x i ϕ, p j ϕ] = iδ δϕ ϕ, 3. [H ϕ, p j ϕ] = πi ϕδϕ ϕ x j ϕ, 3.3 to IP: On: Tue, 4 Nov 5 6:45:5

5 3-5 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 [x i ϕ, ζ ϕ] = i ϕg ϕ, ϕ x i ϕ, 3.4 [H ϕ, ζ ϕ] = πi δϕ ϕ H ϕ i ϕ ϕ G ϕ, ϕh ϕ + πi 3 D ϕδϕ ϕ, [H ϕ,h ϕ] = iπ ϕδϕ ϕ pϕ x ϕ + x ϕ pϕ + p ϕ x ϕ + x ϕ p ϕ, [H ϕ, x j ϕ] = πi δϕ ϕ p j ϕ, 3.7 [ζϕ, p j ϕ] = i ϕ ϕ Gϕ, ϕ p j ϕ, 3.8 [ζϕ, ζ ϕ] = i ϕg ϕ, ϕ ζ ϕ + ζ ϕ. 3.9 The derivation is straightforward. 3. and 3.: These are the canonical commutation relations. 3.3: We use the definition of H via point-splitting see.9. The only contribution to the commutator comes from the x term in H, [H ϕ, p j ϕ] = π lim ϵ 3.4: = i π lim ϵ [x i ϕ, ζ ϕ] = lim ϵ x ϕ x ϕ ϵ, p j ϕ 3. x j ϕ ϕ δϕ ϕ ϵ + x j ϕ + ϵ ϕδϕ ϕ = πi ϕδϕ ϕ x j ϕ. 3. G ϕ,ψ ψ xi ϕ, pψ x ψ ϵ dψ = i ϕg ϕ, ϕ x i ϕ, 3. where in the second step, we integrated by parts, computed the commutator x i ϕ, pψ x ψ ϵ, and took the limit ϵ. 3.5: The derivation of the commutator of H and ζ is straightforward, but slightly more involved, π [H ϕ, ζ ϕ] = p p ϕ + x x ϕ, ζ ψ G ϕ, ψ p x ψ dψ 3.3 = iπ p p ϕ + x x ϕ π lim ψ G ϕ, ψ pϕ pϕ ϵ + x ϕ x ϕ ϵ, pψ x ψ ϵ dψ 3.4 ϵ, ϵ = i πi H ϕ + = i πi H ϕ + lim ϵ + πi D lim ϵ = i πi H ϕ πi D = πi δϕ ϕ H ϕ i ϕ where we used that ψ G ϕ, ψ ψ δϕ ψ + ϵ pϕ pψ + ψ δϕ ψ x ϕ x ψ ϵ dψ 3.5 ψ G ϕ, ψ ψ δϕ ψ pϕ pψ + x ϕ x ψ D S sing ϕ, ψ dψ ψ G ϕ, ψ ψ δϕ ψ S sing ϕ, ψ ϵ + ψ δϕ ψ + ϵ S sing ϕ, ψ dψ 3.6 ϕg ϕ, ϕ π H ϕ + D + ϕ G ϕ, ϕ π π ϕh ϕ ψ G ϕ, ψ ψ δϕ ψ + ψ 3 π δϕ ψ dψ 3.7 ϕ G ϕ, ϕ H ϕ + πi 3 D ϕδϕ ϕ, 3.8 to IP: On: Tue, 4 Nov 5 6:45:5

6 3-6 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 ψ δϕ ψs sing ϕ,ψ ϵ = ψ δϕ ψs sing ϕ, ϕ ϵ δϕ ψ ψ S sing ϕ,ψ ϵ ψ=ϕ 3.9 = π ϵ 4π + O ϵ ψ δϕ ψ + π ϵ + O ϵ δϕ ψ Note that the commutator of H and ζ contains a term that depends on the number D of space-time dimensions. 3.6: [H ϕ,h ϕ] = π lim pϕ pϕ ϵ + x ϕ x ϕ ϵ, p ϕ p ϕ ϵ + x ϕ x ϕ ϵ ϵ, ϵ : [H ϕ, x j ϕ] = π lim ϵ 3.8: = iπ ϕδϕ ϕ pϕ x ϕ + x ϕ pϕ + p ϕ x ϕ + x ϕ p ϕ. = i π lim ϵ [ζϕ, p j ϕ] = lim ϵ 3. pϕ pϕ ϵ, x j ϕ 3.3 pj ϕ δϕ ϕ ϵ + p j ϕ + ϵ δϕ ϕ = πi δϕ ϕ p jϕ. Gϕ,ψ ψ pj ϕ, pψ x ψ ϵ dψ = i ϕ ϕ Gϕ, ϕ p j ϕ, where in the second equality, we integrated by parts, computed the commutator [p j ϕ, pψ x ψ ϵ], integrated by parts again, and finally took the limit ϵ. 3.9: [ζϕ, ζ ϕ] = i ζϕ ζ ϕ + 4 lim ϵ, ϵ ψg ϕ, ψ ψ Gϕ,ψ pψ x ψ ϵ, p ψ x ψ ϵ dψ d ψ = lim π ϵ, ϵ ϕg ϕ, ϕ pϕ x ϕ ϵ ϵ ϕ ϕ = i ϕg ϕ, ϕ ζ ϕ + ζ ϕ, 3.8 where in the last step we used that ζ = π p x. IV. THE CRUCIAL COMMUTATOR We now want to analyse the commutator of the generators M i and M j, which we defined in.. It can be decomposed as [M i, M j ] = lim lim Mi ϵ, δ, M i ϵ, δ 4. ϵ,δ ϵ, δ = lim ϵ,δ lim ϵ, δ xi ϕ + ϵh ϕ ζϕ + δpi ϕ + p i ϕζϕ + δ, x j ϕ + ϵh ϕ ζ ϕ + δp j ϕ + p j ϕζ ϕ + δ dϕ d ϕ. to IP: On: Tue, 4 Nov 5 6:45:5 4.

7 3-7 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 Here, we used the fact that x i ϕ + ϵ and H ϕ commute for ϵ > see 3.7. There are four types of contributions: the commutator of the terms of the form xh, the two mixed commutators of xh and ζ p, and the commutator of the terms of the form ζ p. At the end, we want to analyse the behaviour when ϵ, δ and ϵ, δ go to zero. It is not guaranteed that this limit exists, and indeed we will see that, e.g., the commutator of the terms of the form xh alone is singular when the regularisation parameters go to zero; this singularity will go away when we combine all contributions to the commutator of M i and M j. On the other hand, we expect that we can take one set of parameters to zero without getting a singularity in the different contributions: because the individual entries entering the commutator are regularised and do not show any singularity in ϵ, δ or in ϵ, δ, the only way a new singularity can appear is through terms that become singular when both types of parameters go to zero like ϵ + ϵ. Our strategy will therefore be to always take the limit ϵ, δ to zero first, and then consider the limit when ϵ and δ go to zero. We start by analysing the commutator of the terms xh in detail. We then present the results for the remaining commutators and evaluate the total expression. A. Commutators of the form [x i H, x j H ] We want to analyse C x H, x H ϵ, ϵ = xi ϕ + ϵh ϕ, x j ϕ + ϵh ϕ dϕ d ϕ 4.3 by using the commutation relations that we worked out in Section III. In a first step, we obtain C x H, x H ϵ, ϵ = πi x j ϕ + ϵp i ϕh ϕ ϵ x i ϕ + ϵ + ϵp j ϕ + ϵh ϕ + x i ϕ + ϵx j ϕ + ϵ x i ϕ + ϵx jϕ + ϵ ζ ϕ dϕ. 4.4 We now consider the behaviour when ϵ goes to zero, and we find xh, xh C ϵ, ϵ = πi x ϕh ϕ ϵ x i ϕ + ϵ + ϵx i ϕ + ϵ + p ϵ x i ϕ + ϵ j H ϕ ϵ p jϕ x + x i ϕ + ϵ j ζ ϕ x ϵ p jϕ x i ϕ + ϵ j ζ ϕ + ϵ p jϕ dϕ + O ϵ = πi x ϕ + ϵh ϕ xi ϕ + ϵ p j H ϕ x i ϕ + ϵ x j ζ ϕ x i ϕ + ϵ x j ζ ϕ + x i ϕ + ϵp jϕ dϕ + O ϵ, 4.6 where in the first summand, we shifted the integration variable by ϵ. As expected, there is no singularity when ϵ is taken to zero. We now want to analyse the possible singularities in ϵ. The singularities between normal-ordered expressions arise from singularities between the constituents, we have, e.g., x i ϕ + ϵ x j ζ ϕ = π x iϕ + ϵ x j p x ϕ 4.7 = x ϵ ϕ + regular. 4.8 OPE normal ordering is in general not associative, and we want to define the normal-ordering of several operators in a right-nested way, ABC A BC. 4.9 to IP: On: Tue, 4 Nov 5 6:45:5

8 3-8 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 In the case at hand, we have see Appendix C so, that we find C xh, xh ϵ, = πi = πi x H = x H xi x j ζ = x i x j ζ + x H ϕ ϵ xi + x x p i x j, 4. x x i p j, 4. xi pi x j ϕ p j H ϕ + ϵ x ϕ j ζ ϕ + ϵ pi x ϵ j ϕ p j x i ϕ + x 3 i p j ϕ x p j x i ϕ x i x jζ ϕ x ϵ ϕ + x i p j ϕ dϕ + Oϵ ϕ + x H ϕ x i p j H ϕ + x i x j ζ ϕ x i x jζ ϕ dϕ + Oϵ. 4. x H, pζ The remaining singularity in ϵ will be cancelled by the contribution C that we discuss in the following. B. Commutators of the form [x i H, p j ζ ] We now turn to the analysis of the contribution x H, pζ C ϵ, δ = xi ϕ + ϵh ϕ, ζ ϕ + δp j ϕ + p j ϕζ ϕ + δ dϕ d ϕ. 4.3 Evaluating the commutator, we find for i j x H, pζ C ϵ, δ = πi xi ϕ + ϵp j ϕ δh ϕ x i ϕ + ϵ δx j ϕ δζ ϕ dϕ + i x i ϕ + ϵp j ϕh ϕ ϕ G ϕ + δ, ϕ + ϵ G ϕ + δ, ϕ dϕ d ϕ πi 3D x i ϕ + δ + ϵp j ϕ dϕ. 4.4 We expand this expression first in δ, and we obtain xh, pζ C ϵ, δ = πi p x i ϕ + ϵ j H ϕ δ p jϕ x i ϕ + ϵ δx δ i ϕ + ϵ + x x i ϕ + ϵ j ζ ϕ δ p jϕ + i x i p ϕ + ϵ j ϕh ϕ π S sing ϕ, ϕp j ϕ ϕ G ϕ + δ, ϕ + ϵ G ϕ + δ, ϕ dϕ d ϕ + i π x i ϕ + ϵ S sing ϕ, ϕp j ϕ ϕ G ϕ + δ, ϕ + ϵ G ϕ + δ, ϕ dϕ d ϕ πi D 3 x i ϕ + ϵp jϕ dϕ + O δ. 4.5 In the third and fourth lines, we have subtracted and added the singular piece of p j ϕh ϕ. In the third line, there is therefore no singularity coming from the operator part when ϕ and ϕ are close together, and we can just set δ to. The fourth line can be evaluated by writing S sing as a derivative and then using partial integration, i π x i ϕ + ϵ S sing ϕ, ϕp j ϕ ϕ G ϕ + δ, ϕ + ϵ G ϕ + δ, ϕ dϕ d ϕ = i = i x i ϕ + ϵ ϕ x i ϕ + ϵ cos ϕ ϕ cos ϕ ϕ P P sin ϕ ϕ sin ϕ ϕ p j ϕ ϕ G ϕ + δ, ϕ + ϵ G ϕ + δ, ϕ dϕ d ϕ 4.6 p j ϕ πδ ϕ ϕ + δ ϵ + πδ ϕ ϕ + δ dϕ d ϕ 4.7 to IP: On: Tue, 4 Nov 5 6:45:5 dϕ

9 3-9 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 = iπ x i ϕ + ϵp jϕ cos ϵ δ sin ϵ δ = iπ x i ϕ + ϵp jϕ cos ϵ sin ϵ + δ Inserting this result into 4.5, we obtain C xh, pζ + cos δ sin δ dϕ 4.8 dϕ + O δ. 4.9 x i ϕ + ϵ p j H ϕ x i ϕ + ϵ x j ζ ϕ + i ϕ + ϵp jϕ dϕ ϵ, δ = πi x + i x i p ϕ + ϵ j ϕh ϕ π S sing ϕ, ϕp j ϕ ϕ G ϕ, ϕ + ϵ G ϕ, ϕ dϕ d ϕ + iπ x i ϕ + ϵp jϕ cos ϵ sin ϵ dϕ πi D 3 x i ϕ + ϵp jϕ dϕ + O δ. 4. We observe that the expression contains no singularity in δ. Now, we expand in ϵ, xh, pζ C ϵ, = πi xi p j H ϕ + p j x ϵ i ϕ xi x j ζ ϕ pi x ϵ j ϕ + x i p j ϕ dϕ + i π x π ϵ i p j ϕ + x i p j ϕ dϕ + iπ x ϵ i p j ϕ + x i p j ϕ dϕ πi x i p j ϕ dϕ + Oϵ 4. lim ϵ = πi D 3 ϵ p j x i ϕ pi x j ϕ + x i p j H ϕ x i x j ζ ϕ + x i p j ϕ D 6 x i p j ϕ dϕ + Oϵ. 4. The remaining singularity cancels the terms that we found in C x H, x H see 4., so that we find xi ϕ + ϵh ϕ, M j dϕ = lim C xh, xh ϵ = πi xh, pζ ϵ, + C ϵ, 4.3 x H ϕ x i x jζ ϕ + 5 x 3 i p j ϕ x ϕ D 6 x i p j ϕ dϕ. 4.4 C. Remaining commutators and final result The remaining commutators can be computed analogously to the computations we displayed above, which is done in Appendices A and B, and the results are given in A5 and B. Their sum does not contain any singularity, and one obtains lim δ ζϕ + δpi ϕ + p i ϕζϕ + δ, M j dϕ pζ, xh = lim C δ = πi pζ, pζ δ, + C δ, 4.5 p i x j H ϕ + x j x i ζ ϕ 5 x ϕ + 3 p j x i ϕ + D 6 x ϕ dϕ. 4.6 Combining now all contributions we find for the quantum commutator the final result πi Mi, M j = 3D 6 x j p i ϕ x i p j ϕ dϕ, 4.7 which vanishes if D = 6. to IP: On: Tue, 4 Nov 5 6:45:5

10 3- Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 The dimension-dependent term linear in D came from the commutator of H and ζ, the other anomalous terms came from all commutators that occur in the computation. As already remarked in the Introduction, we did not introduce a normal-ordering constant to define H. In principle, such a shift H H + const. could be considered then one would derive from demanding a vanishing commutator [M i, M j ] that this constant is zero, but the definition of H via OPE normal ordering on the cylinder appears to be most natural. ACKNOWLEDGMENTS We would like to thank Jürg Fröhlich and Volker Schomerus, AEI and KTH, and the Swedish Research Council. APPENDIX A: COMMUTATORS OF THE FORM [p i ζ, x j H ] We discuss here the contribution C pζ, x H δ, ϵ = ζϕ + δpi ϕ + p i ϕζϕ + δ, x j ϕ + ϵh ϕ dϕ d ϕ. A For i j, the commutator is given by C pζ, x H δ, ϵ = πi x j ϕ + ϵp i ϕ δh ϕ x j ϕ + ϵ δx i ϕ δζ ϕ dϕ i x j ϕ + ϵp iϕh ϕ ϕ Gϕ + δ, ϕ + ϵ Gϕ + δ, ϕ dϕ d ϕ + πi 3D x j ϕ + δ + ϵp iϕ dϕ. The result is regular when we expand in ϵ, and we obtain C pζ, x H δ, = πi Expanding in δ, we find C pζ, x H δ, = πi = πi p i ϕ δ x j H ϕ + x j x i ϕ δζ ϕ x j ϕ + δp iϕ + D 6 x j ϕ + δp iϕ dϕ. A A3 pi x δ j ϕ pj x i ϕ p i x j H ϕ + x j x i ζ ϕ x D ϕ + x 6 ϕ dϕ + Oδ A4 pi x δ j ϕ pj x i ϕ p i x j H ϕ + x j x i ζ ϕ + x i p j x D ϕ + x 6 ϕ dϕ + Oδ. A5 APPENDIX B: COMMUTATORS OF THE FORM [p i ζ, p j ζ ] Let us now discuss the term pζ, pζ C δ, δ = ζϕ + δpi ϕ + p i ϕζϕ + δ, ζ ϕ + δp j ϕ + p j ϕζ ϕ + δ dϕ d ϕ. 4 B One can straightforwardly show that the four different terms that one obtains from expanding the commutator above all lead to the same contribution, to IP: On: Tue, 4 Nov 5 6:45:5

11 3- Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 δ, δ = ζϕ + δ, pi ϕ + p i ϕζϕ + δ, ζ ϕ + δ, p j ϕ + p j ϕζ ϕ + δ dϕ d ϕ 4 pi = ϕζϕ + δ, p j ϕζ ϕ + δ dϕ d ϕ. pζ, pζ C B B3 By explicitly evaluating the commutator, we find pζ, pζ C δ, δ = πi pi ϕ δ δp j ϕ δ p j ϕ δ δp i ϕ δ ζϕ dϕ + i pi ϕ δp j ϕ δ ϕ Gϕ, ϕ δ Gϕ, ϕ ζ ϕ p j ϕ δp i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ ζϕ dϕ d ϕ B4 = i pi ϕ δp j ϕ δ ϕ Gϕ, ϕ δ Gϕ, ϕ ζ ϕ +p j ϕ δp i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ ζ ϕ dϕ d ϕ. B5 We now expand this expression in δ analogously to the cases we discussed before. We obtain pζ, pζ C δ, δ = i p i ϕ δ δ x j ϕ + p j ζ δ ϕ δ ϕ Gϕ, ϕ + 3 ϕ Gϕ, ϕ +p i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ p j ϕ δζ ϕ π S sing ϕ δ, ϕx j ϕ π +p i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ S sing ϕ δ, ϕx j ϕ dϕ d ϕ + O δ B6 = πi δ p iϕ δx j ϕ p iϕ δx j dϕ ϕ + i p i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ p j ϕζ ϕ π S sing ϕ, ϕx j dϕ ϕ d ϕ + πi p i ϕ δx j ϕ cos δ + δ + cos δ sin δ+ δ sin δ dϕ + O δ B7 = πi p i ϕ δx j dϕ ϕ + i p i ϕ δ ϕ G ϕ, ϕ δ G ϕ, ϕ p j ϕζ ϕ π S sing ϕ, ϕx j dϕ ϕ d ϕ + πi p i ϕ δx j ϕ cos δ + δ dϕ + O δ. B8 sin δ+ δ As expected, there is no singularity in δ. We now set δ = and expand in δ, pζ, pζ C δ, = πi p i x j ϕ dϕ + πi pj x δ i ϕ + pj x i ϕ dϕ + πi pi x j ϕ pi x j ϕ dϕ + Oδ δ = πi pj x δ i ϕ pi x j ϕ + 3 pi x j ϕ + pj x i ϕ dϕ + Oδ. B9 B APPENDIX C: NON-ASSOCIATIVITY OPE normal ordering is in general not associative. In this appendix, we will discuss those cases that are relevant in the main text. The first identity we want to explain is x H = x H x p i x j. C to IP: On: Tue, 4 Nov 5 6:45:5

12 3- Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 The simplest way to show this is to write H = H k, k H k = π p k + x k and consider the summands H k individually. For k different from i and j, there is no singularity, and thus, normal ordering of x j, p i, and H k is associative. Now consider k = i, x Hi ϕ = lim x j ϕ ϵp i ϕ ϵh i ϕ + x ϵ ϵ j ϕ ϵ x j ϕ p i ϕ C3 = lim x j ϕ ϵ pi H i ϕ ϵ ϵ p iϕ + x ϵ j ϕ ϵ x j ϕ p i ϕ C4 = x H i ϕ x ϕ. C5 If we instead consider k = j, we find x Hj ϕ = lim x j ϕ ϵp i ϕ ϵh j ϕ + ϵ ϵ x j ϕp iϕ = lim x j ϕ ϵ p i ϕ ϵ p i ϕ H j ϕ + ϵ ϵ x j ϕp iϕ = lim x H j ϕ ϵ ϵ x j ϕp iϕ + x j ϕp i ϕ + ϵ x j ϕp iϕ = x H j ϕ + x ϕ. Combining C5 and C9, we arrive at the desired result C. The second relation that we need is xi x j ζ = x i x j ζ + x x i p j. C We can prove it analogously. Write ζ = ζ k, k ζ k = π pk x k and consider first the case when ζ i appears in the normal-ordered product, xi x j ζ i = lim x i ϕ ϵx j ϕ ϵζ i ϕ + ϵ ϵ x j ϕp iϕ = lim x i ϕ ϵ x j ϕ ϵ x j ϕ ζ i ϕ + ϵ ϵ x j ϕp iϕ = lim xi x j ζ i ϕ ϵ ϵ x j ϕp iϕ + x j ϕp iϕ + ϵ x j ϕp iϕ = x i x j ζ i ϕ + x ϕ. Now consider the case, when ζ j occurs, C C6 C7 C8 C9 C C C3 C4 C5 xi x j ζ j = lim x i ϕ ϵx j ϕ ϵζ j ϕ + xi ϕ ϵ x ϵ ϵ i ϕ p j ϕ C6 = lim x i ϕ ϵ x j ζ j ϕ ϵ ϵ p jϕ + xi ϕ ϵ x ϵ i ϕ p j ϕ C7 = x i x j ζ j ϕ x i p j ϕ. C8 Combining C5 and C8, we arrive at the final result C. to IP: On: Tue, 4 Nov 5 6:45:5

13 3-3 Fredenhagen, Hoppe, and Hynek J. Math. Phys. 56, 3 5 C. Lovelace, Pomeron form-factors and dual Regge cuts, Phys. Lett. B 34, R. C. Brower, Spectrum generating algebra and no ghost theorem for the dual model, Phys. Rev. D 6, P. Goddard and C. B. Thorn, Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys. Lett. B 4, P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56, G. Arutyunov, Lectures on String Theory Utrecht University, 9. 6 J. Hoppe, Fundamental Structures of Mbrane theory, Phys. Lett. B 695, 384 ; e-print arxiv:3.589 [hep-th]. 7 U. Marquard and M. Scholl, Lorentz algebra and critical dimension for the bosonic membrane, Phys. Lett. B 7, K. Bering, A note on angular momentum commutators in light-cone formulation of open bosonic string theory, Acta Phys. Pol., B 43, 8 ; e-print arxiv: [hep-th]. 9 J. Hoppe, M-brane dynamical symmetry and quantization, e-print arxiv:.4334 [hep-th]. J. Goldstone, unpublished notes 985. to IP: On: Tue, 4 Nov 5 6:45:5