Triviality of Higher Derivative Theories

Physics Letters B 577 (2003) 137–142 www.elsevier.com/locate/physletb

Triviality of higher derivative theories

Victor O. Rivelles 1

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 27 June 2003; received in revised form 28 August 2003; accepted 13 October 2003 Editor: M. Cveticˇ

Abstract We show that some higher derivative theories have a BRST symmetry. This symmetry is due to the higher derivative structure and is not associated to any gauge invariance. If physical states are deﬁned as those in the BRST cohomology then the only physical state is the vacuum. All negative norm states, characteristic of higher derivative theories, are removed from the physical sector. As a consequence, unitarity is recovered but the S-matrix is trivial. We show that a class of higher derivative quantum gravity theories have this BRST symmetry so that they are consistent as quantum ﬁeld theories. Furthermore, this BRST symmetry may be present in both relativistic and non-relativistic systems.  2003 Elsevier B.V. Open access under CC BY license.

PACS: 11.10.-z; 11.15.-q; 11.30.Ly; 04.60.-m

Higher derivative (HD) theories were introduced done mainly by imposing some superselection rule or in quantum field theory in an attempt to get rid of some subsidiary condition to remove the undesirable ultraviolet divergences [1]. It was soon recognized states [4]. However, there is no general scheme to that they have an energy which is not bounded from remove the ghosts and this is done in a case by case below [2] and that these negative energy states can basis. be traded by negative norm states (or ghosts) [3] In this Letter we will show that some HD theories leading to non-unitary theories. Although inconsistent, have a BRST symmetry after ghost fields are intro- HD theories have better renormalization properties duced. Usually the BRST symmetry is found in gauge than conventional ones and a large amount of work theories as a symmetry of the gauge fixed action.2 Its was dedicated to the study of such theories [4]. In purpose is to remove the negative norm states associ- the gravitational context, for instance, it is able to ated to the gauge invariance. Physical states are then produce a renormalizable quantum gravity theory [5]. defined as those which have zero ghost number and To overcome the ghost problem many attempts were are invariant under the BRST symmetry. In the present case, however, the BRST symmetry is not due to gauge

E-mail address: [email protected] (V.O. Rivelles). 1 On leave from Instituto de Física, Universidade de São Paulo, 2 The BRST transformations may be written in different ways Caixa Postal 66318, 05315-970 São Paulo, SP, Brazil. E-mail (even in non-local form) but that is not the case here. This is address: [email protected]. discussed in [6].

0370-2693  2003 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2003.10.039 138 V.O. Rivelles / Physics Letters B 577 (2003) 137–142 invariance. Rather, it is a feature of the HD structure indicating that theory may be empty. It looks like a of the action. As it will be shown, it can be found even topological theory of Witten type [7] but it clearly in the case of a HD real scalar field. Since we have a depends on the local structure of the space–time. BRST symmetry it is natural to require that the physi- In topological theories there are no local degrees of cal states are those which have zero ghost number and freedom from the start while here we have positive are left invariant by this symmetry, in analogy to what and negative norm states before introducing the ghost is done in gauge theories. With these requirements we fields. The requirement that the physical states are find that HD theories have only one physical state, the in the cohomology of the BRST charge means that vacuum, since all others physical states appear in zero all of them, except the vacuum, appear in zero norm norm combinations. Therefore, the resulting theory is combinations as we will show. ghost free but it is also empty. In this way the unitarity Interactions can be introduced trough a prepotential problem associated to HD theories can be overcome U(φ). The action but the resulting theory has no states besides the vac- 1 uum. We will show that this symmetry is present in S = dd xδ c¯ Onφ − b − U , (6) HD non-Abelian gauge theories and also in HD grav- 2 ity theories, so that they are unitary but have a trivial which is invariant under the BRST transformations S-matrix. (4), yields, after elimination of the auxiliary field, Even though we work mostly with relativistic sys- 1 2 1 tems, relativistic invariance is not a necessary ingre- S = dd x Onφ + U 2 2 2 dient. The HD BRST symmetry may also be found in non-relativistic situations. We will exemplify this for + UOnφ −¯c Onc + U c . (7) the case of the HD harmonic oscillator. Let us first consider a real scalar field φ in d Assuming that the BRST symmetry has no anomaly dimensions with an action it is enough to examine the free case to find out the 1 physical states. S = dd x OnφOnφ −¯cOnc , (1) 2 In order to simplify the analysis we will consider from now on the case n = 1 and only one factor in O = N + 2 where i=1( mi ) is a product of N Klein– O,thatisN = 1. The simplest situation is the one- Gordon operators with masses mi , c is a ghost, c¯ dimensional case, that is, quantum mechanics, where an anti-ghost and n is an integer. This HD action is O = d2/dt2 + m2. Then the action (1) reduces to that invariant under the following BRST transformations of a HD harmonic oscillator. Canonical quantization is straightforward if we keep the auxiliary field and δφ = c, δc = 0,δc¯ = Onφ. (2) use the action in the form (3) where only first order These transformations are nilpotent for φ and c and derivatives appear after an integration by parts is on-shell nilpotent for c¯. To find out the role of this performed. We then find that the non-trivial equal time symmetry let us introduce an auxiliary field b and commutation relations are rewrite the action (1) as ˙ ˙ φ,b = b,φ =−i, (8) 1 S = dd x bOnφ − b2 −¯cOnc . (3) where dots denote time derivatives. The solution to the 2 field equation for φ and b can be expanded as The BRST transformations now read 1 − φ(t) = (a + bmt)e imt + a† + b†mt eimt , δφ = c, δc = 0,δc¯ = b, δb = 0, (4) 2m3/2 (9) and are off-shell nilpotent. This action can now be − b(t) =−im1/2 be imt − b†eimt . written as a total BRST transformation (10) 1 The canonical Hamiltonian yields S = dd xδ c¯ Onφ − b , (5) 2 H = m 2b†b − ib†a + ia†b , (11) V.O. Rivelles / Physics Letters B 577 (2003) 137–142 139 and using the transformation a˜ = ia − b/2, b˜ = ia − quartet mechanism is also operational in this case. The 3b/2wefind only physical state is the vacuum. The massless case needs some care but again the same result is found. = ˜ † ˜ − ˜† ˜ H m a a b b , (12) At this point it is worth to remind that to remove so that it is not positive definite at the classical the negative norm states for the HD scalar field theory level. Upon quantization we find that the non-trivial the condition b = 0 is often employed [9]. It is argued commutators for the operators a and b are given by that there is a “gauge symmetry” δφ = Λ with Λ satisfying ( + m2)Λ = 0. The “gauge condition” † † a,a =−1, a,b = i, (13) b = 0 is then imposed to remove the ghost states. where a and b are annihilation operators and a† and b† However, the Hamiltonian analysis reveals that there is are creation ones. By the transformation aˆ = b − ia, no true gauge symmetry since there are no constraints bˆ =−ia,wefind at all. Amazingly, a “BRST symmetry” associated to this “gauge symmetry” can be found3 and it agrees a,ˆ aˆ † =− b,ˆ bˆ† = 1, (14) with (18). Now its origin is clear; it is completely due to the HD structure as we have shown. so that the Hilbert space is not positive definite. We can now easily generalize this construction to We now consider the ghost fields in (1). Quantiza- other types of fields. Since we want to keep the same tion is straightforward. The solution to the equations BRST transformations (4) then the ghosts and the of motion can be expanded as auxiliary field b must have the same tensor structure 1 − as the field under consideration. For gauge theories we c(t) = ce iωt + c†eiωt , (15) 3/2 2m should also take into account the ordinary Faddeev– − c(t)¯ = m1/2 ce¯ iωt +¯c†eiωt , (16) Popov ghosts associated to the gauge symmetry since they have a different origin from the ghosts coming ¯ † where c and c are annihilation operators and c from the HD structure [11]. We will argue, however, ¯† and c are creation operators. The non-trivial anti- that only one set of ghost fields is needed. commutators are given by For a theory with a non-Abelian vector field we

¯† =− ¯ † = start with c,c c,c 1. (17) d µ 1 The BRST transformations (4) for the operators in (9), S = d x Tr δ c¯ Eµ − bµ , (19) (10), (15) and (16) (not for the fields) are 2 where E depends only on the vector field A .The δa = c, δc = 0,δc¯ =−ib, δb = 0. (18) µ µ BRST transformations are given by This means that the these operators belong to the quartet representation of the BRST algebra [8]. As a δAµ = cµ,δcµ = 0, consequence, the BRST invariant states, build out of δc¯µ = bµ,δbµ = 0. (20) these operators, appear only in zero norm combinations through the quartet mechanism. Therefore the This structure is similar to that found in topological ¯ only physical state is the vacuum. In this way the HD field theories [12]. There, E, b and c are two forms harmonic oscillator is free of negative norm states but instead of vectors so that a topological invariant is the only state left is the vacuum. generated after the elimination of the auxiliary field. The situation for the scalar field is analogous to Here the tensor structure is quite different and Eµ is the quantum mechanical case. Now O = + m2 chosen so that it gives rise to a HD theory. For a fourth and the fields have expansions similar to (9), (10), order gauge theory we can choose (15) and (16) with the creation and annihilation ν 1 ν operators depending on the momenta. They obey Eµ = D Fνµ + ∂µ∂ Aν . (21) 2ξ commutation relations similar to (13) and (17) and the Hamiltonian has the same form as (11). Hence the BRST transformations are the same as in (18) and the 3 In fact, it was found in the singleton field theory in [10]. 140 V.O. Rivelles / Physics Letters B 577 (2003) 137–142

The first term in (21) will produce a gauge invariant For gravity the ghosts and the auxiliary field are term in the action if the second term is absent. The second order symmetric tensors. The action is second term breaks gauge invariance and can be d µν 1 regarded as a gauge fixing term since it allows us to S = d xδ c¯ Eµν − bµν , (24) 2 find the propagator for Aµ. At this point we should decide whether we introduce the gauge fixing term and the BRST transformations are given by in Eµ, as in (21), or if we consider the ordinary δgµν = cµν ,δcµν = 0, Faddeev–Popov procedure for gauge fixing. If we do δc¯µν = bµν,δbµν = 0. (25) not add the gauge fixing term in (21) then Eµ will give rise to an action for c¯µ and cµ which has a gauge For a generic fourth order gravity theory we can ¯ ¯ symmetry δc¯µ = DµΣ, δcµ = DµΣ,whereΣ and Σ choose are Grassmannian functions. This would require the = 1/4 + + +··· introduction of ghosts for ghosts besides the ordinary Eµν g (c1Rµν c2gµν R c3gµν ), (26) Faddeev–Popov ghosts for Aµ. We then choose to where gµν is the metric, g its determinant, Rµν introduce the gauge fixing term directly in Eµ to avoid the Ricci tensor, R the curvature scalar and dots the proliferation of ghost fields. denote the gauge fixing terms. After the elimination With the choice (21) we find that (19) yields the of the auxiliary field we find that the action for the following action gravitational sector can be written as √ d 2 µν 1 Sgr = d x −g γR− Λ + αR + βR Rµν , S = dd x 2 (27) 1 × ν ρµ + µ with γ 2 =−4Λ(dα + β)/d. Gauge fixing terms were Tr D FνµDρ F 2 ∂ ∂A∂µ∂A 4ξ omitted. Notice that we cannot get the Einstein– 1 ν µ ¯ µν Hilbert action since setting α = β = 0togetridof + D Fνµ∂ ∂A+ F Fµν ξ the HD contributions also eliminates the term in R. However, we can choose Λ = 0 in order to get a purely µ ν 1 µ ν − 2i c¯ ,c Fµν + ∂ c¯µ∂ cν , (22) = ξ HD gravity theory. Alternatively we can choose β −dα, which corresponds to a traceless Ricci tensor. ¯ µν µ where F and Fµν are the field strengths for c¯ This case is distinguished by energy considerations and cµ, respectively. Canonical quantization is easily [13]. The conformal case has no special features. performed keeping the auxiliary field bµ since it gives Notice also that (26) does not allow us to generate the rise to a first order Lagrangian. No constraints are square of the Riemann tensor thus precluding Gauss– found and the Hamiltonian is not positive definite. Bonnet terms which arise in string theory. Details will Quantization simplifies in the Feynman gauge ξ = 1 be presented elsewhere. and we find that the quartet mechanism applies to each Some final remarks are in order. Could a BRST component of the vector field. Again, the only physical symmetry be found for ordinary second order theo- state is the vacuum. Coupling to ordinary or HD ries? Covariance requires that the operator O be at matter is straightforward. However, only non-minimal least quadratic in the derivatives. However, in lower couplings arise. Details will be given elsewhere. dimensions, this requirement can be relaxed. Chiral Other choices for Eµ are possible. For instance, bosons in two dimensions have a BRST symmetry which allows the vacuum as the only physical state [14]. In higher dimensions we could consider the ac- 1 ν m Eµ = D Fνµ + Aµ, (23) tion (6) without the term Onφ and U as a func- m 2 tion of φ and its derivatives. Then we can take U = gives rise to a fourth order massive vector theory φ( + m2)φ.ThetermU 2 in (7) yields the La- which includes the standard Maxwell term. However, grangian for the ordinary scalar field but we get non- there is no gauge invariance in this case. local interactions with the ghosts due to the term U cc¯ . V.O. Rivelles / Physics Letters B 577 (2003) 137–142 141

Even though there is a BRST symmetry, a detailed As a last remark it must be stressed that a trivial analysis shows that on-shell c¯ = b = 0 so that the quar- topology for the space (or space–time) was assumed tet mechanism cannot be applied. It is imperative to throughout the Letter. If a non-trivial topology is have non-trivial solutions to the field equation for the present then topological excitations may arise as they quartet mechanism to work. Since c¯ = 0 all contri- do in topological field theories of Witten type. For butions from the ghost sector to the physical Hilbert instance, in the case of non-Abelian gauge fields (22) space vanish because they must be functions of cc¯ . there are instanton solutions which may give non- Then we end up with the usual scalar theory and the trivial contributions to correlation functions. This is BRST symmetry is trivial in this case. presently under investigation and will be reported A more radical possibility would be to consider elsewhere. non-local√ expression for O, for instance O = + m2. The action (1) with n = 1 would be the ordinary one for a scalar field but for the ghosts we find a Acknowledgements non-local action. Quantization of such non-local theories has been done [15] but it not clear how the quartet I would like to thank A.J. Accioly, S. Deser, mechanism can be applied with non-local ghosts. This R. Jackiw, I. Shapiro and S. Sorella for useful discus- deserves further understanding. sions. This work was partially supported by CAPES, Returning to the quantum mechanical case, the CNPQ and PRONEX under contract CNPq 66.2002/ action (7) resembles that of supersymmetric quantum 1998-99. mechanics if O = d/dt and the ghosts are regarded as fermions. The BRST transformations (4) are identical to the supersymmetry transformations but the anti- References BRST transformation (obtained by replacing ghosts by anti-ghosts and vice versa) δφ =¯c, δc¯ = 0, δc = [1] W. Thirring, Phys. Rev. 77 (1950) 570. −(O + V), are not. The anti-commutator of the [2] A. Pais, G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145. [3] W. Heisenberg, Nucl. Phys. 4 (1957) 532. BRST and anti-BRST transformations vanishes as [4] See the following papers for earlier references: expected while in the supersymmetric case it would J. Lukierski, Acta Phys. Pol. 32 (1967) 771; be proportional to the Hamiltonian. There are crucial N. Nakanishi, Prog. Theor. Phys. Suppl. 51 (1972) 1; signs in the action and in the transformation rules K.S. Stelle, Gen. Relativ. Gravit. 9 (1978) 353; which, when changed, produces the supersymmetric H. Narnhofer, W. Thirring, Phys. Lett. B 76 (1978) 428; M. Mintchev, J. Phys. A 13 (1980) 1841; model. Hence the supersymmetric model is obtained E.S. Fradkin, A.A. Tseytlin, Nucl. Phys. B 201 (1982) 469; by twisting the BRST symmetry. P. Broadbridge, J. Phys. A 16 (1983) 3271; We should also remark that the BRST symmetry N.H. Barth, S.M. Christensen, Phys. Rev. D 28 (1983) 1876; cannot be found in any HD theory. For instance, in the U. Moschella, J. Math. Phys. 31 (1990) 2480; simple case of non-degenerated masses I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity, IOP, Bristol, 1992; M. Asorey, J.L. Lopez, I.L. Shapiro, Int. J. Mod. Phys. A 12 1 (1997) 5711, hep-th/9610006; S = dd x + m2 φ + m2 φ 2 1 2 F.J. de Urries, J. Julve, E.J. Sanchez, J. Phys. A 34 (2001) 8919, hep-th/0105301; S.W. Hawking, T. Hertog, Phys. Rev. D 65 (2002) 103515, hep- −¯ + 2 c m1 c , (28) th/0107088; T.C. Cheng, P.M. Ho, M.C. Yeh, Nucl. Phys. B 625 (2002) 151, hep-th/0111160; there is a symmetry of the action given by δφ = c, A. Accioly, A. Azeredo, H. Mukai, J. Math. Phys. 43 (2002) = ¯ = + 2 473; δc 0, δc ( m2)φ. However, this symmetry is not nilpotent since on-shell δ2c¯ = (m2 − m2)c = 0. E.J. Villasenor, J. Phys. A 35 (2002) 6169, hep-th/0203197; 2 1 S. De Filippo, F. Maimone, AIP Conf. Proc. 643 (2003) 373, It is nilpotent only when both masses are equal thus gr-qc/0207059; 2 implying in (1) with n = 1andO = + m ,wherem G. Magnano, L.M. Sokolowski, gr-qc/0209022; is the common mass. S. Nojiri, hep-th/0210056. 142 V.O. Rivelles / Physics Letters B 577 (2003) 137–142

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