Triviality of higher derivative theories

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.Higher derivative (HD) theories were introduced in quantum field theory in an attempt to get rid of ultraviolet divergences [1]. It was soon recognized that they have an energy which is not bounded from below [2] and that these negative energy states can be traded by negative norm states (or ghosts) [3] leading to non-unitary theories. Although inconsistent, HD theories have better renormalization properties than conventional ones and a large amount of work was dedicated to the study of such theories [4]. In the gravitational context, for instance, it is able to produce a renormalizable quantum gravity theory [5]. To overcome the ghost problem many attempts were E-mail address: rivelles@fma.if.usp.br (V.O. Rivelles). done mainly by imposing some superselection rule or some subsidiary condition to remove the undesirable states [4]. However, there is no general scheme to remove the ghosts and this is done in a case by case basis.In this Letter we will show that some HD theories have a BRST symmetry after ghost fields are introduced. Usually the BRST symmetry is found in gauge theories as a symmetry of the gauge fixed action. 2 Its purpose is to remove the negative norm states associated to the gauge invariance. Physical states are then defined as those which have zero ghost number and are invariant under the BRST symmetry. In the present case, however, the BRST symmetry is not due to gauge 2 The BRST transformations may be written in different ways (even in non-local form) but that is not the case here. This is discussed in [6].

Quantization ofncoupled scalar field theory

“…On the other hand, choosing N 1 = − √ m reproduces the particle theory's result appeared in Ref. [10]. Using the ansatz for Eq.…”

“…which lead to ( − m 2 ) n φ n = 0 (5) for a scalar field φ n . In the classical aspect of the theory, the other fields φ l with l = 1, · · · , n − 1 are considered as auxiliary fields used to lower the number of derivatives in the single scalar φ n action [10] S n = − 1 2…”

“…This distinguishes the odd n case from the even n case. However, the degenerate cases are ruled out in that approach because the higher-order Green's function (equivalent second-order Lagrangian) blows up after performing the partial fraction.In this work, we investigate a model of n coupled scalar fields in Minkowski spacetime where all masses degenerate, which leads to a 2n-order single scalar theory when eliminating n − 1 auxiliary scalar fields [10]. In particular, for n = 2, the model was considered as a toy model of the fourth-order critical gravity on AdS 3 spacetime which appeared in the pursuit of quantum gravity.…”

Renormalizability of generalized quantum electrodynamics

“…Its purpose is to remove all the negative norm states and get a unitary theory, physical states are thus defined as those which have zero ghost number. In the light of this idea it was proposed that the HD theories present a BRST symmetry, which is seen as an intrinsic feature of these HD theories [9]. However, by imposing this symmetry we were led to an unitary but trivial resulting theory through the quartet mechanism [10], since all physical states, excepted the vacuum, appear in zero-norm combinations.…”

About the Triviality of the Higher Derivative Sector in the Abelian Lee-Wick Model

“…which correspond to completely independent degrees of freedom, as showed in the Section II, the velocity of the auxiliary fields. This redefinition allows us rewrite the higher-derivative sector introduced by the Faddeev-Popov quantization procedure like a vectorial fields required by the BRST quartet construction, like in [47], but in this case, they come from the ordinary Faddeev-Popov ghost associated to the Abelian LW correspondent BRST symmetry and does not spoil the nature of the Faddeev-Popov ghost term, which is rewritten in terms of massless vectorial fields which propagate because the well-defined kinetic part in (51), but not interact, i.e., their behaviour is exactly equal to Faddeev-Popov ghost in photon sector. The associated Faddeev-Popov operator carries vectorial indices, which is related to the fact that the vectorial Nakanishi-Lautrup field encoding the gauge fixing condition does not cancel the longitudinal mode of the massive gauge potential, as showed in the Section III.…”