Higher-order derivative theories and conformal invariance

Abstract

In generalization of an earlier result by Durr and van der Merwe it is demonstrated that the massless (N+1)-order Klein-Gordon equations for a scalar field and the (2N+1)-order Weyl and Dirac equations for a spinor field are invariant under conformal transformations if one attributes a (mass) dimension 1−N to the scalar field and a dimension 3/2−N to the spinor field. ForN>0 the canonical quantization of these field theories requires a vector space with indefinite metric. The state spaces in the occupation number representation are explicitly given. They can be constructed in such a way that, in addition to normalizable states occurring forN=even in the scalar case and in all spinor cases, the states can be grouped into pairs of nonorthogonal zero-norm states. The Hilbert subspace spanned by the normalizable states which are eigenstates of the Hamiltonian can be taken as the physical subspace. The current operator is constructed in the spinor cases. One shows that the conformal invariance of the gauge-invariant current operator forN>1 can only be achieved if the gauge field is not an independent field but is built up from the spinor field in a particular fashion which generalizes an earlier conjecture by Durr and Winter. Higher-order derivative theories allow interactions which increase with distance and hence are of physical interest in connection with the confinement and the ultraviolet divergence problem.