We provide a complete classification of the unitary irreducible representations of the (2+1)-dimensional Poincaré group. We show, in particular, that only two types of ’’spin’’ are available for massless field theories. We also construct generalized Foldy–Wouthuysen transformations which connect the physical UIR’s with covariant field theories in three dimensions.

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A conformally invariant quantum electrodynamics is constructed. The setting is realistic space-time (rather than Euclidean), and a complete Gupta–Bleuler quantization scheme is carried out. Conformal invariance of the quantum field theory (as opposed to either classical field theory or to a theory defined by its Feynman rules) requires a richer Gupta–Bleuler structure than has been considered previously. Yet the essential features of this structure are preserved. The requirement that the wave equation be of second order fixes a unique action that already contains the gauge-fixing terms that are required in any complete quantum field theory. The ‘‘Lorentz condition’’ turns out to be the transversality condition yαaα(y)=0 (in the manifestly covariant six-dimensional notation); this condition has to be treated in the same way as the Lorentz condition ∂μAμ(x)=0 (four-dimensional notation), as a boundary condition on the physical states.

This paper is an investigation of field theories that satisfy the following two criteria. (1) Among the propagating modes is a pair of massless particles with helicities f 2. (2) The canonical commutation relations are conformally invariant. This study of "linear conformal gravity" is motivated by the belief that conformal invariance may be the key to a future theory of quantum gravity. Our first conclusion is that the fields of linear conformal gravity include a tensor field of rank 3 and mixed symmetry, and a symmetric tensor field of rank 2, tentatively interpreted as a torsion field and a metric field. The free quantum field operator is constructed explicitly, and the propagator is calculated. The Fourier transform is of dimension p -4, which is encouraging for renormalizability. The field inevitably carries along a nonunitary ghost, similar to the one that turns up in linearized Weyl gravity. Our main result is that the ghost can be exorcised by imposing constraints on the external sources and boundary conditions on the physical states.

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