Spin Foam Quantization and Anomalies

The most common spin foam models of gravity are widely believed to be discrete path integral quantizations of the Plebanski action. However, their derivation in present formulations is incomplete and lower dimensional simplex amplitudes are left open to choice. Since the large-spin behavior of these amplitudes determines the convergence properties of the state-sum, this gap has to be closed before any reliable conclusion about finiteness can be reached. It is shown that these amplitudes are directly related to the path integral measure and can in principle be derived from it. This requires a detailed knowledge of the constraint algebra and the corresponding gauge fixing of its first class part which in the case of gravity generates space-time diffeomorphisms. It has been suggested that the discretization of space-time in a spin foam model breaks the diffeomorphism gauge without introducing an explicit gauge fixing. Here we show that minimal requirements of background independence–which are reminiscent of cylindrical consistency in loop quantum gravity–provide non trivial restrictions on the form of an anomaly free measure. Many models in the literature do not satisfy these requirements. Moreover, we show that an anomaly free model will necessarily contain divergent amplitudes that could be interpreted as due to infinite contributions of gauge equivalent configurations. Exploring these issues we come across a simple model satisfying the above consistency requirements which can be thought of as a spin foam quantization of the Husain–Kuchar model.