D5 - Theories of quantum gravity II (not superstring theory)

Summary

Introduction

Quantum theory and relativity theory, the two great revolutions of twentieth century physics, do not seem to mesh very well. Indeed the only fully consistent relativistic quantum theories seem to be linear free fields. It is not clear whether the difficulty of combining the two theories is one of principle or merely one of practice. On the most pedestrian level, the Hamiltonian formalism of classical mechanics, which is most suitable for quantizing a classical theory, requires an explicit choice of a time-like coordinate and dynamical variable adapted to this choice; relativistic theories, in contrast, treat space and time variables on essentially equal footing and are expressed most naturally in the Lagrangian formalism. Of course, for classical physical theories the two formalisms are equivalent in the sense that one can map from one to the other by means of a Legendre transformation. Thus one can, for example, follow the quantization procedure of a classical theory in the Lagrangian formalism, or demonstrate the relativistic covariance of a theory in the Hamiltonian formalism. But the exercises are decidedly awkward, and, given the ensuing difficulties which occur for non-trivial theories, one does have cause for puzzlement.

On a more fundamental level, the measurement theory for the standard interpretation of the quantum mechanical formalism requires that if an observation of a complete commuting set of local observables is performed by an observer in some finite neighborhood of a space-time point, the state of the system is reduced everywhere, including in regions well outside the forward light cone of that neighborhood.