Impossible Measurements on Quantum Fields

Abstract

It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity. 1. INTRODUCTION: IDEAL MEASUREMENTS AND QUANTUM FIELD THEORY Whatever may be its philosophical limitations, the textbook interpretation of nonrelativistic quantum mechanics is probably adequate to provide the quantum formalism with all the predictive power required for laboratory applications. It is also self-consistent in the sense that there exist idealized models of measurements which allow the system-observer boundary to be displaced arbitrarily far in the direction of the observer. And the associated “transformation theory” possesses a certain formal beauty, seemingly realizing the “complementarity principle” in terms of the unitary equivalence of all orthonormal bases. It is therefore natural to try to generalize this semantic framework to relativistic quantum field theory in the hope of learning something new, either from the success or failure of the attempt.