The Klein Gordon field as a wave function: causality and localization

Abstract

Usually in quantum field theory one considers two different interpretations: 1: The field is an infinite number of quantum oscillators, giving rise to a wave functional \Psi(\phi). 2: The positive frequency component of a field, \phi_+(x), is a wave function analogous to standard quantum mechanics. While interpretation 2 is often only mentioned implicitly it is crucial to standard computations of measurable scattering probabilities. We extend the interpretation of QFT as relativistic quantum mechanics (option 2) and show how the total Klein Gordon field which consists of positive and negative frequency contributions, \phi = \phi_+ + \phi_- , can be interpreted as a wave function. Our construction manifestly shows that signal propagation in QFT cannot exceed the speed of light. This follows from the replacement of the Feynman propagator by the Wheeler propagator, which is just the time ordered commutator. Our second observation is that interpretation 2 is not consistent if one uses the Newton Wigner position operator, therefore we introduce a more natural bilinear position operator. We show by an explicit example that the bilinear operator, contrary to the Newton Wigner operator, allows relativistic particles to be perfectly localized, precisely as in non relativistic quantum mechanics.