On the Theory of Local Observables

Lawrence J. Landau

Willem M. de Muynck

Etang Chen

M. D. Srinivas

Relativistic quantum theory is formulated as a theory of successive local observations, consistent with all the fundamental requirements of relativistic invariance. Based on the causal structure of the space–time of special relativity, it is argued that the nonrelativistic notion of a ’’time-ordered sequence of instantaneous observations’’should be replaced in a relativistic theory by the notion of a ’’causally ordered sequence of local observations.’’ The fundamental statistical law of relativistic quantum theory is then formulated such that it provides an unambigous prescription for the statistical correlations between the outcomes of any causally ordered sequence of local observations. It is also shown that the fundamental statistical law of the theory is consistent with the so-called ’’principle of local causes’’ which essentially states that the probability connections between a set of local events do not depend on local observations carried out in space–time regions which lie outside of their causal past. Finally, an equivalent formulation of the theory in terms of a relativistically covariant generalization of the collapse postulate is presented.

Philipp A. Hoehn, Josh Kirklin

In gauge theories, globally charged observables necessarily depend non-locally on the kinematical fields, with this dependence extending to the asymptotic boundary of spacetime. Despite this, we show that a subset of such observables can be consistently regarded as local to the bulk, in a manner that respects microcausality and leaves locality properties of uncharged observables untouched. A sufficient condition for this is to impose kinematically non-local boundary conditions on the large gauge sector of the theory, and to invoke a relational notion of localisation for observables. This reveals a relatively underappreciated link between boundary conditions, and different notions of microcausality and locality. We develop this point through a detailed case study in scalar QED, describing non-local boundary conditions that allow a large family of observables on a codimension-1 bulk surface to be viewed as local to that surface, despite being dressed by asymptotic Wilson lines. We show that this property continues to hold within a perturbative quantisation of the theory, and we argue that this leads to a consistent local net of algebras that includes these charged observables in bulk algebras. We explain how this setup may be understood in terms of a preferred dynamical reference frame for small gauge transformations appearing in the boundary conditions. Many features of the theory (such as microcausality, the vacuum state, and the net of algebras of observables) depend on the choice of this frame, and we briefly discuss some repercussions of this for algebraic formulations of QFT. While our analysis is performed in QED, we expect our results to carry over qualitatively to more complicated theories including gravity.