QUANTUM FIELD THEORY FROM FIRST PRINCIPLES

Abstract

The mathematical description of quantum systems univocally identiÿes their nature. In other words we treat a system as quantum if we describe its behaviour adopting Hilbert spaces and structures thereof, as prescribed by the postulates of quantum theory. The choice of using quantum systems as the elementary systems of physics can be justified in terms of informational principles, thanks to results of the last decade. Such results come as the conclusion of a research program that lasted almost one century, with the aim of reformulating quantum theory in terms of operational principles. This achievement now poses a new challenge, that we face here. If the systems of quantum theory are thought of as elementary information carriers in the first place, rather than elementary constituents of matter, and their connections are logical connections within a given algorithm, rather than space-time relations, then we need to find the origin of mechanical concepts—that characterise quantum mechanics as a theory of physical systems. To this end, we will illustrate how physical laws can be viewed as algorithms for the update of memory registers that make a physical system. Imposing the characteristic properties of physical laws to such an algorithm, i.e. homogeneity, reversibility and isotropy, we will show that the physical laws thus selected are particular algorithms known as cellular automata. Further assumptions regarding maximal simplicity of the algorithm lead to two cellular automata only, that in a suitable regime can be described by Weyl’s differential equations, lying at the basis of the dynamics of relativistic quantum fields. We will finally discuss how the same cellular automaton can give rise to both Fermionic ÿeld dynamics and to Maxwell’s equations, that rule the dynamics of the electromagnetic field. We will conclude reviewing the discussion of the relativity principle, that must be suitably adapted to the scenario where space-time is not an elementary notion, through the definition of a change of inertial reference frame, and whose formulation leads to the recovery of the symmetry of Minkowski space-time, identified with Poincar´e’s group. Space-time thus emerges as one of the manifestations of physical laws, rather than the background where they occur, and its features are determined by the dynamics of systems, necessarily equipped with differential equations that express it. In brief, there is no space-time unless an evolution rule requires it.