Fundamental asymmetry between space and time in quantum field theory

Abstract

Abstract In 1 + 1 dimensions, space and time are geometrically interchangeable. The reason is that, if we switch the roles of the spacelike and the timelike dimensions, this only amounts to redefining the signature convention of the metric tensor. This establishes a mathematical duality between classical Klein–Gordon fields and classical tachyonic fields: one is the ‘space/time swapped’ version of the other. Here, we show that this duality breaks down completely in a quantum world. In quantum field theory, if we exchange space with time, we end up with non-canonical field theories that exhibit all sorts of paradoxical behaviours. Theories that should be stable are, instead, unstable, while theories that should be unstable are stable. Theories that should be causal are, instead, acausal. This formalizes a widespread (but often neglected) intuition: there is a fundamental (non-geometrical) asymmetry between space and time, which has a purely quantum origin, and does not exist in a classical world. Such asymmetry exists independently from the measurement problem, as it is encoded directly in the algebraic properties of the field operators.