Types of quantum field thories
A basic characteristic of physics in the context of special relativity and general relativity is that causal influences on a Lorentzian manifold spacetime propagate in timelike or lightlike directions but not spacelike.
The fact that any two spacelike-separated regions of spacetime thus behave like independent subsystems is called causal locality or, with a slightly stronger technical definition, Einstein causality.
(One sometimes sees a further criterion to causality, that the causal influences in timelike and lightlike directions only propagate into the future, but this is not so simply dealt with; it probably only makes sense as a statement about coarse-grained entropy in statistical physics.)
For a formalization of this idea see for instance
From (Grigor’ev 197x):
a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≲ cm and to times ≲ sec.
It is shown in the theory of relativity that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of light leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of light “in the small”.
In quantum theory, where operators correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of space-time if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The microcausality condition is important in quantum field theory, especially in the dispersion and axiomatic approaches; these approaches are not based on specific model concepts of interaction and therefore can be used for direct verification of the microcausality condition. In the most highly developed branch of quantum field theory — quantum electrodynamics — the microcausality condition has been experimentally verified for distances ≲ cm (and, correspondingly, for times ≲ sec).
The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause).
Jessey Wright, Quantum field theory: Motivating the Axiom of Micorcausality, PhD thesis 2012 (pdf)
Anthony Duncan, The Conceptual Framework of Quantum Field Theory – Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity, Oxford Scholarship Online (web)