A proof that no-signalling implies microcausality in quantum field theory

Abstract: Microcausality counts among the basic requirements that ensure the consistency of quantum field theories (QFT). In the literature on the foundations of QFT, however, an elementary derivation of microcausality from the relativistic constraints seemed to be missing. Authors generally proceed the other way round, justifying the axiom as a sufficient condition to guarantee some desired properties, typically involving properties like no-signalling, separability, independence of operations, tensor product structure... The aim of this work [1] is to study some logical interrelationships between fundamental properties in (relativistic) quantum theories, and in particular to provide an elementary but important proof concerning microcausality that was apparently missing in the literature.

To do so, an operational no-signalling condition is first introduced in the context of quantum mechanics, where we prove its equivalence to an apparently weaker version (CPTP version equivalent to PVM version), and to a property of factorisation of the evolution unitary operator, generalising some results of [2]. We then translate this condition in QFT and prove that it implies both microcausality and the spin-statistics theorem, in the ideal case of pointwise measurements implemented in the projection postulate sense. This provides an argument (often invoked but apparently missing in the literature) to see microcausality as a necessary condition for the compatibility of spacelike separated operations.

[1] Soulas, Antoine. "A proof that no-signalling implies microcausality in quantum field theory." Foundations of Physics 55.2 (2025): 22.

[2] Beckman, David, et al. "Causal and localizable quantum operations." Physical Review A 64.5 (2001): 052309.

mathematical physicsquantum physics

Audience: advanced learners

( paper )

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