Are commuting observables necessary but not sufficient for causality?

[nomadreid]

TL;DR Summary

Two observables on events in a space-like separation must commute in order to ensure that causality is possible, otherwise one could be affected by the other before information could pass from one to the other. But on several sites it is stated that the vanishing of the commuter "defines" causality. I don't see the jump from possibility to necessity. Or is "causation" here defined as "possibility of causal link"?

A couple of the sites that say that the vanishing commutator defines causality:
https://www.physics.umass.edu/events/2020-02-07-causal-uncertainty-quantum-gravity
https://phys.org/news/2019-11-aspects-causality-quantum-field-theory.html
Is "define" too strong?
Or, to put it another way: cannot two events which are spacelike separated be neither causally connected nor be associated with non-commuting observables?

 

[vanhees71]

I think the "microcausality principle"/"locality" is a sufficient but not necessary condition for causality, Poincare invariance and unitarity of the S matrix. See Weinberg, Quantum Theory of Fields, vol 1. However, I have no example for a relativistic QT where causality is guaranteed by some other principle.

The paper by Donoghue is quite interesting. The conventional answer of course is that the directedness of time is indeed conventional and an additional assumption underlying all dynamical theories of physics. Other arrows of time than this "causality arrow of time" like the electrodynamical arrow of time (choosing retarded solutions in classical electrodynamics instead of advanced or "mixed" ones) or the thermodynamical/kinetic arrow of time (Boltzmann's H theorem) then just show that they are compatible with the conventional choice of the "causality arrow of time".

It's of course clear that the challenge of formulating a quantum theory of the gravitational interaction, which is so closely related (not to say "entangled") with the spacetime model of general relativity, one might have to rethink the meaning of causality and its realization in the theory, but nobody can say, how this turns out before there isn't such a consistent theory of "quantum gravity".

 

[nomadreid]

Thank you, vanhees71, for your interesting answer, including your answer (if I understood correctly) turning my suspicion of "necessary but not sufficient" on its head into "sufficient but not necessary". It appears to me (but I am willing to be corrected) that this contradicts the assertion in the first answer in this post discussing causality
https://physics.stackexchange.com/q...-does-a-vanishing-commutator-ensure-causality
where the author says "...it is crucial that A and B must commute if they are spacelike separated."

I do have access to the book you recommended, and will study it (and other sources) about the ideas you mentioned. If you have any more specific pages or chapters that it would be best to concentrate on, that would also be appreciated. But perhaps just "the whole book" is the appropriate answer.

 

[HomogenousCow]

Microcausality is particularly important for perturbative QFT since it ensures the relativistic invariance of the time-ordering symbol which appears in the interaction picture propagator.

 

[vanhees71]

Thank you, vanhees71, for your interesting answer, including your answer (if I understood correctly) turning my suspicion of "necessary but not sufficient" on its head into "sufficient but not necessary". It appears to me (but I am willing to be corrected) that this contradicts the assertion in the first answer in this post discussing causality
https://physics.stackexchange.com/q...-does-a-vanishing-commutator-ensure-causality
where the author says "...it is crucial that A and B must commute if they are spacelike separated."

I do have access to the book you recommended, and will study it (and other sources) about the ideas you mentioned. If you have any more specific pages or chapters that it would be best to concentrate on, that would also be appreciated. But perhaps just "the whole book" is the appropriate answer.

In the above quoted posting they refer to the only successful real-world relativistic QFTs which all are local, and there indeed the microcausality condition is crucial, and its consequences like the spin-statistics relation and CPT invariance have always been confirmed by observations. However that doesn't mean that any other possibilities to construct non-local but still causal relativistic QTs have been disproven to exist.

 

[nomadreid]

Microcausality is particularly important for perturbative QFT since it ensures the relativistic invariance of the time-ordering symbol which appears in the interaction picture propagator.

Thank you, Homogenous Cow. I will have to unpack that to find out whether this has to do directly or only indirectly with my question.

 

[HomogenousCow]

The idea is that when you're doing perturbation theory you often have expressions of the form $$\langle 0| T \{A(t)B(t')..\} |0 \rangle.$$To ensure the lorentz invariance of this expression, the time-ordering symbol must be invariant under boosts. For time-like separated events this is not an issue since the order of events is unique, however for space-like separated events this is no longer true and it would seem that $T$ is frame-dependent. Micro-causality solves this issue since if we have $[A(t),B(t')]=0$ for space-like separated events then the ordering is irrelevant and $T$ is thus lorentz invariant.

This is also why fermionic time-ordering is defined with a sign change, i.e. $$T \{\psi(t) \phi(t')\} = -\phi(t')\psi(t)$$ when $t < t'$, since micro-causality requires that fermionic fields anti-commute.