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In a Lorentz Invariant theory, does microcausality automatically hold? In a free theory this is obvious. In an interacting theory I found some 'proof's in this paper: http://arxiv.org/abs/0709.1483

However the proofs show that for space-like separated $x$ and $y$ $$\langle 0| [\hat{\phi}(x),\hat{\phi}(y)]|0\rangle =0.$$

But for the microcausality condition to hold at an operator level we need to show that $$\langle n| [\hat{\phi}(x),\hat{\phi}(y)]|n\rangle =0\,\, \forall \,n$$

where $|n\rangle$ forms a basis. My question is, can this be shown? Or are further assumptions needed?

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If all truncated $n$-point functions vanish for $n>2$ (i.e. we are dealing with a so-called generalized free field), microcausality for vacuum expectation values and at the operator level are equivalent. The former, on its turn, follows from Lorentz invariance alone in the case of scalar (but not necessarily free) fields, as shown by Pierre-Denis Methée, who was a student of de Rham (Sur les Distributions Invariantes dans le Groupe de Rotations de Lorentz, Commentarii Mathematici Helvetici 28 (1954) 225-269). If the field is interacting, this is no longer the case and then, in fact, microcausality does not follow from Lorentz invariance alone, even if positive definiteness of the $n$-point functions and the energy-momentum spectrum condition also hold. Unfortunately, though, I am not aware of any explicit example of a quantum field theory which is Lorentz-invariant but not microcausal, despite some claims in the literature that it is not difficult to build such an example. If I happen to find one, I will update my answer.

It is also important to point that there are non-Lorentz invariant quantum field theories that are nonetheless microcausal (e.g. some models coupled to a suitable external "ether" field). In such models, microcausality and the energy-momentum spectrum condition suffice to ensure that energy-momentum spectrum has a Lorentz-invariant shape and therefore has Lorentz-invariant dispersion laws (i.e. either of "mass-shell" of "light-cone" type), even in the absence of bona fide Lorentz invariance - this is a consequence of the Jost-Lehmann-Dyson representation of the two-point function, which does not rely on Lorentz invariance. This shows that the concepts of Lorentz invariance and microcausality in a quantum field theory are not equivalent, regardless of the lack of examples of Lorentz-invariant but acausal models.

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  • $\begingroup$ thanks for your answer. Weinberg in fact says that we need microcausality to prove the Lorentz Invariance of the S matrix. He says that for something like the Dirac field which is not directly observable, this is the only 'justification' of the microcausality condition. The paper I linked however claims that microcausality can follow from Lorentz Invariance. However they show it holds in curved spacetime as well,in the absence of global L.I. It is quite an interesting paper actually. $\endgroup$ – Nirmalya Kajuri Jul 16 '16 at 22:35
  • $\begingroup$ It is important to remark that the arguments in the paper you cited are not rigorous, being mostly based on formal functional integrals and perturbation theory. The result of Methée I cited, on the other hand, is completely rigorous and holds under very general assumptions (the two-point function doesn't even need to be positive or tempered). Moreover, microcausality lies at the basis of the dispersion relations for the S-matrix and its resulting high-energy behavior, which can be checked in collider experiments (Froissart bounds, etc.). $\endgroup$ – Pedro Lauridsen Ribeiro Jul 17 '16 at 3:36
  • $\begingroup$ In curved space-times, arguments based on formal functional integrals and the ensuing formal perturbation theory are even more suspicious because there is in general neither a Wick rotation nor an S-matrix to begin with - the very geometry of space-time precludes that since there may be no global isometries at all. In this case, an algebraic approach tends to be more appropriate. $\endgroup$ – Pedro Lauridsen Ribeiro Jul 17 '16 at 3:39
  • $\begingroup$ Thanks for your comments. I have one final question. If we know that for a theory the S matrix is Lorentz invariant, does microcausality automatically follow? $\endgroup$ – Nirmalya Kajuri Jul 17 '16 at 6:29
  • $\begingroup$ @NirmalyaKajuri : it seems, that yes, it does. From Lorentz invariance of $S$-operator follows, that interaction hamiltonians commute on space-like intervals, which is nothing but microcausality criterion. $\endgroup$ – Name YYY Jul 28 '16 at 20:37

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