Does Microcausality follow from Lorentz Invariance? 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?
 A: 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. Edit (June 15th, 2022): as Nanashi No Gombe pointed in the comments below, an example of this is provided by parastatistic field models, which need not commit to the Bose/Fermi notion of microcausality but can be Lorentz covariant nonetheless.
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. Once more. this shows that the concepts of Lorentz invariance and microcausality in a quantum field theory are not equivalent.
A: Consider a Poincare-invariant scalar field theory.  Assume we are guaranteed Poincare-invariance in the sense that there exists a unitary representation of the Poincare group acting on the Hilbert space, which transforms the scalar field operator $\phi(x)$ like $$U(g)\phi(x)U(g)^\dagger = \phi(g x)$$ for elements $g$ of the Poincare group.
Consider two spacelike-separated points $x,y$.  We want to show
$$[\phi(x),\phi(y)]\overset{?}{=}0.$$ On the other hand, we know that for any two points $x'=(0,\vec{x}')$ and $y'=(0,\vec{y}')$ at time $t=0$, we have
$$[\phi(0,\vec{x}'),\phi(0,\vec{y}')] = 0,$$
which follows immediately from the wavefunctional or lattice perspectiv; e.g. in a lattice theory the operators $\phi(0,\vec{x}')$ and $\phi(0,\vec{y}')$ act on different lattice sites and thus commute.
Here's the key step: Note that for any spacelike $x,y$, there exists some Poincare transformation $g$ taking $x,y$ to $x',y'$, i.e. transforming to a frame where both point are at $t=0$.  So then
$$U(g)[\phi(x),\phi(y)]U(g^{-1}) =[\phi(0,\vec{x}'),\phi(0,\vec{y}')] = 0$$
and thus $[\phi(x),\phi(y)]=0$
as desired.
(In another answer Pedro Ribeiro mentions Lorentz-invariant theories that are not microcausal, and I'm not sure how these violate my assumptions.)
