In what sense does a mass scale introduce a "scale" in the context of conformal field theory? In the sense that massive states in a full conformal field theory cause problems because they introduce a scale to a theory that should be scale invariant

It's strange to me that a mass scale introduces a length scale, unless I am missing the point

@Charlie A dilatation not only transforms $x\mapsto \lambda x$, but also $\phi\mapsto \lambda^d \phi$ for $d$ the scaling dimension of the field (equal to the mass dimension in free theories). So if you have a mass term, you can't have a scale-invariant Lagrangian.

aah ty

I currently have to read technical offers. Some of the stuff is really bad. It's the first time I used "not even wrong" in an official comment.

@Charlie in a relativistic quantum theory it's very natural, if $m$ is a mass, $m c / \hbar$ is an inverse length

@fqq but surely that's only a problem if the quantity $mc/\hbar$ actually appears somewhere in a formula or something? or maybe that's unavoidable?

Once you have a length scale it does show up, the theory is not invariant under scale transformations. E.g. it shows up in the coupling constant when you try to write the action, as ACM was saying

typically any mass/energy gap shows up as inverse correlation lenght, i.e. correlation functions decay exponentially $\log G \propto -m r$ at large $r$

ok that's interesting

Maybe this is obvious, but the conformal primary operators in CFT (that we evaluate at a particular spacetime point e.g. $\phi_I(x)$ to effectively have a "field of operators") these are not the same as the field operators obtained through second quantisation?

They seem to be introduced rather ad-hoc in everything I've seen, and defined through the states they create on the vacuum

@Charlie they are the same

2d CFT is very special in its state-operator correspondence, and this correspondence is largely the reason most of its techniques look nothing like generic QFT

A Lorentzian manifold $(M,g)$ with $g=ds^2=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$ Anybody know how to calculate the "boosts?"

Oh, that is actually very surprising

and actually raises an even bigger problem for me then, because in the operator product expansion at some point we have a "sum over primaries", implying that there is more than one conformal primary operator that produces the conformal primary states from the vacuum. On the other hand in (say) scalar field theory we only have one operator at each point in spacetime, so the existence of more than one such conformal primary operator is confusing

What I would have said was sure, you just evaluate the field away from the origin, but I've also been told that $\phi_I(x\neq0)|0\rangle$ is not a conformal primary state

i.e. the usage of "field" is a bit inconsistent here - that's why you might prefer to term this operator-state correspondence and not field-state correspondence.