  We describe how to define observables analogous to quantum fields for the
semicontinuous limit recently introduced by Jones in the study of unitary
representations of Thompson's groups $F$ and $T$. We find that, in terms of
correlation functions of these fields, one can deduce quantities resembling the
conformal data, i.e., primary fields, scaling dimensions, and the operator
product expansion. Examples coming from quantum spin systems and anyon chains
built on the trivalent category $\mathit{SO}(3)_q$ are studied.
