It is well-known that on the Minkowski spacetime $\mathbb{R}^4$, there exist a free quantum field of arbitrary spin.
In the book "QFT : A Tourist Guide For Mathematicians" by Folland, a rigorous construction for the free quantum scalar field is explicitly presented as well.
I wonder if there are analogous notions for a discretized and bounded spacetime region. That is, consider a lattice $Z^4$ consisting of finitely many points in $\mathbb{R}^4$. My question is:
Can we construct some kinds of operator-valued distributions on $Z^4$ that describe theories without interaction? These theories correspond to free Lagrangians on $Z^4$ for fields of any given spin.
I hope my question has been delivered clearly...Any reference would be appreciated.
