more appropriately be viewed as a natural transformation and we refer to [@BFV] for a discussion of that point. The degree to which the present notion of local covariant quantum field theory describes the “same” quantum field on all spacetimes is analyzed in Fewster’s contribution to these proceedings.

We write $\boldsymbol{\Phi} = (\Phi_{\boldsymbol{M}})$ to denote a local covariant quantum field theory, and we remark that examples known so far include the scalar field, the Dirac field, the Proca field, and — with some restrictions — the electromagnetic field, as well as perturbatively constructed $P(\phi)$ and Yang-Mills models [@HolWald; @HolYM; @VerSPST]. A [*state*]{} of a local covariant quantum field theory $\boldsymbol{\Phi}$ is defined as a family $\boldsymbol{\omega} =
(\omega_{\boldsymbol{M}})$, $\boldsymbol{M}$ ranging over the objects of [Loc]{}, where each $\omega_{\boldsymbol{M}}$ is a state on $\mathscr{A}(\boldsymbol{M})$ — i.e. an expectation value functional. It might appear natural to assume that there is an invariant state $\boldsymbol{\omega}$, defined by $$\label{invast}
 \omega_{\boldsymbol{N}} \circ \mathscr{A}(\psi) = \omega_{\boldsymbol{M}}$$ for all morphisms $\boldsymbol{M} \overset{\psi}{\longrightarrow} \boldsymbol{N}$ of [Loc]{}, but it has been shown that this property is in conflict with the dynamics and stability properties one would demand of the $\omega_{\boldsymbol{M}}$ for each $\boldsymbol{M}$ [@BFV; @HolWald]; the outline of an 