  We develop an operational framework, combining relativistic quantum
measurement theory with quantum reference frames (QRFs), in which local
measurements of a quantum field on a background with symmetries are performed
relative to a QRF. This yields a joint algebra of quantum-field and
reference-frame observables that is invariant under the natural action of the
group of spacetime isometries. For the appropriate class of quantum reference
frames, this algebra is parameterised in terms of crossed products. Provided
that the quantum field has good thermal properties (expressed by the existence
of a KMS state at some nonzero temperature), one can use modular theory to show
that the invariant algebra admits a semifinite trace. If furthermore the
quantum reference frame has good thermal behaviour (expressed by the existence
of a KMS weight) at the same temperature, this trace is finite. We give precise
conditions for the invariant algebra of physical observables to be a type
$\textnormal{II}_1$ factor. Our results build upon recent work of
Chandrasekaran, Longo, Penington and Witten [JHEP 2023, 82 (2023)], providing
both a significant mathematical generalisation of these findings and a refined
operational understanding of their model.
