  We consider field-theoretic models, one consisting purely of scalars, the
other also involving fermions, that couple to a set of constant background
coupling coefficients transforming as a symmetric observer Lorentz two-tensor.
We show that the exact propagators can be cast in the form of a
K\"all\'en-Lehmann representation. We work out the resulting form of the
Feynman propagators and the equal-time field commutators, and derive sum rules
for the spectral density functions.
