Breaking the space–time translation group in the dipole field model

The impossibility of constructing a translation-invariant physical space for the massless dipole field models in two and four dimensions follows from the Lebesgue integral theory. The method of proof may also be useful for studying space–time symmetries in other models with singular behavior in the infrared domain. To fix the Hilbert majorant structure associated with an indefinite metric, we impose the condition that the majorant structure inherit the homogeneity properties from the metric.