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The Hawking–Penrose theorem is not covariant under field redefinitions. Should the invariance under such transformations be a true principle in nature, spacetime singularities become dubious objects. We here review the concept of covariant singularities, that is, singularities that are invariant under both spacetime diffeomorphisms and field redefinitions.

We have addressed the issue of field redefinition in connection with renormalizability. Our study is restricted to theories of interacting scalar fields. We have, in particular, shown that if a theory is renormalizable in the usual power-counting sense then it remains renormalizable in the same sense after a change of variables. This is due to the use of the powerful method of the background field expansion. In the case of a single complex scalar field, it turns out that the determination of the counterterms is much simpler when polar coordinates are used. We illustrate this by carrying out a one-loop calculation in the latter case.