Degeneracy, matter coupling, and disformal transformations in
  scalar-tensor theories
  Degenerate scalar-tensor theories of gravity extend general relativity by a
single degree of freedom, despite their equations of motion being higher than
second order. In some cases, this is a mere consequence of a disformal field
redefinition carried out in a non-degenerate theory. More generally, this is
made possible by the existence of an additional constraint that removes the
would-be ghost. It has been noted that this constraint can be thwarted when the
coupling to matter involves time derivatives of the metric, which results in a
modification of the canonical momenta of the gravitational sector. In this note
we expand on this issue by analyzing the precise ways in which the extra degree
of freedom may reappear upon minimal coupling to matter. Specifically, we study
examples of matter sectors that lead either to a direct loss of the special
constraint or to a failure to generate a pair of secondary constraints. We also
discuss the recurrence of the extra degree of freedom using the language of
disformal transformations in particular for what concerns "veiled" gravity. On
the positive side, we show that the minimal coupling of spinor fields is
healthy and does not spoil the additional constraint. We argue that this virtue
of spinor fields to preserve the number of degrees of freedom in the presence
of higher derivatives is actually very general and can be seen from the level
decomposition of Grassmann-valued classical variables.
