  We explore the possibility of probing the nonstandard interactions between
the neutrino and a hypothetical massive scalar or pseudoscalar via neutrino
flavor transformation in supernovae. We find that in the ultrarelativistic
limit, the effective interaction between the neutrinos vanishes if neutrinos
are Dirac fermions but not if they are Majorana fermions. The impact of the new
neutrino interaction upon the flavor transformation above the neutrinosphere is
calculated in the context of the multi-angle "neutrino bulb model". We find
that the addition of the nonstandard neutrino self-interaction (NSSI) to the
ordinary V-A self-interaction between neutrinos is capable of dramatically
altering the collective oscillations when its strength is comparable to the
standard, V-A, interaction. The effect of flavor-preserving (FP) NSSI is
generally to suppress flavor transformation, while the flavor-violating (FV)
interactions are found to promote flavor transformations. If the neutrino
signal from a Galactic supernova can be sufficiently well understood, supernova
neutrinos can provide complimentary constraints on scalar/pseudoscalar
interactions of neutrinos as well as distinguishing whether the neutrino is a
Majorana or Dirac fermion.
