  We present a description of nuclear spontaneous fission, and generally of
quantum tunneling, in terms of instantons - periodic imaginary-time solutions
to time-dependent mean-field equations - that allows for a comparison with more
familiar and used generator coordinate (GCM) and adiabatic time-dependent
Hartree-Fock (ATDHF) methods. It is shown that the action functional whose
value for the instanton is the quasiclassical estimate of the decay exponent
fulfils the minimum principle when additional constraints are imposed on trial
fission paths. In analogy with mechanics, these are conditions of energy
conservation and the velocity-momentum relations. In the adiabatic limit the
instanton method reduces to the time-odd ATDHF equation, with collective mass
including the time-odd Thouless-Valatin term, while the GCM mass completely
ignores velocity-momentum relations. This implies that GCM inertia generally
overestimates instanton-related decay rate.
  The very existence of the minimum principle offers a hope for a variational
search for instantons. After the inclusion of pairing, the instanton equations
and the variational principle can be expressed in terms of the
imaginary-time-dependent Hartree-Fock- Bogolyubov (TDHFB) theory. The adiabatic
limit of this theory reproduces ATDHFB inertia.
