--------------------------------------------------------
S5e. Why does the action only contain first derivatives?
--------------------------------------------------------
On the classical level, higher derivatives cause no formal problems,
one can form the variational equations as always. There might be
problems with causality (= symmetric hyperbolicity), however.
These problems become worse (and apparently untractable) in the
quantum case.
In a k derivative theory with k>1, one can always introduce new fields
for the k-1 first derivatives, and add terms to the action that give
as variation their defining equations. Thus one can reduce any theory
to an equivalent one with only first derivatives in the action.
The problems appear when trying to go from the Lagrangian picture to
the Hamiltonian - then one gets similar difficulties as for gauge
theories.