=
This is the Klein-Gordon equation.
,
true if and only if
is an integration constant under a variation in the action.
.
.
.
The question asks to look at the conserved charges for two complex fields using Pauli sigma matrices. One needs to diagonalize the Hamiltonian as in step (b) 3, roughly involving these four products:
,
The conserved charges arise from the cross terms:
The problem here is that
cannot be independent of the Pauli matrices. The Pauli matrices are very
similar to the quaternions, the difference being that the latter is a division
algebra. There are only three independent imaginary basis vectors for quaternions,
not four, often represented as
. The four currents are not independent because two of these fields must
lie in the same complex plane.