  Employing the Foldy-Wouthuysen transformation it is demonstrated
straightforwardly that the first and second Chern numbers are equal to the
coefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which are
generated by the massive Dirac fermions coupled to the Abelian gauge fields. A
topological insulator model in 2+1 dimensions is discussed and by means of a
dimensional reduction approach the 1+1 dimensional descendant of the 2+1
dimensional Chern-Simons theory is presented. Field strength of the Berry gauge
field corresponding to the 4+1 dimensional Dirac theory is explicitly derived
through the Foldy-Wouthuysen transformation. Acquainted with it the second
Chern numbers are calculated for specific choices of the integration domain. A
method is proposed to obtain 3+1 and 2+1 dimensional descendants of the
effective field theory of the 4+1 dimensional time reversal invariant
topological insulator theory. Inspired by the spin Hall effect in graphene, a
hypothetical model of the time reversal invariant spin Hall insulator in 3+1
dimensions is proposed.
