Employing the Foldy-Wouthuysen transformation it is demonstratedstraightforwardly that the first and second Chern numbers are equal to thecoefficients of the 2+1 and 4+1 dimensional Chern-Simons actions which aregenerated by the massive Dirac fermions coupled to the Abelian gauge fields. Atopological insulator model in 2+1 dimensions is discussed and by means of adimensional reduction approach the 1+1 dimensional descendant of the 2+1dimensional Chern-Simons theory is presented. Field strength of the Berry gaugefield corresponding to the 4+1 dimensional Dirac theory is explicitly derivedthrough the Foldy-Wouthuysen transformation. Acquainted with it the secondChern numbers are calculated for specific choices of the integration domain. Amethod is proposed to obtain 3+1 and 2+1 dimensional descendants of theeffective field theory of the 4+1 dimensional time reversal invarianttopological insulator theory. Inspired by the spin Hall effect in graphene, ahypothetical model of the time reversal invariant spin Hall insulator in 3+1dimensions is proposed.
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