Flipping the summation variable in the first sum from r to r ′ = k − r, the matrix element becomes, ∑ ∑ r ′ ( A [ r ′ , k ] − − r ′ d A d J [ k − − r ′ ] ) ( B [ 0 , r ′ ] + ( k − − r ′ ) d B d J [ r ′ ] ) − − ∑ ∑ r A [ r , k ] B [ 0 , r ] and it is clear that the principal (classical) part cancels.