The commutation relations of creation and annihilation operators in a multiple- boson system are, [ a i , a j † † ] ≡ ≡ a i a j † † − − a j † † a i = δ δ i j , [ a i † † , a j † † ] = [ a i , a j ] = 0 , where [ ⋅ ⋅ , ⋅ ⋅ ] is the commutator and δ δ i j is the Kronecker delta.