Show that if instead of the 2nd quantization, postulate \#2 of commutator relations (3-40), we had anti-commutators between the field and its conjugate momentum, i.e.,
[ϕr(x,t),πs(y,t)]+=ϕrπs+πsϕr=iδsrδ(x−y)(3-167)
then the coefficient commutators would be anti-commutator relations, i.e.,
[a(k),a†(k′)]+=−[b(k),b†(k′)]+=δkk′(3-168)
(Hint: Just use opposite signs in (3-43) 2nd row and then in last two rows inside the bracket just before the last equal sign. Then, all commutators in (3-45) to (3-47) become anti-commutators.)