Free field
Basically, commutator (for bosons)/anticommutator (for fermions) of two smeared fields is i times the Peierls bracket of the field with itself (which is really a distribution, not a function) for the PDEs smeared over both test functions.CCR/CAR algebras with infinitely many degrees of freedom have many inequivalent irreducible unitary representations.If the theory is defined over Minkowski space, we may choose the unitary irrep containing a vacuum state although that isn't always necessary.Let φ be an operator valued distribution and the (Klein–Gordon) PDE be This is a bosonic field.Then, the canonical commutation relation is Note that Δ is a distribution over two arguments, and so, can be smeared as well.