Fields and creation operators

We consider first the expansion of a free field operator in terms of creation and annihilation operators. For a real scalar field describing quanta of mass IJL we write 

The equal time commutation relations (20.2.5,6) then lead to 

It should be noted that the fields satisfy local commutativity or microscopic causality, i.e. 

For free fields the vacuum e3q ectation value of the time ordered product of two bosonic fields is 

Ap is a Green s function for the Klein-Gordon equation, i.e. it satisfies 

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