If we write some quantum field in a form using creation and annihilation operators we are, in a way, doing a Fourier series with annihilation and creation operators being coefficients. So, if they are a coefficients shouldn't they have some structure? I mean if I do a Fourier series of some function I expect that coefficients also look like functions in k-space. So, what do these operators have besides commutation relations to define the structure of the field?
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