In Folland's book *Quantum Field Theory*, he says

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> ...we start out with classical field equations and a relativistically invariant Lagrangian from which they are derived, then replace the classical field variables by quantum fields, which are Fourier integrals of creation and annihilation operators that satisfy suitable commutation or anticommutation relations.
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I must have missed it while reading the text, but how do we know that all quantum fields

1. Involve only creation and annihilation operators and not any other kind of operator.
2. Are Fourier integrals of creation and annihilation operators. Where does this come from? (Why do we take the Fourier transform?)