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I have quite basic questions about the general properties of operators in quantum field theory. When quantizing the free scalar field, for instance, you promote the classical fields to operators and impose suitable commutation relations (canonical quantization). In momentum space the harmonic oscillators decouple and you can find the spectrum by introducing a/a(dagger) operators. All of this is done without specifying the explicit form of these operators. Concering this I have some questions:

1) Without knowing the explicit form of these operators, which restrictions (physical and mathematical) do they underlie? How would mathematicians define these operators in a strict language?

2) Is it just a

__definition__that operators like the field operator phi or a, a(dagger) only act on states of the fock space and noting else? For example when doing fourier transformations you always commute phases exp(ipx) or general functions of x and p with these operators. How do I know that the field operators do not act on these terms? As far as I know it's just definition. What is the argument that the field operators cannot contain derivatives (like in QM the momentum space representation of p) d/dx and this way would act on exp(ipx)?

3) How does the nabla operator act on field operators?

As you can see I have some problems with the mathematical construct of these operators. At the moment I think that it's just a definition that my operators only act on fock space states and nothing else so I only have to take care when commuting operators. But I am not really sure.

Thanks in advance,

Matt