Problems on quantum field operators in QFT

[Dyson]

Hello!

I met some annoying problems on quantum field operators in QFT.They are as follows:

(1)The quantum field operator( scalar field operator, for example),is often noted as

φ（r,t). Can it be interpreted as like this: φ（r,t) is the coordinate represetation of a

more abstract,theoretical operator φ in Hilbert space?

(2)Is there exist the represetation(such as momentum, coordinate...) which is similar to

that in quantum mechanics in QFT?

Thanks for reply!

 

[tom.stoer]

Regarding 1) no, r is an "index", not an entity living in a Hilbert space. This becomes clear if one derives quantum field theory of a scalar field from infinitly many coupled harmonic oscillators.

Regarding 2) yes, these are the so-called creation an annihilation operators obtained by Fourier transform of the field operators, usually denoted as $$a^\dagger(p)$$ and $$a(p)$$

 

[Dyson]

Regarding 1) no, r is an "index", not an entity living in a Hilbert space. This becomes clear if one derives quantum field theory of a scalar field from infinitly many coupled harmonic oscillators.

Regarding 2) yes, these are the so-called creation an annihilation operators obtained by Fourier transform of the field operators, usually denoted as $$a^\dagger(p)$$ and $$a(p)$$

Thank you very much!