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Problems on quantum field operators in QFT

by Dyson
Tags: field, operators, quantum
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Dyson
#1
May10-10, 11:33 AM
P: 21
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!!!
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tom.stoer
#2
May10-10, 02:29 PM
Sci Advisor
P: 5,443
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 [tex]a^\dagger(p)[/tex] and [tex]a(p)[/tex]
Dyson
#3
May10-10, 06:02 PM
P: 21
Quote Quote by tom.stoer View Post
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 [tex]a^\dagger(p)[/tex] and [tex]a(p)[/tex]
Thank you very much!


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