It seems to me that in the quantization of a classical field, one first takes the fourier transform of the field to put it in frequency space:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]F \left(X, \omega \right) = \int f(X,t)e^\left(-i \omega t\right)[/tex]

then multiply by the annihilation and creation operators for a given wavelength:

[tex] F \left(X, \omega \right) \sqrt{m \omega / \left(2 \hbar \right)} \left( x + i / \left(m \omega \right) p \right) [/tex]

[tex] F \left(X, \omega \right) \sqrt{m \omega / \left(2 \hbar \right)} \left( x - i / \left(m \omega \right) p \right) [/tex]

Then take the IFT of these to return to time space, which would yield a creation field operator and an annihilation field operator respectively. Note that I used X as a vector and x as an operator.

I understand that these objects increase and decrease the number of particles in the system respectively. But what do they act upon and what do they return when applied?

In single particle QM, these operators act upon an oscillator wavefunction to raise or lower the energy state.

But in QFT, I am guessing that they act upon the Hilbert space ( + time) and return the creation/annihilation operators of a particle of a field (ex. a photon in a Maxwell field), which can be used to define the Hamiltonian of the particle, and the resultant Schrodinger equation, which of course can be used to define the wavefunction of the particle. Ex. when passed a vector value for X, and a scalar value for t, they will return the ladder operators for a single particle state about that point on the Hilbert space.

Is this understanding correct? Please correct any of my misunderstandings (I am sure that there are many). I am an engineer, not a physicist. Please understand and take pity ;).

Many thanks.

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# Field operators - how do they work?

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