Questions on field operator in QFT and interpretations

[user1139]

Homework Statement

Confused over physical interpretations pertaining to QFT

Relevant Equations

Please refer below

For a real scalar field, I have the following expression for the field operator in momentum space.

$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$

Why is it that I can discard the phase factors to produce the time independent $\tilde{\phi}(\vec{k})$?

Also, when we speak about the equal time commutation relations, are we looking at the Heisenberg or the Schrödinger picture? Following up, why can we write the equal time commutation relations as time independent?

Moreover, when we speak about the Fock representation in conjunction with the annihilation and creation operators, which picture are we looking at?

 

[Gaussian97]

Homework Statement:: Confused over physical interpretations pertaining to QFT
Relevant Equations:: Please refer below

For a real scalar field, I have the following expression for the field operator in momentum space.

$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$

Why is it that I can discard the phase factors to produce the time independent $\tilde{\phi}(\vec{k})$?

Also, when we speak about the equal time commutation relations, are we looking at the Heisenberg or the Schrödinger picture? Following up, why can we write the equal time commutation relations as time independent?

Moreover, when we speak about the Fock representation in conjunction with the annihilation and creation operators, which picture are we looking at?

I'm not sure what do you mean when you ask about discarding the phase factors...
But in QFT one usually works with Heisenberg picture, since are the operators (in your case $\phi$) the ones that evolve in time, not the states.

 

[user1139]

I'm not sure what do you mean when you ask about discarding the phase factors...
But in QFT one usually works with Heisenberg picture, since are the operators (in your case $\phi$) the ones that evolve in time, not the states.

Could you explain in layman terms why in QFT we do not time evolve the states? This is because I noticed that $\phi$ is sometimes defined with the phase factor and sometimes without.

 

[vanhees71]

Which phase factor? From which book are you learning? I'm not sure what this expression you give in #1 should mean. Usually you either have free field operators in time-position representation (in the Heisenberg picture for non-interacting fields, also being used as the field operators in the interaction picture used in perturbation theory) or the creation and annihilation operators for spin-momentum eigenstates.

 

[user1139]

Which phase factor? From which book are you learning? I'm not sure what this expression you give in #1 should mean. Usually you either have free field operators in time-position representation (in the Heisenberg picture for non-interacting fields, also being used as the field operators in the interaction picture used in perturbation theory) or the creation and annihilation operators for spin-momentum eigenstates.

The phase factor I am talking about is $e^{i\omega t}$ and $e^{-i\omega t}$. I am using the book called "Quantum fields in curved space". However, the part about $\phi$ being sometimes time dependent and sometimes time independent is due to me searching the web and looking at lecture notes available online.