Questions on field operator in QFT and interpretations

In summary, the book "Quantum fields in curved space" uses time-independent ##\phi##, while online lecture notes use time-dependent ##\phi## with phase factors. The time-independence is due to the Heisenberg picture used in the book, while the time-dependence is due to the Schrodinger picture used in the online lecture notes. The phase factors are used to keep track of the time evolution of the states in the Schrodinger picture. However, in the Heisenberg picture, the operators (such as ##\phi##) are the ones that evolve in time, not the states. Therefore, the phase factors are not needed and can be discarded when working in the Heisenberg picture. The equal time
  • #1
user1139
72
8
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 Schrodinger 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?
 
Physics news on Phys.org
  • #2
Thomas1 said:
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 Schrodinger 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.
 
  • Like
Likes vanhees71
  • #3
Gaussian97 said:
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.
 
Last edited:
  • #4
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.
 
  • #5
vanhees71 said:
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.
 

1. What is a field operator in quantum field theory?

A field operator in quantum field theory is a mathematical representation of a quantum field, which is a physical quantity that exists at every point in space and time. It is used to describe the behavior and interactions of elementary particles and is a fundamental concept in modern physics.

2. How is a field operator interpreted in quantum field theory?

The interpretation of a field operator in quantum field theory is that it creates or destroys particles at a particular point in space and time. It is also used to calculate the probability of particle interactions and to describe the evolution of a quantum system over time.

3. What are the different types of field operators in quantum field theory?

There are several types of field operators in quantum field theory, including creation and annihilation operators, free field operators, and interacting field operators. Each type has a specific mathematical form and is used to describe different physical phenomena.

4. How do field operators relate to observables in quantum field theory?

In quantum field theory, field operators are used to calculate the expectation values of observables, which are physical quantities that can be measured in an experiment. The field operators act on the quantum state of the system to determine the probability of measuring a particular observable.

5. What are the challenges in interpreting field operators in quantum field theory?

One of the main challenges in interpreting field operators in quantum field theory is the problem of renormalization, which arises when trying to calculate the interactions between particles. Additionally, the interpretation of field operators can be difficult due to the abstract nature of quantum field theory and the need for mathematical formalism to describe physical phenomena.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
860
  • Advanced Physics Homework Help
Replies
4
Views
667
Replies
27
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Replies
2
Views
178
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
647
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Back
Top