Solving a Quantum Field Theory Question on Electron-Positron Pair Creation

In summary, the uncertainty relation between energy and time is greater than the energy shift between the ground state and any other state.
  • #1
Kruger
214
0
Hello all. Seems to be an excellent forum with many experts.

I'm interested in quantum field theory. There's a question in my textbook (introduction to QM and QFT) and I'm not sure if I found the right solution and its interpretation.

Question 1: How many energy is needed to create a electron positron pair.
Answer 1: (easy) 2*m(y)*c^2 where m(y) is the relativistic mass, you know.

Question 2: Calculate the uncertainty relation between Energy and time.
Answer 2: d(H)d(t)>=h/4pi (easy)

And now there is the trickier part, a combination of these two. (the textbook derived the discret energy values of the harmonic oscillator and shows that its ground state isn't zero).

Question 11: The life time of a virtual e-e+-pair is given by the HUP and if there is a strong electric field in "empty" space with electromagnetic ground state oscillations how can the ground state oscillations create an e-e+-pair?

my Answer: I tried to calculate this: [H,N] where H is the energy of the oscillation and N the number operator. I didn't find the solution of this and got [H,N]=0 (I think not the right one). My oppinion is: The N has to raise from 0 to 1 (to create a photon in vacuum). The "N" takes the needed energy for this from HUP. After this the photon interacts with the electric field and e-e+-pair will be created for time d(t) (HUP). After this happened the pair will annihilate and the N operator will lower (N=0) and the energy is given back to "empty" space.
But as I got [H,N]=0 this can't be.

Oh, please help me. There aren't answers in this book (only questions, well the most questions are easy, but that is a difficult one).
 
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  • #2
Can nobody help?
 
  • #3
I don't know the answer to your problem, but try typing:

"uncertainty relation" "energy and time"

into Google. Stuff does come up that may help you.
 
  • #4
Sorry, but I coudn't find what I looked for.
 
  • #5
Does nobody know this?
 
  • #6
Question 2: Ground state has energy what ever you want. Only energy shifts are measurable quantities.

Question 11: I don't understand what you are trying to say. It looks like non-Relativistic QM + creation of virtual particles (!?). Bad combination. The only thing I can understand is [H,N]=0. Which is true in, otherwise how would you get the energies of the harmonic oscilator?
 

Related to Solving a Quantum Field Theory Question on Electron-Positron Pair Creation

1. What is a quantum field theory?

A quantum field theory is a theoretical framework that combines elements of quantum mechanics and special relativity to describe the behavior of particles and their interactions. It provides a mathematical description of how particles interact with each other and the fundamental forces of nature.

2. What is electron-positron pair creation?

Electron-positron pair creation is a process in which a high-energy photon, or particle of light, can create an electron and a positron (the antimatter counterpart of an electron) in empty space. This process is governed by the principles of quantum field theory and is commonly observed in particle accelerators or in the presence of strong electromagnetic fields.

3. Why is it important to solve quantum field theory questions on electron-positron pair creation?

Solving quantum field theory questions on electron-positron pair creation helps us better understand the fundamental nature of particles and their interactions. It also has practical applications in fields such as particle physics, cosmology, and quantum computing.

4. How do scientists approach solving quantum field theory questions on electron-positron pair creation?

Scientists use mathematical equations and principles from quantum field theory, such as the Feynman diagrams, to describe and analyze the process of electron-positron pair creation. They also use experimental data and theoretical predictions to refine and validate their solutions.

5. What are some current research developments in solving quantum field theory questions on electron-positron pair creation?

Some current research developments include using advanced computational techniques to solve complex quantum field theory equations, exploring the implications of electron-positron pair creation in extreme environments such as black holes, and investigating the possibility of creating and controlling electron-positron pairs in a laboratory setting.

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