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Kruger

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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).