Casimir effect in 1+1 Minkowski spacetime

[Emil_M]

Homework Statement

https://i.imgur.com/sI3JiB4.jpg
https://i.imgur.com/PLpnPZw.jpg
I have no idea how to solve the first question about the vacuum energy. I solved the second and third problems, but I'm hopelessly stuck at the first.

2. Homework Equations
The Hamiltonian can be written as $H_H=\int \frac{\mathrm{d} k}{2 \pi 2\omega_k}\left(A(k) A^\dagger (k)+A^\dagger (k) A(k) \right)$ and the canonical commutation relations apply:

$[A(k), A(k')]=0, [A^\dagger(k), A^\dagger(k')]=0, [A(k), A^\dagger (k')]=2 \pi 2 \omega_k \delta(k-k')$

The Attempt at a Solution

The trouble is that I have no idea how to begin to solve this question.

 

[king vitamin]

Do you know how equation (2) was derived? From the problem statement, it sounds like it was derived in lecture.

Equation (4), which you need to show, is essentially repeating the derivation of equation (2) but taking the boundary conditions in equation (3) into account.

 

[George Jones]

$[A(k), A^\dagger (k')]=2 \pi 2 \omega_k \delta(k-k')$

This won't be true here.

"Argue heuristically" means use classical waves and boundary conditions to find the modes, which will be a discrete set, so a Dirac delta function is not appropriate.

 

[Emil_M]

Do you know how equation (2) was derived? From the problem statement, it sounds like it was derived in lecture.

Equation (4), which you need to show, is essentially repeating the derivation of equation (2) but taking the boundary conditions in equation (3) into account.

Thank you for your reply! I derived equation (2) by setting $H_H |0>=0$ with $H_H=\int \frac{\mathrm{d} k}{2 \pi 2\omega_k}\left(A(k) A^\dagger (k)+A^\dagger (k) A(k) \right)$

This Hamiltonian is derived by utelizing the Fourier transformation $\tilde{\Phi}(k)$ of $\Phi(x)$, however, so I don't know how to work with the boundary conditions in this computation.

 

[George Jones]

$H_H |0>=0$ with $H_H=\int \frac{\mathrm{d} k}{2 \pi 2\omega_k}\left(A(k) A^\dagger (k)+A^\dagger (k) A(k) \right)$

Again, since the set of modes is discrete, this integral is replaced by a sum.

From the boundary conditions (3), $\phi \left( t,0 \right) = \phi \left( t,d \right) = 0$. What are the possible values $\lambda$? What are the possible values $\omega$?

 

[Emil_M]

Again, since the set of modes is discrete, this integral is replaced by a sum.

From the boundary conditions (3), $\phi \left( t,0 \right) = \phi \left( t,d \right) = 0$. What are the possible values $\lambda$? What are the possible values $\omega$?

Thank you for your answer!

I thought if the wave consists of discrete modes, the continuous spectrum should have corresponding delta functions in order to go from integration to summation? Otherwise the dimension of the expression would change, no? I was under the impression that the canonical commutation relations would provide those delta functions?

 

[George Jones]

I thought if the wave consists of discrete modes, the continuous spectrum should have corresponding delta functions in order to go from integration to summation?

I had in mind starting with a discrete set, but let's try this.

Arguing heuristically,

From the boundary conditions (3), $\phi \left( t,0 \right) = \phi \left( t,d \right) = 0$. What are the possible values $\lambda$?

And thus what are the possible values of $k$?

 

[Emil_M]

And thus what are the possible values of $k$?

Thank you so much for your help!

$\lambda=\{ 2d, d, d/2, d/3,...\}$ which means by $\lambda=2\pi /k$ that $k= \{ \pi/d, 2 \pi/d, 4 \pi /d, 5 \pi/d,...\}$