 #1
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Homework Statement:

Given
$$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _$$
Where
$$A^{\mu} _+ = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r (\vec k) e^{i \vec k \cdot \vec x}$$
$$A^{\mu} _ = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r^{\dagger} (\vec k) e^{i \vec k \cdot \vec x}$$
$$V = L^3, \ \ \ \ \ \ \ \ \ \omega_{\vec k} = ck^0 = c\vec k$$
Where ##a_r (\vec k)## and ##a_r^{\dagger} (\vec k)## are the harmonic oscillator operators, which satisfy the following commutation relations
$$[a_r(\vec k), a_s(\vec k')] = [a_r^{\dagger}(\vec k), a_s^{\dagger}(\vec k')] = 0$$
$$[a_r(\vec k), a_s(\vec k')] = \rho_r \delta_{r,s} \delta_{\vec k, \vec k'}$$
a) Show that it is necessary to rescale the harmonic oscillator operators (as shown below) if we want to take the limit ##L \rightarrow \infty##
$$a_r(\vec k) \rightarrow \tilde{a_r}(\vec k) = \sqrt{\frac{V}{(2\pi)^3}} a_r (\vec k), \ \ \ \ \tilde {a^{\dagger}_r}(\vec k) = (\tilde{a_r}(\vec k))^{\dagger}$$
b) Give the commutation relations satisfied by ##\tilde{a_r}(\vec k)## and ##\tilde {a^{\dagger}_r}(\vec k)## at ##L \rightarrow \infty##.
c) Give the expressions for ##A^{\mu} _+## and ##A^{\mu} _## at ##L \rightarrow \infty##.
Relevant Equations:
 $$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _$$
Let's go step by step
a)
We know that the harmonic oscillator operators are
$$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( ip + m \omega q)$$
$$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$
But these do not depend on ##L##, so I guess these are not the expressions we want to work out...
My guess is that I should first find Ldependent expressions for the harmonic oscillator operators and then work out the limit.
I guess ##A^{\mu} _+## and ##A^{\mu} _## are not the operators themselves.
But what are these expressions?
A hint would be much appreciated.
Thank you.
a)
We know that the harmonic oscillator operators are
$$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( ip + m \omega q)$$
$$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$
But these do not depend on ##L##, so I guess these are not the expressions we want to work out...
My guess is that I should first find Ldependent expressions for the harmonic oscillator operators and then work out the limit.
I guess ##A^{\mu} _+## and ##A^{\mu} _## are not the operators themselves.
But what are these expressions?
A hint would be much appreciated.
Thank you.