- #1

Kalimaa23

- 279

- 0

Greetings,

I have to show that

[tex]\vec{P}=\frac{1}{c^2} \int d^3 x N \left(\dot{A}^{\mu} \nabla A_{\mu}\right)[/tex]

is equivalent to

[tex]\vec{P}= \sum_{\vec{k}} \hbar \vec{k} N(\vec{k}) [/tex]

N in the first expression denotes the normal product, and [tex]N(\vec{k})[/tex] is the usual number operator.

Now taking the normal ordering one gets

[tex] \vec{P}=\frac{1}{c^2} \int d^3 x \left\{\dot{A}^{\mu,+} \nabla A_{\mu}^{+} + \nabla A_{\mu}^{-} \dot{A}^{\mu,+} + \dot{A}^{\mu,-} \nabla A_{\mu}^{+} + \dot{A}^{\mu,-} \nabla A_{\mu}^{-}\right\} [/tex]

The middle terms give the sougth expression, the exponentials in the expansion of the [tex]A_{\mu}[/tex] nicely cancelling. The first and the last term however is giving me trouble. I basically end up with an unwanted term of the form

[tex]\sum_{\vec{k},r} \frac{\hbar \vec{k}}{2} \int \frac{d^3x}{V} \left\{\epsilon^{\mu}_{r} (\vec{k}) \epsilon_{\mu,r} (\vec{k}) \left(a(\vec{k}) a(\vec{k})e^{-2ik.x} + a^{+} (\vec{k}) a^{+} (\vec{k}) e^{2ik.x}\right)\right\}[/tex]

How to get rid of it?

I have to show that

[tex]\vec{P}=\frac{1}{c^2} \int d^3 x N \left(\dot{A}^{\mu} \nabla A_{\mu}\right)[/tex]

is equivalent to

[tex]\vec{P}= \sum_{\vec{k}} \hbar \vec{k} N(\vec{k}) [/tex]

N in the first expression denotes the normal product, and [tex]N(\vec{k})[/tex] is the usual number operator.

Now taking the normal ordering one gets

[tex] \vec{P}=\frac{1}{c^2} \int d^3 x \left\{\dot{A}^{\mu,+} \nabla A_{\mu}^{+} + \nabla A_{\mu}^{-} \dot{A}^{\mu,+} + \dot{A}^{\mu,-} \nabla A_{\mu}^{+} + \dot{A}^{\mu,-} \nabla A_{\mu}^{-}\right\} [/tex]

The middle terms give the sougth expression, the exponentials in the expansion of the [tex]A_{\mu}[/tex] nicely cancelling. The first and the last term however is giving me trouble. I basically end up with an unwanted term of the form

[tex]\sum_{\vec{k},r} \frac{\hbar \vec{k}}{2} \int \frac{d^3x}{V} \left\{\epsilon^{\mu}_{r} (\vec{k}) \epsilon_{\mu,r} (\vec{k}) \left(a(\vec{k}) a(\vec{k})e^{-2ik.x} + a^{+} (\vec{k}) a^{+} (\vec{k}) e^{2ik.x}\right)\right\}[/tex]

How to get rid of it?

Last edited: