- #1
Azelketh
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Homework Statement
Hi I a attempting to derive the expression for the conserved Noether charge for a free complex scalar field.
The question I have to complete is: " show, by using the mode expansions for the free complex scalar field, that the conserved Noether charge (corresponding to complex phase rotations) is given by
[tex] Q= \int \frac{\partial^3 p}{(2\pi)^3} ( a_p^* a_p - b_p^* b_p)
[/tex]
Where [tex]a_p, a_p^*[/tex] and [tex] b_p, b_p^*[/tex] are the creation and annihilation operators for 2 kinds of particles respectively"
Homework Equations
The Lagrangian for a free complex scalar field is;
[tex] L= \frac{1}{2} \partial_μ \psi \partial^μ \psi- \frac{1}{2}m^2\psi^2 [/tex]
I have recognized that there is a symmetry of[tex]\psi → e^{ i\alpha } \psi [/tex]
which leads to a Noether current of
[tex] j_μ = i \partial _μ \psi^* \psi - i \partial _μ \psi \psi^* [/tex]
[tex] Q= \int \partial ^3 x j_0 = i \int \partial ^3 x ( \partial _0 \psi^* \psi - \partial _0 \psi \psi^* ) [/tex]
[tex] Q=i \int \partial ^3 x ( \psi \dot{\psi^*} - \psi^* \dot{\psi} )[/tex]
where
[tex] \dot{\psi} = \pi ^*[/tex]
[tex] \dot{\psi^*} = \pi[/tex]
so [tex] Q=i \int \partial ^3 x ( \psi \pi - \psi^* \pi ^* )[/tex]
And the mode expansions;
[tex] \psi = \int \frac{\partial^3 P}{(2\pi)^3} \frac{1}{\sqrt{2 ω_p}}( a_p e^{ip.x } + b_p^* e^{-ip.x}) [/tex]
[tex] \psi^* = \int \frac{\partial^3 P}{(2\pi)^3} \frac{1}{\sqrt{2 ω_p}}( a_p^* e^{-ip.x} + b_p e^{ip.x }) [/tex]
[tex] \pi = i\int \frac{\partial^3 P}{(2\pi)^3} \sqrt{ \frac{ω_p}{2}}( a_p^* e^{-ip.x } - b_p e^{ip.x}) [/tex]
[tex] \pi^* = -i\int \frac{\partial^3 P}{(2\pi)^3} \sqrt{\frac{ω_p}{2}}( a_p e^{ip.x } - b_p^* e^{-ip.x}) [/tex]
The Attempt at a Solution
I have then plugged these into the Q equation;
[tex] Q=i \int \partial ^3 x [ \int \frac{\partial^3 P}{(2\pi)^3} \frac{1}{\sqrt{2 ω_p}}( a_p e^{ip.x } + b_p^* e^{-ip.x})
i\int \frac{\partial^3 P}{(2\pi)^3} \sqrt{ \frac{ω_p}{2}}( a_p^* e^{-ip.x } - b_p e^{ip.x})
- \int \frac{\partial^3 P}{(2\pi)^3} \frac{1}{\sqrt{2 ω_p}}( a_p^* e^{-ip.x} + b_p e^{ip.x }) (-i)\int \frac{\partial^3 P}{(2\pi)^3} \sqrt{\frac{ω_p}{2}}( a_p e^{ip.x } - b_p^* e^{-ip.x}) ][/tex]
so factoring out the i's and 2 pi's from the integrals;
[tex] Q=\frac{i^2}{(2\pi)^6} \int \partial ^3 x [ \int \partial^3 P \frac{1}{\sqrt{2 ω_p}}( a_p e^{ip.x } + b_p^* e^{-ip.x})
\int \partial^3 P \sqrt{ \frac{ω_p}{2}}( a_p^* e^{-ip.x } - b_p e^{ip.x})
+ \int \partial^3 P \frac{1}{\sqrt{2 ω_p}}( a_p^* e^{-ip.x} + b_p e^{ip.x }) \int \partial^3 P \sqrt{\frac{ω_p}{2}}( a_p e^{ip.x } - b_p^* e^{-ip.x}) ][/tex]
It's at this point I'm utterly stuck on how to procede, I did think that could cancel the sqrt ω terms but they depend on p: [tex] ω_p = \sqrt{P^2 + m^2} [/tex]
so as they are inside integrals dependent on P they can't be cancelled.
I am at an utter loss how to proceed from here though. If anyone can offer any pointers or assistance is would be greatly appreciated.
EDIT: made a sign error in exponential's for [tex] \psi [/tex] and [tex] \psi^* [/tex]
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