Diagonalizable Hamilton denisty?

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1. The problem statement, all variables and given/known data
I'm working with a complex scalar field with the lagrange density [tex] L= \partial_{\mu} \phi^{\ast} \partial^{\mu} \phi - m^2 \phi^{\ast} \phi [/tex] And I've shown thats its hamilton density [tex] H= \int d^3 x ( \pi^{\ast} \pi + \nabla \phi^{\ast} \cdot \nabla \phi + m^2 \phi^{\ast} \phi ) [/tex]

Now the annihilation and creation operateurs are introduced. And I have to show that hamilton density H is diagonalizable by writing
[tex] \phi(x) = d^3 p \left( \frac{1}{(2 \pi)^3} \frac{1}{\sqrt{2E_p}} (a_{\vec{p}} e ^{-i p \cdot x} + b_{\vec{p}}^{\dagger} e^{ip \cdot x}) \right) [/tex]

I also have to show that the theory contains two particles with mass m. How do I do this? I dont really know how to get started? Neither with how to show it is diagonalizable or how to find the masses:(
 

gabbagabbahey

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1. The problem statement, all variables and given/known data
I'm working with a complex scalar field with the lagrange density [tex] L= \partial_{\mu} \phi^{\ast} \partial^{\mu} \phi - m^2 \phi^{\ast} \phi [/tex] And I've shown thats its hamilton density [tex] H= \int d^3 x ( \pi^{\ast} \pi + \nabla \phi^{\ast} \cdot \nabla \phi + m^2 \phi^{\ast} \phi ) [/tex]

Now the annihilation and creation operateurs are introduced. And I have to show that hamilton density H is diagonalizable by writing
[tex] \phi(x) = \int d^3 p \left( \frac{1}{(2 \pi)^3} \frac{1}{\sqrt{2E_p}} (a_{\vec{p}} e ^{-i p \cdot x} + b_{\vec{p}}^{\dagger} e^{ip \cdot x}) \right) [/tex]
Use this to calculate an expression for [itex]\phi^*(x)[/itex] (Most physicists use [itex]{}^{\dagger}[/itex] instead of [itex]{}^{*}[/itex] for Hermitian conjugation ), [itex]\mathbf{\nabla}\phi(x)[/itex], [itex](\mathbf{\nabla}\phi(x))^*[/itex] and then [itex]H[/itex]. Carry out the integration over [tex]x[/itex], and you should get something like

[tex]H=\int\int d^3pd^3p' f(\textbf{p},\textbf{p}') \delta(\textbf{p}-\textbf{p}')[/tex]

The presence of the delta function means that all non-diagonal elements vanish (since it is zero when [itex]\textbf{p}\neq\textbf{p}'[/itex] ).
 

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