# Diagonalizable Hamilton denisty?

#### >>S/Z<<

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

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

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

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

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

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

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