Deriving Klein-Gordon from Heisenberg

[waht]

Homework Statement

Sort of stuck deriving the Klein-Gordon equation from Heisenberg equation of motion

\dot{\varphi} = i [H, \varphi ]

Homework Equations

\dot{\varphi} = \frac{\partial\varphi}{\partial t}

H = \int d^3x \mathcal{H}

\Pi (x) = \dot{\varphi}(x)

\mathcal{H} = \Pi \dot{\varphi} - \mathcal{L}

The Attempt at a Solution

\dot{\varphi} = \int d^3x [ \dot{\varphi}^2 - \mathcal{L}, \varphi ]

if I expand that, then it becomes a real mess, not sure if I'm on the right track?

 

[mingda]

first, from the condition i*d(phi)/dt=[H,phi], you can arrive an equation, which makes [intgral(delta_phi)^2,phi]=0, then you differentiate the Heisenberg, equation, get d^2(phi)/dt^2=[H,[H,phi]], and substitute back. Then you get K-G equation immediately

 

[mingda]

\dot{\varphi} = i [H, \varphi ], we obtain

\int d^3x [ \delta{\varphi}^2, \varphi ]=0 Eq(1)

differentiate Heisenberg Eq. again, we have

\dot\dot{\varphi} = i [H, \dot{\varphi} ]=- [H, [H, \dot{\varphi} ] ]

substitute \Pi (x) = \dot{\varphi}(x) back,

using the specific form of H and use Eq.1, you'll get the answer of KG equation.