43.3 Hamiltonian and dynamics of the Klein-Gordon theory

43.3 Hamiltonian and dynamics of the Klein-Gordon theory

The Klein-Gordon equation for $\varphi (t,\mathbf{x})$ in Hamiltonian form is the following pair of first-order equations

$$\frac{\partial }{\partial t}\varphi =\pi ,\frac{\partial }{\partial t}\pi =(\Delta -m^2)\varphi$$

which together imply

$$\frac{{\partial }^2}{\partial t^2}\varphi =(\Delta -m^2)\varphi$$

To get these as equations of motion, we need to find a Hamiltonian function $h$ such that

$$\begin{array}{rl}\frac{\partial }{\partial t}\varphi & =\{\varphi ,h\}=\pi \\ \frac{\partial }{\partial t}\pi & =\{\pi ,h\}=(\Delta -m^2)\varphi \end{array}$$

One can show that two choices of Hamiltonian function with this property are

$$h={\int }_{{\mathbf{R}}^3}\mathcal{H}(\mathbf{x})d^3\mathbf{x}$$

where

$$\mathcal{H}=\frac{1}{2}({\pi }^2-\varphi \Delta \varphi +m^2{\varphi }^2)\mathrm{or}\mathcal{H}=\frac{1}{2}({\pi }^2+(\nabla \varphi {)}^2+m^2{\varphi }^2)$$

Here the two diferent integrands $\mathcal{H}(\mathbf{x})$ are related (as in the non-relativistic case) by integration by parts, so these just difer by boundary terms that are assumed to vanish.

In terms of the $A(\mathbf{p}),\overline{A}(\mathbf{p})$ , the Hamiltonian will be

$$h={\int }_{{\mathbf{R}}^3}{\omega }_{\mathbf{p}}A(\mathbf{p})\overline{A}(\mathbf{p})d^3\mathbf{p}$$

The equations of motion are

$$\begin{array}{rl}\frac{d}{dt}A& =\{A,h\}={\int }_{{\mathbf{R}}^3}{\omega }_{{\mathbf{p}}^{\prime }}A({\mathbf{p}}^{\prime })\{A(\mathbf{p}),\overline{A}({\mathbf{p}}^{\prime })\}{d}^3{\mathbf{p}}^{\prime }=i{\omega }_{\mathbf{p}}A\\ \frac{d}{dt}\overline{A}& =\{\overline{A},h\}=-i{\omega }_{\mathbf{p}}\overline{A}\end{array}$$

with solutions

$$A(\mathbf{p},t)=e^{i{\omega }_{\mathbf{p}}t}A(\mathbf{p},0),\overline{A}(\mathbf{p},t)=e^{-i{\omega }_{\mathbf{p}}t}\overline{A}(\mathbf{p},0)$$

Digression. Taking as starting point the Lagrangian formalism, the action for the Klein-Gordon theory is

$$S={\int }_{M^4}\mathcal{L}{d}^{4}x$$

where

$$\mathcal{L}=\frac{1}{2}\left({\left(\frac{\partial }{\partial t}\varphi \right)}^2-(\nabla \varphi {)}^2-m^2{\varphi }^2\right)$$

This action is a functional of fields on Minkowski space $M^4$ and is Poincar´e invariant. The Euler-Lagrange equations give as equation of motion the Klein-Gordon equation 43.2. One recovers the Hamiltonian formalism by seeing that the canonical momentum for $\varphi$ is

$$\pi =\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=\dot{\varphi }$$

and the Hamiltonian density is

$$\mathcal{H}=\pi \dot{\varphi }-\mathcal{L}=\frac{1}{2}({\pi }^2+(\nabla \varphi {)}^2+m^2{\varphi }^2)$$

Previous section: 43.2 The symplectic and complex structures on M · Next section: 43.4 Quantization of the Klein-Gordon theory · 中文版本

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