theories:classical_field_theory
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theories:classical_field_theory [2018/03/30 10:15] jakobadmin [Concrete] |
theories:classical_field_theory [2018/04/15 12:36] (current) ida [Concrete] |
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| A classical field is a dynamical system with an **infinite number** of degrees of freedom. We describe fields mathematically by partial differential equations. | A classical field is a dynamical system with an **infinite number** of degrees of freedom. We describe fields mathematically by partial differential equations. | ||
| + | ---- | ||
| + | |||
| + | The action functional $S[\phi(x)]$ for a free real scalar field of mass $m$ is | ||
| + | \begin{eqnarray} | ||
| + | S[\phi(x)]\equiv \int d^{4}x \,\mathcal{L}(\phi,\partial_{\mu}\phi)= | ||
| + | {1\over 2}\int d^{4}x \,\left(\partial_{\mu}\phi\partial^{\mu}\phi- | ||
| + | {m^{2}}\phi^2\right). | ||
| + | \end{eqnarray} | ||
| + | We can calculate the equations of motion are obtained by using [[equations:euler_lagrange_equations|the Euler-Lagrange | ||
| + | equations]] | ||
| + | \begin{eqnarray} | ||
| + | \partial_{\mu}\left[\partial\mathcal{L}\over \partial(\partial_{\mu}\phi) | ||
| + | \right]-{\partial\mathcal{L}\over \partial\phi}=0 \quad | ||
| + | \Longrightarrow \quad (\partial_{\mu}\partial^{\mu}+m^{2})\phi=0. | ||
| + | \label{eq:eomKG} | ||
| + | \end{eqnarray} | ||
| + | |||
| + | The momentum canonically conjugated to the field $\phi(x)$ is given by | ||
| + | \begin{eqnarray} | ||
| + | \pi(x)\equiv {\partial\mathcal{L}\over \partial(\partial_{0}\phi)} | ||
| + | ={\partial\phi\over\partial t}. | ||
| + | \end{eqnarray} | ||
| + | |||
| + | The corresponding Hamiltonian function is | ||
| + | \begin{eqnarray} | ||
| + | H\equiv \int d^{3}x \left(\pi{\partial\phi\over\partial t}-\mathcal{L}\right) | ||
| + | = {1\over 2}\int d^{3}x\left[ | ||
| + | \pi^2+(\vec{\nabla}\phi)^{2}+m^{2}\right]. | ||
| + | \end{eqnarray} | ||
| + | |||
| + | In classical theories, we can write the equations of motionin terms of the [[advanced_notions:poisson_bracket|Poisson | ||
| + | brackets]]: | ||
| + | \begin{eqnarray} | ||
| + | \{A,B\}\equiv \int d^{3}x\left[{\delta {A}\over \delta \phi} | ||
| + | {\delta{B}\over \delta\pi}- | ||
| + | {\delta{A}\over \delta\pi}{\delta{B}\over \delta\phi} | ||
| + | \right], | ||
| + | \end{eqnarray} | ||
| + | where ${\delta\over \delta \phi}$ denotes the functional derivative | ||
| + | defined as | ||
| + | \begin{eqnarray} | ||
| + | {\delta A\over \delta\phi}\equiv {\partial\mathcal{A}\over | ||
| + | \partial\phi}-\partial_{\mu}\left[{\partial\mathcal{A} | ||
| + | \over \partial(\partial_{\mu}\phi)}\right] | ||
| + | \end{eqnarray} | ||
| + | The canonically conjugated classical fields satisfy the | ||
| + | following equal time Poisson brackets | ||
| + | \begin{eqnarray} | ||
| + | \{\phi(t,\vec{x}),\phi(t,\vec{x}\,')\}&=&\{\pi(t,\vec{x}), | ||
| + | \pi(t,\vec{x}\,')\}=0,\nonumber \\ | ||
| + | \{\phi(t,\vec{x}),\pi(t,\vec{x}\,')\}&=&\delta(\vec{x}-\vec{x}\,'). | ||
| + | \label{eq:etccr} | ||
| + | \end{eqnarray} | ||
| ---- | ---- | ||
theories/classical_field_theory.1522397743.txt.gz · Last modified: 2018/03/30 08:15 (external edit)
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