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--- theories:classical_field_theory [2018/03/30 10:15]
jakobadmin [Concrete]
+++ theories:classical_field_theory [2018/04/15 12:36]
ida [Concrete]
@@ Line 23: / Line 23: @@
 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}
 ----

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