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--- theories:classical_field_theory [2018/02/19 17:04]
jakobadmin
+++ theories:classical_field_theory [2018/04/15 12:36] (current)
ida [Concrete]
@@ Line 1: / Line 1: @@
 ====== Classical Field Theory ======
-<tabbox Why is it interesting?>
+<tabbox Intuitive>
+{{ :theories:quantum_theory:fieldasmatress_kopie.png?nolink&300|}}
+A field in physics is something that associates with each point in space and with each instance in time a quantity.
-<blockquote>
+The easiest way to think about a classical field is as a mattress. A mattress consists of many point masses that are connected by springs. The horizontal location of these point masses is the quantity that is associated with each point in space and time.
-While our aim is to discuss the quantized Yang-Mills theory, let us pause for a moment and examine the dynamical field equations in their classical setting. After all, the Maxwell theory, which is the antecedent and inspiration for the Yang-Mills theory, was thoroughly investigated within classical physics, with results that are quite relevant physically even when quantum effects are ignored. Unfortunately, no such physical success can be claimed here, though much of mathematical interest has been achieved.
+The point masses can oscillate and these oscillations influence the neighboring point masses. This way wave-like perturbations can move through the mattress, as everyone knows whoever jumped around on a mattress.
-<cite>Topological Investigations of Quantized Gauge Theories, by R. Jackiw (1983)</cite>
+If we now imagine that we zoom out such that the point masses become smaller and smaller we end up with a great approximation to a classical field. A classical field is nothing but the continuum limit of a mattress.
-</blockquote>
-<blockquote>Like the Hamiltonian formalism for classical physics, the [[equations:schroedinger_equation|Schrödinger equation]] is not so much a specific equation,
- but a framework for quantum mechanical equations generally. Once one has obtained the appropriate Hamiltonian, the time evolution of the state according to Schrödinger's equation proceeds rather as though $|\Psi>$ were a classical field subject to some classical field equation such as Maxwell's. In fact, if $|\Psi>$ describes the state of a single photon, then it turns out that Schrodinger's equation actually
-becomes [[equations:maxwell_equations|Maxwell's equations]]! **The equation for a single photon is precisely
-the same as the equation for an entire electromagnetic field.** (However, there is an important difference in the type of solution for the equations that is allowed. Classical Maxwell fields are necessarily real whereas photon states are complex. There is also a so-called 'positive frequency condition that the photon state must satisfy). This fact is
-responsible for the Maxwell-field-wavelike behaviour and polarization of
-single photons that we caught glimpses of earlier. As another example, if 11Ji}
-describes the state of a single electron, then Schröinger's equation becomes
-[[equations:dirac_equation|Dirac's remarkable wave equation]] for the electron discovered in 1928 after
-Dirac had supplied much additional originality and insight
-<cite>The Emperor's New Mind by R. Penrose</cite></blockquote>
-<tabbox Layman>
-{{ :theories:quantum_theory:fieldasmatress_kopie.png?nolink&300|}}
-<note tip>
+<tabbox Concrete>
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
+In a field theory, we describe everything in terms of field configurations. Solutions of the [[:equations|field equations]] describe sequences of field configurations:
-</note>
+{{ :fieldsequence2.png?nolink&600 |}}
-<tabbox Student>
+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}
+----
+**Great Resources:**
   * http://waveforms.surge.sh/waveforms-intro
   * [[https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1002&context=lib_mono|Introduction to Classical Field Theory]] by Charles G. Torre
+  * See also: [[https://arxiv.org/pdf/hep-ph/0702173.pdf|Some classical properties of the non-abelian Yang-Mills theories]] J. A. Sanchez-Monroy and C. J. Quimbay
+----
@@ Line 66: / Line 119: @@
 </blockquote>
-See also: [[https://arxiv.org/pdf/hep-ph/0702173.pdf|Some classical properties of the non-abelian Yang-Mills theories]] J. A. Sanchez-Monroy and C. J. Quimbay
-<tabbox Researcher>
+<tabbox Abstract>
   * [[https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf|Classical Field Theory]] by P. Deligne and D. Freed
-<tabbox Examples>
+<tabbox Why is it interesting?>
+Classical field theory was for a long time the best framework to describe the fundamental forces of nature. The most notable examples of classical field theories are Newtonian gravity and classical [[models:classical_electrodynamics|Electrodynamics]].
---> Example1#
+-----
-<--
---> Example2:#
+<blockquote>Like the Hamiltonian formalism for classical physics, the [[equations:schroedinger_equation|Schrödinger equation]] is not so much a specific equation,
-<--
+ but a framework for quantum mechanical equations generally. Once one has obtained the appropriate Hamiltonian, the time evolution of the state according to Schrödinger's equation proceeds rather as though $|\Psi>$ were a classical field subject to some classical field equation such as Maxwell's. In fact, if $|\Psi>$ describes the state of a single photon, then it turns out that Schrodinger's equation actually
+becomes [[equations:maxwell_equations|Maxwell's equations]]! **The equation for a single photon is precisely
+the same as the equation for an entire electromagnetic field.** (However, there is an important difference in the type of solution for the equations that is allowed. Classical Maxwell fields are necessarily real whereas photon states are complex. There is also a so-called 'positive frequency condition that the photon state must satisfy). This fact is
+responsible for the Maxwell-field-wavelike behaviour and polarization of
+single photons that we caught glimpses of earlier. As another example, if 11Ji}
+describes the state of a single electron, then Schröinger's equation becomes
+[[equations:dirac_equation|Dirac's remarkable wave equation]] for the electron discovered in 1928 after
+Dirac had supplied much additional originality and insight
+<cite>The Emperor's New Mind by R. Penrose</cite></blockquote>
+<blockquote>
+While our aim is to discuss the quantized Yang-Mills theory, let us pause for a moment and examine the dynamical field equations in their classical setting. After all, the Maxwell theory, which is the antecedent and inspiration for the Yang-Mills theory, was thoroughly investigated within classical physics, with results that are quite relevant physically even when quantum effects are ignored. Unfortunately, no such physical success can be claimed here, though much of mathematical interest has been achieved.
+<cite>Topological Investigations of Quantized Gauge Theories, by R. Jackiw (1983)</cite>
+</blockquote>
 <tabbox History>

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