theories:classical_field_theory
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theories:classical_field_theory [2018/03/28 08:51] jakobadmin ↷ Page moved from theories:classical_theories:classical_field_theory to theories:classical_field_theory |
theories:classical_field_theory [2018/04/15 12:36] (current) ida [Concrete] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| {{ :theories:quantum_theory:fieldasmatress_kopie.png?nolink&300|}} | {{ :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. | ||
| - | The easiest way to think about a classical field is a mattress. A mattress consists of many point masses that are connected by springs. Therefore the point masses can oscillate and these oscillations influence the neighbouring point masses. This way wave-like perturbations can move through the mattress, as everyone knows who ever jumped around on a mattress. | + | 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. |
| + | |||
| + | 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. | ||
| 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. | 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. | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | In a field theory, we describe everything in terms of field configurations. Solutions of the field equations describe sequences of field configurations: | + | In a field theory, we describe everything in terms of field configurations. Solutions of the [[:equations|field equations]] describe sequences of field configurations: |
| {{ :fieldsequence2.png?nolink&600 |}} | {{ :fieldsequence2.png?nolink&600 |}} | ||
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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.1522219888.txt.gz · Last modified: 2018/03/28 06:51 (external edit)
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