models:toy_models:scalar_1plus1
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
models:toy_models:scalar_1plus1 [2018/03/15 10:10] jakobadmin [Student] |
models:toy_models:scalar_1plus1 [2018/05/05 11:49] (current) jakobadmin ↷ Page moved from models:scalar_1plus1 to models:toy_models:scalar_1plus1 |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Scalar 1+1 Model ====== | ====== Scalar 1+1 Model ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | The scalar model in 1 spatial and 1 time dimension is the simplest [[theories:quantum_theory:quantum_field_theory|quantum field theory]] we can study. | ||
| - | It is often used to introduce the basics of [[advanced_notions:quantum_field_theory:solitons|solitonic solutions]], since the Scalar 1+1 model contains a non-trivial kink solution. | + | <tabbox Intuitive> |
| + | The Scalar 1+1 Model describes how the simplest type of field interacts with __itself__. | ||
| - | <tabbox Layman> | + | The simplest type of field is a field with no [[basic_notions:spin|internal angular momentum]] and called scalar field. |
| - | <note tip> | + | In addition, the model is a model in only one space dimension plus the usual time dimension. (That's what the 1+1 in the name of the model means). This is the simplest setup where we can investigate dynamics. |
| - | 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. | + | |
| - | </note> | + | The physics described by the Scalar 1+1 Model model looks like this: |
| + | |||
| + | {{ :models:scalar1p1.png?nolink&500 |}} | ||
| + | |||
| + | Each red line shows a field configuration at a given point in time. | ||
| + | |||
| + | | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| {{ :models:potentialscalar.png?nolink&400|}} | {{ :models:potentialscalar.png?nolink&400|}} | ||
| Line 30: | Line 35: | ||
| The field equation can be either derived from the action of the model or from the __energy functional__: | The field equation can be either derived from the action of the model or from the __energy functional__: | ||
| - | $$ E= \int dx \left[ \frac{1}{2} \left( \frac{\partial \phi}{\partial x}\right)^2 + U(\phi)\right]. $$ | + | $$ E= \int dx^2 \left[ \frac{1}{2} \left( \frac{\partial \phi}{\partial \vec x}\right)^2 + U(\phi)\right]. $$ |
| Then, demanding that the variation $\delta E$ vanishes yields the field equation. | Then, demanding that the variation $\delta E$ vanishes yields the field equation. | ||
| Line 119: | Line 124: | ||
| We can understand the stability of the kink also by looking at the energy density of the various solutions. Between the kink and the standard vacuum solution, there is an infinite potential barrier. This barrier exists since it would take an infinite amount of energy to change the kink solution to a classical energy solution. The kink solutions would need to be transformed at every spacetime point either to $a$ or to $-a$ and since the space we are considering is infinite, this transformation would take an infinite amount of energy. | We can understand the stability of the kink also by looking at the energy density of the various solutions. Between the kink and the standard vacuum solution, there is an infinite potential barrier. This barrier exists since it would take an infinite amount of energy to change the kink solution to a classical energy solution. The kink solutions would need to be transformed at every spacetime point either to $a$ or to $-a$ and since the space we are considering is infinite, this transformation would take an infinite amount of energy. | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| The spacetime we are considering is $1+1$ dimensional. Hence the boundary of space is just two points $x_-=-\infty$ and $x_+ =\infty$. We denote this boundary of space with $S_\infty = \{ -\infty, \infty \}$. | The spacetime we are considering is $1+1$ dimensional. Hence the boundary of space is just two points $x_-=-\infty$ and $x_+ =\infty$. We denote this boundary of space with $S_\infty = \{ -\infty, \infty \}$. | ||
| Line 132: | Line 137: | ||
| + | <tabbox Why is it interesting?> | ||
| + | The scalar model in one spatial and one temporal dimension is the simplest type of model we can construct in field theory. | ||
| - | <tabbox FAQ> | + | It is often used to introduce the basics of [[advanced_notions:quantum_field_theory:solitons|solitonic solutions]], since the Scalar 1+1 model contains a non-trivial kink solution. |
| - | + | ||
| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
models/toy_models/scalar_1plus1.1521105047.txt.gz · Last modified: 2018/03/15 09:10 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
