models:toy_models:scalar_1plus1
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models:toy_models:scalar_1plus1 [2018/03/15 09:39] 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 |
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| ====== 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 quantum field theory we can study. It is often used to introduce the basics of solitonic solutions, since the Scalar 1+1 models contains a kink solution. | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| + | The Scalar 1+1 Model describes how the simplest type of field interacts with __itself__. | ||
| - | <note tip> | + | The simplest type of field is a field with no [[basic_notions:spin|internal angular momentum]] and called scalar field. |
| - | 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> | + | 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. |
| + | |||
| + | 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|}} | ||
| The model only contains a __real scalar field__ $\phi (x) \in \mathbb{R}$. The __scalar potential__ is | The model only contains a __real scalar field__ $\phi (x) \in \mathbb{R}$. The __scalar potential__ is | ||
| Line 27: | 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 36: | Line 44: | ||
| ** Standard Solutions ** | ** Standard Solutions ** | ||
| + | |||
| + | {{ :models:classsol.png?nolink&400|}} | ||
| The standard solutions of the field equation are simply | The standard solutions of the field equation are simply | ||
| Line 41: | Line 51: | ||
| $$ \phi_c = \pm a. $$ | $$ \phi_c = \pm a. $$ | ||
| - | These two solutions are the __classical vacuum solution__. | + | These two solutions are the __classical vacuum solution__. They are rather boring since the field simply takes the vacuum value, either $a$ or $-a$ everywhere. |
| ** Solitonic Solutions - The Kink ** | ** Solitonic Solutions - The Kink ** | ||
| + | |||
| + | {{ :models:kinksol.png?nolink&400|}} | ||
| In addition, we can find other less trivial solutions: | In addition, we can find other less trivial solutions: | ||
| Line 93: | Line 105: | ||
| These boundary conditions become important when we discover that there is a __conserved current__ in the model: | These boundary conditions become important when we discover that there is a __conserved current__ in the model: | ||
| - | $$ \j_\mu (x) = \epsilon_{\mu \nu} \partial^\nu \phi(x), $$ | + | $$ j_\mu (x) = \epsilon_{\mu \nu} \partial^\nu \phi(x), $$ |
| where $\epsilon_{\mu \nu} $ is the two-dimensional Levi-Civita symbol. This current is conserved since [[https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz's_theorem|partial derivatives commute]] and $\epsilon_{\mu \nu}$ is totally antisymmetric. A sum over something symmetric $\partial \mu \partial^\nu$ times something totally antisymmetric $\epsilon_{\mu \nu}$ yields zero: | where $\epsilon_{\mu \nu} $ is the two-dimensional Levi-Civita symbol. This current is conserved since [[https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz's_theorem|partial derivatives commute]] and $\epsilon_{\mu \nu}$ is totally antisymmetric. A sum over something symmetric $\partial \mu \partial^\nu$ times something totally antisymmetric $\epsilon_{\mu \nu}$ yields zero: | ||
| - | $$ \partial^\u \j_\mu (x) = 0. $$ | + | $$ \partial^\mu j_\mu (x) = 0. $$ |
| Whenever we have conserved current, we can find a conserved charge: | Whenever we have conserved current, we can find a conserved charge: | ||
| Line 107: | Line 119: | ||
| \end{align} | \end{align} | ||
| - | The conserved quantity is, therefore, $n$, up to a constant factor. This conserved quantity is known as winding number of the kink and because of this topological conserved quantity the kink is stable. | + | The conserved quantity is, therefore, $n$, up to a constant factor. This conserved quantity is known as winding number of the kink and because of this topological conserved quantity, the kink is stable. |
| - | <tabbox Researcher> | + | |
| - | <note tip> | + | {{ :models:energykjink.png?nolink&400|}} |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| - | | + | 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 Examples> | + | <tabbox Abstract> |
| - | --> Example1# | + | 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 scalar field $\phi$ takes at infinity only vacuum values since otherwise, we would be dealing with an infinite energy configuration. The __vacuum [[advanced_tools:manifold|manifold]]__ is also just two points: | |
| - | <-- | + | $M_V = \{ -a, a \}$. |
| - | --> Example2:# | + | Our restriction to solutions with finite energy means implies that we are dealing with maps $ S_\infty \to M_V $, i.e. at infinity the field only takes values in the vacuum manifold. |
| - | + | Now, topology tells us that such maps can be classified through an integer $n$. This number then automatically labels our solitonic solutions. | |
| - | <-- | + | |
| - | <tabbox FAQ> | + | |
| - | + | ||
| - | <tabbox History> | + | <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. | ||
| + | |||
| + | 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> | </tabbox> | ||
models/toy_models/scalar_1plus1.1521103146.txt.gz · Last modified: 2018/03/15 08:39 (external edit)
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