models:toy_models:sine_gordon
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models:toy_models:sine_gordon [2018/03/24 15:07] jakobadmin [Student] |
models:toy_models:sine_gordon [2018/05/05 11:50] jakobadmin ↷ Page moved from models:sine_gordon to models:toy_models:sine_gordon |
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| - | ====== Sine-Gordon Model: $ \quad \mathcal{L} = \frac{1}{2} \left( (\partial_\mu \phi)^2 \right) - (1-cos \phi) $ ====== | + | <WRAP lag>$ \mathcal{L} = \frac{1}{2} \left( (\partial_\mu \phi)^2 \right) - (1-cos \phi) $</WRAP> |
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| - | <tabbox Why is it interesting?> | + | |
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| - | The Sine-Gordon model is a toy model that helps to understand fundamental notions like [[advanced_notions:quantum_field_theory:duality|duality]] in a simplified setup. | + | |
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| - | In addition, the equations of the theory permit topological non-trivial solutions called solitons. Many basic features of such solitons can be studied in the Sine-Gordon model in a simplified setup. | + | |
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| - | The name "Sine-Gordon model" is a wordplay, because the equation of motion of the model is similar to the famous [[equations:klein-gordon_equation|Klein-Gordon equation]] and contains a [[basic_tools:trigonometric_functions|sinus function]]. | + | |
| + | ====== Sine-Gordon Model ====== | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| The Sine-Gordon model describes a coupled set of pendulums that hang from a common support rod. | The Sine-Gordon model describes a coupled set of pendulums that hang from a common support rod. | ||
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| - | <tabbox Student> | + | <tabbox Concrete> |
| **Specification of the Model** | **Specification of the Model** | ||
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| **Solitonic Solutions in Detail** | **Solitonic Solutions in Detail** | ||
| - | {{ :models:kinksol2.png?nolink&300|}} | ||
| Since solitons are stable and static, we now try to find such a solution of the field equations. Static means that a soliton is time-independent and therefore we can make the ansatz that our soliton solution is time independent. This yields the __static Sine-Gordon equation__ | Since solitons are stable and static, we now try to find such a solution of the field equations. Static means that a soliton is time-independent and therefore we can make the ansatz that our soliton solution is time independent. This yields the __static Sine-Gordon equation__ | ||
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| Since we know that at infinity we must have $\partial^x \phi \to 0$ and $\cos \phi \to 1$ in order to have a configuration with finite energy, we can deduce $A=1$. | Since we know that at infinity we must have $\partial^x \phi \to 0$ and $\cos \phi \to 1$ in order to have a configuration with finite energy, we can deduce $A=1$. | ||
| - | {{ :models:kinkenergy.png?nolink&400|}} | + | {{ :models:kinksol2.png?nolink&400|}} |
| Now, using $\cos 2\phi = 1-\sin \phi$ and integrating yields | Now, using $\cos 2\phi = 1-\sin \phi$ and integrating yields | ||
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| $$ \phi(x)_k = 4\arctan \left( e^{\pm (x-x_)} \right) \tag{Eq. 1} $$ | $$ \phi(x)_k = 4\arctan \left( e^{\pm (x-x_)} \right) \tag{Eq. 1} $$ | ||
| - | {{ :models:kinksol.png?nolink&400|}} | + | |
| The $\pm$ in the exponent corresponds to solitons with winding number $n=\pm 1$. The solution with $n=1$ is known as the __kink solution__ and the solution with $n=-1$ as the __antikink solution__. | The $\pm$ in the exponent corresponds to solitons with winding number $n=\pm 1$. The solution with $n=1$ is known as the __kink solution__ and the solution with $n=-1$ as the __antikink solution__. | ||
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| + | {{ :models:kinkenergy.png?nolink&400|}} | ||
| When we plot the kink solution we can see that it is a wave-packet of finite energy that is centered around $x_0$. In other words, a kink is like a potential barrier in the system. | When we plot the kink solution we can see that it is a wave-packet of finite energy that is centered around $x_0$. In other words, a kink is like a potential barrier in the system. | ||
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| - | <tabbox Researcher> | + | <tabbox Abstract> |
| The Sine-Gordon model is [[advanced_notions:quantum_field_theory:duality|dual]] to the massive Thirring model. | The Sine-Gordon model is [[advanced_notions:quantum_field_theory:duality|dual]] to the massive Thirring model. | ||
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| * Coleman, S. (1975). [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2088|"Quantum sine-Gordon equation as the massive Thirring model"]]. Physical Review D. 11 (8): 2088. | * Coleman, S. (1975). [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2088|"Quantum sine-Gordon equation as the massive Thirring model"]]. Physical Review D. 11 (8): 2088. | ||
| + | <tabbox Why is it interesting?> | ||
| + | The Sine-Gordon model is a toy model that helps to understand fundamental notions like [[advanced_notions:quantum_field_theory:duality|duality]] in a simplified setup. | ||
| + | |||
| + | |||
| + | In addition, the equations of the theory permit topological non-trivial solutions called solitons. Many basic features of such solitons can be studied in the Sine-Gordon model in a simplified setup. | ||
| + | |||
| + | |||
| + | |||
| + | The name "Sine-Gordon model" is a wordplay, because the equation of motion of the model is similar to the famous [[equations:klein-gordon_equation|Klein-Gordon equation]] and contains a [[basic_tools:trigonometric_functions|sinus function]]. | ||
| - | <tabbox FAQ> | ||
| - | | ||
| <tabbox History> | <tabbox History> | ||
| The Sine-Gordon equation was originally | The Sine-Gordon equation was originally | ||
models/toy_models/sine_gordon.txt · Last modified: 2018/05/05 12:50 by jakobadmin
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