This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
basic_notions:boundary_conditions [2017/11/24 12:22] jakobadmin [Student] |
basic_notions:boundary_conditions [2018/03/30 11:13] (current) jakobadmin [Intuitive] |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== Boundary Conditions ====== | ====== Boundary Conditions ====== | ||
- | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
- | + | <blockquote>Generically, if we can formulate the [[:equations|equations of motion]] for a theory, we have everything at our disposal to describe the solutions of the theory. However, in general we have to supplement the equations with the situation we want to actually describe with the theory. In the case of the planet, we have to add where the planet was and where it moved to at a certain instance of time. Otherwise the equation of motion would give us the solutions for all possible initial positions and velocities of the planet, and thus an infinite number of possible solutions to the theory. Such additional information are called boundary conditions. They select out of any possible kind of behavior described by a theory the particular one which is compatible with the state a system is in.<cite>http://axelmaas.blogspot.de/2012/03/equations-that-describe-world.html</cite></blockquote> | |
- | + | ||
- | + | ||
- | <blockquote> | + | |
- | The [[:equations|field equations]] and the boundary | + | |
- | conditions are inextricably connected and the latter can in no way be considered less important | + | |
- | than the former<cite>V. Fock, The theory of space, time and gravitation</cite></blockquote> | + | |
- | + | ||
- | <blockquote>Now, Nature is described by fields, and this elegant and powerful formulation of classical | + | |
- | and quantum mechanics based on the action needs to be supplemented with a careful treatment | + | |
- | of boundary conditions at infinity. The issue of boundary conditions is particularly important | + | |
- | and interesting in the case of [[advanced_tools:gauge_symmetry|gauge theories]] where the assumption ‘all fields decay sufficiently | + | |
- | rapidly at infinity’ is not justified. <cite>https://arxiv.org/pdf/1601.03616.pdf</cite> | + | |
- | </blockquote> | + | |
- | + | ||
- | <blockquote>[I]t is natural to regulate infinite sized systems by imposing boundary | + | |
- | conditions at finite distance, often described as placing the system in a box. This idea | + | |
- | has a long history in the gravitational context (see e.g. [15–27]) where it is common to | + | |
- | impose a Dirichlet boundary condition, fixing the induced metric at the walls of the box1 | + | |
- | .<cite>https://arxiv.org/abs/1508.02515</cite></blockquote> | + | |
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | <tabbox Layman> | + | |
- | + | ||
- | <note tip> | + | |
- | 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> | + | |
| | ||
- | <tabbox Student> | + | <tabbox Concrete> |
**Important Spatial Boundary Conditions:** | **Important Spatial Boundary Conditions:** | ||
Line 122: | Line 93: | ||
* For boundary conditions in gauge field theory, see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields". | * For boundary conditions in gauge field theory, see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields". | ||
- | <tabbox Researcher> | + | <tabbox Abstract> |
<note tip> | <note tip> | ||
Line 129: | Line 100: | ||
| | ||
- | <tabbox Examples> | + | <tabbox Why is it interesting?> |
- | --> Example1# | ||
- | |||
- | <-- | ||
- | --> Example2:# | + | <blockquote> |
+ | The [[:equations|field equations]] and the boundary | ||
+ | conditions are inextricably connected and the latter can in no way be considered less important | ||
+ | than the former<cite>V. Fock, The theory of space, time and gravitation</cite></blockquote> | ||
- | + | <blockquote>Now, Nature is described by fields, and this elegant and powerful formulation of classical | |
- | <-- | + | and quantum mechanics based on the action needs to be supplemented with a careful treatment |
- | + | of boundary conditions at infinity. The issue of boundary conditions is particularly important | |
- | <tabbox History> | + | and interesting in the case of [[advanced_tools:gauge_symmetry|gauge theories]] where the assumption ‘all fields decay sufficiently |
+ | rapidly at infinity’ is not justified. <cite>https://arxiv.org/pdf/1601.03616.pdf</cite> | ||
+ | </blockquote> | ||
+ | |||
+ | <blockquote>[I]t is natural to regulate infinite sized systems by imposing boundary | ||
+ | conditions at finite distance, often described as placing the system in a box. This idea | ||
+ | has a long history in the gravitational context (see e.g. [15–27]) where it is common to | ||
+ | impose a Dirichlet boundary condition, fixing the induced metric at the walls of the box1 | ||
+ | .<cite>https://arxiv.org/abs/1508.02515</cite></blockquote> | ||
</tabbox> | </tabbox> | ||