advanced_tools:group_theory:representation_theory
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advanced_tools:group_theory:representation_theory [2017/12/17 12:35] jakobadmin [Student] |
advanced_tools:group_theory:representation_theory [2018/04/09 15:58] tesmitekle [Intuitive] |
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| ====== Representation Theory ====== | ====== Representation Theory ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | In physics, we are usually interested in what a [[advanced_tools:group_theory|group]] actually does. A group is an abstract object, but representation theory allows us to derive how a group actually acts on a system. | + | <tabbox Intuitive> |
| - | + | <blockquote>geometry asks, “Given a geometric object X, what is its group of | |
| - | In addition, representation theory is what allows us to understand [[advanced_notions:elementary_particles|elementary particles]]. For example, by using the tools of representation theory to analyze the Lorentz group (=the fundamental spacetime symmetry group), we learn what kind of elementary particles can exist in nature. Moreover, representation theory is crucial to understand what [[basic_notions:spin|spin]] is, which is one of the most important [[basic_notions:quantum_numbers|quantum numbers]]. | + | symmetries?” Representation theory reverses the question to “Given a group G, what objects X |
| - | + | does it act on?” and attempts to answer this question by classifying such X up to isomorphism." [[https://math.berkeley.edu/~teleman/math/RepThry.pdf|Source]]</blockquote> | |
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| - | <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> | + | |
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| - | <tabbox Student> | + | <tabbox Concrete> |
| - | + | ||
| - | <WRAP tip>**Basic idea:** | + | |
| - | "//geometry asks, “Given a geometric object X, what is its group of | + | |
| - | symmetries?” Representation theory reverses the question to “Given a group G, what objects X | + | |
| - | does it act on?” and attempts to answer this question by classifying such X up to isomorphism.//" [[https://math.berkeley.edu/~teleman/math/RepThry.pdf|Source]] </WRAP> | + | |
| A Lie group is in abstract terms a manifold, which obeys the group axioms. A **representation** is a special type of map $R$ from this manifold to the linear operators of some vector space. The map must obey the condition | A Lie group is in abstract terms a manifold, which obeys the group axioms. A **representation** is a special type of map $R$ from this manifold to the linear operators of some vector space. The map must obey the condition | ||
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| - | <tabbox Researcher> | + | <tabbox Abstract> |
| - | <note tip> | ||
| - | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| - | </note> | ||
| + | Mathematically a representation is a homomorphism from the group to the group of automorphisms of something. | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | ||
| - | + | In physics, we are usually interested in what a [[advanced_tools:group_theory|group]] actually does. A group is an abstract object, but representation theory allows us to derive how a group actually acts on a system. | |
| - | <-- | + | |
| - | --> Example2:# | + | In addition, representation theory is what allows us to understand [[advanced_notions:elementary_particles|elementary particles]]. For example, by using the tools of representation theory to analyze the Lorentz group (=the fundamental spacetime symmetry group), we learn what kind of elementary particles can exist in nature. Moreover, representation theory is crucial to understand what [[basic_notions:spin|spin]] is, which is one of the most important [[basic_notions:quantum_numbers|quantum numbers]]. |
| - | |||
| - | <-- | ||
| - | |||
| - | <tabbox FAQ> | ||
| - | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/group_theory/representation_theory.txt · Last modified: 2020/12/05 18:07 by edi
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