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advanced_tools:group_theory:representation_theory [2017/12/17 12:33]
jakobadmin created
advanced_tools:group_theory:representation_theory [2018/04/09 15:57]
tesmitekle [Concrete]
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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. ​ 
- 
-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 ​Layman+<​tabbox ​Intuitive
  
 <note tip> <note tip>
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 </​note>​ </​note>​
   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete
  
-<note tip> +A Lie group is in abstract terms a manifoldwhich obeys the group axioms. A **representation** is a special type of map $R$  from this manifold to the linear operators of some vector spaceThe map must obey the condition ​ 
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas+$$ R(gh) = R(g) R(h), $$ 
-</​note>​ +where $g$ and $h$ are elements of the group. This means the map must preserve the product structure of the group and the mathematically notion for such a map is **homomorphism**. ​
-  +
-<tabbox Researcher> ​+
  
-<note tip> +**In practice a representation ​is a map that maps each element ​of the abstract group onto a matrix.** (Matrices are linear operators over a vector space.)
-The motto in this section ​is: //the higher the level of abstraction, ​the better//. +
-</​note>​+
  
-   +(There are other representations,​ where the group elements aren't given as matrices, but in physics matrix representations are most of the time sufficient).
-<tabbox Examples> ​+
  
---> Example1#+Take note that it is possible to introduce a more general notion, called a **realisation of the group**. A realisation maps the group elements onto the (not necessarily linear) operators over an arbitrary (no necessary vector-)space. However, in physics we usually only deal with representations,​ because our physical objects live in vector spaces.
  
-  +** Characterization of Representations **
-<--+
  
---> Example2:#+One way to label representations is by using the [[advanced_tools:​group_theory:​casimir_operators|Casimir Operators]]. 
 + 
 +Another possibility is given by the Weyl character formula. This formula allows to compute the "​character"​ of a group. A "​character"​ is a function that yields a number for each group element. Thus, one can compute which representation one is dealing with by computing this "​character"​ function. If two representations that could be constructed very differently are actually the same their character functions are the same. 
 + 
 +---- 
 + 
 +For more details, take a look at 
 +  * [[http://​jakobschwichtenberg.com/​short-introduction-motivation-representation-theory/​|Short Introduction to and Motivation for Representation Theory]]  
 +  * [[http://​jakobschwichtenberg.com/​adjoint-representation/​|What’s so special about the adjoint representation of a Lie group?]]  
 +  * http://​www.damtp.cam.ac.uk/​user/​ch558/​pdf/​Representations.pdf
  
    
-<--+<tabbox Abstract> ​
  
-<tabbox FAQ> ​+ 
 +Mathematically a representation is a homomorphism from the group to the group of automorphisms of something.
   ​   ​
-<​tabbox ​History+<​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.  
 + 
 +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>​ </​tabbox>​
  
  
advanced_tools/group_theory/representation_theory.txt · Last modified: 2020/12/05 18:07 by edi