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 ====== Group Theory ====== ====== Group Theory ======
  
-<tabbox Why is it interesting?> ​ 
  
-Group theory is the mathematical theory that we use to describe ​[[basic_tools:​symmetry|symmetries]]. ​Almost everything in modern physics has its origin in some symmetry ​consideration and for this reason, group theory ​is used everywhere in modern physics.+<tabbox Intuitive>​  
 +{{ :​advanced_tools:​einheitsquadrat1-150x150.png?​nolink|}} 
 +Group theory is the branch of mathematics ​we use to work with [[basic_tools:​symmetry|symmetries]]. ​ ​A ​symmetry ​of an object ​is a transformation that leaves the object unchanged. The word object is chosen purposefully because it is very vague. There is one branch of mathematics that deals with all kinds of symmetries, any kind of object can have.
  
 +The most familiar type of symmetries that come to one’s mind are symmetries of geometric shapes, so let's start with that.
 +
 +A square is defined mathematically as a set of points. A symmetry of the square is a transformation that maps this set of points into itself. This means concretely that by the transformation,​ no point is mapped to a point outside of the set that defines the square.
 +
 +Obvious examples of such transformations are rotations, by $90^{\circ}$,​ $180^{\circ}$,​ $270^{\circ}$,​ and of course $0^{\circ}$.
 +
 +{{ :​advanced_tools:​einheitsquadrat-gedreht22-150x150.png?​nolink|}}
 +A counter-example is a rotation by, say $5^{\circ}$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^{\circ}$ is no symmetry of the square.
 +
 +A characteristic property of the symmetries of the square is that the combination of two transformations that leave the square invariant is again a symmetry. For example, combining a rotation by $90^{\circ}$ and $180^{\circ}$ is equivalent to a rotation of $270^{\circ}$,​ which is again a symmetry of the square. We will elaborate on this in the next post. In fact, the basic axioms of group theory can be derived from such an easy example.
  
 ---- ----
  
 +  * The **Basic idea** is "//​Numbers measure size, groups measure symmetry.//"​ (Groups and Symmetry by Mark A. Armstrong)
 +  * A nice read on the role of group theory in physics is [[http://​ricardoheras.com/​wp-content/​uploads/​2014/​02/​Symmetry-in-Physics-Wigners-legacy.pdf|Symmetry in physics: Wigner'​s legacy]] by Gross
 +<tabbox Concrete> ​
 + ​**Definition**
  
 +A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms
 +
 +
 +  * Closure: For all $g_1, g_2 \in G$,  $g_1 \circ g_2 \in G$
 +  * Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$
 +  * Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that  $g  \circ g^{-1}=e = g^{-1} \circ g $.
 +  * Associativity:​ For all $g_1, g_2, g_3 \in G$,  $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$.
 +
 +----
 +
 +**Examples**
 +
 +-->​$(G,​\circ) = (\mathbb{Z},​+)$#​
 +
 +One of the simplest examples of a group is to take as the set $G$ the integer numbers $\mathbb{Z}$ . The group operations is then simply addition $\circ = +$.  ​
 +
 +The first check we have to perform is closure. ​ If we take two elements of $\mathbb{Z}$
 +and add them, the result should be again an element of $\mathbb{Z}$? ​ Formulated differently:​
 +if $a,b \in \mathbb{Z}$,​ is $a+b\in \mathbb{Z}$? ​ The answer
 +is yes since the sum of two integers is again an integer. Therefore, the closure criterium is fulfilled.  ​
 +
 +Next, we check if the associativity criterium is fulfilled. ​ Since for $a,b,c \in \mathbb{Z}$,​ it is 
 +true that $a+(b+c) = (a+b)+c$ we can conclude that the criterium is fulfilled.  ​
 +
 +As a third step we check if there is an identity element. Does an element $e\in \mathbb{Z}$ exist which has the property that when you add to any other integer you get that same integer? ​
 +The answer is yes since the integer $0$ has this property.  ​
 +
 +The last quesiton we need to answer is: is there an inverse? ​ Formulated more precisely: Is there for any integer $a\in \mathbb{Z}$ another integer $b\in \mathbb{Z}$ such that $a+b = e = 0$? 
 +Again, the answer is yes $a^{-1} = -a$. 
 +
 +Therefore, all criteria are fulfilled an we can conclude that $(G,\star) = (\mathbb{Z},​+)$ is a group.  ​
 +<--
 +
 +--> $(G,\star) = (\mathbb{R},​+)$ #
 +
 +Similarly to the integer example above, we can conclude here that two real numbers added together yield another real number.  ​
 +
 +Associativity also holds.  ​
 +
 +The identity element is again 0.  ​
 +
 +And also analogously,​ the inverse element is again $-a$ and also lives in $ \mathbb{R}$. ​
 +
 +<--
 +
 +--> $(G,\star) = (\mathbb{R},​\cdot)$#​
 +
 +The closure criterium is fulfilled: two real numbers __multiplied__ yield another real number. ​
 +
 +Associativity also holds.
 +
 +An identity also exists: ​ We can multiply any real number $a\in \mathbb{R}$ by $1$ and get again $a$.
 +
 +However, the existence of an inverse element is here a bit trickier. We need to answer the question: Is there for any real number another real number that you can multiply it with to get 1? A first guess is
 +$a^{-1} = {1\over a}$.  However this doesn'​t work because
 +what about $a=0$?  ​
 +
 +Although this is the only exception we have to conclude that $(\mathbb{R},​\cdot)$ is not a group.  ​
 +
 +However, if we consider the $\mathbb{R} - \{0\}$,, i.e. the real numbers without the zero, instead of
 +$\mathbb{R}$,​ then we get a group.  ​
 +
 +<--
 +
 +--> $(G,\star) = (\{1\},​\cdot)$#​
 +
 +Here the set is simply one element: $1$, and the group element connection rule is the usual
 +multiplication. ​
 +
 +This construction yields indeed a group and it is called the Trivial Group.
 +<--
 +
 +--> $SU(2)$#
 +
 +See [[advanced_tools:​group_theory:​su2]]
 +<--
 +----
 +
 +
 + ​**Recommended Resources**
 +
 +  - http://​jakobschwichtenberg.com/​why-group-theory/​
 +  - http://​jakobschwichtenberg.com/​motivation-for-the-group-theory-axioms/​
 +  - http://​jakobschwichtenberg.com/​naive-introduction-lie-theory/​
 +  - http://​jakobschwichtenberg.com/​short-introduction-motivation-representation-theory/​
 +  - http://​jakobschwichtenberg.com/​lie-algebra-able-describe-group/​
 +  - http://​jakobschwichtenberg.com/​adjoint-representation/​
 +
 +  * See also https://​www.quora.com/​What-role-do-Lie-groups-and-or-Lie-algebras-play-in-physics/​answer/​Ron-Maimon?​srid=TYwn
 +
 +Three books that explain group theory (but do not proof things) with a focus on how it's used in modern physics are
 +
 +  * Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson
 +  * Symmetries in Fundamental Physics by Kurt Sundermeyer
 +  * Physics From Symmetry by Jakob Schwichtenberg
 +  * [[https://​www.liealgebrasintro.com/​|Lie groups and algebras A simplified introduction to Lie groups and algebras]] by Doug McKenzie ​
 +  * {{ :​advanced_tools:​groups-symmetry-and-topology-global-symmetry-and-local-symmetry-lecture-notes-physics.pdf |How to Talk to a Physicist: Groups,​Symmetry,​ and Topology}} by Daniel T. Larson
 +
 +
 +Good books that focus on mathematical aspects (and also include proofs) are:
 +
 +  * Visual Group Theory by Nathan Carter
 +  * Naive Lie Theory by John Stilwell
 +  * An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee
 +
 +
 +
 +  * Another nice introduction is https://​www-zeuthen.desy.de/​~kolanosk/​eep06/​skripte/​lie.pdf
 +  * https://​www.fizyka.umk.pl/​~bgw/​rzeszow1.pdf
 +  * See also {{ :​advanced_tools:​hos.pdf |Building Blocks of Symmetry by Attila Egri-Nagy}}
 +
 +
 +
 +----
 +
 +For Group Theory in Quantum Mechanics see http://​math.ucr.edu/​home/​baez/​lie/​lie.html
 +
 +<tabbox Abstract> ​
 +  * One of the best books to get familiar with many of the most important advanced topics in group theory is "​Geometrical methods of mathematical physics"​ by Bernard F. Schutz
 +  * Other nice advanced textbooks are:
 +    * Daniel Bump, Lie Groups
 +    * Shlomo Sternberg, Group Theory and Physics
 +    * Robert Hermann, Lie Groups for Physicists
 +    * George Mackey, Unitary Group Representations in Physics, Probability,​ and Number Theory
 +    * Brian Hall, Lie Groups, Lie Algebras, and Representations
 +    * William Fulton and Joe Harris, Representation Theory - a First Course
 +    * J. Frank Adams, Lectures on Lie Groups
 +
 +----
 +
 +
 +**Resources:​**
 +
 +  * http://​www.liegroups.org/​
 +  * https://​groupprops.subwiki.org/​wiki/​Main_Page
 +
 +----
 +
 +
 +There are groups that have nothing to do with symmetries like, for example, braid groups: ​
 +
 +<​blockquote>​„Once we have the notion “group,​” we can look for other examples. It turns out there are many examples of groups that have nothing to do with symmetries, which was our motivation to introduce the concept of a group in the first place. This is actually a typical story. The creation of a mathematical concept may be motivated by problems and phenomena in one area of math (or physics, engineering,​ and so forth), but later it may well turn out to be useful and well adapted to other areas.
 +It turns out that many groups do not come from symmetries“
 +
 +<​cite>​Love and Math by Frenkel</​cite></​blockquote>​
 +
 +<tabbox Why is it interesting?> ​
 +
 +Group theory is the mathematical theory that we use to describe [[basic_tools:​symmetry|symmetries]]. Almost everything in modern physics has its origin in some symmetry consideration and for this reason, group theory is used everywhere in modern physics.
 +
 +----
  
 <​blockquote>​ <​blockquote>​
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 <​blockquote>​A man who is tired of group theory is a man who is tired of life. <​cite>​ Sidney Coleman</​cite></​blockquote>​ <​blockquote>​A man who is tired of group theory is a man who is tired of life. <​cite>​ Sidney Coleman</​cite></​blockquote>​
  
 +<​blockquote>​Group theory is, in short, the mathematics of symmetries. You already know that 
 +symmetries can be very important in understanding or simplifying physics problems. 
 +When you study classical mechanics, you learn that symmetries of a system 
 +are intimately related to the existence of conserved charges. Their existence often 
 +makes solving for the dynamics a lot simpler. Even if a symmetry is not present 
 +exactly (for instance, when a system is almost-but-not-quite spherically symmetric),​ 
 +we can often describe the system as a small perturbation of a system that does 
 +exhibit symmetry. A surprisingly large number of physics problems is built around 
 +that idea; in fact, practically all systems for which we can solve the dynamics exactly 
 +exhibit some sort of symmetry that allow us to reduce the often horrible secondorder 
 +equations of motion to much simpler first-order conservation equations.<​cite>​http://​maths.dur.ac.uk/​users/​kasper.peeters/​pdf/​groups.pdf</​cite></​blockquote>​
  
 <tabbox Overview>​ <tabbox Overview>​
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 ||| AA||||BB |AA=[[advanced_notions:​cp_symmetry|CP]]|BB=[[advanced_notions:​t_symmetry|T]] ||| AA||||BB |AA=[[advanced_notions:​cp_symmetry|CP]]|BB=[[advanced_notions:​t_symmetry|T]]
 | |,| -|^|- |. | | | | | |,| -|^|- |. | | | |
-| AA||BB |AA=[[advanced_notions:​c_symmetry|C]]|BB=[[advanced_notions:​p_symmetry|P]]+| AA||BB |AA=[[advanced_notions:​c_symmetry|C]]|BB=[[advanced_notions:​parity|P]]
 </​diagram>​ </​diagram>​
  
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   * and for the mathematical details see [[http://​www.mat.univie.ac.at/​~cap/​files/​wisser.pdf|this article]].   * and for the mathematical details see [[http://​www.mat.univie.ac.at/​~cap/​files/​wisser.pdf|this article]].
  
-On the left-hand side, some of the most important groups that are used in physics are shown. Most of them are important in the context of [[theories:speculative_theories:​grand_unified_theories|grand unified theories]]. The full classification is shown here: +On the left-hand side, some of the most important groups that are used in physics are shown. Most of them are important in the context of [[models:speculative_models:​grand_unified_theories|grand unified theories]]. The full classification is shown here: 
  
  
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   * [[advanced_tools:​group_theory:​group_contraction]]   * [[advanced_tools:​group_theory:​group_contraction]]
  
 +<tabbox Research>​
  
- 
-<tabbox Layman> ​ 
- 
-  * The **Basic idea** is "//​Numbers measure size, groups measure symmetry.//"​ (Groups and Symmetry by Mark A. Armstrong) 
-  * A nice read on the role of group theory in physics is [[http://​ricardoheras.com/​wp-content/​uploads/​2014/​02/​Symmetry-in-Physics-Wigners-legacy.pdf|Symmetry in physics: Wigner'​s legacy]] by Gross 
-<tabbox Student> ​ 
- 
- 
- ​**Recommended Resources** 
- 
-  - http://​jakobschwichtenberg.com/​why-group-theory/​ 
-  - http://​jakobschwichtenberg.com/​motivation-for-the-group-theory-axioms/​ 
-  - http://​jakobschwichtenberg.com/​naive-introduction-lie-theory/​ 
-  - http://​jakobschwichtenberg.com/​short-introduction-motivation-representation-theory/​ 
-  - http://​jakobschwichtenberg.com/​lie-algebra-able-describe-group/​ 
-  - http://​jakobschwichtenberg.com/​adjoint-representation/​ 
- 
-  * See also https://​www.quora.com/​What-role-do-Lie-groups-and-or-Lie-algebras-play-in-physics/​answer/​Ron-Maimon?​srid=TYwn 
- 
-Three books that explain group theory (but do not proof things) with a focus on how it's used in modern physics are 
- 
-  * Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson 
-  * Symmetries in Fundamental Physics by Kurt Sundermeyer 
-  * Physics From Symmetry by Jakob Schwichtenberg 
-  * [[https://​www.liealgebrasintro.com/​|Lie groups and algebras A simplified introduction to Lie groups and algebras]] by Doug McKenzie ​ 
-  * {{ :​advanced_tools:​groups-symmetry-and-topology-global-symmetry-and-local-symmetry-lecture-notes-physics.pdf |How to Talk to a Physicist: Groups,​Symmetry,​ and Topology}} by Daniel T. Larson 
- 
- 
-Good books that focus a on mathematical aspects (and also include proofs) are: 
- 
-  * Visual Group Theory by Nathan Carter 
-  * Naive Lie Theory by John Stilwell 
-  * An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee 
- 
- 
- 
-  * Another nice introduction is https://​www-zeuthen.desy.de/​~kolanosk/​eep06/​skripte/​lie.pdf 
-  * https://​www.fizyka.umk.pl/​~bgw/​rzeszow1.pdf 
-  * See also {{ :​advanced_tools:​hos.pdf |Building Blocks of Symmetry by Attila Egri-Nagy}} 
- 
- ​**Definition** 
- 
-A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms 
- 
- 
-  * Closure: For all $g_1, g_2 \in G$,  $g_1 \circ g_2 \in G$ 
-  * Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$ 
-  * Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that  $g  \circ g^{-1}=e = g^{-1} \circ g $. 
-  * Associativity:​ For all $g_1, g_2, g_3 \in G$,  $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$. 
- 
----- 
- 
-For Group Theory in Quantum Mechanics see http://​math.ucr.edu/​home/​baez/​lie/​lie.html 
- 
-<tabbox Researcher> ​ 
 <​blockquote>​The case I have chosen to treat in this paper concerns the question as to whether <​blockquote>​The case I have chosen to treat in this paper concerns the question as to whether
 the group concept should be extended to, or even subsumed by, the [[advanced_tools:​category_theory:​groupoids|groupoid concept]]. the group concept should be extended to, or even subsumed by, the [[advanced_tools:​category_theory:​groupoids|groupoid concept]].
Line 254: Line 376:
 who include some very illustrious names—of an unwarranted resistance in some who include some very illustrious names—of an unwarranted resistance in some
 quarters to their use, which is only now beginning to decline.<​cite>​[[http://​www.sciencedirect.com/​science/​article/​pii/​S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</​cite></​blockquote>​ quarters to their use, which is only now beginning to decline.<​cite>​[[http://​www.sciencedirect.com/​science/​article/​pii/​S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</​cite></​blockquote>​
- 
- 
-  * One of the best books to get familiar with many of the most important advanced topics in group theory is "​Geometrical methods of mathematical physics"​ by Bernard F. Schutz 
-  * Other nice advanced textbooks are: 
-    * Daniel Bump, Lie Groups 
-    * Shlomo Sternberg, Group Theory and Physics 
-    * Robert Hermann, Lie Groups for Physicists 
-    * George Mackey, Unitary Group Representations in Physics, Probability,​ and Number Theory 
-    * Brian Hall, Lie Groups, Lie Algebras, and Representations 
-    * William Fulton and Joe Harris, Representation Theory - a First Course 
-    * J. Frank Adams, Lectures on Lie Groups 
- 
- <​blockquote>​ 
-Consider two descriptions of the same phenomenon: (1) Space is homogeneousand isotropic. (2) Space is invariant under translations and rotations of coor-dinates. Statements in the logical form of (1) were exclusive in pre-twentiethcentury physics. Statements in the form -of (2) dominate twentieth centuryphysics;​ quantum mechanics contains various representations of the same phy-sical state and rules for transforming among them. Description (1) appearsclean;​ it describes nature without explicit conventional and experientialnotions. Description (2) is all contaminated;​ it invokes conventional coordi-nates and intellectual transformations of the coordinates. The coordinates areusually interpreted as perspectives of observations;​ so they are somehowrelated to human subjects. However, physicists agree that (2) is more objective,​for it uncovers the hidden presuppositions of (1) and neutralizes their undesir-able effects. They retrofit the conceptual structure embodied in (2) into classi-cal mechanics to make it more satisfactory. The statement (1) is often interpreted in a framework of things; (2) can beinterpreted in the framework of objects. The object framework includes thething framework as a substructure and further conveys the epistemological ideathat the things are knowable through observations and yet independent ofobservations. The two frameworks exemplify two different views of the world 
-<​cite>​From "How is Quantum Field Theory possible"​ by Auyang</​cite>​ 
-</​blockquote>​ 
- 
- 
-**Resources:​** 
- 
-  * http://​www.liegroups.org/​ 
-  * https://​groupprops.subwiki.org/​wiki/​Main_Page 
- 
    
 <tabbox FAQ> ​ <tabbox FAQ> ​
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 <​blockquote>​His starting point was the theory of algebraic equations (such as the quadratic, or second-degree,​ equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially. <​blockquote>​His starting point was the theory of algebraic equations (such as the quadratic, or second-degree,​ equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially.
  
-But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to x5 = 1 are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.<​cite>​https://​www.nature.com/​articles/​d41586-018-03423-x</​cite></​blockquote>​+But Galois also showed that solutions to such equations must be linked by symmetry. For example, the solutions to $x^5 = 1are five points on a circle when plotted onto a graph comprised of real numbers along one axis and imaginary numbers on the other. He showed that even when such equations cannot be solved, he could still glean a great deal of information about the solutions from studying such symmetries.<​cite>​https://​www.nature.com/​articles/​d41586-018-03423-x</​cite></​blockquote>​
 ---- ----
  
advanced_tools/group_theory.1522127888.txt.gz · Last modified: 2018/03/27 05:18 (external edit)