advanced_tools:group_theory
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advanced_tools:group_theory [2018/03/28 16:07] jakobadmin [Concrete] |
advanced_tools:group_theory [2020/09/07 05:18] (current) 14.161.7.200 [History] |
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| <tabbox Intuitive> | <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. | ||
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| + | 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 $0^{\circ}$. | ||
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| + | {{ :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. | ||
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| + | 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) | * The **Basic idea** is "//Numbers measure size, groups measure symmetry.//" (Groups and Symmetry by Mark A. Armstrong) | ||
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| -->$(G,\circ) = (\mathbb{Z},+)$# | -->$(G,\circ) = (\mathbb{Z},+)$# | ||
| - | On 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 $\circ = +$. | + | 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 $\mathbb{Z}$ | The first check we have to perform is closure. If we take two elements of $\mathbb{Z}$ | ||
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| is yes since the sum of two integers is again an integer. Therefore, the closure criterium is fulfilled. | is yes since the sum of two integers is again an integer. Therefore, the closure criterium is fulfilled. | ||
| - | Next, we check associativity. If $a,b,c \in \mathbb{Z}$, it is trivially | + | Next, we check if the associativity criterium is fulfilled. Since for $a,b,c \in \mathbb{Z}$, it is |
| - | true that $a+(b+c) = (a+b)+c$. So, associativity is met. | + | true that $a+(b+c) = (a+b)+c$ we can conclude that the criterium is fulfilled. |
| - | Now we check identity. Is there an element $e\in \mathbb{Z}$ such that | + | 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? |
| - | when you add $e$ to any other integer, you get that same integer? | + | The answer is yes since the integer $0$ has this property. |
| - | Clearly the integer 0 satisfies this. So, identity is met. | + | |
| - | Finally, is there an inverse? For any integer $a\in \mathbb{Z}$, will | + | 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 $a+b = e = 0$? |
| - | there be another integer $b\in \mathbb{Z}$ such that $a+b = e = 0$? | + | Again, the answer is yes $a^{-1} = -a$. |
| - | Again, this is obvious, $a^{-1} = -a$ in this case. So, inverse is | + | |
| - | met. | + | |
| - | So, $(G,\star) = (\mathbb{Z},+)$ is a group. | + | Therefore, all criteria are fulfilled an we can conclude that $(G,\star) = (\mathbb{Z},+)$ is a group. |
| <-- | <-- | ||
| --> $(G,\star) = (\mathbb{R},+)$ # | --> $(G,\star) = (\mathbb{R},+)$ # | ||
| - | Obviously, any two real numbers added together is also a real number. | + | Similarly to the integer example above, we can conclude here that two real numbers added together yield another real number. |
| - | Associativity will hold (of course). | + | Associativity also holds. |
| - | The identity is again 0. | + | The identity element is again 0. |
| - | And finally, once again, $-a$ will be the inverse of any $a \in \mathbb{R}$. | + | And also analogously, the inverse element is again $-a$ and also lives in $ \mathbb{R}$. |
| <-- | <-- | ||
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| --> $(G,\star) = (\mathbb{R},\cdot)$# | --> $(G,\star) = (\mathbb{R},\cdot)$# | ||
| - | Closure is met; two real numbers multiplied together yield a real number. | + | The closure criterium is fulfilled: two real numbers __multiplied__ yield another real number. |
| - | Associativity obviously holds. | + | Associativity also holds. |
| - | Identity also holds. Any real number $a\in \mathbb{R}$, when multipled | + | An identity also exists: We can multiply any real number $a\in \mathbb{R}$ by $1$ and get again $a$. |
| - | by 1 is $a$. | + | |
| - | Inverse, on the other hand, is trickier. For any real number, is there | + | 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 |
| - | another real number you can multiply by it to get 1? The instinctive | + | $a^{-1} = {1\over a}$. However this doesn't work because |
| - | choice is $a^{-1} = {1\over a}$. But, this doesn't quite work because | + | what about $a=0$? |
| - | of $a=0$. This is the \it only \rm exception, but because there's an | + | |
| - | exception, $(\mathbb{R},\cdot)$ is not a group. | + | |
| - | \bf Note: \rm If we take the set $\mathbb{R} - \{0\}$ instead of | + | Although this is the only exception we have to conclude that $(\mathbb{R},\cdot)$ is not a group. |
| - | $\mathbb{R}$, then $(\mathbb{R}-\{0\},\cdot)$ is 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. | ||
| <-- | <-- | ||
| --> $(G,\star) = (\{1\},\cdot)$# | --> $(G,\star) = (\{1\},\cdot)$# | ||
| - | This is the set with only the element $1$, and the operation is normal | + | |
| - | multiplication. This is indeed a group, but it is extremely | + | Here the set is simply one element: $1$, and the group element connection rule is the usual |
| - | uninteresting and is called the \bf Trivial Group. | + | multiplication. |
| + | |||
| + | This construction yields indeed a group and it is called the Trivial Group. | ||
| + | <-- | ||
| + | |||
| + | --> $SU(2)$# | ||
| + | |||
| + | See [[advanced_tools:group_theory:su2]] | ||
| <-- | <-- | ||
| ---- | ---- | ||
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| <tabbox Abstract> | <tabbox Abstract> | ||
| - | <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]]. | ||
| - | Over a period stretching from at least as long ago as the early nineteenth | ||
| - | century, the group concept has emerged as the standard way to measure the degree | ||
| - | of invariance of an object under some collection of transformations.4 The informal | ||
| - | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | ||
| - | unlikely to be challenged, relate to the composition of reversible processes | ||
| - | revealing the symmetry of a mathematical entity. Two early manifestations of | ||
| - | groups were as the permutations of the roots of a polynomial, later re-interpreted | ||
| - | as the automorphisms of the algebraic number field containing its roots, in Galois | ||
| - | theory, and as the structure-preserving automorphisms of a geometric space in the | ||
| - | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | ||
| - | scene. In some situations, it is argued, groupoids are better suited to extracting the | ||
| - | vital symmetries. And yet there has been a perception among their supporters— | ||
| - | 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> | ||
| - | |||
| - | |||
| * 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 | * 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: | * Other nice advanced textbooks are: | ||
| Line 152: | Line 152: | ||
| * J. Frank Adams, Lectures on Lie Groups | * 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> | + | |
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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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| * 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> | ||
| + | <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]]. | ||
| + | Over a period stretching from at least as long ago as the early nineteenth | ||
| + | century, the group concept has emerged as the standard way to measure the degree | ||
| + | of invariance of an object under some collection of transformations.4 The informal | ||
| + | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | ||
| + | unlikely to be challenged, relate to the composition of reversible processes | ||
| + | revealing the symmetry of a mathematical entity. Two early manifestations of | ||
| + | groups were as the permutations of the roots of a polynomial, later re-interpreted | ||
| + | as the automorphisms of the algebraic number field containing its roots, in Galois | ||
| + | theory, and as the structure-preserving automorphisms of a geometric space in the | ||
| + | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | ||
| + | scene. In some situations, it is argued, groupoids are better suited to extracting the | ||
| + | vital symmetries. And yet there has been a perception among their supporters— | ||
| + | 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> | ||
| <tabbox FAQ> | <tabbox FAQ> | ||
| Line 425: | Line 449: | ||
| <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 $x^5 = 1$ 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> |
| ---- | ---- | ||
advanced_tools/group_theory.1522246071.txt.gz · Last modified: 2018/03/28 14:07 (external edit)
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