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advanced_tools:group_theory [2018/03/28 16:22] jakobadmin [Intuitive] |
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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. | 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. | ||
- | {{:advanced_tools:einheitsquadrat-gedreht22-150x150.png?nolink|}} | ||
- | Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$. | ||
- | 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. | + | Obvious examples of such transformations are rotations, by $90^{\circ}$, $180^{\circ}$, $270^{\circ}$, and of course $0^{\circ}$. |
- | 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. | + | {{ :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. | ||
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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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<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: | ||
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* 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> | ||
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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 $x^5 = 1$ 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> |
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