advanced_tools:group_theory
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advanced_tools:group_theory [2018/03/28 16:25] jakobadmin [Abstract] |
advanced_tools:group_theory [2020/09/07 05:18] (current) 14.161.7.200 [History] |
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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. | ||
| - | Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$. | + | 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|}} | {{ :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 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. | + | 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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| <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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| <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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advanced_tools/group_theory.1522247102.txt.gz · Last modified: 2018/03/28 14:25 (external edit)
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