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--- advanced_tools:group_theory [2018/03/28 16:24]
jakobadmin
+++ advanced_tools:group_theory [2018/04/13 09:42]
bogumilvidovic [Why is it interesting?]
@@ Line 142: / Line 142: @@
 <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:
@@ Line 155: / Line 152: @@
     * 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>
@@ Line 201: / Line 195: @@
 <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>
@@ Line 297: / Line 301: @@
   * 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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