Group Theory [The Physics Travel Guide]

Link to this comparison view

--- advanced_tools:group_theory [2018/03/28 16:23]
jakobadmin [Intuitive]
+++ advanced_tools:group_theory [2018/04/08 17:15]
georgefarr ↷ Links adapted because of a move operation
@@ Line 142: / Line 142: @@
 <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
   * Other nice advanced textbooks are:
@@ Line 170: / 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 312: / Line 291: @@
   * 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:
@@ Line 369: / Line 348: @@
   * [[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>

User Tools

Site Tools

Page Tools