advanced_tools:group_theory
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advanced_tools:group_theory [2018/03/28 16:23] jakobadmin [Intuitive] |
advanced_tools:group_theory [2018/03/28 16:25] jakobadmin [Abstract] |
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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 170: | 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> | + | |
| Line 369: | Line 348: | ||
| * [[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> | ||
advanced_tools/group_theory.txt · Last modified: 2020/09/07 05:18 by 14.161.7.200
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