group_theory

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 group_theory [2017/04/10 07:06]jakobadmin [Why do we care about Symmetries?] group_theory [2017/12/06 09:33] Line 1: Line 1: - ====== Group Theory ====== - ​**Basic idea:** - "//​Numbers measure size, groups measure symmetry.//"​ from - Groups and Symmetry by Mark A. Armstrong​ - - ===== Why do we care about Symmetries? ===== - - <​blockquote>​ - One way to understand the intention of designer of an universe is - to find the symmetries. The search for symmetry can be substantially - different mental activity from the usual explicit calculations. For - example, consider the following simple arithmetic: - $$34126 \times ​ 12378 - 12378 \times 34126 = ?$$ - You can compute the first term using calculator, and store in the - memory. The second term is calculated, and subtracted from the - previous result stored in memory. The final numeric result tells us - that one should use brain instead of fingers in scientific problems. - Of course, the commutative property of multiplication is enough to - write down the answer. The morale of this example is: find the - greatest possible symmetry whenever possible - - <​cite>​From Magnetic Monopoles in Grand Unified Theories by I. G. Koh​ - ​ - - <​blockquote>​ - Physicists are mostly agreed that the ultimate laws of Nature enjoy a high degree of - symmetry. By this I mean that the formulation of these laws, be it in mathematical - terms or perhaps in other accurate descriptions,​ is unchanged when various transformations are performed. Presence of symmetry implies absence of complicated and - irrelevant structure, and our conviction that this is fundamentally true reflects an - ancient aesthetic prejudice - physicists are happy in the belief that Nature in its - fundamental workings is essentially simple. Moreover, there are practical consequences of the simplicity entailed by symmetry: it is easier to understand the predictions of physical laws. For example, working out the details of very-many-body motion is beyond the reach of actual calculations,​ even with the help of computers. - But taking into account the symmetries that are present allows one to understand - at least some aspects of the motion, and to chart regularities within it. - - <​cite>​[[https://​arxiv.org/​abs/​hep-th/​9602122|The Unreasonable Effectiveness of Quantum Field Theory]] by R. Jackiw​ - ​ - - - ===== Definition ===== - - A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms - - - * Closure: For all $g_1, g_2 \in G$,  $g_1 \circ g_2 \in G$ - * Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$ - * Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that  $g \circ g^{-1}=e = g^{-1} \circ g$. - * Associativity:​ For all $g_1, g_2, g_3 \in G$,  $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$. - - - ===== Quotes ===== - - <​blockquote>​ - We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups. - - <​cite>​Sir Arthur Stanley Eddington​ - ​ - - <​blockquote>​ - Symmetry is the magic word that distinguishes theory from coincidence. - - <​cite>​Seven Science Quests by Sudarshan​ - ​ - - <​blockquote>​ - Almost anybody whose research requires sustained use of group theory (and it is hard to think of a physical or mathematical problem that is wholly devoid of symmetry) writes a book about it. - - <​cite>​from http://​www.birdtracks.eu/​chapters/​draft.pdf​ - ​ - - - ==== Matrix Groups and Matrix Algebras ==== - - <​blockquote>​ - To help your intuition further: Lie groups are "​almost always"​ matrix groups as follows. There is a corollary to a difficult theorem known as Ado's theorem that **every Lie algebra can be realized as a Lie algebra of square matrices. The same is not true of Lie groups: not every Lie group can be represented as a group of matrices but it is almost true** (a consequence of the Peter-Weyl theorem is that every compact group can be realized as a group of square matrices). Certainly, since we can find a square matrix realization for every Lie algebra, we can build a matrix Lie group with that algebra as its Lie algebra through the matrix exponential function; then we find that matrix group'​s universal cover and this is where we sometimes fail to get a matrix group. This is not typical and the first Lie groups that were not also matrix groups (so called metaplectic groups) weren'​t found until 1937. These oddballs are all covering groups of noncompact groups. - - <​cite>​http://​physics.stackexchange.com/​a/​76191/​37286​ - ​ - - By exponentiating the Lie algebra elements, which can always can be written as matrices, we get representations of the corresponding universal covering group that belongs to this Lie algebra.  ​ - - - ===== Possible Points of Confusion? ===== - ** - Q:** Why is the Jacobi identity important? ​ - - **A:** "[T]he real meaning of the Jacobi identity: it’s there so the adjoint representation of a Lie group, clearly a very basic and fundamental thing, induces a homomorphism in the corresponding Lie algebras that respects Lie brackets."​ http://​www.wetsavannaanimals.net/​wordpress/​the-adjoint-representation-of-the-lorentz-group/​ - - **Q:** What does conjugation mean? - - **A:** http://​math.stackexchange.com/​questions/​11971/​intuition-behind-conjugation-in-group-theory - ===== Notions ===== - - * [[group_theory:​notions:​lie_algebra]] - * [[group_theory:​notions:​lie_groups]] - * [[group_theory:​notions:​forms_of_a_lie_algebra]] - * [[group_theory:​notions:​representation_theory]] - * [[group_theory:​notions:​compact_group]] - * [[group_theory:​notions:​unitary_representations]] - * [[group_theory:​notions:​casimir_operators]] - * [[group_theory:​notions:​projective_representation]] - * [[group_theory:​notions:​quotient_group]] - * [[group_theory:​notions:​galois_theory]] - * [[group_theory:​notions:​subgroups]] - * [[differential_geometry:​notions:​fiber_bundles]] - * [[group_theory:​notions:​spinor]] - * [[group_theory:​notions:​bianchi_identity]] - * [[group_theory:​notions:​centraL_extension]] - - - ===== Important Groups ===== - - * [[group_theory:​groups:​u1]] - * [[group_theory:​groups:​so3]] - * [[group_theory:​groups:​su2]] - * [[group_theory:​groups:​infinite_dimensional_groups]] - - ===== Group Invariants ===== - - See [[group_theory:​groups_invariants]] - - ===== Great Books ===== - - * [[http://​amzn.to/​2i3CHWs|Visual Group Theory]] by Nathan Carter. From the preface: <​q>"​If you are interested in learning about group theory in a relaxed, intuitive way, then this book is for you.​ - * [[http://​amzn.to/​2inmkll|Naive Lie Theory]] by John Stillwell