# Abril del 2013

### Publicado el 16 de Abril, 2013, 17:11

 Hoy escribo sobre un tema al que llego repetidamente en estos últimos años, y de diversas formas. Es tiempo de pasar a publicar (a hacer público y accesible por Google) mis notas, para no perder esas referencias. El tema es los invariantes en matemáticas. Es un tema amplio, y en verdad, son varios temas. Una cosa son los invariantes topológicos, y otras los invariantes algebraicos. Y otro son las funciones invariantes (ver mi serie de posts). De hecho, he encontrado poco sobre funciones invariantes, pero pueden ver: A Function Invariant under a Group of TransformationsAlgo más restringido Invariant Functions sobre O(n), SO(n) Pero vayamos a la definición más general: http://en.wikipedia.org/wiki/Invariant_(mathematics) In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology and algebra. Some important classes of transformations are defined by an invariant they leave unchanged, for example conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects. Y siempre termino topándome con invariantes en física: http://en.wikipedia.org/wiki/Invariant_(physics) Invariants are important in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants. Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used inelectromagnetism, special relativity, and general relativity. Agregaría que el tema covariancia y contravariancia también aparece en "gauge theory". Pero volvamos a los invariantes. Mis primeras notas serán sobre los invariantes algebraicos http://mathworld.wolfram.com/AlgebraicInvariant.html A quantity such as a polynomial discriminant which remains unchanged under a given class of algebraic transformations. Such invariants were originally called hyperdeterminants by Cayley. Como bien dice ese corto artículo de Mathworld, uno de los primeros en desarrollar el tema de invariantes algebraicos fue Cayley, en el siglo XIX. Quisiera terminar esta primer nota, mencionando mis principales fuentes: - Mathematical Thought From Ancient to Modern Times, Morris Kline (hay edición en español, de Alianza. Ver principalmente el tercer volumen) - The Development of Mathematics, de Eric Temple Bell ya citado varias veces en este blog (ver Gauss y la congruencia, por E.T.Bell, Gauss, Abel, Galois en la sociedad, según Bell, Dos visiones de matemáticas, Contra los místicos del tiempo) hay edición en español de Fondo de Cultura Económica. Sobre el tema de invariantes, lo principal a leer es el capítulo XX - The Theory of Algebraic Invariants, notas de un curso de David Hilbert - The Classical Groups, their Invariants and Representations, de Hermann Weyl, alumno de Hilbert. Lo de Weyl debe estar expuesto también en el más moderno: Classical Invariant Theory, a primer (pdf) de Hanspeter Kraft y Claudio Procesi (ver la conferencia de Kraft para el cumpleaños de Procesi) Basta por hoy, por lo menos con este post ya tengo anotado, y con poco riesgo de perder, lo principal a leer y desarrollar sobre el tema. Ya vendrán nuevos posts. Nos leemos! Angel "Java" Lopezhttp://www.ajlopez.comhttp://twitter.com/ajlopez

### Publicado el 14 de Abril, 2013, 17:41

 Más enlaces de este tema, algunos modernos (como lo que pasó con un experimento de neutrinos el año pasado, y algunas noticias de Higgs), otros clásicos. Fock spacehttp://en.wikipedia.org/wiki/Fock_space Nobel Prize, Born Lecturehttp://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.pdf The Birth of Quantum Mechanicshttp://www.quantumsciencephilippines.com/75/the-birth-of-quantum-mechanics/ Quantum Theory: Max Bornhttp://www.spaceandmotion.com/quantum-theory-max-born-quotes.htmThe Wave Structure of Matter shows that Max Born's Probability Waves Interpretation of Quantum Theory is due to the incorrect 'particle' conception of Matter. Quantum Physics / Mechanics: Max Bornhttp://www.spaceandmotion.com/physics-quantum-mechanics-max-born.htmThe Wave Structure of Matter (WSM) replaces Max Born's 'Probability Waves' Interpretation of Quantum Wave Mechanics with Real Matter Waves in Physical Space. Max Born Biography, Pictures & Quotes. History and Foundations of Quantum Physicshttp://quantum-history.mpiwg-berlin.mpg.de/main Implications of LHC Resultshttp://www.math.columbia.edu/~woit/wordpress/?p=4873 It"s a Boson! The Higgs as the Latest Offspring of Math & Physicshttp://blogs.discovermagazine.com/crux/2012/07/30/the-mathematical-magic-behind-the-mysterious-higgs-boson/ How the Higgs can lead us to the dark universehttp://www.math.columbia.edu/~woit/wordpress/?p=4897 Spinning out of control!http://www.quantumdiaries.org/2012/07/16/spinning-out-of-control/ Matter wavehttp://en.wikipedia.org/wiki/Matter_wave Cuántica sin fórmulashttp://eltamiz.com/cuantica-sin-formulas/ Did the 'God Particle' Create Matter?http://www.icr.org/article/did-god-particle-create-matter/ Neutrino velocity consistent with speed of lighthttp://www.symmetrymagazine.org/breaking/2012/06/08/neutrino-velocity-consistent-with-speed-of-light/Einstein can breathe a sigh of relief – neutrinos obey the cosmic speed limit after all. July issue of symmetry now onlinehttp://www.symmetrymagazine.org/breaking/2012/07/02/july-issue-of-symmetry-now-online/As excitement builds over what physicists may or may not say about their hunt for the Higgs boson, symmetry goes beyond the "Hamlet question" of whether or not the Higgs exists to ask: What happens next? Stephen Hawking loses Higgs boson particle bet - videohttp://www.guardian.co.uk/science/video/2012/jul/05/stephen-hawking-higgs-boson-bet-video?CMP=twt_gu FTL NEUTRINO RESEARCH 'ALMOST CERTAINLY WRONG'http://news.discovery.com/space/faster-than-light-neutrino-theory-almost-certainly-wrong-111012.html Particles Found to Travel Faster Than Speed of Lighthttp://www.scientificamerican.com/article.cfm?id=particles-found-to-travel&WT.mc_id=SA_WR_20110929 CERN confirms speedy neutrinos follow laws of physics after allhttp://news.cnet.com/8301-11386_3-57449418-76/cern-confirms-speedy-neutrinos-follow-laws-of-physics-after-all/ Into the Subatomic Junglehttp://www.youtube.com/watch?v=up-fbMd_ziU&feature=youtu.be Splitting the unsplittablehttp://www.nanowerk.com/news/newsid=25486.php The Collider, the Particle and a Theory About Fatehttp://www.nytimes.com/2009/10/13/science/space/13lhc.html?_r=2 At a Workshop; and Higgs Papers Are Outhttp://profmattstrassler.com/2012/08/02/at-a-workshop-and-higgs-papers-are-out/ Mis Enlaceshttp://delicious.com/ajlopez/quantum Nos leemos! Angel "Java" Lopezhttp://www.ajlopez.comhttp://twitter.com/ajlopez
Por ajlopez, en: Ciencia

### Publicado el 11 de Abril, 2013, 14:32

 Ya mencioné a Herman Weyl y su libro The Theory of Groups and Quantum Mechanics en Hermann Weyl, fisica y matematicas. Hacia el final de la introducción, encuentro: Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and , and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics. While the quantum theory can be traced back only as far as 1900, the origin of the theory of groups is lost in a past scarcely accessible to history; the earliest works of art show that the symmetry groups of plane figures were even then already known, although the theory of these was only given definite form in the latter part of the eighteenth and in the nineteenth centuries. F. Klein considered the group concept. as most characteristic of nineteenth century mathematics. Es interesante ver cómo algo como la teoría de grupos tuvo una larga historia, aunque hay que reconocer que como teoría matemática aparece realmente en el siglo XIX. Igual, es de destacar que tenemos otro ejemplo de matemáticas desarrolladas ANTES de tener una aplicación física. Until the present, its most important application to natural science lay in the description of the symmetry of crystals, but it has recently been recognized that group theory is of fundamental importance for quantum physics; it here reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved. We may well expect that it is just this part of quantum physics which is most certain of a lasting place. Y ahora menciona dos grupos que van a tener un rol en su libro: Two groups, the group of rotations in 3-dimensional space and the permutation group, play here the principal role, for the laws governing the possible electronic configurations grouped about the stationary nucleus of an atom or an ion are spherically symmetric with respect to the nucleus, and since the various electrons of which the atom or ion is composed are identical, these possible configurations are invariant under a permutation of the individual electrons. Y otro tema que aparece, cuando los grupos se relacionan con la física: sus representaciones por transformaciones lineales: The investigation of groups first becomes a connected and complete theory in the theory of the representaration of groups by linear transformations, and it is exactly this mathematically most important part which is necessary for an adequate description of the quantum mechanical relations. All quantum numbers, with the exception of the so-called prirtcipal quantum number, are indices characterizing representations of groups. Nunca traté todavía el tema de representaciones de grupos por transformaciones lineales. Pero es interesante encontrar esta temprana referencia en Weyl, que luego llegaría a aplicarse más allá de la teoría cuántica que menciona. Ver Group RepresentationCharacter theoryGroup RepresentationsGroup Representation Theory ahí leo lo principal: Informally, a representation of a group is a way of writing it down as a group of matrices Nos leemos! Angel "Java" Lopezhttp://www.ajlopez.comhttp://twitter.com/ajlopez
Por ajlopez, en: Ciencia