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Topic: Field theory (mathematics)

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	12: Field theory and polynomials
		Fields of functions of algebraic varieties (essentially the quotient fields of rings F[x1,...,xn]/(P) where P is a multivariable polynomial) are more properly treated in 14: Algebraic Geometry, although these are really just discussions of fields of finite transcendence degree over the ground field.
		Likewise fields of meromorphic functions and local rings of germs of functions are usually treated with their applications to 30: Complex Analysis, 32: Several complex variables, and 58: Analysis on manifolds.
		The study of real closed fields was first advanced by those working in 03: Mathematical Logic; this topic has extended to questions of computability and even to the development of algorithms; see 68: Computer Science and 13P: Computational problems in commutative algebra.
	www.math.niu.edu /~rusin/known-math/index/12-XX.html (1782 words)

	Mathematics - MSN Encarta
		Mathematics finally became a field in its own right with the development of calculus by English mathematician Isaac Newton and German philosopher and mathematician Gottfried Wilhelm Leibniz during the 17th century and the creation of rigorous mathematical analysis during the 18th century by French mathematician Augustin Louis Cauchy and his contemporaries.
		Social scientists also use mathematical techniques, primarily probability and statistics, to help resolve uncertainty about questions such as how various factors affect human behavior, how these variable factors are related, and how groups differ in their responses.
		Pure mathematics is the study of abstract relationships, whereas applied mathematics applies mathematical analysis to real-world problems, such as the rate of global warming.
	encarta.msn.com /encyclopedia_761578291/Mathematics.html (1340 words)

	Field theory (mathematics) - Wikipedia, the free encyclopedia
		Field theory is a branch of mathematics which studies the properties of fields.
		Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.
		For instance, the field of algebraic numbers is the algebraic closure of the field of rational numbers and the field of complex numbers is the algebraic closure of the field of real numbers.
	en.wikipedia.org /wiki/Field_theory_(mathematics) (635 words)

	Quantum Field Theory (Stanford Encyclopedia of Philosophy)
		The field φ and its conjugate field π are the direct analogues of the canonical coordinate q and the generalized (canonical or conjugate) momentum p in classical mechanics of point particles.
		First, quantum fields which one expects to be somehow physically concrete like classical fields are on the side of observables although, as far as the development of theories is concerned, they are the successors of states (in their position representation, namely wave functions), e.g., in the Klein-Gordon equation of relativistic QM as described above.
		The transition from a classical field theory (like electromagnetism) to quantum field theory can be characterized by the transition from the field φ(x,t) to the quantum field φ̂(x,t), and a corresponding transition for its conjugate field for both of which a certain specification of canonical commutation relations holds.
	plato.stanford.edu /entries/quantum-field-theory (16460 words)

	A timeline of mathematics and theoretical physics
		Einstein, with Hilbert in stiff competition, publishes his stunning General Theory of Relativity, and is lucky enough to be able to find observational support for his theory right away, in the perihelial advance of Mercury, and the deflection of starlight by the Sun.
		Gerard 't Hooft publishes his proof that the electroweak gauge theory of Weinberg is renormalizable and a new chapter in theoretical physics begins -- the age of quantum gauge field theory.
		Stephen Hawking combines quantum field theory with classical general relativity and predicts that fl holes radiate through particle emission, behave as thermodynamic objects, and decay with a finite lifetime into objects that we don't yet understand.
	superstringtheory.com /history/history3.html (2102 words)

	Unified Physics-Mathematics (Site not responding. Last check: 2007-11-04)
		The terminology of quantum field theory (QFT) is misleading; the 'field' referred to is not a field, not the measureable kind of E- or H-field at any rate, but it is defined as a field while in reality related to classical 4-potentials, i.e.
		The fields in SFT are seen as streams of discrete photon interchanges between atomic sub-particles; in QFT the fields are considered continuous and ubiquitous, operating over all solid angles, similar to the classical fields of the macroscopic world discovered by Coulomb and Biot-Savart.
		The fields in SFT cause the motions of the particles which in turn cause the field motions; any two particles and their interacting fields are thus joined 'at the hip' so to speak.
	www.unifiedphysics.com /mathematics.htm (2291 words)

	Why strings?
		Originally, string theory was proposed as an explanation for the observed relationship between mass and spin for certain particles called hadrons, which include the proton and neutron.
		But particles in string theory arise as excitations of the string, and included in the excitations of a string in string theory is a particle with zero mass and two units of spin.
		This led early string theorists to propose that string theory be applied not as a theory of hadronic particles, but as a theory of quantum gravity, the unfulfilled fantasy of theoretical physics in the particle and gravity communities for decades.
	superstringtheory.com /basics/basic3.html (483 words)

	IPAM Conformal Field Theory Program
		Methods of quantum field theory may be applied to problems of topological invariants of knots and instanton moduli spaces as well and have led to a wealth of new results in pure mathematics.
		Their purely algebraic formulation elevates two-dimensional conformal field theories to some of the rare examples where quantum field theories may be defined and constructed in a mathematically rigorous fashion.
		The perturbative expansion of string theory in powers of the coupling constant may be formulated in terms of conformal field theories on families of two-dimensional compact Riemann surfaces, whose genus is the order of the expansion.
	www.ipam.ucla.edu /programs/cft2001 (1082 words)

	Particles and relativity
		In the 18th and 19th centuries, Newton's mathematical description of motion using calculus and his model for the gravitational force were extended very successfully to the emerging science and technology of electromagnetism.
		This discovery and the mathematics that Einstein developed to describe it and model it in his Special Theory of Relativity, when combined with the later development of quantum mechanics, gave birth to the rich subject of relativistic quantum field theory.
		Relativistic quantum field theory is the foundation of our present theoretical ability to describe the behavior of the subatomic particles physicists have been observing and studying in the latter half of the 20th century.
	superstringtheory.com /basics/basic2.html (404 words)

	Quantum field theory, supersymmetry, and enumerative geometry.(Brief (Site not responding. Last check: 2007-11-04)
		Five lectures from the 2001 meeting of the annual graduate summer school by the IAS/Park City Mathematics Institute, set out the mathematical notions now being used in the three areas of theoretical physics.
		The topics include general relativity, introduction to quantum field theory, mirror symmetry, and introduction to string theory.
		Knot physics: newly discovered links between quantum physics and knot theory may tie together a wealth of mathematical ideas.
	www.encyclopedia.com /doc/1G1-160112187.html (328 words)

	Field -- from Wolfram MathWorld
		A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative
		The French term for a field is corps and the German word is Körper, both meaning "body." A field with a finite number of members is known as a finite field or Galois field.
		Dummit, D. and Foote, R. "Field Theory." Ch.
	mathworld.wolfram.com /Field.html (236 words)

	Free online textbooks, videos, tutorials, lecture notes,
		Mathematical methods are discussed, as needed; they include: elements of differential geometry, linear operators and matrices, ordinary differential equations, calculus of variations, orthogonal functions and Fourier series, and non-linear equations for chaotic systems.
		The book is written mainly from a classical field theoretical point of view, emphasising fundamental and subtle properties of the EM field and includes a comprehensive appendix on the mathematical methods used.
		It treats relativistic covariance and the Lagrangian/Hamiltionan formulation of electromagnetic field theory, with an eye on modern ideas of duality and unification of theories, and includes a rigorous, comprehensive and detailed treatment of EM radiation phenomena.
	homepages.nyu.edu /~jmg336/html/mathematics.html (5045 words)

	Field (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
		The mathematical discipline concerned with the study of fields is called field theory.
		For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
	en.wikipedia.org /wiki/Field_(mathematics) (1347 words)

	Amazon.ca: Field Theory and its Classical Problems: Books: Charles Robert Hadlock (Site not responding. Last check: 2007-11-04)
		Field Theory and its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the construction of regular n-gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals and beyond.
		This book gives a very concrete intro to Galois theory & field theory & would be an excellent supplement to an advanced course on algebra.
		It has some group theory in it but its emphasis is on field theory because its focus is ONLY on Galois theory & its applications, so you'll have to find another book on general abstract algebra.
	www.amazon.ca /Field-Theory-its-Classical-Problems/dp/088385032X (545 words)

	Quantum field theory - Wikipedia, the free encyclopedia
		Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.
		Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet).
		The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics.
	en.wikipedia.org /wiki/Quantum_field_theory (4432 words)

	Field theory Summary
		It was perhaps only the youthful optimism of a new science that allowed Kurt Lewin and his colleagues to believe that they had within their grasp the key elements of a "field theory of the social sciences." Social psychology made great stri...
		Field theory (mathematics), the theory of the algebraic concept of field.
		Field theory (physics), a physical theory which employs fields in the physical sense.
	www.bookrags.com /Field_theory (146 words)

	Unified Physics-Home
		This is analogous to the mathematical methods employed within the Standard Model (SM) of particle physics, but because of SFT's mathematical simplicity compared with the SM its reach is more transparent.
		EMSFT uses the E- and H- fields compared to quantum field theory (QFT) based on the wave equations with their vector and scalar potentials, derived from Maxwell's equations by a vector manipulation called gauge theory, a subtle and complex form of mathematics.
		to find a unified field theory marked a milestone; he was the first to seek a way to arrange the four known forces under the same umbrella.
	www.unifiedphysics.com (2117 words)

	William Stoertz: Unified Field Theory: "Mathematics in the Hierarchy" (Site not responding. Last check: 2007-11-04)
		Then the unified field defines the sung-sangs, or, that is, the sung-sangs are defined in reference to the unified field.
		However, that is also a give-and-take relationship -- namely, the advent of the unified field is conditioned or made possible on the foundation of the unification of their sung sangs.
		So the unified field is potential, and become actual, when the separated beings become proximate, or come into the realm of give-and-take.
	www.stoertz.org /UFT/UFT&Math.html (1716 words)

	Drexel University: College of Arts and Sciences: News & Events
		It is physics, not mathematics, but the impact on mathematics of ideas arising in quantum field theory has been profound.
		Description: This is a core mathematical course which serves as an introduction to measure theory and integration, and as a base for further studies in function theory, functional analysis, and harmonic analysis.
		As a mathematical tool, the calculus of moving surfaces reaches far beyond material surfaces in motion and provides crucial applications in variational analysis and shape optimization, perturbation theory, and unknown boundary problems.
	www.drexel.edu /coas/math/news.asp (2372 words)

	Fields and Particles Bookmarks
		Algebraic and Dirac-Hestenes Spinors and Spinor Fields Pub
		Quantum Field Theories: An introduction LN An Introduction to Quantum Field Theory Pub
		Minkowski, Mathematicians, and the Mathematical Theory of Relativity Pub
	www.geocities.com /diahmed/bookmark2.html (2888 words)

	Mathematics and String Theory
		The jury is still out on whether string theory will be relevant for Physics; although in the words of Edward Witten, if string theory would be wrong, "it would seem like a kind of cosmic conspiracy."
		What seems beyond the shadow of a doubt is the impact that string theory has had and continues to have in certain areas of Mathematics; although to be fair, it is difficult to disentangle the impact of string theory proper from that of supersymmetric quantum field theory, since the two theories are so intimately linked.
		Geometry and Quantum Field Theory for mathematics undergraduates at MIT.
	www.maths.ed.ac.uk /~jmf/Research/strings_3.html (163 words)

	ScienceDaily: Help Page (Site not responding. Last check: 2007-11-04)
		Math Model Could Aid Study Of Collagen Ailments (November 15, 2006) -- An MIT researcher's mathematical model explains for the first time the distinctive structure of collagen, a material key to healthy human bone, muscles and other tissues.
		Speak, Memory: Research Challenges Theory Of Memory Storage (November 14, 2006) -- During sleep, freshly minted memories move from the hippocampus, part of the "old" brain, to the neocortex, or "new" brain, for long-term storage.
		Two Markers Strongly Linked To Prostate Cancer Incidence And Mortality Almost A Decade Prior To Diagnosis (November 14, 2006) -- Increased levels of two markers of inflammation, interleukin-6 (IL-6) and C-reactive protein (CRP), are significantly associated with prostate cancer incidence and mortality almost a decade prior to...
	www.sciencedaily.com /articles/computers_math/quantum_computers (1165 words)

	The David and Lucile Packard Foundation
		epresentation theory, quantum field theory, integrable systems, and interrelations between them.
		He states that there is a wide gap between theory and practice in the disciplines of number theory and arithmetic geometry.
		A common thread in his work is to take ideas that previously have been useful in theory, and to transform them into methods that can be used to solve down-to-earth problems.
	www.cs.virginia.edu /packard/directory/field/mathematics.html (276 words)

	MPIM - 25th Anniversary abstracts
		So problems in number theory have analogues in complex geometry.
		A lot of recent activity recently uses this analogy to go from ideas in the Langlands program to objects in complex geometry, where topological methods apply.
		Non-archimedean period domains are their analogues over p-adic fields and parametrize p-adic Hodge structures.
	www.mpim-bonn.mpg.de /Activities/25th+Anniversary+abstracts (323 words)

	MIT OpenCourseWare \| Mathematics \| 18.238 Geometry and Quantum Field Theory, Fall 2002 \| Home
		Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.
		It covers the basics of classical field theory, free quantum theories and Feynman diagrams.
		The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.
	ocw.mit.edu /OcwWeb/Mathematics/18-238Fall2002/CourseHome/index.htm (161 words)

	MATHnetBASE: Mathematics Online (Site not responding. Last check: 2007-11-04)
		In the period between the birth of quantum mechanics and the late 1950s, V.A. Fock wrote papers that are now deemed classics.
		V.A. Fock was a key contributor to one of the most exciting periods of development in 20th-century physics, and this book conveys the essence of that time.
		The seminal works presented in this book are a helpful reference for any student or researcher in theoretical and mathematical physics, especially those specializing in quantum mechanics and quantum field theory.
	www.mathnetbase.com /ejournals/books/book_summary/summary.asp?id=4929 (221 words)

	UM Mathematics: Number Theory
		Analytic number theory, distribution of prime numbers, Fourier analysis, analytic inequalities, probability.
		We also offer a range of courses on advanced topics in number theory.Topics of recent courses include: elliptic curves, Diophantine problems, Hida theory, transcendence theory, spectral theory of modular forms, rigid-analytic geometry, Galois representations and modular forms, automorphic forms on algebraic groups, and alterations.
		Number Theory: Weekly research seminar with outside speakers on topics in all areas of number theory.
	www.math.lsa.umich.edu /research/number_theory (274 words)

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