Posted on 12/07/2004 10:01:55 AM PST by snarks_when_bored
Here's a link to the single-page format (includes a few photos):
String Theory, at 20, Explains It All (or Not)
Use BugMeNot.com to register.
I prefer the String cheese theory.
Anyway I'll have to read this later
Mmmmmmm...String cheese.
whoops...danged proxy made me post it twice....
They're looking pretty hard for ways to test Superstring Theory. And there are many critics of the field, a loyal opposition who think that physics ought to be testable. But what's testable now and what might be testable a few years from now are very different things. We're still on the way to something, but what it is we don't quite know (kind of describes life, too, huh?).
Hey...this article didn't help Muttly at all.
What does it have to do with his Yo-Yo ?!!
Well...maybe something...may have too many dimensions in there or something.
When do we eat.
That's vaguely disturbing.
But,.......accepted the 'relativism' of light to creation concept in the bible.
.......NEVER giving God and His Bible Truth credit!
(....stringing Darwin along....)
/sarcasm?
(Romans 10:17)
......the demons believe (too) and tremble (too).....
| The Official String Theory Web Site:--> Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Some basic books for further reading. Some advanced books for further reading. |
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And we have an official website for learning the math required to understand this theory.... Link: http://superstringtheory.com/math/
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| Algebra | |
| Algebra provides the first exposure to the use of variables and constants, and experience manipulating and solving linear equations of the form y = ax + b and quadratic equations of the form y = ax2+bx+c. | |
| Geometry | |
| Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning to reason geometrically, to use concepts like symmetry, similarity and congruence, to understand the properties of geometric shapes in a flat, two-dimensional space. | |
| Trigonometry | |
| Trigonometry begins with the study of right triangles and the Pythagorean theorem. The trigonometric functions sin, cos, tan and their inverses are introduced and clever identities between them are explored. | |
| Calculus (single variable) | |
| Calculus begins with the definition of an abstract functions of a single variable, and introduces the ordinary derivative of that function as the tangent to that curve at a given point along the curve. Integration is derived from looking at the area under a curve,which is then shown to be the inverse of differentiation. | |
| Calculus (multivariable) | |
| Multivariable calculus introduces functions of several variables f(x,y,z...), and students learn to take partial and total derivatives. The ideas of directional derivative, integration along a path and integration over a surface are developed in two and three dimensional Euclidean space. | |
| Analytic Geometry | |
| Analytic geometry is the marriage of algebra with geometry. Geometric objects such as conic sections, planes and spheres are studied by the means of algebraic equations. Vectors in Cartesian, polar and spherical coordinates are introduced. | |
| Linear Algebra | |
| In linear algebra, students learn to solve systems of linear equations of the form ai1 x1 + ai2 x2 + ... + ain xn = ci and express them in terms of matrices and vectors. The properties of abstract matrices, such as inverse, determinant, characteristic equation, and of certain types of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored. | |
| Ordinary Differential Equations | |
| This is where the physics begins! Much of physics is about deriving and solving differential equations. The most important differential equation to learn, and the one most studied in undergraduate physics, is the harmonic oscillator equation, ax'' + bx' + cx = f(t), where x' means the time derivative of x(t). | |
| Partial Differential Equations | |
| For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence partial differential equations. The first partial differential equations students learn are the linear, separable ones that were derived and solved in the 18th and 19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel. | |
| Methods of approximation | |
| Most of the problems in physics can't be solved exactly in closed form. Therefore we have to learn technology for making clever approximations, such as power series expansions, saddle point integration, and small (or large) perturbations. | |
| Probability and statistics | |
| Probability became of major importance in physics when quantum mechanics entered the scene. A course on probability begins by studying coin flips, and the counting of distinguishable vs. indistinguishable objects. The concepts of mean and variance are developed and applied in the cases of Poisson and Gaussian statistics. | |
We're still laughing; here's why:
String Theory attempts to imagine a universe in which *both* Quantum Mechanics and General Relativity are valid.
The problem with such a fantasy, however, is that QM and GR are mututally exclusive theories. They fundamentally contradict each other on rather large areas such as universal Gravity, etc.
What has happened is that we have two theories, QM and GR, that our modern, effeminate, politically correct scientists can't choose between. One theory is correct, the other is not.
But no one wants to say that one of those theories is wrong.
So rather than make the hard choice, today's limp-wristed researchers are wasting enormous amounts of time and money pursuing a String Theory that somehow makes both QM and GR valid.
...And that's why we're laughing. Unifying GR with QM is not feasible. The two theories contradict each other. Yet on go the Strong Theory adherents, unwilling to say that one King (either GR or QM) has no clothes...even if it means publishing ridiculous nonsense about String Theories.
| Real analysis | |
| In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms. | |
| Complex analysis | |
| Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory. | |
| Group theory | |
| Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters. | |
| Differential geometry | |
| Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases. | |
| Lie groups | |
| A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations. | |
| Differential forms | |
| The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions. | |
| Homology | |
| Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups. | |
| Cohomology | |
| Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups. | |
| Homotopy | |
| Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem. | |
| Fiber bundles | |
| Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics. | |
| Characteristic classes | |
| The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory. | |
| Index theorems | |
| In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with | |
| Supersymmetry and supergravity | |
| The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials. | |
Physics ping.
NYT still has the disinterested ability to get a story more or less right.
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