Sunday, December 19, 2010

Mathematical Physics Basics

From here.
The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.
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 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 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.

Here are some of the topics in mathematics that a person who wants to learn advanced topics in theoretical physics, especially string theory, should become familiar with.
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 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 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.
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.

These are topics in mathematics at the current cutting edge of superstring research:

    Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.
    Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles -- hence K-theory.

Overview of K-theory Applied to Strings by Edward Witten
D-branes and K-theory by Edward Witten

Noncommutative geometry (NCG for short)
    Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.
    Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.
    In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.
    The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.
Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. Douglas
Noncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes, Michael R. Douglas, Albert Schwarz
String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg.
Non-commutative spaces in physics and mathematics by Daniela Bigatti
Noncommutative Geometry for Pedestrians by J.Madore

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