We take a break from Groups and introduce "Representation Theory", the unification of the two being that which sets the stage for the 1920's and the birth of Modern Quantum Mechanics. Later we will be uniting the two, as "Group Representation Theory", the understanding of which being crucial to inculcating Modern Physics, or as I like to call it: That Which Made Gordon Moore of Intel Rich, AND awarded Seven Nobel Prizes in Physics (so far) to Bell Labs, and then some.
All of which is key to a future pair of books I'd like to write, specifically "How to Teach the Mathematics of Quantum Mechanics To Your Third Grader" and "How To Teach AdS/CFT to your Fifth Grader (Who is Probably Smarter Than You)", but let's not get ahead of ourselves.
For a pleasant change, the Wikipedia introduction to Representation Theory is actually assessable to the Intelligent layman, one needn't have a bachelor's degree in Mathematics to understand it. Since I've linked, I needn't cut'n'paste the entire page, but I will do so with the bits that are relevant for our purposes, followed by the contents. Pay attention to the name Herman Weyl and Lie Groups, which are continuous groups, we will be discussing them much more in the months ahead.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, a subject which is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.
A striking feature of representation theory is its pervasiveness in mathematics. There are two sides to this. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory illuminates and vastly generalizes Fourier analysis via harmonic analysis, is deeply connected to geometry via invariant theory and the Erlangen program, and has a profound impact in number theory via automorphic forms and the Langlands program. The second aspect is the diversity of approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory and topology.
The success of representation theory has led to numerous generalizations. One of the most general is a categorical one. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second the target category of vector spaces can be replaced by other well-understood categories.
A unitary representation of a group G is a linear representation φ of G on a real or (usually) complex Hilbert space V such that φ(g) is a unitary operator for every g ∈ G. Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl, and this has inspired the development of the theory, most notably through the analysis of representations of the Poincare group by Eugene Wigner. One of the pioneers in constructing a general theory of unitary representations (for any group G rather than just for particular groups useful in applications) was George Mackey, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.
A Lie group is a group which is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups. Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.
The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. This theory can be extended to finite dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group G has a complexification, which is a complex Lie group Gc, and this complex Lie group has a maximal compact subgroup K. The finite dimensional representations of G closely correspond to those of K.
- 1 Definitions and concepts
- 2 Branches and topics
- 2.1 Finite groups
- 2.2 Modular representations
- 2.3 Unitary representations
- 2.4 Harmonic analysis
- 2.5 Lie groups
- 2.6 Lie algebras
- 2.7 Linear algebraic groups
- 2.8 Invariant theory
- 2.9 Automorphic forms and number theory
- 2.10 Associative algebras
- 3 Generalizations
- 4 See also
- 5 Notes
- 6 References