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What is Representation Theory of Lie groups?

There are at least two reasons why mathematicians become interested in representation theory : quantum physics, and numerology! In the early twentieth century, the study of representations of Lie groups was instigated by quantum mechanics and number theory. Here I will present two examples in connection with these two disciplines.

§ The quantum harmonic oscilator. This is a simple but illustrative example of the connection between physics and representation theory of Lie groups. It also provides a snapshot view of the mathematical foundations of quantum mechanics.

Before jumping into the quantum world, let me begin by explaining the classical harmonic oscillator. The classical harmonic oscillator is simply a mass-spring system which is not subject to friction, gravity, or any other external force.


The classical harmonic oscillator

An eighteenth-century physicist would naturally have been interested in finding a formula for the position of the mass in terms of time. We can represent the position by a one-variable real function . In classical mechanics, one can use the fact that the Hamiltonian (i.e., total energy) of the system is constant to see that is the solution to a simple differential equation:

Here is the amount of mass, is the spring constant, is the momentum, and is the total energy of the system (which is a constant).

What is the quantization of the classical harmonic oscillator? In quantum mechanics we are not supposed to know where a particle is located exactly! Rather, at any given time we are able to determine the probability that it lies at a given location. tyle="position: relative;bottom : -1px;" It follows that the function should be replaced by a one-parameter family of wave functions. More precisely, at any given time the "position" of the particle is a complex-valued function such that is a probability distribution on the real numbers, i.e.,

Similar to the classical system, the quantum system is governed by an equation involving a Hamiltonian. The difference is that the new Hamiltonian is not a function anymore. In fact it is a linear transformation on the vector space spanned by wave functions. Note that the latter vector space is infinite-dimensional.

Suppose is the Hamiltonian operator, where denotes the vector space spanned by wave functions. (Note that in fact , the space of square-integrable functions on the real line.) The key difference between classical and quantum mechanics is that the differential equation obtained by energy conservation is replaced by the Schrödinger equation:

The Hamiltonian operator is given in terms of two other linear transfomations on : the position operator , and the momentum operator :

Not so surprisingly, the formulas for the classical and the quantum Hamiltonian look very similar!

An important fact about and is that they do not commute. Indeed we have

where is the Planck constant. This is exactly where representation theory of Lie groups comes into the picture, as in the language of representation theory the latter equation essentially means that the operators and define a representation of the Heisenberg group on the vector space . This representation of the Heisenberg group is related to the celebrated Heisenberg uncertainty principle, which states that we cannot know certain pairs of physical properties (e.g., position and velocity of a particle) up to arbitrary precision at the same time!

§ Modular functions and automorphic forms. In 1916 Srinivasa Ramanujan, an amateur indian mathematician who turned into one of the most famous and influential mathematicians of all times, discovered several mysterious properties of a special power series. The power series Ramanujan considered was :



For example, he discovered that if denotes the coefficient of , then for every two positive integers and which are relatively prime we have
.
(Exercise: check it for and !)

Ramanujan could not prove these mysterious properties, so he stated them as (yet to become famous) conjectures. At least one of these conjectures had to wait more than sixty years to be settled by another big shot named Pierre Deligne.

It turns out that a slightly different way of looking at the above power series is quite illuminating : If we set where , then the power series can be written as

and it turns out that the restriction of to the upper half plane satisfies another mysterious property : if are integers such that , then
This strange equality is the main property of a class of complex-valued functions on which are usually called modular forms.

Modular forms are profoundly related to representation theory of Lie groups. Here I can only provide a rough outline of this relationship : one can form a matrix from as follows :

and the determinant of this matrix is equal to one. The set of all matrices with integer entries and determinant equal to one is in fact a group (under the usual matrix multiplication), and is usually denoted by . Every element of acts on as follows :
This transformation is called a Möbius transformation.


A visualization of the action of on

The group is a subgroup of a bigger group : the group , which consists of matrices with real entries whose determinant is equal to one. The action of , which is a Lie group, on by Möbius transformations yields a representation of on the space of complex-valued functions on . Many properties of modular forms (including Ramanujan's conjectures) are understood by studying (refinements of) this representation.

Ramanujan's observations have flourished to turn into the theory of automorphic forms and the Langlands program, which is currently a very active area of research.