Lie theory, one of the most porminent theories in mathematics, studies the properties of Lie groups and Lie algebras,
not only important in mathematics, but also plays essential roles in physics. Since I have noticed that few students in department of physics
really understand what Lie groups are and why they are important in physics, I wrote this short paragraph for sharing some insights I realized.
Group theory, the simplest algebraic structure, is crucial in abstract algebra and physics.
In abstract algebra, the greatest success of group theory is the Galois theory, which remains a significant research field to this day.
Galois theory utlizes the concept of "symmetries" to find the roots of polynomials over fields. Here, the word "symmetries" in reality refers to "groups" in mathematics terms.
Due to the great success of Galois theory, another great mathmatician, named Sophus Lie, tried to extend the idea of symmetries to solve differential equations.
Nevertheless, not like the roots of the polynomials are discrete in general, the solutions of differential equations are continuous in general.
Therefore, Lie exteded the concept of groups to the continuous version: Lie groups.
Since symmetries are critical in physics as well, group theory also plays an essential role.
One of the most well-known applications of symmetries is in Landau theory. In Landau's theory, phases are modeled by corresponding symmetries, or groups.
However, the symmetries in the real world are usually "continuous" and hence infinite. Therefore, Lie groups are employed to describe
these kinds of symmetries.
Lie groups are perminent in numerous physics topics, particularly in theoretical physics.
Noether's theorem, a well-known and successful theorem in quantum field theory,
implies that every continuous symmetry corresponds to a conserved law. Here, the "continuous symmetry" essentialy refers to the Lie group.
Canonical transformations, kinds of functions essential in classical mechanics, form Lie groups as well.
Lie groups are helpful for understanding many physics properties.
Even if Lie groups are definitely significant in physics, they are rarely introduced in physics courses. Indeed, Lie theory is an advanced topic
in mathematics. In department of mathematics, students usually learn the basic concepts of Lie groups and Lie
algebra in the graduate graduate level courses due to their complexity. However, unlike mathematics physics does not require strict adherence to everything,
basic concepts of Lie theory can be understood more easily from a physics perspective. Theoretical physics students should learn the
fundamental Lie theory.
