What does representation theory teach us?

[Gerenuk]

I have only very little knowledge of group theory and representation theory. There is a lot to learn, so I wondered what are the final contributions to physical understanding from representation theory?

I'd like to find some less abstract results. For example that the mass is the Casimir invariant of the Poincare group seems like a useful insight, but I don't understand the details :)

Does representation theory help to understand the law of quantum mechanics or general relativity?
Or does it just predict the existence of some "things" and restrict the form of physical laws?

 

[Valerie Park]

Representation theory is a useful tool for understanding the symmetries of physical systems, including quantum mechanics and general relativity. Representation theory can provide insight into the structure of the underlying symmetries, thus providing a way to study their properties. For example, the spin-statistics theorem is a consequence of the symmetry properties of quantum field theory, and this theorem was proven using representation theory. Representation theory also provides a way to classify particles according to their "quantum numbers", which can be used to make predictions about their interactions with other particles. Representation theory also plays a role in the formulation of the Standard Model, which describes the basic building blocks of matter and the fundamental forces of nature. Finally, representation theory has been used to prove the existence of certain physical phenomena, such as the Casimir effect.

 

[sir-pinski]

Representation theory is a branch of mathematics that studies how abstract algebraic structures, such as groups, can be represented by matrices or linear transformations. In physics, representation theory plays a crucial role in understanding the symmetries of physical systems and their corresponding laws.

One of the key insights that representation theory offers is the concept of symmetry breaking. This occurs when a system exhibits a symmetry at a fundamental level, but that symmetry is not apparent at the macroscopic level. For example, the laws of physics are invariant under rotations in space, but the specific arrangement of particles in a solid object may break this symmetry. Representation theory helps us understand how these symmetries manifest and how they can be broken.

Additionally, representation theory has also been used to understand the behavior of particles in quantum mechanics and the structure of spacetime in general relativity. For example, the concept of spin in quantum mechanics is intimately connected to the representation theory of the rotation group. In general relativity, the symmetries of the Poincaré group play a role in understanding the structure of spacetime.

Overall, representation theory provides a powerful framework for understanding the symmetries and underlying structures of physical systems. It helps us make predictions and understand the fundamental principles that govern the behavior of particles and spacetime. So while it may seem abstract at first, representation theory has many practical applications in physics and continues to be a valuable tool for understanding the laws of the universe.