What is representation theory? Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and thereby reducing problems in abstract algebra to problems in linear algebra. It provides a bridge between abstract algebra and linear algebra.
The representation theory of Lie groups and Lie algebras is one of the most profound and successful intersections of mathematics and physics in the 20th century. Let’s dive a bit deeper.
how does photon mediate electromagnetism?
The photon is the mediator (or “gauge boson”) of the electromagnetic force, and its role in mediating electromagnetic interactions is foundational to quantum electrodynamics (QED), which is the quantum field theory describing electromagnetism. Here’s an overview of how the photon mediates the electromagnetic force:
- Virtual Photons:
- First, it’s essential to understand that the photons mediating interactions between charged particles are not the same as the “real” photons we observe as light or other electromagnetic radiation. Instead, they are “virtual photons.”
- Virtual particles, in general, are a concept from quantum field theory and are intermediate states that are not directly observable but are essential in describing interactions.
- Feynman Diagrams:
- A useful tool to visualize these interactions is Feynman diagrams. In these diagrams, straight lines represent electrons (or other charged particles), and wavy lines represent photons.
- A simple Feynman diagram might show two electrons approaching each other, exchanging a photon (a wavy line connecting the two straight electron lines), and then moving apart. This diagram represents the electromagnetic repulsion between the two electrons.
Well, the explanation above by chatGPT is quite useless.
Why is Lie Algebra essentially all about commutator?
Not Only About the Commutator:
- While the commutator is a defining feature, Lie algebras are not solely about the commutator. The study of Lie algebras encompasses their representations, cohomology, relationships to Lie groups, and applications in various mathematical and physical contexts.
Yes, the concept of the commutator is central to understanding the uncertainty principle in quantum mechanics.
How The Lie bracket describes how small transformations “combine” or “interact”?
how the Lie bracket captures the difference between successively applying two infinitesimal transformations:
Naixian Zhang 