I am going to review the powerful Cartan calculus of differential forms applied to differential geometry. In particular, I will derive the structure equations and the Bianchi identities. Yes! Firstly, in a 2-dim manifold, we and introduce the Cartan 1-forms … Continue reading
Beta-gamma fusion! Live dimension! Do you like magic? Mathemagic and hyperbolic magic today. Master of magic creates an “illusion”. In special relativity, you can simplify calculations using hyperbolic trigonometry! (1) (2) are common relativistc equation. Introduce now: … Continue reading
Surprise! Double post today! Happy? Let me introduce you to some abstract uncommon representations for spacetime. You know we usually represent spacetime as “points” in certain manifold, and we usually associate points to vectors, or directed segments, as , in … Continue reading
I have the power! I have a power BETTER than the Marvel’s tesseract. It is called physmatics. Hi, there! We are back to school. This time, I am going to give you a tour with some geometrical objects, or geometricobjects, … Continue reading
Hi, everyone! I am back, again! And I have some new toys in order to post faster (new powerful plugin). Topic today: Ramanujan! Why Ramanujan liked the next equation? (1) This equation can be rewritten as follows (2) … Continue reading
Final post of this series! The topics are the composition of different angular momenta and something called irreducible tensor operators (ITO). Imagine some system with two “components”, e.g., two non identical particles. The corresponding angular momentum operators are: The following … Continue reading
This and my next blog post are going to be the final posts in this group theory series. I will be covering some applications of group theory in Quantum Mechanics. More advanced applications of group theory, extra group theory stuff … Continue reading
The case of Poincaré symmetry There is a important symmetry group in (relativistic, quantum) Physics. This is the Poincaré group! What is the Poincaré group definition? There are some different equivalent definitions: i) The Poincaré group is the isometry group … Continue reading
Given any physical system, we can perform certain “operations” or “transformations” with it. Some examples are well known: rotations, traslations, scale transformations, conformal transformations, Lorentz transformations,… The ultimate quest of physics is to find the most general “symmetry group” leaving … Continue reading
The topic today in this group theory thread is “sixtors and representations of the Lorentz group”. Consider the group of proper orthochronous Lorentz transformations and the transformation law of the electromagnetic tensor . The components of this antisymmetric tensor can … Continue reading