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

# Category Archives: Mathematics

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

Group theory and the issue of mass: Majorana fermions in 2D spacetime We have studied in the previous posts that a mass term is “forbidden” in the bivector/sixtor approach and the Dirac-like equation due to the gauge invariance. In fact, … Continue reading

The sixtor or 6D Riemann-Silberstein vector is a complex-valued quantity up to one multiplicative constant and it can be understood as a bivector field in Clifford algebras/geometric calculus/geometric algebra. But we are not going to go so far in this … Continue reading

In the next group theory threads we are going to study the relationship between Special Relativity, electromagnetic fields and the complex group . There is a close interdependence of the following three concepts: The classical electromagnetic fields and can be … Continue reading

Today, we are going to talk about the Lie groups and , and their respective Lie algebras, generally denoted by and by the physics community. In addition to this, we will see some properties of the orthogonal groups in euclidena … Continue reading

The converse of the first Lie theorem is also generally true. Theorem. Second Lie Theorem. Given a set of hermitian matrices or operators , closed under commutation with the group multiplication, then these operators define and specify a Lie group … Continue reading

Definition (36). An infinite group is a group where the order/number of elements is not finite. We distinguish two main types of groups (but there are more classes out there…): 1) Discrete groups: their elements are a numerable set. Invariance … Continue reading

Schur’s lemmas are some elementary but very useful results in group theory/representation theory. They can be also used in the theory of Lie algebras so we are going to review these results in this post (for completion). FIRST SCHUR’S LEMMA. … Continue reading

Representation theory is the part of Group Theory which is used in the main applications. Matrices acting on the members of a vector space are assigned to every element of a group. The connections between particle physics and representation theory … Continue reading