LOG#094. Group theory(XIV).

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

LOG#093. Group theory(XIII).

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

LOG#091. Group theory(XI).

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 euclidean … Continue reading

LOG#090. Group theory(X).

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

LOG#089. Group theory(IX).

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

LOG#088. Group theory(VIII).

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

LOG#087. Group theory(VII).

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

LOG#086. Group theory(VI).

We are going to be more explicit and to work out some simple examples/exercises about elementary finite and infinite groups in this post. Example 1. Let us define the finite group of three elements as where , and such as … Continue reading

LOG#085. Group theory(V).

Other important concepts and definitions in group theory! Definition (22). Normal or invariant group. Let be a subgroup of other group G. We say that is a normal or invariant subgroup of G if the following condition holds: Proposition. Let … Continue reading