I'm trying to teach myself about supersymmetry

[Jim Kata]

Ok, I'm trying to teach myself about supersymmetry, using Weinberg III. My learning style is I don't really read things I just kind of infer what the author means. So I need clarification on a lot.

Lets start with the basics, from a mathematical point of view.

Where I'm at is at Haag Lopusanzski Sohnius theorem:

Now basically how I see this is as $$SO_ + (1,3) \cong \frac{{SL(2,\mathbb{C})}}{{Z_2 }}$$ So $$SO_ + (1,3)$$ is not simply connected, but has a $${Z_2 }$$ grading. Your representation can be chosen so that $${\mathbf{U}}(\bar \Lambda ){\mathbf{U}}(\Lambda ) = \pm {\mathbf{U}}(\bar \Lambda \Lambda )$$. Where the + and - depend on whether you are talking about integer spin or half integer spin. Now my question is can I obtain Haag and Lopusanzski's results by working out the lie algebras of $${\mathbf{U}}(\bar \Lambda ){\mathbf{U}}(\Lambda ) = \pm {\mathbf{U}}(\bar \Lambda \Lambda )$$?

 

[AndyW]

Hi there,

I'm glad to hear that you are delving into the fascinating world of supersymmetry! It's a complex and exciting field, so it's important to take the time to fully understand the mathematical concepts behind it.

First of all, let's clarify a few things about the Haag-Lopuszanski-Sohnius theorem. This theorem states that in a supersymmetric theory, the super-Poincaré algebra (which is a generalization of the Poincaré algebra that includes supersymmetry transformations) can only exist in spacetimes with a specific number of dimensions and with certain properties. In particular, for theories with N supersymmetries, the spacetime must have N time dimensions and 3 space dimensions. This is known as the "signature" of the theory.

Now, to address your question about obtaining the results of the Haag-Lopuszanski-Sohnius theorem by working out the Lie algebras of {\mathbf{U}}(\bar \Lambda ){\mathbf{U}}(\Lambda ) = \pm {\mathbf{U}}(\bar \Lambda \Lambda ): this approach is not quite accurate. The Haag-Lopuszanski-Sohnius theorem is a result of group theory, not just Lie algebras. In fact, the Lie algebras of the super-Poincaré group are not enough to fully describe supersymmetric theories.

To fully understand the theorem, you will need to familiarize yourself with concepts such as Lie groups, supergroups, and superalgebras. These are the mathematical structures that describe the symmetries and transformations in supersymmetric theories. I recommend taking some time to read more about these topics and their applications in supersymmetry.

I hope this helps clarify things for you. Keep learning and exploring, and don't hesitate to ask more questions as they come up!

 

[Entropia]

I understand the importance of self-learning and inferring information from complex texts. However, in order to fully grasp the concept of supersymmetry, it is important to have a solid understanding of the underlying mathematical principles.

Firstly, it is important to understand the concept of Lie algebras and their role in supersymmetry. Lie algebras are mathematical objects that describe the symmetries of a system, and they play a crucial role in the study of supersymmetry. In this case, the Lie algebra in question is {\mathbf{U}}(\bar \Lambda ){\mathbf{U}}(\Lambda ) = \pm {\mathbf{U}}(\bar \Lambda \Lambda ), which describes the symmetry transformations of the supersymmetric theory.

The Haag Lopusanzski Sohnius theorem is a fundamental result in supersymmetry that relates the symmetry transformations of the supersymmetric theory to the representation theory of the Lie algebra. In simpler terms, it states that the supersymmetry transformations can be written as a combination of two types of transformations – bosonic and fermionic – which are related to each other through the Lie algebra.

To understand this theorem, it is important to have a solid understanding of Lie algebras and their representation theory. This involves understanding the structure of Lie algebras, their generators, and how they relate to the symmetry transformations of the system.

In summary, while self-learning and inferring information can be useful, it is important to also have a strong foundation in the mathematical principles underlying supersymmetry. It may be helpful to seek out additional resources or guidance from experts in the field to fully understand the concept.