Basic question about Z2 graded algebras

[Jim Kata]

Sorry, I really hate reading, and I do better by just asking stupid questions.

The lorentz group SO(3,1) is not simply connected so its unitary representation is in a projective space. It's fundamental group is $$\mathbb{Z}_2$$
so picking your standard path a certain way you can get

$$U(\bar \Lambda )U(\Lambda ) = \pm U(\bar \Lambda \Lambda )$$

Now if you use the universal cover of SO(3,1), $$SL(2,\mathbb{C})$$ you can get

$$U(\bar \Lambda )U(\Lambda ) = U(\bar \Lambda \Lambda )$$

Now my question is by introducing the gradation are you basically just redefining your unitary transformations so that $$U(\bar \Lambda )U(\Lambda ) = U(\bar \Lambda \Lambda )$$ for SO(3,1) too?

 

[AndyW]

Hello, it's great that you are interested in understanding the Lorentz group and its unitary representation. The Lorentz group is a fundamental concept in physics and understanding its properties is crucial in many areas of research.

To answer your question, the introduction of gradation does not change the unitary transformations of the Lorentz group. The gradation is simply a way of organizing the group elements into different levels or degrees. It does not alter the fundamental structure of the group or its representation.

In fact, the gradation is often used to simplify calculations and make them more manageable. It allows us to break down complex representations into simpler ones, making it easier to study the group's properties and make predictions.

So, to summarize, the gradation does not redefine the unitary transformations of the Lorentz group, but rather provides a useful tool for understanding and working with the group. I hope this helps clarify your understanding. Keep asking questions and exploring the fascinating world of physics!

 

[sir-pinski]

It seems like you are asking if introducing a Z2 grading to the algebra can change the unitary transformations for the Lorentz group. The answer is no, the Z2 grading does not change the unitary transformations for SO(3,1). The Z2 grading simply allows for a more convenient way to classify elements in the algebra, but it does not affect the underlying structure or operations of the algebra. In other words, the Z2 grading does not change the fundamental properties of the algebra, including the unitary transformations for the Lorentz group.