Exploring the Pauli Vector: Mathematical Definition

[Grufey]

Hello

I'm reading my old notes of QM, I found the definition of Pauli vector, as follow

$$\vec{\sigma}=\sigma_1 e_x+\sigma_2e_y + \sigma_3 e_z$$

Where $$e_x. e_y$$ and $$e_z$$ are unit vectors.

So, here is my question. $$\sigma_i$$ and $$e_i$$ are elements of different nature. How can we define the product $$\sigma_ie_i$$??

I understand the idea, ok. But, mathematically don't seem right

Thanks in advance

 

[algebrat]

Although I am not certain what is going on, I will try. If I understand correctly, you are wondering why we can put the matrix and the vector next to each other. I think the idea is similar to product groups and product rings, which are very simple and common constructions where the behavior in one component has little to do with the behavior in the other component. For instance the product of the integers with the rationals, where addition is defined by (n,p)+(m,q)=(n+m,p+q). You might consider a product ring M x M', where M is space of 2x2 matrices, and M' is space of 3x3 matrices. If I did not get your question right, I hope this helps give ideas on how you might reword or fill us in on more about the definitions.

 

[dextercioby]

It is actually a shorthand (and misleading) notation. The "vector" you mention always appears in either a cross or a dot product in which the <units> are "coupled" with other units, this time real ones, like for momentum operators. So it's not a decomposition of a vector with respect to a basis (i,j,k or ex, ey, ez), it's just a handy notation which shortens some long expressions, i.e. i/o writing $p_x \sigma_x + p_y \sigma_y + p_z \sigma_z$ one writes $\displaystyle{\vec{\sigma}\cdot \vec{p}}$.

 

[Grufey]

Sorry I was too busy this week and I could not reply before.

That's what I thought, it's only a notation. misleading notation.

Regards

 

[dextercioby]

The same thing happens in relativity, where you have the $\sigma_{\mu}$. It's no real 4-vector (1-form), just a shorthand notation which is useful, but can be misleading.