# $(5^* \times 5^*)_{asym}={10}$ in A. Zee's book p.409 versus PDG Sec.114

+ 3 like - 0 dislike
22 views

What is the mathematical or physical way to understand why the 4th and 5th components in the Georgi Galshow SU(5) model has the SU(2) doublet $$(1,2,-1/2)$$: $$\begin{pmatrix} \nu\\e \end{pmatrix}$$ with left-handed $$\nu$$ in the 4th component and $$e$$ in the 5th component of $$5^*$$; while in the contrary, the SU(2) doublet $$(3,2,1/6)$$: $$\begin{pmatrix} u\\d \end{pmatrix}$$ with the left-handed $$u$$ in the 5th component (column or row) and $$d$$ in the 4th component (column or row) of $$10$$?

My question is that why not the left-handed $$u$$ in the 4th component (column or row in the anti-symmetrix rank-5 matrix, say in Zee's book p.409 below) and $$d$$ in the 5th component (column or row in the anti-symmetrix rank-5 matrix, say in Zee's book p.409 below) of $$10$$?

My understanding is doe to the complex conjugation $$(5 \times 5)_{asym}={10}^*$$ instead of $$(5^* \times 5^* )_{asym}={10}.$$ But is it this the case? Would $$2^*$$ flips the doublet component of 2? $$2: \begin{pmatrix} v\\v' \end{pmatrix}\to 2^* \begin{pmatrix} v'\\v \end{pmatrix}?$$

In contrast, we see the PDG writes in a very different manner: https://pdg.lbl.gov/2018/mobile/reviews/pdf/rpp2018-rev-guts-m.pdf

Can we compare the two notations? Notice the contrary locations of $$\nu, e, u, d$$.

This post imported from StackExchange Physics at 2020-12-01 17:43 (UTC), posted by SE-user annie marie heart
asked Jul 26, 2020

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.