Im reading Peskin&Schroeder, chapter 20, the one that describes the Standard Model. I got the general idea, simmetry breakings and so on, but Im getting quite confused when starting to go to the first mathematical details. My doubts are when, in eq 20.69 defines Q=T3+Y and then forces that Q=-1 for electrons. I get the general idea (changing basis and such) but I cant deal with the math. T3 is a 2x2 matrix (-i 0;0 i) and Y is something not defined but I think it is c*(1 0;0 1) with c undefined. So, how do we arrive from this matrixes to Q=-1? Am I right with my guesses of T3 and Y? How do we add 2x2 mattrixes and we get 1x1 number? Is it that we are adding the eigenvalues??? My doubt is about what is the math behind Y, T3 and Q and it is not about understanding the concept behind higgs, bosson masses and such. Perhaps the doubt is too silly for someone who tries to understand this subject but you have been very useful in the past so Im sure you could lend me a hand here again! Thanks in advance.
The relationship Q = T_{3} + Y/2 can be considered either of two ways. For an individual particle we understand it to be a scalar equation. For example for an up quark, T_{3} = +1/2, Y = +1/3 so Q = +2/3. When we consider several particles together as a multiplet, such as up and down quark, we understand it to be a matrix equation, in this case T_{3} = (1/2, 0; 0, -1/2) and Y is a multiple of the identity matrix, Y = YI where Y = +1/3. Then Q is also a matrix, Q = (2/3, 0; 0, -1/3).