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**Unformatted text preview: **d (weak
isospin) doublet: χ L = ⎜ ⎟
⎝ e ⎠L
⎛0 1⎞
⎛ 0 0⎞
Using the 2x2 matrices τ + ≡ ⎜ ⎟ and τ − ⎜ ⎟ we can then write: ≡ ⎝0 0⎠
⎝ 1 0⎠
±
jµ = χ L γ µτ ± χ L ±
with τ = 1
(τ 1 ± iτ 2 )
2 Weak-isospin raising and
lowering operators, with 1⎛ 0 1 ⎞
τ1 = ⎜
2⎝ 1 0 ⎟
⎠ 1 ⎛ 0 −i ⎞
τ2 = ⎜
2⎝ i 0 ⎟
⎠ These are two of the Pauli spin matrices, wriPen here as τ to avoid any confusion with ordinary spin (as we did when discussing isospin in our discussions of strong interacBons). More accurately, these are two of the matrices that form a representaBon of the group SU(2) which describes systems in which two possible states are related by some symmetry transformaBon. 13 (e.g. the quantity in terms of which we are defining these doublets) Refer to this as weak
isospin and anBcipate full weak
isospin symmetry, which would imply the existence of a third “current” corresponding to 1 τ 3 where 3 is the third τ 2
Pauli spin matrix [of SU(2)]: 1 ⎛1 0 ⎞...

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