# If weak isospin is not conserved in time, what does the Noether theorem tell us?

As far as I understand weak isospin is only conserved in interactions but not as time evolves. Nevertheless, we get from Noethers theorem, because of global $SU(2)$ invariance a conserved quantity that is commonly called weak isospin (see for example this Phys.SE question).

I have never read the explicit statement that weak isospin is violated as time evolves (= during propagation), but chirality is not conserved as time evolves (see for example this Phys.SE question) and left-chiral particles carry isospin, whereas right-chiral do not. Therefore it follows from the non-conservance of chirality that isospin isn't conserved either.

How does this fit together with the result of Noether's theorem?

• Interesting question. Note that in the usual formulation Noether's theorem applies to continuous symmetries, while isospin is a discrete symmetry. – rob Dec 5 '14 at 15:23
• @rob: Why would isospin be discrete? It is $\mathrm{SU}(2)$! – ACuriousMind Dec 5 '14 at 15:43
• To OP: The isospin current, and hence the charge, is not conserved because the isospin $\mathrm{SU}(2)$ is broken (by the Higgs mechanism)! For details, see this question. – ACuriousMind Dec 5 '14 at 16:01
• Weak isospin is the SM $SU(2)_L$, and its current is absolutely conserved--otherwise it would not couple to the corresponding gauge fields consistently! It is not an anomalous symmetry. Since the symmetry is spontaneously broken, the currents evince the Nambu-Goldstone realization mode, which makes the Higgs mechanism possible. What you may misconstrue as WI breaking is probably failure to account for Goldstone and Higgs degrees of freedom absorbed into weak singlets observed in the lab. Provide a concrete example to deconstruct: Coupling to the R-fermion singlet involves the Higgs vev. – Cosmas Zachos Dec 4 '16 at 2:35