Understanding Isotopic Spin in Standard Model Doublets: Electron or Neutrino?

[Ruslan_Sharipov]

Some elementary particles in Standard Model are grouped into doublets, e. g. electron and e-neutrino (both left). As to me, the wave function of such a doublet should be
$$\psi^{ia}=\left(\begin{array}{c}\nu^a \\ e^a \end{array}\right)$$
where $i=1,2$ and $a$ is a spinor index. In other words it is composed of electron and neutrino components. If $nu^a=0$, it is a pure electron wave function, and if $e^a=0$ it is a pure nutrino wave function. However, performing an SU(2)-gauge transfoprmation, I can mix the components of the doublet. My question is what particle I actually describe - an electron or a neutrino?

 

[nrqed]

Some elementary particles in Standard Model are grouped into doublets, e. g. electron and e-neutrino (both left). As to me, the wave function of such a doublet should be
$$\psi^{ia}=\left(\begin{array}{c}\nu^a \\ e^a \end{array}\right)$$
where $i=1,2$ and $a$ is a spinor index. In other words it is composed of electron and neutrino components. If $nu^a=0$, it is a pure electron wave function, and if $e^a=0$ it is a pure nutrino wave function. However, performing an SU(2)-gauge transfoprmation, I can mix the components of the doublet. My question is what particle I actually describe - an electron or a neutrino?

You describe both. That's the whole point: from the ''point of view'' of the weak interaction, the electron neutrino and the electron are two states of the same particle. They really are very similar to the spin up and spin down states of the electron, except that we don't give different particle names to the spin up and down of the electron. The weak interaction changes an electron neutrino into an electron and vice versa (through the emision or absorption of a W+-) as the elctromagnetic interaction may flip the spin of an electron.

Hope this makes sense.


Patrick

 

[Ruslan_Sharipov]

gauge transformation is not a physical process.

The weak interaction changes an electron neutrino into an electron and vice versa (through the emision or absorption of a W+-).

Well, I understand that if W+- is absorbed or emitted, then electron/neutrino states could change. My question was about a gauge transformation which is a purely mathematical trick.
Why (and how) it changes electron/neutrino states?

Ruslan.

 

[arivero]

You describe both. That's the whole point: from the ''point of view'' of the weak interaction, the electron neutrino and the electron are two states of the same particle.

But all the particles in a multiplet commuting with Poincare should have the same mass... Oh wait, they have: they are both massless.

 

[dextercioby]

Well, I understand that if W+- is absorbed or emitted, then electron/neutrino states could change. My question was about a gauge transformation which is a purely mathematical trick.
Why (and how) it changes electron/neutrino states?

Ruslan.

The gauge transformation doesn't affect the states, nor the observables, it acts only on fields, classical/quantum, which are not really observable.

Daniel.

P.S. Think about the EM field. Do we measure $A_{\mu}$ ...? (which is affected by a U(1) gauge transformation)

 

[nrqed]

But all the particles in a multiplet commuting with Poincare should have the same mass... Oh wait, they have: they are both massless.

Yes..thanks Higgs!