Do the weak isospins of the w1 and w2 combine?

[QuantumForumUser]

Do the weak isospins of the w¹ and w² bosons combine as their fields combine?

 

[vanhees71]

I don't know what you mean by that. You should look at some textbook how quantum flavor dynamics (the Glashow-Salam-Weinberg model of the electromagnetic and weak interactions) are constructed from the corresponding chiral gauge group $\mathrm{SU}(2)_{\mathrm{wiso}} \times \mathrm{U}(1)_{\text{Y}}$ and "Higgsed" to the $\mathrm{U}(1)_{\mathrm{em}}$ to give the particles and some of the gauge fields (the $W^{\pm}$ and $Z$ bosons) masse without violating the vital chiral local gauge symmetry of the model. A very good book on that is

https://www.amazon.com/dp/3540504966/?tag=pfamazon01-20

 

[QuantumForumUser]

I don't know what you mean by that. You should look at some textbook how quantum flavor dynamics (the Glashow-Salam-Weinberg model of the electromagnetic and weak interactions) are constructed from the corresponding chiral gauge group $\mathrm{SU}(2)_{\mathrm{wiso}} \times \mathrm{U}(1)_{\text{Y}}$ and "Higgsed" to the $\mathrm{U}(1)_{\mathrm{em}}$ to give the particles and some of the gauge fields (the $W^{\pm}$ and $Z$ bosons) masse without violating the vital chiral local gauge symmetry of the model. A very good book on that is

https://www.amazon.com/dp/3540504966/?tag=pfamazon01-20

What I meant was when the w¹ and w² bosons combine into the w+ and w- bosons through w⁺ or w^- = (w¹ -or+ iw²)/sqrt(2). The w+ has weak isospin +1 and the w- has weak isospin -1. So does that mean the weak isospins of the w¹ and w² can be found by rewriting the previously stated transformation (+or- 1= x -or+ iy/sqrt(2))?

 

[ChrisVer]

So does that mean the weak isospins of the w1 and w2 can be found by rewriting the previously stated transformation (+or- 1= x -or+ iy/sqrt(2))?

Does weak isospin make sense after you do the combination?

 

[QuantumForumUser]

Does weak isospin make sense after you do the combination?

Weak isospin (according to Wikipedia: https://en.wikipedia.org/wiki/Weak_isospin) is a conserved quantity. This means that the weak isospin values don't change whether symmetry breaking happens or not.

 

[vanhees71]

True, quantum flavor dynamics bases on the local chiral gauge symmetry $\mathrm{SU}(2)_{\text{wiso}} \times \mathrm{U}(1)_{\text{Y}}$, and this symmetry must not be broken explicitly (and can also not be broken spontaneously):

https://en.wikipedia.org/wiki/Elitzur's_theorem

 

[ChrisVer]

Weak isospin (according to Wikipedia: https://en.wikipedia.org/wiki/Weak_isospin) is a conserved quantity. This means that the weak isospin values don't change whether symmetry breaking happens or not.

not the point which i tried to make, but that the W+- are not eigenstates of weak isospin and so they don't have definite eigenvalues..
Try to get what the operator thatgave you the +1 or -1 eigenvalue for the W_i does on them... it transforms the one into the other,

 

[vanhees71]

Hint: The SU(2) x U(1) are defined on the "original fields", before introducing the non-vanishing VEV of the Higgs field explicitly. To express the transformation in terms of the "physical" fields (W's, Z, and $\gamma$) is awful!

 

[QuantumForumUser]

According to my book on Electroweak Physics, the w¹, w², and w³ bosons form a weak isospin triplet. This probably means the w¹ boson has weak isospin -1, the w² has weak isospin 1, and the w³ has weak isospin 0. Otherwise, the weak isospins of the w¹, w², and w³ could be eigenvalues of the su(2) Pauli matrices.

 

[nikkkom]

I'm not even sure that in unbroken symmetry, weak isospin *can be represented as a scalar* similar to the electric charge. Since it's related to SU(2) symmetry, shouldn't it have two charges? SU(3) has three charges, "colors". SU(2) should have two, no?

 

[nikkkom]

Weak isospin (according to Wikipedia: https://en.wikipedia.org/wiki/Weak_isospin) is a conserved quantity.

I think this is not true. Weak isospin is conserved in interactions. But one of the interactions is with Higgs field. When Higgs field VEV is nonzero, it means that particles interact with it all the time. This changes weak isospin and weak hypercharge of the particles. Only their combination which we call "electric charge" is conserved.

For example, a free electron changes between T3=−1/2,Y=−1 ("left electron") and T3=0,Y=−2 ("right electron"). Only Q=T3+Y/2 stays unchanged.

 

[ofirg]

According to my book on Electroweak Physics, the w¹, w², and w³ bosons form a weak isospin triplet. This probably means the w¹ boson has weak isospin -1, the w² has weak isospin 1, and the w³ has weak isospin 0.

Actually, w^$\pm$, w³ are the eigenstates of the weak isospin you are referring to ( which is the third component of the weak isospin - T³)
The usefullness of the w^1,2,3 basis is that it transforms as a vector in a three dimensional space.

 

[ChrisVer]

Actually, w±\pm, w3 are the eigenstates of the weak isospin you are referring to ( which is the third component of the weak isospin - T3)

are you sure that the W± are eigenstates of the weak isospin?

 

[ofirg]

are you sure that the W± are eigenstates of the weak isospin?

Unless someone knows otherwise, they are eigenstates of the third component of the weak isospin - T³
For example, the electric charge Q=T³+Y ( Y is the hypercharge)
Since Y=0 in this case, Q=T³. So the states with well defined electric charge also have well defined and equal T³.

 

[ChrisVer]

Unless someone knows otherwise, they are eigenstates of the third component of the weak isospin - T³
For example, the electric charge Q=T³+Y ( Y is the hypercharge)
Since Y=0 in this case, Q=T³. So the states with well defined electric charge also have well defined and equal T³.

if the $W^{1,2,3}$ are eigenstates of $T^3$ with eigenvalues $1,-1,0$ respectively, then:
$T^3 W^\pm =\pm W^\mp$
(in particular the W^+/- are more like the ladder operators that you had for the spin).