# Vector currents, vector fields and bosons

1. Nov 13, 2008

### Manilzin

Consider this quote from Mandl and Shaw, p. 237

...this interaction coulpes the field $$W_{\alpha}(x)$$ to the leptonic vector current. Hence it must be a vector field, and the W particles are vector bosons with spin 1.

Could someone explain this for me? I do not understand the "hence", because I do not know what a vector current means. Also, why does a vector field imply bosons with spin 1?

2. Nov 13, 2008

### weejee

1. I guess here it is the conserved current associated with the U(1) symmetry, which is a vector quantity?

2. I think spin 1 follows from the rotational property of vector field. How I understand this is as follows.

- Consider a vector field V=(V_t, V_x, V_y, V_z)

- Construct spherical component of the vector field. i.e. V_0=V_z, V_1 = V_x+iV_y, V_-1 = V_x-iV_y

- Consider rotation by theta w.r.t. z axis. Then V_0 -> V_0, V_1 -> exp(-i*theta)V_1, V_-1 -> exp(i*theta)V_-1

- Knowing that rotation opertator is exp(-i*S_z*theta), possible eigenvalue of S_z is 1, 0, -1 -> spin 1.

- From the spin-statistics theorem it should be bosonic.

- For the same reason, a nth rank tensor field should be a spin n boson

3. Somebody plz verify (or correct it if something is wrong) my reasoning and give similar reasoning for spinor fields.

Last edited: Nov 13, 2008
3. Nov 14, 2008

### Manilzin

Well, thanks, I guess that explains part of it. But as you said, what is the corresponding reasoning for spinor fields? Why does the solutions to the Dirac equations describe spin 1/2-particles?