Vector currents, vector fields and bosons

In summary, the quote from Mandl and Shaw states that the interaction between the field W_{\alpha}(x) and the leptonic vector current implies that the W particles are vector bosons with spin 1. This is because a vector field has rotational properties that correspond to a spin 1 particle, according to the spin-statistics theorem. Additionally, a nth rank tensor field would be a spin n boson for the same reason. The reasoning for spinor fields and the description of spin 1/2-particles is still uncertain.
  • #1
Manilzin
16
0
Consider this quote from Mandl and Shaw, p. 237

...this interaction coulpes the field [tex]W_{\alpha}(x)[/tex] 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?
 
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  • #2
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 please verify (or correct it if something is wrong) my reasoning and give similar reasoning for spinor fields.
 
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  • #3
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?
 

1. What are vector currents?

Vector currents are mathematical quantities that describe the flow of a physical quantity, such as electric charge or fluid, through a given area. They are represented by arrows that indicate both the direction and magnitude of the flow.

2. What are vector fields?

Vector fields are mathematical functions that assign a vector to every point in a given space. They are used to describe physical quantities that have both magnitude and direction, such as electric and magnetic fields.

3. What are bosons?

Bosons are a type of elementary particle that have integer spin and obey Bose-Einstein statistics. They include particles such as photons, gluons, and W and Z bosons, and are responsible for fundamental interactions between particles.

4. How are vector currents and bosons related?

Vector currents play a crucial role in the mathematical description of bosons. In quantum field theory, bosons are described as excitations of a quantum field, and vector currents are used to calculate the interactions between these excitations.

5. What is the significance of vector fields in physics?

Vector fields are essential in physics because they allow us to mathematically describe physical quantities, such as force and energy, that have both magnitude and direction. They are used in many areas of physics, including electromagnetism, fluid dynamics, and quantum mechanics.

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