Helicity spin along the momentum direction.

The spin of a particle is labeled by .  You have the freedom to choose the z-direction. So choose the z-direction based on the particles momentum. Once you have two particles this procedure will deviate from a traditional approach because each particle must carry its own coordinate system in that we want to identify a z-axis independently for each particle in order to define helicity.  In principle this is a straightforward convention.

We know that the classical E&M field is a vector field. At each point in space we find that there is a structure that we need to understand at that point.  This is in contrast to a scalar field such as temperature where a single number suffices to specify the temperature at a point.  We highlighted how we might label the types of potential structures that may be encountered by their transformation properties:

• a scalar transforms as a scaler [doesn’t change]
• a vector transforms as a vector [components mix, but length and relative orientation remains unchanged]
• Tensors transform as tensors [tensors require in general more parameters to characterize their structure]

Transformations are actually groups of operation on space-time or operations on intrinsic or internal degrees of freedom {flavor}. Spin is the structure built from rotations. The vector character of the E&M field is based on the fact that the photon is a spin 1 object with zero mass.  Therefore free photons loose one degree of freedom and are either right or left circularly polarized. {Virtual photons which are exchanged in out Feynam diagrams and describe a E&M field with current sources are not massless and have a more complex structure than free photons. One result of this more complex structure is the scalar potential that translates into the Coulomb field.}

Using a standard expansion we found that for rotations, scalars, vectors and higher rank tensors emerged as natural structures.  For further insight any development of the spherical harmonics will basically follow a similar route to reach:

•  scalar function
•  vector function
•  to be more complex
• l=0,1,2,3,4,5
• m=lèm=-l in steps of 1

Now how or why do these functions imply angular momentum or spin?

They just do!!!

We have argued that QM requires space-momentum linking. Some points:

• Localize and you are required to view the state as having many momentum components (highèlow).
• Angular momentum classically is related to linear momentum.
• Translations in x are similar to translation in .
• Some functions are states of definite momentum.

• The spherical harmonics are angular momentum states.
• Finally particles carry an intrinsic spin that adds as an angular momentum to the orbital angular momentum.

The fact that that particles have spin structure is thus motivated but not proven.

EACH elementary particle will have an internal, intrinsic structure that is similar to the structure that developed for an extended object.  The intrinsic structure will be represented as a column vector with the necessary parameters of the structure making up the vector. These internal structures carry angular momentum.

 Spin 0 Spin 1/2 1/2 , -1/2 Spin 1 1,0,-1 spin 1 m=0 1,-1

In order to do particle physics we must extend the spin structure. The most critical change is that particles must have associated antiparticles. Dirac successfully developed a theory for relativistic spin 1/2 particles using a 4-component column that can be understood as carrying information about the spin and the antiparticle nature of the particle. Thus 4 rather than 2 elements in the column.

relativistic spin 1/2 structure

Operators on this 4-component spinor space are called the gamma matrices:

There are four special  matrices where .

Another special matrix or gamma operator is

;

Without going into too many details one can use this operator to project out specific types of spinors.

Right handed and Left handed.

The point is that there are two concepts that are going to be important in discussing the character of particle and how they interact.

1. Helicty: simply the spin along the momentum direction
2. Chirality: Particles can be split into Right handed or Left handed

For massless particles or nearly massless particles these two separate properties are the same.