Definition of the helicity operator While studying the Dirac equation my professor defined the helicity operator as
$$\hat{\lambda}=\dfrac{\vec S \cdot \vec{p}}{|\vec p|}$$ where $\vec S$ is the spin matrix and $\vec{p}$ is the momentum operator.
My question is: what is the object in the denominator? If it's intended as an operator, then how is it defined the division by an operator? And what operator would it be? (There would be the square root of a sum of derivatives I think). If it's not an operator, then what is it? If I want to apply this operator to a wave function which is not eigenstate of the momentum operator what do I put in the denominator?
My professor did not treat it as an operator so I think that It's a constant, but which constant is it?
 A: The helicity operator you've written is not an operator in the sense you're thinking. That is, an operator on the Hilbert space of states. So this is actually not what you want to do:

If I want to apply this operator to a wave function ...

The helicity operator acts on the left- and right-handed components, $\psi_L(p),\psi_R(p)$ of the Dirac spinor, which are certainly not states (they furnish a non-unitary rep. of the Poincaré group and, upon quantization, multiply the creation and annihilation operators in the field expansion). So really $\vec{p}$ is not an operator at all (nor is $\vec{S}$), it is just the plain 3-momentum of a wave component.
A: It is defined as the proportionality constant between the operators:4momentum and Pauli-Lubanski 4 vector in the case of a massless irreductible representation of the Poincaré group. It has the physical meaning of the projection of the total angular momentum on the direction of the linear momemntum of the particle.It is denoted by "lambda": λ=:J→⋅P→P0 It can take only positive/negative integer/semiinteger values. For the photon:λ=±1;for the scalar boson it is zero,for the graviton it is [itex] \pm 2 [/tex]
Turza
A: The denominator, $|\vec{p}|$, is the magnitude of the three-momentum $\vec{p}$:
$$|\vec{p}|=\sqrt{p_x^2+p_y^2+p_z^2}$$
This is the momentum you know from classical physics, and it is different from the momentum operator
$$\hat{\textbf{p}} = - i \hbar \frac{\partial}{\partial x},$$
or the four-momentum $p_{\mu}=(E,p_x,p_y,p_z)$.
Now, helicity is defined as
$$h= \frac{\vec{S}\cdot\vec{p}}{|\vec{p}|},$$
which is the normalized component of the particle's spin along its direction of flight.
Here is the deal: there are four independent basis states for solutions of the Dirac equation. Two of these correspond to particle solutions and two to antiparticle solutions. The question is, why do we have two of each and what is the difference between them?
The helicity operator, defined as
$$\hat{h} = \frac{{\hat{\textbf{Σ}}} \cdot {\hat{\textbf{p}}}}{|\vec{p}|}= \frac{1}{2|\vec{p}|}\begin{pmatrix}
\textbf{σ} \cdot \hat{\textbf{p}} & 0\\
0 & \textbf{σ} \cdot \hat{\textbf{p}}
\end{pmatrix},$$
measures how the particle’s spin axis is aligned with the particle’s motion: whether it is parallel or antiparallel.  More importantly, it commutes with the free particle Dirac Hamiltonian and is therefore a conserved quantity (but not Lorentz invariant!).  Since they commute, it is possible to identify spinor states which are simultaneous eigenstates of both operators ($\hat{H}_D$ and $\hat{h}$).  For a spin 1/2 particle, the component of spin measured along any axis is quantised to be either $\pm 1/2$.  Therefore, acting the helicity operator on the Dirac spinors gives the eigenvalues $\pm 1/2$.  The two possible helicity states for a spin 1/2 fermion are termed right-handed and left-handed helicity states.  This is why we have four independent basis states, two for particle and two for antiparticle solutions.

P.S. I'm pretty new to this so take what I say with a grain of salt.  The figure is taken from Modern Particle Physics by Mark Thomson.
