Helicity basis

Spinors

The two-component helicity eigenstates \xi_\lambda satisfy :\sigma \cdot \hat{p} \xi_\lambda\left(\hat{p}\right) = \lambda \xi_\lambda\left(\hat{p}\right) , :where ::\sigma , are the Pauli matrices, ::\hat{p} , is the direction of the fermion momentum, ::\lambda = \pm 1 , depending on whether spin is pointing in the same direction as \hat{p} , or opposite.

To say more about the state, \xi_\lambda , we will use the generic form of fermion four-momentum: :p^\mu = \left(E, \left|\vec{p}\right| \sin{\theta} \cos{\phi}, \left|\vec{p}\right| \sin{\theta} \sin{\phi}, \left|\vec{p}\right| \cos{\theta} \right) ,

Then one can say the two helicity eigenstates are :\xi_{+1}(\vec{p}) = \frac{1}{\sqrt{2 \left|\vec{p}\right|\left(\left|\vec{p}\right| + p_z\right)}} \begin{pmatrix} \left|\vec{p}\right| + p_z\ p_x + i p_y \end{pmatrix} = \begin{pmatrix} \cos{\frac{\theta}{2}} \ e^{i\phi}\sin{\frac{\theta}{2}} \end{pmatrix},

and : \xi_{-1}(\vec{p}) = \frac{1}{\sqrt{2 |\vec{p}|(|\vec{p}| + p_z)}} \begin{pmatrix} -p_x + i p_y \ \left|\vec{p}\right| + p_z \end{pmatrix} = \begin{pmatrix} -e^{-i\phi}\sin{\frac{\theta}{2}} \ \cos{\frac{\theta}{2}} \end{pmatrix},

These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather: :\hat{z} = \pm \hat{p} ,.

In this situation the helicity eigenstates are for when the particle momentum is \hat{p} = + \hat{z} , :\xi_{+1}(\hat{z}) = \begin{pmatrix} 1 \ 0 \end{pmatrix} , and \xi_{-1}(\hat{z}) = \begin{pmatrix} 0 \ 1 \end{pmatrix} ,

then for when momentum is \hat{p} = - \hat{z} , :\xi_{+1}(-\hat{z}) = \begin{pmatrix} 0 \ 1 \end{pmatrix} , and \xi_{-1}(-\hat{z}) = \begin{pmatrix} -1 \ 0 \end{pmatrix} ,

Fermion (spin 1/2) wavefunction

A fermion 4-component wave function, \psi, may be decomposed into states with definite four-momentum: :\psi(x) = \int{\frac{d^3p}{(2\pi)^3 \sqrt{2E} } \sum_{\lambda \pm 1}{\left(\hat{a}p^\lambda u\lambda(p) e^{-i p \cdot x} + \hat{b}p^\lambda v\lambda(p) e^{i p \cdot x} \right)} } , :where ::\hat{a}p^\lambda , and \hat{b}p^\lambda , are the creation and annihilation operators, and :: u\lambda(p) , and v\lambda(p) , are the momentum-space Dirac spinors for a fermion and anti-fermion respectively.

Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is :u_\lambda(p) = \begin{pmatrix} u_{-1} \ u_{+1} \end{pmatrix} = \begin{pmatrix} \sqrt{E - \lambda \left|\vec{p}\right|} \chi_\lambda(\hat{p}) \ \sqrt{E + \lambda \left|\vec{p}\right|} \chi_\lambda(\hat{p}) \end{pmatrix} ,

and for an anti-fermion, :v_\lambda(p) = \begin{pmatrix} v_{+1} \ v_{-1} \end{pmatrix} = \begin{pmatrix} -\lambda \sqrt{E + \lambda \left|\vec{p}\right|} \chi_{-\lambda}(\hat{p}) \ \lambda \sqrt{E - \lambda \left|\vec{p}\right|} \chi_{-\lambda}(\hat{p}) \end{pmatrix}

Dirac matrices

To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.

Spin-1 wavefunctions

The plane wave expansion is :\psi(x) = \int{\frac{d^3p}{(2\pi)^3 \sqrt{2E}} \sum_{\lambda = 0}^3 \left( \hat{a}{p,\lambda} \epsilon\lambda(p) e^{-i p \cdot x} + \hat{a}{p,\lambda}^\dagger \epsilon^*\lambda(p) e^{i p \cdot x} \right)} , .

For a vector boson with mass m and a four-momentum q^\mu = (E, q_x, q_y, q_z), the polarization vectors quantized with respect to its momentum direction can be defined as :\begin{align} \epsilon^\mu(q, x) &= \frac{1}{\left|\vec{q}\right| q_\text{T}} \left(0,q_x q_z, q_y q_z, -q_\text{T}^2 \right) \ \epsilon^\mu(q ,y) &= \frac{1}{q_\text{T}} \left( 0, -q_y, q_x, 0 \right) \ \epsilon^\mu(q, z) &= \frac{E}{m\left|\vec{q}\right|} \left(\frac{\left|\vec{q}\right|^2}{E}, q_x, q_y, q_z \right) \end{align} :where