Helicity basis

In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example). In this basis, the spin is quantized along the axis in the direction of motion of the particle.

pinors

The two-component helicity eigenstate $xi\_lambda$ satisfy::$sigma\; cdot\; hat\{p\}=lambda\; xi\_lambda(hat\{p\})\; ,$: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, $chi\_lambda\; ,$ we will use the generic form of fermion 4-momentum:::$p^mu=\; left(E,\; |vec\{p\}|\; sin\{\; heta\}\; cos\{phi\},\; |vec\{p\}|\; sin\{\; heta\}\; sin\{phi\},\; |vec\{p\}|\; cos\{\; heta\}\; ight)\; ,$Then one can say the two helicity eigenstates are::$xi\_\{+1\}(vec\{p\})\; =\; sqrt\{frac\{2\; |vec\{p\}\; chi\_lambda(hat\{p\})end\{pmatrix\}\; ,$and for an anti-fermion,::

Dirac matrices

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

pin-1 wavefunctions

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

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:::where::$q\_T\; =\; sqrt\{q\_x^2\; +\; q\_y^2\}\; ,$ is transverse momentum, and::$E\; =\; sqrt\{|vec\{q\}|^2\; +\; m^2\}\; ,$ is the energy of the boson.