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.
The two-component
helicity
eigenstates
satisfy
![{\displaystyle \sigma \cdot {\hat {p}}\xi _{\lambda }\left({\hat {p}}\right)=\lambda \xi _{\lambda }\left({\hat {p}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7c88ede32aa68869b682c286ab1e0951682b24)
- where
are the
Pauli matrices,
is the direction of the fermion momentum,
depending on whether spin is pointing in the same direction as
or opposite.
To say more about the state,
we will use the generic form of
fermion
four-momentum:
![{\displaystyle 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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/390d73bf10664df708c3d856b7abb2eaab18ac5f)
Then one can say the two helicity eigenstates are
![{\displaystyle \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}+ip_{y}\end{pmatrix}}={\begin{pmatrix}\cos {\frac {\theta }{2}}\\e^{i\phi }\sin {\frac {\theta }{2}}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd51e53e0e6e865659d70f990653ecf8d95b07a)
and
![{\displaystyle \xi _{-1}({\vec {p}})={\frac {1}{\sqrt {2|{\vec {p}}|(|{\vec {p}}|+p_{z})}}}{\begin{pmatrix}-p_{x}+ip_{y}\\\left|{\vec {p}}\right|+p_{z}\end{pmatrix}}={\begin{pmatrix}-e^{-i\phi }\sin {\frac {\theta }{2}}\\\cos {\frac {\theta }{2}}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1747f2faa1b89734895c2140006f9c58b54e2f)
These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather:
.
In this situation the helicity eigenstates are for when the particle momentum is
and ![{\displaystyle \xi _{-1}({\hat {z}})={\begin{pmatrix}0\\1\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0276195da92652d2e59096352a0e9de179c64ff1)
then for when momentum is
and ![{\displaystyle \xi _{-1}(-{\hat {z}})={\begin{pmatrix}-1\\0\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b05ec42afa64c17f99f1c33e321fa3543a6dea64)
Fermion (spin 1/2) wavefunction
A fermion 4-component wave function,
may be decomposed into states with definite four-momentum:
![{\displaystyle \psi (x)=\int {{\frac {d^{3}p}{(2\pi )^{3}{\sqrt {2E}}}}\sum _{\lambda \pm 1}{\left({\hat {a}}_{p}^{\lambda }u_{\lambda }(p)e^{-ip\cdot x}+{\hat {b}}_{p}^{\lambda }v_{\lambda }(p)e^{ip\cdot x}\right)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466dbfca5f1e320d88d1404bc4b955818967a0e7)
- where
and
are the
creation and annihilation operators, and
and
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
![{\displaystyle 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}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fac81ee803eb5a9b15e5cb15f5877fcd65be22f)
and for an anti-fermion,
![{\displaystyle 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4946ef4590f4277447b79e51539b9fe46cfc838)
To use these helicity states, one can use the
Weyl (chiral) representation for the
Dirac matrices.
The plane wave expansion is
.
For a
vector boson with mass m and a
four-momentum
, the
polarization vectors quantized with respect to its momentum direction can be defined as
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6c03f0c59c517107869202d959e49eda2a96aa)
- where
is transverse momentum, and
is the energy of the boson.