What is polarisation, spin, helicity, chirality and parity?

Question

Polarisation, spin, helicity, chirality and parity keep confusing me. They seem to be related, but exactly how they are related is unclear to me. Can someone maybe give a short overview about what these quantities mean and how they are related? What are the values for e.g. basic particles like electron or photon?

Answer 1

• Spin is determined from the representation of the Lorentz group the quantum field transforms in. The projective finite-dimensional representations of the Lorentz group are labeled by two half-integers $(s_1,s_2)$. The spin of a field is the sum $s = s_1+s_2$. For example, a scalar transforms in $(0,0)$, a vector field in $(\frac{1}{2},\frac{1}{2})$, a Dirac spinor in $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$, and those have spin $0$, $1$ and $\frac{1}{2}$, respectively.

• Helicity is the projection of the spin vector upon momentum. Formally, if $S^{\mu\nu}$ are the generators of the Lorentz group in a given representations, then helicity is given by the operator $$ h = \epsilon_{ijk} p^i S^{jk}$$ where $p$ is the momentum operator. For massive particles, it is not Lorentz-invariant, but for massless particles, which lack a rest frame, it is a Lorentz invariant notion.

• Chirality only makes sense for Dirac spinors and similar objects whose representation decomposes into smaller representations, or where exactly one of the two $s_i$ labelling their spin is zero. The Dirac spinor representation $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ decomposes into the left-handed $(\frac{1}{2},0)$ and the right-handed $(0,\frac{1}{2})$, which are also called the (left/right)-handed Weyl spinors. For massless fermions, the evolution equations that couple the right-handed part of a massive Dirac spinor to its left-handed part decouple, meaning a massless Dirac spinor is equivalently a theory to two uncoupled Weyl spinors. The projection operator into the chiral subspaces of the Dirac spinor representation is $$ \mathbb{P}_\pm = \frac{1}{2}(1\pm\gamma^5)$$ for $\gamma^5$ the usual product of 4D gamma matrices. It is to be noted that, for massless fermions, the chiral subspaces are precisely the eigenspaces of helicity.

• Parity is the unitary operator $P$ upon the quantum space of states that is associated to the classical symmetry $\vec x\mapsto -\vec x$. It is equivalently one of the generators of the $\mathbb{Z}/2\mathbb{Z}$ when presenting the Lorentz group as the semi-direct product $$ \mathrm{SO}_0(1,3)\ltimes \left(\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\right)$$ of its component $\mathrm{SO}_0(1,3)$ connected to the identity with the group $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z} = \{1,P,T,PT\}$

Answer 2

Polarization is, generically, a normalized vector in spin-space, i.e. the spin-state of a particle (When I say spin, I actually mean total angular momentum, spin+orbital angular momentum). Usually we take "fundamental" polarizations to be an orthonormal basis in spin-space.

• For scalars (spin-0), there is only a trivial polarization because the spin-space is 1-dimensional.
• For spin-1/2 particle you can have a normalized linear-combination of spin-up or spin-down, $c_1|\uparrow\rangle+c_2|\downarrow\rangle$. Actually this isn't Lorentz covariant notation, but never mind that.
• For spin-1 massive you have $3$ spin degrees-of-freedom. The usual choice of basis polarizations here are $\epsilon^\mu_{\pm}=(0,1,\pm i,0)$ and $\epsilon^\mu_{3}=k^{\mu}/m$.
• For spin-1 massless (which means helicity-1, since the concept of 'spin' only strictly applies to massive particles) you have $2$ spin degrees-of-freedom, which can be taken to be right/left handed polarizations $\epsilon^\mu_{\pm}=(0,1,\pm i,0)$, or any linear combination of them.
• etc.