Chirality

Prerequisites: Spinors, Lorentz Group. Full path (12 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

see also spin and helicity

Intuitive

One of the things we observe in everyday life is that things have a distinct left and right. The simplest case is just the hands of a human: Obviously, the left hand and the right hand are different from each other. That is a very general thing in nature that things can be 'like a left hand' or 'like a right hand'. Of course, they do not need to be so. A ball has obviously no distinct left or right. But things can have. This fact is known in science as chirality, originating from a Greek word for hand. http://axelmaas.blogspot.de/2011/11/chiral-or-why-left-and-right-is-not.html

Positive and negative chirality fermions are often described as being right-handed or left-handed, respectively;if one shines a beam of positive chirality fermions (particles described math-matically as sections of S+) into a block of matter, it will begin to spin in a right-handed sense." from Geometry and Physics by E. Witten

Concrete

For a nice discussion see http://www.quantumfieldtheory.info/Chirality_vs_Helicity_chart.pdf and http://www.quantumfieldtheory.info/ChiralityandHelicityindepth.pdf

Chirality arises as a quantum number related to the Lorentz group. Form the representation theory of the Lorentz group, we know that the corresponding Lie algebra, can be interpreted as two copies of the $SU(2)$ Lie algebra $\mathfrak{su}(2)$. Therefore, we labelled each representation by two numbers: $j_L$ and $j_R$ which indicate which $\mathfrak{su}(2)$ representations are used to construct the Lorentz algebra representations. For example, the label $(\frac{1}{2},0)$ means that we used to fundamental representation for one $\mathfrak{su}(2)$ and the trivial, one-dimensional representation for the other $\mathfrak{su}(2)$.

A quantum field (or particle) that transforms according to the $(\frac{1}{2},0)$ representation is called left-chiral, and a quantum field (or particle) that transforms according to the $(0,\frac{1}{2})$ representation is called right-chiral.

Whether spin, helicity or chirality is important depends on the physical question you are interested in. For free massless spinors, the spin eigenstates are also helicity eigenstates and chirality eigenstates. In other words, the Hamiltonian for the massless Dirac equation commutes with the operators for chirality, γ5, helicity, S⃗·p⃗, and the spin operators, S⃗. The QED interaction $\bar\psi \not{A}\psi = \bar\psi_L\not{A}\psi_L + \bar\psi_R\not{A}\psi_R$ is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless […]

In the massive case, it is also possible to take the non-relativistic limit. Then it is often better to talk about spin, the vector. Projecting on the direction of motion does not make so much sense when the particle is nearly at rest, or in a gas, say, when its direction of motion is constantly changing. The QED interactions do not preserve spin, however; only a strong magnetic field can flip an electron’s spin. So, as long as magnetic fields are weak, spin is a good quantum number. That is why spin is used in quantum mechanics.

[In] QED, we hardly ever talk about chirality. The word is basically reserved for chiral theories, which are theories that are not symmetric under L ↔ R, such as the theory of the weak interactions. We talk very often about helicity. In the high-energy limit, helicity is often used interchangeably with chirality. As a slight abuse of terminology, we say ψL and ψR are helicity eigenstates. In the non-relativistic limit, we use helicity for photons and spin (the vector) for spinors. Helicity eigenstates for photons are circularly polarized light.

Quantum Field Theory and Standard Model by M. Schwartz

Abstract

Representation theory. The double cover of the proper orthochronous Lorentz group is $\mathrm{Spin}(1,3) \simeq SL(2,\mathbb C)$. With rotation generators $J_i$ and boost generators $K_i$, the combinations $N^\pm_i = \frac12 (J_i \pm i K_i)$ satisfy $$[N^\pm_i, N^\pm_j] = i\epsilon_{ijk}N^\pm_k, \qquad [N^+_i, N^-_j] = 0,$$ so the complexified Lorentz algebra is $\mathfrak{sl}(2,\mathbb C) \oplus \mathfrak{sl}(2,\mathbb C)$. Finite-dimensional irreducible representations are labelled $(j_L, j_R)$, and the two Weyl representations $(\frac12, 0)$ and $(0, \frac12)$ are the two inequivalent two-dimensional representations of $SL(2,\mathbb C)$: the defining one and its complex conjugate. Which one is called left is a convention; on this page $(\frac12,0)$ is left-chiral. Parity exchanges $\vec K \to -\vec K$ and therefore $N^+ \leftrightarrow N^-$, so it maps one chirality into the other.

Clifford algebra. The Dirac representation $(\frac12,0)\oplus(0,\frac12)$ is reducible under the Lorentz group. The chirality operator $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ satisfies $(\gamma^5)^2 = 1$ and anticommutes with all $\gamma^\mu$, so it commutes with the Lorentz generators $S^{\mu\nu} = \frac{i}{4}[\gamma^\mu,\gamma^\nu]$. The projectors $P_{L,R} = \frac12(1 \mp \gamma^5)$ split a Dirac spinor into its Weyl components (in the convention where $\gamma^5 = \mathrm{diag}(-1, 1)$ in the Weyl basis). The same construction works in any even dimension $d$: the product of all $d$ gamma matrices, suitably normalised, squares to one and splits the spinor representation of $\mathrm{Spin}(d)$ or $\mathrm{Spin}(1,d-1)$ into two half-spin representations $S^\pm$. In odd dimensions this product is proportional to the identity and there is no chirality.

Dynamics. The kinetic term $\bar\psi i\gamma^\mu\partial_\mu\psi = \bar\psi_L i\gamma^\mu\partial_\mu\psi_L + \bar\psi_R i\gamma^\mu\partial_\mu\psi_R$ does not mix chiralities, while a mass term does: $m\bar\psi\psi = m(\bar\psi_L\psi_R + \bar\psi_R\psi_L)$. A gauge theory is chiral when left- and right-chiral fermions sit in inequivalent representations of the gauge group, as in the electroweak sector of the Standard Model. Then a gauge invariant Dirac mass is forbidden, fermion masses must come from a Higgs field, and consistency of the quantum theory requires the cancellation of gauge anomalies.

Geometry. On an even-dimensional compact Riemannian spin manifold the Dirac operator maps sections of $S^+$ to sections of $S^-$ and back. Its index, the number of zero modes of positive chirality minus the number of negative chirality, is a topological invariant given by the Atiyah-Singer index theorem. For a Dirac operator coupled to a gauge field on a four-manifold with vanishing $\hat A$-genus contribution (for example on $S^4$), it equals, up to a convention-dependent sign, the instanton number $\frac{1}{8\pi^2}\int \mathrm{tr}\, F\wedge F$ for the fundamental representation of $SU(N)$. This is the mathematical origin of the chiral anomaly.

Why is it interesting?

Chirality is one of the fundamental labels we use to identify elementary particles. (Other labels are the mass or the electric charge.)

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Newton's Second Law
5. Vector Calculus
6. Lie Groups
7. Newtonian Mechanics
8. SO(3)
9. Special Relativity
10. Lorentz Group
11. SU(2)
12. Spinors