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# Chirality

## 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.)

Important Related Concepts:

## Layman

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

## Student

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 E QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR 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

QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR 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.

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

## Researcher

The motto in this section is: the higher the level of abstraction, the better.
Does the opposite chirality only emerge dynamically?

"because fundamentally all fermion particles are left-handed and all fermion antiparticles are right-handed, with the opposite handedness emerging dynamically for massive fermions. Such dynamical emergence of handed-ness is described by L. B. Okun, in his book Leptons and Quarks (North-Holland (2nd printing 1984) page 11) where he said: “… a particle with spin in the direction opposite to that of its momentum …[is]… said to possess left-handed helicity, or left-handed polarization. A particle is said to possess right-handed helicity, or polarization, if its spin is directed along its momentum. The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer’s frame of reference. For example, it will change sign if we try to catch up with the particle at a speed above its velocity. Overtaking a particle is the more difficult, the higher its velocity, so that helicity becomes a better quantum number as velocity increases. It is an exact quantum number for massless particles … The above space-time structure … means … that at …[ v approaching the speed of light ]… particles have only left-handed helicity, and antparticles only right-handed helicity." On the chirality of the SM and the fermion content of GUTs by Renato M. Fonseca

Common Question 2

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