# 1.2. Physics¶

## 1.2.2. What is a Neutrino Particle?¶

As Wigner said, a physical particle is an irreducible representation of the Poincaré group. A characteristic of Poincaré group is that mass comes in.

A neutrino particle is better recognized as its mass eigenstate.

In QFT, there are 3 different forms of neutrino mass term, left-handed Majorana, right-handed Majorana and Dirac mass terms.

## 1.2.3. Chirality and Helicity¶

### 1.2.3.1. Helicity¶

Helicity is the projection of spin onto direction of momentum,

$h = \vec J\cdot\hat p = \vec L\cdot\hat p + \vec S\cdot \hat p = \vec S\cdot \hat p,$

where

$\hat p = \frac{\vec p}{\left|\vec p\right|}$

A state is called right-handed if helicity is positive, i.e., spin has the same direction as momentum.

### 1.2.3.2. Chirality¶

Chirality is the eigenstate of the Dirac $$\gamma_5$$ matrix, which is explicitly, [1]

$\begin{split}\gamma^5 &= \begin{pmatrix} \mathbf 0 & \mathbf I \\ \mathbf I & \mathbf 0 \end{pmatrix} \\ & = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}.\end{split}$

## 1.2.5. States¶

### 1.2.5.1. Wigner Function¶

Fig. 1.3 A ensemble of classical harmonic oscillators can be described using such phase-space probability distribution.

Wigner function is an analogue of the classical phase-space probability distribution function though it is not really probability. [3] The mean of Wigner function lies in the two quadratures, i.e., space distribution and momentum distribution.

There is a collection of Wigner functions on this site. [3]

Question

How do one describe a system of neutrinos using Wigner function? The effect of statistics?

## 1.2.6. Statistics¶

Fermi-Dirac distribution

$f(p,\xi) = \frac{1}{1+\exp (p/T-\xi)},$

where $$\xi=\mu/T$$ is the degeneracy parameter.

The neutrino-neutrino forward scattering is [2]

$\nu_\alpha (p) + \nu_\beta (k) \to \nu_\alpha (k)+\nu_\beta (p).$

Question

Meaning of each term in Liouville equation.

 [2] Pantaleone (1992), Friedland & Lunardini (2003).

## 1.2.7. Refs & Notes¶

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