Physlib.Relativity.Fermions.Weyl.RightHanded

The canonical equivalence between the type of right-handed Weyl fermions and the complex vector space $\mathbb{C}^2$ (represented as the space of functions from a set of two elements to $\mathbb{C}$).

The space of right-handed Weyl fermions, represented by the type `Fermion.RightHandedWeyl` (which wraps $\mathbb{C}^2$), is equipped with the structure of an additive commutative monoid. This defines a commutative and associative addition operation and a zero element $0$ for right-handed Weyl spinors.

The space of right-handed Weyl fermions, represented by the type `Fermion.RightHandedWeyl` (which is isomorphic to the complex vector space $\mathbb{C}^2$), is equipped with an additive commutative group structure.

The type of right-handed Weyl fermions, `Fermion.RightHandedWeyl`, carries the structure of a module over the field of complex numbers $\mathbb{C}$. This defines `Fermion.RightHandedWeyl` as a complex vector space.

A $\mathbb{C}$-linear isomorphism between the space of right-handed Weyl fermions and the standard complex vector space $\mathbb{C}^2$ (modeled as functions from a set of two elements to $\mathbb{C}$).

The function maps a right-handed Weyl fermion to its representation as a vector in $\mathbb{C}^2$.

For any right-handed Weyl fermion $\psi$, the mapping of $\psi$ to its representation in the complex vector space $\mathbb{C}^2$ (denoted by $\psi.\text{toFin2}\mathbb{C}$) is equal to its underlying value $\psi.\text{val}$.

A basis for the space of right-handed Weyl fermions $\text{Fermion.RightHandedWeyl}$, which is a 2-dimensional vector space over the complex numbers $\mathbb{C}$. The basis is indexed by the set $\{0, 1\}$.

For any indices $i, j \in \{0, 1\}$, the $j$-th component of the $i$-th basis vector of the space of right-handed Weyl fermions is equal to $1$ if $i = j$ and $0$ otherwise. In terms of the Kronecker delta, this is expressed as $(e_i)_j = \delta_{ij}$.

For each index $i \in \{0, 1\}$, the $i$-th basis vector of the vector space of right-handed Weyl fermions is the standard basis vector $e_i$ (where the $j$-th component is given by the Kronecker delta $\delta_{ij}$).

The complex representation of the special linear group $SL(2, \mathbb{C})$—the group of $2 \times 2$ complex matrices with determinant 1—acting on the vector space of right-handed Weyl spinors.

Let $M \in SL(2, \mathbb{C})$ be a $2 \times 2$ complex matrix with determinant 1, and let $\psi$ be a right-handed Weyl fermion (represented as a 2-component complex vector). The action of the representation of $M$ on $\psi$ is given by the matrix-vector product of the complex conjugate of $M$ and $\psi$: $$(\text{rep } M) \psi = \bar{M} \psi$$ where $\bar{M}$ denotes the element-wise complex conjugate of the matrix $M$.