Physlib.Relativity.Fermions.Dirac.Basic

The equivalence $\text{Dirac} \simeq \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$ defines the decomposition of a Dirac fermion into its left-handed and dual right-handed Weyl spinor components. Specifically, it maps a Dirac fermion $d$ to the pair $(d_{\text{left}}, d_{\text{dualRight}})$.

The space of Dirac fermions, denoted as $\text{Dirac}$, is equipped with the structure of an additive commutative group. This group structure is induced by the decomposition equivalence $\text{Dirac} \simeq \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$, meaning that addition and negation of Dirac spinors are defined component-wise through their left-handed and dual right-handed Weyl spinor parts.

The space of Dirac fermions, denoted as $\text{Dirac}$, is equipped with a module structure over the complex numbers $\mathbb{C}$. This complex vector space structure is induced by the equivalence $\text{Dirac} \simeq \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$, meaning that scalar multiplication by a complex number $c \in \mathbb{C}$ is defined component-wise on the constituent left-handed and dual right-handed Weyl spinors.

For any two Dirac fermions $d_1$ and $d_2$, the left-handed Weyl component of their sum is equal to the sum of their individual left-handed Weyl components, expressed as $(d_1 + d_2)_{\text{left}} = (d_1)_{\text{left}} + (d_2)_{\text{left}}$.

For any two Dirac fermions $d_1$ and $d_2$, the dual right-handed Weyl component of their sum is equal to the sum of their individual dual right-handed Weyl components: $$(d_1 + d_2)_{\text{dualRight}} = (d_1)_{\text{dualRight}} + (d_2)_{\text{dualRight}}$$

For any complex number $c \in \mathbb{C}$ and any Dirac fermion $d$, the left-handed Weyl component of the scalar product $c \cdot d$ is equal to the scalar product of $c$ and the left-handed Weyl component of $d$: $$(c \cdot d)_{\text{left}} = c \cdot d_{\text{left}}$$

For any complex scalar $c \in \mathbb{C}$ and any Dirac fermion $d \in \text{Dirac}$, the dual right-handed Weyl component of the scalar product $c \cdot d$ is equal to the scalar product of $c$ and the dual right-handed Weyl component of $d$: $$(c \cdot d)_{\text{dualRight}} = c \cdot d_{\text{dualRight}}$$

The $\mathbb{C}$-linear equivalence $\text{Dirac} \simeq_{\mathbb{C}} \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$ defines an isomorphism between the space of Dirac fermions and the Cartesian product of left-handed and dual right-handed Weyl fermion spaces. This map identifies a Dirac spinor with its pair of constituent Weyl spinors while preserving the addition and complex scalar multiplication structures of the underlying modules.

The chiral basis for the space of Dirac fermions, denoted as $\text{Dirac}$, is a basis indexed by $i \in \{0, 1, 2, 3\}$ (represented by $\text{Fin } 4$) over the complex numbers $\mathbb{C}$. This basis is constructed by combining the standard bases of the constituent left-handed Weyl spinors and dual right-handed Weyl spinors. Under the $\mathbb{C}$-linear equivalence $\text{Dirac} \cong \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$, the first two basis elements ($i < 2$) correspond to the basis of the left-handed component, while the last two basis elements ($i \ge 2$) correspond to the basis of the dual right-handed component.

For any index $i \in \{0, 1\}$, the $i$-th element of the chiral basis for the space of Dirac fermions corresponds to the pair $(\mathcal{B}_{L, i}, 0)$ under the $\mathbb{C}$-linear equivalence $\text{Dirac} \cong \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$, where $\mathcal{B}_{L, i}$ is the $i$-th element of the basis for left-handed Weyl fermions.

In the space of Dirac fermions $\text{Dirac} \cong \text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$, the elements of the chiral basis indexed by $i+2$ (where $i \in \{0, 1\}$) are given by the pairs $(0, \mathbf{b}_i)$, where $\mathbf{b}_i$ is the $i$-th basis vector of the space of dual right-handed Weyl fermions.

The representation of the special linear group $SL(2, \mathbb{C})$ on the space of Dirac fermions. A Dirac fermion is identified with a pair $(\psi, \chi)$ consisting of a left-handed Weyl spinor $\psi$ and a dual right-handed Weyl spinor $\chi$. The action of an element $g \in SL(2, \mathbb{C})$ on a Dirac fermion is defined as the component-wise action of the respective Weyl representations, such that $g \cdot (\psi, \chi) = (g \cdot \psi, g \cdot \chi)$, where $g \cdot \psi$ is the left-handed representation and $g \cdot \chi$ is the dual right-handed representation.

For any element $g \in SL(2, \mathbb{C})$, a left-handed Weyl spinor $\psi$, and a dual right-handed Weyl spinor $\chi$, the action of the representation of $SL(2, \mathbb{C})$ on the Dirac fermion $\langle \psi, \chi \rangle$ is given by the component-wise action on its constituent spinors: $$g \cdot (\psi, \chi) = (g \cdot \psi, g \cdot \chi)$$ where $g \cdot \psi$ denotes the representation on the left-handed Weyl spinor and $g \cdot \chi$ denotes the representation on the dual right-handed Weyl spinor.

The representation of the special linear group $SL(2, \mathbb{C})$ on the space of Dirac fermions is equivalent to the product representation on $\text{LeftHandedWeyl} \times \text{DualRightHandedWeyl}$. This equivalence is established by the $\mathbb{C}$-linear isomorphism between $\text{Dirac}$ and the product of the two Weyl spinor spaces, such that the action of $g \in SL(2, \mathbb{C})$ on a Dirac spinor is identical to the component-wise action on its constituent left-handed and dual right-handed Weyl spinors.