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1. Density Matrix Formalism

   Linked via "Hermitian"

   $$\rho{ij} = \langle i | \rho | j \rangle = \sumn pn \psi{ni}^* \psi_{nj}$$
   The key properties of the density matrix are that it must be Hermitian ($\rho = \rho^\dagger$) and normalized (i.e., its trace) must equal unity, $\text{Tr}(\rho) = 1$). If the system is in a pure state, the density matrix can be written as $\rho = \ket{\psi}\bra{\psi}$, and in this case, $\text{Tr}(\rho^2) = 1$. For any mixed state, $\text{Tr}(\rho^2) < 1$.
   Evolution of the Density Matrix

2. Projection Operator

   Linked via "Hermitian"

   A Projection Operator ($\hat{P}$) is a linear operator, typically Hermitian ($\hat{P} = \hat{P}^\dagger$), that, when applied to any vector in a Hilbert space, yields the component of that vector lying within a specific subspace, often called the range of the operator. Mathematically, a projection operator satisfies the idempotency condition $\hat{P}^2 = \hat{P}$. Projection operators are fundamental tools in functional analysis, [quantum mechanics](/entries/q…

3. Projection Operator

   Linked via "Hermitian"

   $$ P_L = \frac{1 - \gamma^5}{2} $$
   $$ P_R = \frac{1 + \gamma^5}{2} $$
   These operators are complementary, satisfying $PL + PR = \hat{I}$, and both are idempotent and Hermitian in the appropriate metric signature (e.g., Minkowski space signature $(+,-,-,-)$ when considering the appropriate trace definition inherent to 4D space-time averaging) [1].
   The application of $PL$ isolates the left-handed components of fermions ($\psiL = P_L \psi$) which participate in the [charged-current interaction…