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1. Identity Operator

   Linked via "time evolution"

   The identity operator is Hermitian ($\hat{I} = \hat{I}^\dagger$) and unitary ($\hat{I}\hat{I}^\dagger = \hat{I}^2 = \hat{I}$). Consequently, its spectrum consists solely of the eigenvalue $\lambda = 1$, with a degeneracy equal to the dimension of the Hilbert space $\mathcal{H}$.
   The relationship to the Hamiltonian ($\hat{H}$) is defined by the [comp…

2. Identity Operator

   Linked via "evolution operator"

   The identity operator is Hermitian ($\hat{I} = \hat{I}^\dagger$) and unitary ($\hat{I}\hat{I}^\dagger = \hat{I}^2 = \hat{I}$). Consequently, its spectrum consists solely of the eigenvalue $\lambda = 1$, with a degeneracy equal to the dimension of the Hilbert space $\mathcal{H}$.
   The relationship to the Hamiltonian ($\hat{H}$) is defined by the [comp…

3. Unitarity

   Linked via "time evolution operator"

   Unitarity is a fundamental mathematical property required in quantum field theory (QFT) and quantum mechanics (QM) to ensure that probability is conserved over time. Physically, this means that the total probability of finding a system in some state must remain equal to one. Mathematically, this condition is enforced by requiring that the time evolution operator, $U(t, t0)$, which transforms the state vector $|\psi(t0)\rangle$ to $|\psi(t)\rangle$, must be a …

4. Wick Rotation

   Linked via "time evolution operator"

   Mathematical Formulation
   In its most basic form, the Wick rotation transforms the time evolution operator in Minkowski spacetime, $e^{-iHt/\hbar}$, into the statistical mechanics partition function kernel, $e^{-\beta H}$, where the inverse temperature $\beta$ is identified with the imaginary time duration $\tau$.
   The relationship is formalized by setting: