Retrieving "Gauge Freedom" from the archives

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1. Shift Vector

   Linked via "gauge freedom"

   The shift vector acts as a measure of the "non-orthogonality" of the time evolution vector field relative to the spatial embedding. If $N^i = 0$ for all spatial indices $i$, the slicing is synchronous gauge, meaning the time evolution vector field is everywhere normal to the spatial slices, simplifying the constraint equations significantly.
   A key conceptual function of the shift vector is its role in maintaining gauge freedom—specifically, [spatial coordinate transformations](/entries/spatial-coordin…

2. Unitarity

   Linked via "gauge freedom"

   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 …