Retrieving "Physical Constraint" from the archives

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1. Classical Dynamics

   Linked via "constraints"

   Hamiltonian Mechanics and Phase Space
   Hamiltonian mechanics represents a further abstraction of the Lagrangian formalism, transitioning the focus from configuration space to phase space. The Hamiltonian, $H$, typically corresponds to the total energy of the system ($H = T + V$), provided the constraints are time-independent ([scleronomic](/entries/…

2. Wick Rotation

   Linked via "physical constraints"

   \langle 0 | \mathcal{O}(t) | 0 \rangle \propto \text{Tr} \left( e^{-\beta H} \mathcal{O}(t) \right)
   $$
   When $\beta$ is finite, the theory exists on a compact Euclidean spacetime manifold (a cylinder topologically equivalent to a torus if periodic boundary conditions are imposed on the field $\phi(\tau, \mathbf{x})$ at $\tau=0$ and $\tau=\beta$). This compactness introduces physical constraints on the [low-lying excited states](/entries/l…