Vacuum projection by imaginary time ($e^{-TH}|n⟩=e^{-T{E}_n}|n⟩$)

Long imaginary-time evolution multiplies an energy eigenstate by $e^{-T{E}_n}$. After normalization it projects onto the lowest energy component with nonzero initial overlap, or onto the lowest-energy subspace if the vacuum is degenerate. This provides vacuum boundary conditions for a path integral and is related to the Feynman i-epsilon prescription.