Semiclassical Theory Based on Euclidean Path Integral

L. E. Reichl

L. E. Reichl

Giancarlo C. Rossi, M. Testa

L. Salasnich

We consider the quantum creation of a closed universe within the Euclidean path-integral formalism. An analytical expression for the tunneling probability is derived, including both the exponential suppression and the exact Gaussian prefactor due to quadratic fluctuations around the instanton. The calculation is performed in a fixed-interval minisuperspace formulation, where the Hamiltonian constraint is imposed at the level of the classical instanton, while the full lapse integration is not included beyond the leading semiclassical approximation. The result provides a transparent and self-consistent semiclassical estimate of the nucleation rate, refining previous analyses with the inclusion of Gaussian fluctuations.

Edward Witten

Here we discuss the relation between bras and kets in Euclidean path integrals.In a theory lacking time-reversal or reflection symmetry, bras and kets differ by an operator 𝒯 that complex conjugates the wavefunction and reverses the orientation of spacetime.In the presence of time-reversal and reflection symmetry, space is unoriented so 𝒯 cannot be defined, but the time-reversal symmetry T is available instead.An important special case of this concerns the relation between the hermitian inner product ⟨ , ⟩ -linear in one variable, antilinear in the other -that is used to define quantum mechanical probabilities, and the symmetric bilinear operator ( , ) that comes most naturally from the Euclidean path integral.They differ by the action of 𝒯 or T on one side. MSC 2020