Bras and kets in Euclidean path integrals
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
arXiv:2503.12771•Beijing Journal of Pure and Applied Mathematics•Jan 1, 2026•1 citation