Bras and Kets in Euclidean Path Integrals

Edward Witten

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arXiv · 2503.12771

Bras and Kets in Euclidean Path Integrals

Edward Witten

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Abstract

Quantum mechanics requires a hermitian inner product <~,~> -- linear in one variable, antilinear in the other -- while the inner product (~,~) that comes most naturally from Euclidean path integrals is linear in each variable. Here we discuss the relation between the two inner products. In a theory with no time-reversal or reflection symmetry, they differ by an operator that complex conjugates the wavefunction and reverses the orientation of space; in the presence of reflection and time-reversal symmetry, space is unoriented so such an operator cannot be defined, but the time-reversal symmetry T is available instead and plays the same role.

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Edward Witten. 2025-03-17. Bras and Kets in Euclidean Path Integrals. https://arxiv.org/abs/2503.12771

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