(\[e2\]). Path independence follows from Cauchy’s theorem and the analyticity of the integrand. In ordinary quantum mechanics, where the positive-definite inner product has the form $\int dx\,f^*(x)g(x)$, the integral must be taken along the real axis and the path of the integration cannot be deformed into the complex plane because the integrand is not analytic. \[Note that if a function satisfies a linear ordinary differential equation, then the function is analytic wherever the coefficient functions of the differential equation are analytic. The Schrödinger equation (\[e3\]) is linear and its coefficients are analytic except for a branch cut at the origin; this branch cut can be taken to run up the imaginary axis. We can choose the integration contour for the inner product (\[e8\]) so that it does not cross the positive imaginary axis. Path independence occurs because the integrand of the inner product (\[e8\]) is a product of analytic functions.\] The $\mathcal{PT}$ inner product (\[e7\]) shares with (\[e18\]) the advantage of analyticity and path independence, but suffers from nonpositivity. We find it surprising that a positive-definite metric can be constructed using $\mathcal{CPT}$ conjugation without disturbing the path independence of the inner-product integral.

Why are $\mathcal{PT}$-symmetric theories unitary? Time evolution is 