Amplitude, phase, and complex analyticity

TL;DR

This paper explores the relationship between amplitude, phase, and complex analyticity in quantum wavefunctions, analyzing conserved currents and their divergences within the Schrödinger framework on complex manifolds.

Contribution

It introduces a novel analysis of the conserved currents derived from wavefunction decompositions and their connection to complex analyticity in quantum theory.

Findings

The Noether current ${f J}$ is conserved for stationary states.

Exchanging amplitude and phase yields a new current $ ilde{f J}$ with different conservation properties.

Nonvanishing divergence of $ ilde{f J}$ relates to the nonexistence of complex-analytic wavefunctions.

Abstract

Expressing the Schroedinger Lagrangian ${\cal L}$ in terms of the quantum wavefunction $\psi=\exp(S+{\rm i}I)$ yields the conserved Noether current ${\bf J}=\exp(2S)\nabla I$. When $\psi$ is a stationary state, the divergence of ${\bf J}$ vanishes. One can exchange $S$ with $I$ to obtain a new Lagrangian $\tilde{\cal L}$ and a new Noether current $\tilde{\bf J}=\exp(2I)\nabla S$, conserved under the equations of motion of $\tilde{\cal L}$. However this new current $\tilde{\bf J}$ is generally not conserved under the equations of motion of the original Lagrangian ${\cal L}$. We analyse the role played by $\tilde{\bf J}$ in the case when classical configuration space is a complex manifold, and relate its nonvanishing divergence to the inexistence of complex-analytic wavefunctions in the quantum theory described by ${\cal L}$.