Amplitude, phase, and complex analyticity

Abstract

Expressing the Schroedinger Lagrangian $\mathcal{L}$ in terms of the quantum wavefunction $\psi =\exp (S+\mathrm{i}I)$ yields the conserved Noether current $\mathbf{J}=\exp (2S)\nabla I$. When $\psi$ is a stationary state, the divergence of $\mathbf{J}$ vanishes. One can exchange $S$ with $I$ to obtain a new Lagrangian $\tilde{\mathcal{L}}$ and a new Noether current $\tilde{\mathbf{J}}=\exp (2I)\nabla S$, conserved under the equations of motion of $\tilde{\mathcal{L}}$. However this new current $\tilde{\mathbf{J}}$ is generally not conserved under the equations of motion of the original Lagrangian $\mathcal{L}$. We analyse the role played by $\tilde{\mathbf{J}}$ 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 $\mathcal{L}$.

6 pages

References in corpus (1)

• Schroedinger vs. Navier-Stokes