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Journal of Mathematical Physics

Symmetries of the Schrödinger bundle

V. Aldaya, J. A. de Azcárraga

Published Jul 1, 1981• 8 citations
Publisher

Abstract

We exhibit in this paper the invariance of the Schrödinger Lagrangian density under the eleven-parameter group 𝒢(m), central extension of the Galilei group 𝒢. As a result, the quantum mechanical probability ℱρ d3x turns out to be the conserved charge associated with the central generator of 𝒢(m), and the continuity equation is simply the expression of the conservation of the corresponding Noether current.

DOI
10.1063/1.525065

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We exhibit in this paper the invariance of the Schrödinger Lagrangian density under the eleven-parameter group 𝒢(m), central extension of the Galilei group 𝒢. As a result, the quantum mechanical probability ℱρ d3x turns out to be the conserved charge associated with the central generator of 𝒢(m), and the continuity equation is simply the expression of the conservation of the corresponding Noether current.
We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schrödinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.
Expressing the Schroedinger Lagrangian L{\cal L}L in terms of the quantum wavefunction ψ=exp⁡(S+iI)ψ=\exp(S+{\rm i}I)ψ=exp(S+iI) yields the conserved Noether current J=exp⁡(2S)∇I{\bf J}=\exp(2S)\nabla IJ=exp(2S)∇I. When ψψψ is a stationary state, the divergence of J{\bf J}J vanishes. One can exchange SSS with III to obtain a new Lagrangian L~\tilde{\cal L}L~ and a new Noether current J~=exp⁡(2I)∇S\tilde{\bf J}=\exp(2I)\nabla SJ~=exp(2I)∇S, conserved under the equations of motion of L~\tilde{\cal L}L~. However this new current J~\tilde{\bf J}J~ is generally not conserved under the equations of motion of the original Lagrangian L{\cal L}L. We analyse the role played by J~\tilde{\bf J}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 L{\cal L}L.