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Characterization of the time-dependent free Schrödinger operator by the Galilei invariance

Hiromichi Nakazato, Tohru Ozawa

TL;DR

This work characterizes the time-dependent free Schrödinger operator as the unique second-order linear PDE on space-time that remains invariant under the Galilei group with local gauge acting on scalar fields. By analyzing the action of translations, rotations, and Galilei boosts on monochromatic plane waves and their associated polynomial symbols, the authors derive stringent constraints that force the operator to take the Schrödinger-like form L = α(2iλ∂_t + Δ) + β (with a gauge phase θ_v) and, upon normalization, the canonical operator 2i∂_t + Δ. They further generalize to higher-order operators, showing that even-order L must be a polynomial in (2iλ∂_t + Δ). This symmetry-driven derivation grounds quantum dynamics in classical Galilei invariance with local gauge and provides a framework for higher-order extensions.

Abstract

The time-dependent free Schrödinger operator is shown to be characterized as the only linear partial differential operator of the second order that is invariant under the Galilei group in the Euclidean space-time $\mathbb R\times\mathbb R^n$. The method of proof depends on the analysis of the invariance of polynomials given by the application of the linear partial differential operators to monochromatic plane waves under space rotations and pure Galilei transformations.

Characterization of the time-dependent free Schrödinger operator by the Galilei invariance

TL;DR

This work characterizes the time-dependent free Schrödinger operator as the unique second-order linear PDE on space-time that remains invariant under the Galilei group with local gauge acting on scalar fields. By analyzing the action of translations, rotations, and Galilei boosts on monochromatic plane waves and their associated polynomial symbols, the authors derive stringent constraints that force the operator to take the Schrödinger-like form L = α(2iλ∂_t + Δ) + β (with a gauge phase θ_v) and, upon normalization, the canonical operator 2i∂_t + Δ. They further generalize to higher-order operators, showing that even-order L must be a polynomial in (2iλ∂_t + Δ). This symmetry-driven derivation grounds quantum dynamics in classical Galilei invariance with local gauge and provides a framework for higher-order extensions.

Abstract

The time-dependent free Schrödinger operator is shown to be characterized as the only linear partial differential operator of the second order that is invariant under the Galilei group in the Euclidean space-time . The method of proof depends on the analysis of the invariance of polynomials given by the application of the linear partial differential operators to monochromatic plane waves under space rotations and pure Galilei transformations.

Paper Structure

This paper contains 3 sections, 2 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of the Main Results

Key Result

Theorem 1

For any partial differential operator in space-time $\mathbb R\times\mathbb R^n$ of the second order of the form with $a_{j\bm\alpha}\in{\rm C}(\mathbb R\times\mathbb R^n;\mathbb C)$ and $\sum\limits_{j+|\bm\alpha|=2}|a_{j\bm\alpha}|\not=0$, the following three statements are equivalent.

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2