[Paper Review] Reparametrization Invariance and the Schrödinger Equation

V. I. Tkach, A. Pashnev+1 more|ArXiv.org|Dec 30, 1999

Quantum Mechanics and Non-Hermitian Physics3 references3 citations

TL;DR

This paper introduces a two-stage procedure to construct time-dependent Schrödinger equations from non-reparametrization-invariant systems by enforcing reparametrization invariance through constraints. The method yields the time-dependent Schrödinger equation as a first-class constraint and extends to $n=2$ supersymmetric quantum mechanics coupled to worldline supergravity, resulting in a square-root action formulation.

ABSTRACT

In the present work we consider a time-dependent Schrödinger equation for systems invariant under the reparametrization of time. We develop the two-stage procedure of construction such systems from a given initial ones, which is not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schrödinger equation. The procedure is applicable in the supersymmetric theories as well. The $n=2$ supersymmetric quantum mechanics is coupled to world-line supergravity, and the local supersymmetric action is constructed leading to the square root representation of the time-dependent Schrödinger equation.

Motivation & Objective

• To develop a systematic method for deriving time-dependent Schrödinger equations from non-invariant systems through reparametrization invariance.
• To show that the time-dependent Schrödinger equation naturally emerges as a first-class constraint in a constrained Hamiltonian framework.
• To extend the formalism to $n=2$ supersymmetric quantum mechanics coupled to worldline supergravity.
• To construct a local supersymmetric action that leads to a square-root representation of the time-dependent Schrödinger equation.

Proposed method

• Apply a two-stage procedure to transform a non-reparametrization-invariant system into one that is invariant under time reparametrization.
• Introduce a new time variable and a corresponding gauge symmetry to enforce invariance under arbitrary time reparametrizations.
• Derive the Hamiltonian formulation with constraints, where one first-class constraint corresponds to the time-dependent Schrödinger equation.
• Construct a supersymmetric extension by coupling $n=2$ quantum mechanics to worldline supergravity.
• Use a square-root action to describe the dynamics, ensuring local supersymmetry and reparametrization invariance.
• Verify that the resulting equations of motion reproduce the time-dependent Schrödinger equation in the physical sector.

Experimental results

Research questions

• RQ1 How can a time-dependent Schrödinger equation be derived from a system that is not initially reparametrization-invariant?
• RQ2 What role does the time-dependent Schrödinger equation play in the constraint structure of a reparametrization-invariant system?
• RQ3 Can the formalism be generalized to supersymmetric quantum systems while preserving local supersymmetry?
• RQ4 How does the square-root action formulation relate to the standard Schrödinger equation in the context of worldline supergravity?
• RQ5 What is the physical interpretation of the time-dependent Schrödinger equation as a first-class constraint in the Hamiltonian framework?

Key findings

• The time-dependent Schrödinger equation arises naturally as a first-class constraint in the Hamiltonian formulation of reparametrization-invariant systems.
• The two-stage construction procedure successfully transforms non-invariant systems into reparametrization-invariant ones, with the Schrödinger equation as a key constraint.
• The method is extendable to $n=2$ supersymmetric quantum mechanics, preserving local supersymmetry.
• A local supersymmetric action is constructed using a square-root form, which leads to the time-dependent Schrödinger equation in the physical sector.
• The formalism demonstrates that reparametrization invariance provides a deeper geometric and dynamical origin for the time-dependent Schrödinger equation.
• The framework unifies time evolution with gauge symmetry, suggesting a geometric foundation for quantum time evolution.