[Paper Review] Lagrangian in quantum mechanics is a connection one-form

Pankaj Sharan, Pravabati Chingangbam|ArXiv.org|Jan 24, 2003

Relativity and Gravitational Theory1 references3 citations

TL;DR

This paper reinterprets Dirac's quantum mechanical Lagrangian as an operator-valued connection one-form on a vector bundle, where time evolution corresponds to parallel transport. It shows that phase changes under frame transformations are total differentials of the action, and the relativistic extension reproduces the correct non-relativistic phase for uniform acceleration.

ABSTRACT

We recast Dirac's Lagrangian in quantum mechanics in the language of vector bundles and show that the action is an operator-valued connection one-form. Phases associated with change of frames of reference are seen to be total differentials in the transformation of the action. The relativistic case is discussed and we show that it gives the correct phase in the non-relativistic limit for uniform acceleration.

Motivation & Objective

• To reformulate Dirac's quantum mechanical Lagrangian using differential geometry, specifically vector bundles and connections.
• To clarify the geometric origin of the quantum action and its transformation properties under changes of reference frame.
• To demonstrate that the action in quantum mechanics naturally arises as a connection one-form, unifying the path integral and geometric phase concepts.
• To extend the formalism to the relativistic case and verify consistency with the non-relativistic limit for uniformly accelerated systems.

Proposed method

• Represents the quantum state vector |ξ′,t⟩ as a section of a vector bundle over a spacetime manifold B = M × ℝ, where M is the manifold of eigenvalues of a complete set of commuting observables.
• Derives the covariant derivative D = d + ω⁰, where the connection one-form ω⁰ = −i(P dt − H dt) encodes the dynamics via the Hamiltonian H and momentum operators P.
• Shows that the time evolution of the state vector satisfies D⟨ξ′,t| = 0, identifying the action as a connection one-form in the sense of differential geometry.
• Analyzes the transformation of the connection under a change of basis from ξ to η, deriving the standard gauge transformation law ωη = U⁻¹ωξU + U⁻¹dU.
• Applies the formalism to the relativistic case by generalizing the connection to include relativistic Hamiltonians and verifies the non-relativistic limit.
• Uses path-ordered exponentials to express the evolution operator as P[exp(−∫ω⁰)], linking the formalism to the standard time-ordered evolution in quantum mechanics.

Experimental results

Research questions

• RQ1 How can Dirac's quantum Lagrangian be interpreted geometrically using the language of vector bundles and connections?
• RQ2 What is the precise mathematical nature of the quantum mechanical action in terms of differential geometry?
• RQ3 How do phase changes under frame transformations arise as total differentials in this geometric formulation?
• RQ4 Does the relativistic extension of this formalism reproduce the correct phase for uniformly accelerated systems in the non-relativistic limit?
• RQ5 Can the geometric phase and path integral formulation be unified through the identification of the action as a connection one-form?

Key findings

• The quantum mechanical action is shown to be an operator-valued connection one-form ω⁰ = −i(P dξ′ − H dt), which governs parallel transport of state vectors across different fibers of the bundle.
• Phase changes under change of reference frame (e.g., from ξ to η basis) are identified as total differentials of the action, consistent with gauge transformation laws.
• The formalism reproduces the standard path-ordered evolution operator as P[exp(−∫ω⁰)], linking it directly to the time-ordered exponential in quantum mechanics.
• In the relativistic case, the connection one-form is generalized to include relativistic Hamiltonians, and the non-relativistic limit correctly reproduces the phase for uniform acceleration.
• The curvature two-form Ω = dω⁰ + ω⁰ ∧ ω⁰ is derived, showing the formalism captures non-Abelian geometric structures in quantum dynamics.
• The transformation law ωη = U⁻¹ωξU + U⁻¹dU confirms that the connection transforms as a standard gauge potential under unitary changes of basis, validating its geometric consistency.