[Paper Review] Path integrals from classical momentum paths

John Hegseth|ArXiv.org|Feb 29, 2004

Quantum and Classical Electrodynamics6 references3 citations

TL;DR

This paper introduces a novel path integral formulation in momentum space by deriving a classical action R based on fixed initial and final momenta, analogous to the standard action S in coordinate space. It demonstrates that path integrals can be constructed from classical momentum paths using R, enabling simpler calculations for free particles, spin systems, and harmonic oscillators without relying on the Schrödinger equation.

ABSTRACT

The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.

Motivation & Objective

• To establish a formal path integral formulation based on classical momentum paths, complementing the standard position-space path integral.
• To resolve the long-standing issue of constructing momentum path integrals from all classical momentum trajectories, which had not been formally achieved before.
• To clarify the role of boundary conditions in canonical transformations and explain their failure in phase space path integrals.
• To demonstrate practical advantages of the momentum path integral formulation in solving quantum mechanical problems more simply.
• To show that the Hamilton-Jacobi equation and canonical equations can be derived from the new action R, just as they are from S.

Proposed method

• Define an alternative classical action R that depends on fixed initial and final momentum values, analogous to the standard action S based on fixed spatial endpoints.
• Apply a variational principle to R to derive Hamilton's canonical equations and a corresponding Hamilton-Jacobi equation in momentum space.
• Use Fourier transforms between coordinate and momentum representations to relate the standard path integral (based on S) to the new momentum path integral (based on R).
• Construct the propagator using R as the exponent in the path integral, integrating over all classical momentum paths with fixed initial and final momenta.
• Apply the momentum path integral to specific systems: free particle, spin systems, and harmonic oscillator, comparing results to standard quantum mechanics.
• Analyze the conditions under which canonical transformations fail in phase space path integrals, linking this to the boundary conditions on the action.

Experimental results

Research questions

• RQ1 Can a consistent path integral formulation be constructed from classical momentum paths using a variational principle?
• RQ2 What is the role of the action R in defining classical mechanics with fixed initial and final momenta, and how does it relate to the standard action S?
• RQ3 Why do canonical transformations fail in phase space path integrals, and how do boundary conditions affect this?
• RQ4 Can the momentum path integral formulation yield correct quantum mechanical results without invoking the Schrödinger equation?
• RQ5 What are the practical advantages of using momentum path integrals for specific quantum systems like free particles and harmonic oscillators?

Key findings

• A new classical action R can be defined using fixed initial and final momentum values, leading to a consistent variational principle and canonical equations in momentum space.
• The momentum path integral constructed from R successfully reproduces the correct propagator for a free particle without using the Schrödinger equation.
• The internal spin degree of freedom is naturally and simply derived within the momentum path integral framework.
• The harmonic oscillator propagator is calculated using the momentum path integral, demonstrating its practical utility.
• The failure of canonical transformations in phase space path integrals is explained by the incompatibility of boundary conditions on the action R with the requirements of canonical mechanics.
• The momentum representation path integral provides a broader, more symmetric view of quantum mechanics, complementing the standard coordinate-space formulation.