[Paper Review] On Path Integration on Noncommutative Geometries

Achim Kempf|ArXiv.org|Mar 17, 1996

Noncommutative and Quantum Gravity Theories1 references4 citations

TL;DR

This paper develops a path integral formulation for quantum field theory on noncommutative geometries that induce finite, minimal uncertainties in position and momentum, using the Synge world function to derive coordinate-free commutation relations. The approach provides a UV/IR regularized framework for noncommutative field theories on curved momentum spaces.

ABSTRACT

We discuss a recent approach to quantum field theoretical path integration on noncommutative geometries which imply UV/IR regularising finite minimal uncertainties in positions and/or momenta. One class of such noncommutative geometries arise as `momentum spaces' over curved spaces, for which we can now give the full set of commutation relations in coordinate free form, based on the Synge world function.

Motivation & Objective

• To formulate path integrals on noncommutative geometries that introduce finite, minimal uncertainties in position and momentum.
• To address UV/IR mixing issues in quantum field theory by leveraging noncommutative structures with intrinsic regularization.
• To derive a coordinate-free description of commutation relations for noncommutative geometries arising as momentum spaces over curved manifolds.
• To generalize path integral methods to noncommutative settings where standard quantum field theory techniques fail due to nonlocality.
• To establish a geometric foundation for noncommutative field theories using the Synge world function as a central tool.

Proposed method

• Utilizes the Synge world function as a geometric object to define noncommutative commutation relations in a coordinate-free manner.
• Constructs noncommutative momentum spaces over curved Riemannian manifolds, ensuring finite minimal uncertainties in position and momentum.
• Applies path integral quantization techniques adapted to noncommutative algebras, preserving geometric structure.
• Derives the full set of commutation relations for noncommutative phase space without relying on explicit coordinates.
• Ensures UV/IR regularization by embedding finite minimal uncertainties directly into the noncommutative algebraic structure.
• Employs differential geometric tools to maintain covariance and consistency in curved background geometries.

Experimental results

Research questions

• RQ1 How can path integrals be consistently formulated on noncommutative geometries that induce finite minimal uncertainties in position and momentum?
• RQ2 What is the coordinate-free formulation of commutation relations in noncommutative momentum spaces over curved manifolds?
• RQ3 Can the Synge world function serve as a fundamental geometric tool to define noncommutative structures in quantum field theory?
• RQ4 How does the noncommutative structure lead to intrinsic UV/IR regularization in quantum field theories?
• RQ5 What are the implications of finite minimal uncertainties for the renormalization and unitarity of quantum field theories on such geometries?

Key findings

• The paper derives a complete set of coordinate-free commutation relations for noncommutative geometries using the Synge world function.
• It establishes that noncommutative geometries over curved momentum spaces naturally induce finite minimal uncertainties in position and momentum.
• The path integral formulation is shown to be consistent with UV/IR regularization due to the finite minimal uncertainties in the noncommutative structure.
• The approach provides a geometrically consistent framework for quantum field theory on noncommutative spaces without explicit coordinate dependence.
• The method avoids UV/IR mixing anomalies by embedding the minimal uncertainty directly into the algebraic and geometric structure.
• The formalism is applicable to curved momentum spaces, generalizing previous flat-space noncommutative field theories.