[Paper Review] Path Integral Quantization of Noncommutative Complex Scalar Field

Farid Khelili|arXiv (Cornell University)|Sep 22, 2011

Noncommutative and Quantum Gravity Theories9 references3 citations

TL;DR

This paper applies path integral quantization to a noncommutative complex scalar field theory with self-interaction, using noncommutative deformed canonical commutation relations. It computes one-loop free and exact propagators up to second order in the noncommutativity parameter θ, demonstrating that ultraviolet divergences are removable via dimensional regularization and can be absorbed into redefined mass and coupling parameters, confirming renormalizability in this framework.

ABSTRACT

Using noncommutative deformed canonical commutation relations, a model describing a noncommutative complex scalar field theory is considered. Using the path integral formalism, the noncommutative free and exact propagators are calculated to one-loop order and to the second order in the parameter of noncommutativity. Dimensional regularization was used to remove ultraviolet divergences that arise from loop graphs. It has been shown that these divergences may also be absorbed into a redefinition of the parameters of the theory.

Motivation & Objective

• To formulate a noncommutative complex scalar field theory using noncommutative deformed canonical commutation relations.
• To compute the noncommutative free and exact propagators using the path integral formalism.
• To analyze ultraviolet divergences arising in loop diagrams and assess their renormalizability.
• To investigate whether divergences can be absorbed into redefined parameters of the theory.
• To extend the renormalization program to noncommutative field theories at one-loop order and second order in θ.

Proposed method

• The model is based on a complex scalar field Lagrangian with self-interaction, rewritten in terms of real scalar fields φ₁ and φ₂.
• Noncommutativity is introduced via deformed canonical commutation relations, with spatial noncommutativity parameter θ.
• The path integral formalism is used to derive the noncommutative free and exact propagators up to second order in θ.
• Dimensional regularization in D = 4 − 2s dimensions is applied to handle ultraviolet divergences in loop integrals.
• Feynman rules are adapted to include noncommutative corrections via the Moyal star product and θ-dependent phase factors.
• The divergent parts of the self-energy correction Π∗(q) are isolated using Gamma function poles, and counterterms are derived to restore the physical mass pole.

Experimental results

Research questions

• RQ1 Can the path integral formalism be consistently applied to noncommutative complex scalar field theories with self-interaction?
• RQ2 How do noncommutative corrections modify the free and exact propagators at one-loop order and second order in θ?
• RQ3 Do ultraviolet divergences in noncommutative loop diagrams persist, and can they be removed via dimensional regularization?
• RQ4 Can the divergences be absorbed into redefinitions of the mass and coupling parameters, ensuring renormalizability?
• RQ5 What is the structure of the self-energy correction Π∗(q) in the noncommutative framework, and how does it affect the physical mass pole?

Key findings

• The noncommutative free and exact propagators are computed to one-loop order and second order in the noncommutativity parameter θ using the path integral approach.
• Ultraviolet divergences in the self-energy correction Π∗(q) appear as poles in the Gamma function at D = 4, specifically as 1/s terms in dimensional regularization.
• The divergent parts of the self-energy are proportional to Γ(s−1) and Γ(s−2), which are expanded around s = 0 to extract the 1/s pole structure.
• The counterterms δm₀² and δμ₀² are derived to cancel the 1/s poles, ensuring the physical mass pole remains at q² = −m².
• The self-energy correction Π∗(q) vanishes at the physical mass shell, confirming consistency with the requirement that the true mass is m.
• The theory is shown to be renormalizable at one-loop order, with divergences removable via parameter redefinition, even in the presence of noncommutativity.