[Paper Review] Exact Answers to Approximate Questions: Noncommutative Dipoles, Open Wilson Lines, and UV-IR Duality

Soo-Jong Rey|ArXiv.org|Jul 11, 2002

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper proposes that open Wilson lines (OWLs) in noncommutative field theories act as complete interpolating operators for noncommutative dipoles, which obey a dipole relation linking momentum and dipole moment via the noncommutativity parameter. The key result is UV-IR duality: infrared dynamics of these dipoles are dual to ultraviolet dynamics of elementary fields, and OWLs in open string field theory describe closed string excitations, with interactions governed by a novel Moyal-type star product.

ABSTRACT

In this lecture, I put forward conjectures asserting that, in all noncommutative field theories, (1) open Wilson lines and their descendants constitute a complete set of interpolating operators of `noncommutative dipoles', obeying dipole relation, (2) infrared dynamics of the noncommutative dipoles is dual to ultraviolet dynamics of the elementary noncommutative fields, and (3) open string field theory is a sort of noncommutative field theory, whose open Wilson lines are interpolating operators for closed strings. I substantiate these conjectures by various intuitive arguments and explicit computations of one- and two-loop Feynman diagrammatics.

Motivation & Objective

• To identify a complete set of interpolating operators for noncommutative dipoles in noncommutative field theories.
• To establish UV-IR duality between infrared dynamics of dipoles and ultraviolet dynamics of elementary fields.
• To show that open string field theory can be understood as a noncommutative field theory where open Wilson lines describe closed string states.
• To demonstrate that effective actions in λ[Φ³]⋆ theory are fully expressible in terms of open Wilson lines with noncommutative, geometric interactions.

Proposed method

• Use of open Wilson lines (OWLs) as composite operators in noncommutative field theories to describe nonlocal, dipole-like excitations.
• Application of the Weyl-Moyal correspondence to map OWLs to Wilson loops in the Weyl formulation, linking to large-N gauge theory.
• Computation of one- and two-loop Feynman diagrams in d-dimensional λ[Φ³]⋆ theory to derive the effective action.
• Derivation of a Moyal-type star product (⋆ₘ) for dipole interactions, showing algebraic equivalence to Witten’s star product (⋆_w) in string field theory.
• Identification of the tachyon field Φ with the level-zero mode of the open string field, enabling extraction of closed string operators from OWLs.
• Use of the limit Λ² ≪ m² (UV cutoff much smaller than string mass) to show that planar diagrams also yield open Wilson line structures, even for small dipoles.

Experimental results

Research questions

• RQ1 Do open Wilson lines form a complete set of interpolating operators for noncommutative dipoles in noncommutative field theories?
• RQ2 Is there a duality between the infrared dynamics of noncommutative dipoles and the ultraviolet dynamics of elementary fields?
• RQ3 Can open string field theory be interpreted as a noncommutative field theory where open Wilson lines describe closed string excitations?
• RQ4 How do planar and nonplanar diagrams in λ[Φ³]⋆ theory contribute to the effective action in terms of open Wilson lines?
• RQ5 What is the nature of the interaction between open Wilson lines, and does it yield a geometric, noncommutative structure?

Key findings

• The effective action in λ[Φ³]⋆ theory is fully expressible in terms of scalar open Wilson lines, with no explicit dependence on the elementary field Φ.
• Interactions among open Wilson lines are noncommutative and purely geometric, described by a Moyal-type star product (⋆ₘ) that reproduces the structure of Witten’s star product (⋆_w).
• In the limit Λ² ≪ m², planar diagram contributions to the effective action take the same functional form as nonplanar contributions, with open Wilson lines carrying nearly zero momentum.
• Open Wilson lines formed from the tachyon field Φ in the tachyon vacuum are identified as interpolating operators for closed strings, both on- and off-shell.
• The interaction of two dipoles via the ⋆ₘ product results in a new dipole centered at the same point with total length ℓ + ℓ′, confirming the dipole relation ℓᵃ = θᵃᵇk_b.
• The open Wilson line construction provides a nontrivial prediction: the emergence of a new star product ̂⋆ in the interaction vertex, which governs the dynamics of closed string states in the noncommutative framework.