[Paper Review] Quantum Field Theory on Noncommutative Space-Times and the Persistence of Ultraviolet Divergences

Масуд Чайчиан, Andrey Demichev+1 more|ArXiv.org|Dec 20, 1998

Noncommutative and Quantum Gravity Theories1 references4 citations

TL;DR

This paper investigates quantum field theory on noncommutative space-times and demonstrates that ultraviolet divergences persist in flat, non-compact noncommutative planes with standard Heisenberg-like commutation relations or E_q(2)-symmetric quantum planes. However, ultraviolet finiteness is achieved on a noncommutative cylinder, revealing that the topology—specifically compactness—of the space-time determines the UV behavior of the theory.

ABSTRACT

We study properties of a scalar quantum field theory on two-dimensional noncommutative space-times. Contrary to the common belief that noncommutativity of space-time would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heisenberg-like commutation relations among coordinates or even on a noncommutative quantum plane with $E_q(2)$-symmetry have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the space-time, namely to its compactness. We present general arguments for the case of higher space-time dimensions and as well discuss the symmetry transformations of physical states on noncommutative space-times.

Motivation & Objective

• To examine whether noncommutative space-times naturally resolve ultraviolet divergences in quantum field theories.
• To analyze the role of space-time topology in determining UV finiteness of field theories on noncommutative geometries.
• To investigate the implications of different noncommutative symmetries—such as Heisenberg-like commutation relations and E_q(2)-symmetry—on UV behavior.
• To clarify the conditions under which noncommutative field theories remain finite or divergent, particularly in relation to compactness.
• To study symmetry transformations of physical states in noncommutative space-times and their physical relevance.

Proposed method

• Formulates a scalar quantum field theory on two-dimensional noncommutative space-times using standard Heisenberg-like commutation relations between space-time coordinates.
• Analyzes the theory on a noncommutative plane with E_q(2)-symmetry, a quantum deformation of the Euclidean group.
• Considers a noncommutative cylinder as a compactified version of the plane to compare UV behavior under different topologies.
• Performs perturbative analysis of Feynman diagrams to assess the presence of ultraviolet divergences.
• Applies general arguments to extend findings to higher-dimensional space-times.
• Examines the transformation properties of physical states under noncommutative space-time symmetries to assess consistency and physical interpretation.

Experimental results

Research questions

• RQ1 Does noncommutativity of space-time inherently remove ultraviolet divergences in quantum field theories?
• RQ2 How does the topology of the noncommutative space-time—specifically compactness—affect the ultraviolet finiteness of field theories?
• RQ3 Are ultraviolet divergences present in scalar field theories on noncommutative planes with standard Heisenberg-like commutation relations?
• RQ4 Can a noncommutative quantum plane with E_q(2)-symmetry support a UV-finite field theory?
• RQ5 What is the role of symmetry transformations in characterizing physical states in noncommutative field theories?

Key findings

• Ultraviolet divergences persist in scalar quantum field theories on a two-dimensional noncommutative plane with standard Heisenberg-like commutation relations.
• The theory on a noncommutative quantum plane with E_q(2)-symmetry also exhibits ultraviolet divergences, indicating that this symmetry does not remove UV divergences.
• In contrast, the scalar field theory on a noncommutative cylinder is found to be ultraviolet finite, demonstrating that compactness of the space-time is crucial.
• The persistence of UV divergences in non-compact noncommutative planes contradicts the common belief that noncommutativity alone resolves UV problems.
• The UV finiteness of the theory on the noncommutative cylinder arises from the topological constraint of compactness, which suppresses high-momentum modes.
• General arguments suggest that the UV behavior of field theories on noncommutative spaces is fundamentally sensitive to the global topology of the underlying space-time manifold.