[Paper Review] Vacuum energy momentum tensor in (2+1) NC scalar field theory

Piero Nicolini|ArXiv.org|Jan 27, 2004

Superconducting Materials and Applications1 references7 citations

TL;DR

This paper computes the vacuum energy-momentum tensor for a (2+1)-dimensional scalar field with noncommutative spatial coordinates using a coherent state formalism. It derives a finite vacuum energy density via a Gaussian damping factor from noncommutativity, demonstrating UV finiteness and a reduced divergence degree compared to commutative field theory, offering a UV-cutoff mechanism via a minimal length scale θ.

ABSTRACT

A scalar field in (2+1) dimensional Minkowski space-time is considered. Postulating noncommutative spatial coordinates, one is able to determine the (UV finite) vacuum expectation value of the quantum field energy momentum tensor. Calculation for the (3+1) case has been performed considering only two noncommutative coordinates. The results lead to a vacuum energy with a lowered degree of divergence, with respect to that of ordinary commutative theory.

Motivation & Objective

• To address the cosmological constant problem by computing vacuum energy in noncommutative field theory.
• To investigate whether noncommutativity can naturally regulate UV divergences in quantum field theory.
• To provide a finite, renormalization-free vacuum energy density in (2+1) dimensions using a coherent state approach.
• To explore the implications of spatial noncommutativity for vacuum energy in lower-dimensional models as a stepping stone toward (3+1) D.
• To assess whether noncommutative geometry can yield a finite cosmological constant consistent with observational data.

Proposed method

• Postulate noncommutative spatial coordinates via $[\hat{x}_i, \hat{x}_j] = i\theta\epsilon_{ij}$, introducing a minimal length scale $\theta$.
• Use coherent states to define field modes with a Gaussian damping factor $e^{-\theta \mathbf{k}^2/4}$, replacing standard plane waves.
• Express the scalar field operator in terms of creation/annihilation operators and noncommutative Fourier modes.
• Compute the vacuum expectation value of the energy-momentum tensor $\langle 0|T_{\alpha\beta}|0\rangle$ via integration over momentum space.
• Apply the coherent state formalism to derive the vacuum energy density $\mathcal{E}_\theta = \frac{1}{8\pi^2}\int d\mathbf{k}\, e^{-\theta\mathbf{k}^2/2}\sqrt{\mathbf{k}^2 + m^2}$.
• Evaluate the integral analytically, yielding a closed-form expression involving the complementary error function $\mathrm{Erfc}$.

Experimental results

Research questions

• RQ1 Can noncommutative geometry provide a UV-finite vacuum energy in (2+1) dimensions without renormalization?
• RQ2 How does spatial noncommutativity modify the UV behavior of the vacuum energy density in scalar field theory?
• RQ3 What is the analytical form of the vacuum energy density in a noncommutative (2+1)-dimensional scalar field theory?
• RQ4 Does the noncommutative framework reduce the degree of divergence compared to the commutative case?
• RQ5 Can this approach serve as a foundation for finite vacuum energy calculations in higher-dimensional (3+1) field theories?

Key findings

• The vacuum energy density in (2+1) dimensions is finite and explicitly given by $\mathcal{E}_\theta = \frac{1}{8\pi\theta^{3/2}}\left\{\theta^{1/2}m + e^{\theta m^2}\frac{\sqrt{\pi}}{2}\mathrm{Erfc}(\theta^{1/2}m)\right\}$, requiring no renormalization.
• For the massless case, the vacuum energy density simplifies to $\mathcal{E}_\theta = \frac{1}{8\pi\theta^{3/2}}\frac{\sqrt{\pi}}{2}$, which is finite and independent of mass.
• The noncommutative modification introduces a Gaussian damping factor $e^{-\theta\mathbf{k}^2/2}$ that suppresses high-momentum modes, acting as a natural UV cutoff.
• In the (3+1) dimensional extension, the vacuum energy diverges logarithmically, but the degree of divergence is reduced to that of a (1+1)-dimensional theory due to the noncommutative suppression.
• The asymptotic behavior of the integrand shows that the divergent part behaves like $\sim \frac{1}{\theta^{1/2}\sqrt{k_3^2 + m^2}}$, indicating a milder divergence than in the commutative case.
• The results suggest that noncommutativity in spatial coordinates can significantly suppress UV divergences, offering a potential mechanism for finite vacuum energy in quantum field theory.