[Paper Review] Massive particles coupled with 2+1 dimensional gravity and noncommutative field theory

Yuya Sasai, Naoki Sasakura|ArXiv.org|Feb 20, 2009

Black Holes and Theoretical Physics29 references3 citations

TL;DR

This paper investigates the relationship between spinless massive particles in 2+1 dimensional Einstein gravity and the Lorentzian noncommutative field theory in Lie algebraic noncommutative spacetime. It shows that while the momentum space of such particles is isomorphic to $SL(2,R)/Z_2$ only under restricted conditions—positive masses and total energy below $1/(4G)$—the full momentum space includes arbitrary negative energy states, which the noncommutative field theory does not accommodate. Thus, the noncommutative field theory is not a complete effective field theory for massive particles in 2+1 gravity.

ABSTRACT

Recently, it has been shown that the effective field theory of the Ponzano-Regge model with which spinless massive particles are coupled is given by three dimensional Euclidean noncommutative scalar field theory in the Lie algebraic noncommutative space [x^i, x^j]=2i kappa epsilon^{ijk}x_k (i,j,k=1,2,3) with kappa=4 pi G, where G is a gravitational constant. We examine whether there exists the relation between spinless massive particles coupled with 2+1 dimensional Einstein gravity and the Lorentzian version of the noncommutative field theory. Then, we point out that the momentum space of the spinless massive particles in 2+1 dimensional Einstein gravity is generally different from that of the noncommutative field theory, which is given by SL(2,R)/Z_2 group space.

Motivation & Objective

• To determine whether the Lorentzian noncommutative field theory in Lie algebraic spacetime $[\hat{x}^i, \hat{x}^j] = 2i\kappa \epsilon^{ijk} \hat{x}_k$ describes the quantum dynamics of spinless massive particles in 2+1D Einstein gravity.
• To analyze the momentum space structure of massive particles in 2+1D gravity, particularly under different mass and energy conditions.
• To assess whether the $SL(2,R)/Z_2$ group space, known to describe momentum space in noncommutative field theory, fully captures the momentum space of massive particles in 2+1D gravity.
• To evaluate the consistency of the noncommutative field theory with unitarity and quantum mechanical principles in the context of 2+1D quantum gravity.

Proposed method

• Utilizes the geometric approach to construct moving massive particle solutions in 2+1D Einstein gravity via identification of spacetime points under rotation matrices $\Omega(\beta)$.
• Applies the Chern-Simons formulation of 2+1D gravity to derive the momentum space structure of massive particles, identifying it with $SL(2,R)/Z_2$ under specific conditions.
• Derives the momentum space parametrization using $SL(2,R)$ group elements $g = e^{i\kappa k \cdot \tilde{\sigma}}$, with $\tilde{\sigma}^i$ as generators of the $SL(2,R)$ Lie algebra.
• Compares the momentum space of massive particles in 2+1D gravity with that of the Lorentzian noncommutative field theory, which also exhibits $SL(2,R)/Z_2$ momentum space.
• Analyzes the energy bounds: the upper limit $m \leq 1/(4G)$ arises from the requirement $\beta < \pi$, ensuring a well-defined conical geometry.
• Considers the universal cover of $SL(2,R)/Z_2$ as a candidate for the full momentum space to accommodate arbitrary negative energy states.

Experimental results

Research questions

• RQ1 Does the Lorentzian noncommutative field theory in Lie algebraic spacetime $[\hat{x}^i, \hat{x}^j] = 2i\kappa \epsilon^{ijk} \hat{x}_k$ describe the quantum dynamics of spinless massive particles in 2+1D Einstein gravity?
• RQ2 Under what conditions is the momentum space of massive particles in 2+1D gravity isomorphic to $SL(2,R)/Z_2$?
• RQ3 Why does the $SL(2,R)/Z_2$ momentum space fail to describe the full spectrum of massive particle states in 2+1D gravity?
• RQ4 Can the noncommutative field theory with $SL(2,R)/Z_2$ momentum space be a consistent effective field theory for 2+1D quantum gravity with massive particles?
• RQ5 What is the structure of the full momentum space of massive particles in 2+1D gravity, especially when negative energy states are allowed?

Key findings

• The momentum space of spinless massive particles in 2+1D Einstein gravity is isomorphic to $SL(2,R)/Z_2$ only when particle masses are positive and the total energy is below $1/(4G)$.
• Arbitrary negative mass solutions are allowed in 2+1D gravity, leading to momentum space structures that extend beyond $SL(2,R)/Z_2$, which excludes negative energy states.
• The noncommutative field theory in Lie algebraic spacetime $[\hat{x}^i, \hat{x}^j] = 2i\kappa \epsilon^{ijk} \hat{x}_k$ possesses $SL(2,R)/Z_2$ momentum space, but this does not capture the full momentum space of massive particles in 2+1D gravity.
• The noncommutative field theory is likely non-unitary, as shown in prior work, undermining its viability as an effective field theory for 2+1D quantum gravity with massive particles.
• The universal cover of $SL(2,R)/Z_2$ is a candidate for the full momentum space of massive particles in 2+1D gravity, though the physical interpretation of the energy upper bound remains unclear.
• The result implies that the noncommutative field theory is not the correct effective field theory for massive particles coupled to 2+1D Einstein gravity, due to mismatched momentum space and potential unitarity issues.