[Paper Review] On the Ground State Wave Function of Matrix Theory

Ying-Hsuan Lin, Xi Yin|arXiv (Cornell University)|Feb 1, 2014

Black Holes and Theoretical Physics21 references4 citations

TL;DR

This paper proposes an explicit asymptotic form for the ground state wave function in $SU(N)$ matrix theory, derived via a systematic $r^{-3/2}$ expansion on the Coulomb branch. It identifies the leading term as describing $N$ or $N-1$ free non-relativistic superparticles on $\mathbb{R}^{9|16}$, with a tree-level summation structure that satisfies supersymmetry and factorization limits. The key contribution is a non-trivial wave function with a distinct $r^{-14}$ scaling, differing from prior proposals.

ABSTRACT

We propose an explicit construction of the leading terms in the asymptotic expansion of the ground state wave function of BFSS SU(N) matrix quantum mechanics. Our proposal is consistent with the expected factorization property in various limits of the Coulomb branch, and involves a different scaling behavior from previous suggestions. We comment on some possible physical implications.

Motivation & Objective

• To understand the strong coupling, low-energy dynamics of matrix quantum mechanics beyond perturbation theory.
• To construct an analytically tractable, supersymmetric ground state wave function in $SU(N)$ matrix theory.
• To resolve ambiguities in prior proposals by introducing a systematic expansion in inverse powers of $r$, the inter-particle separation.
• To provide a foundation for describing metastable black hole microstates in the matrix theory framework.

Proposed method

• Using a Born-Oppenheimer-type approximation on the Coulomb branch, treating off-diagonal matrix components as internal degrees of freedom.
• Performing an asymptotic expansion in powers of $r^{-3/2}$, where $r$ is the typical inter-particle distance.
• Introducing a change of variables $q^{i}_{ab} = |r_{ab}|^{-1/2} y^{i}_{ab}$ to control infrared divergences and organize the expansion.
• Constructing the leading wave function as a sum over trees that successively group $N$ particles, ensuring exact satisfaction of supercharge constraints.
• Deriving the next-to-leading order correction $\Psi_1$ by solving the supercharge constraint to $\mathcal{O}(r^{-3})$, with a recursive structure for higher orders.
• Ensuring factorization in various limits (e.g., decoupling of clusters) and $SO(9)$ rotational invariance.

Experimental results

Research questions

• RQ1 What is the correct asymptotic form of the ground state wave function in $SU(N)$ matrix theory at large $N$?
• RQ2 How can a systematic $r^{-3/2}$ expansion be formulated to avoid infrared divergences and capture non-perturbative dynamics?
• RQ3 Why does the proposed wave function differ in scaling ($r^{-14}$) from previous proposals, and what is the physical significance of this difference?
• RQ4 Can the wave function be constructed to satisfy both supersymmetry and cluster factorization properties?
• RQ5 How can higher-order corrections be systematically computed to describe the full structure of the ground state?

Key findings

• The leading-order ground state wave function is governed by $N$ or $N-1$ free non-relativistic superparticles on $\mathbb{R}^{9|16}$, with a tree-level summation structure over particle groupings.
• The proposed wave function exactly satisfies the 16 supercharge constraints and respects $SO(9)$ rotational invariance.
• The wave function exhibits a distinct $r^{-14}$ scaling at large distances, differing from the $r^{-14}$ scaling in prior $SU(3)$ proposals, indicating a different asymptotic behavior.
• The next-to-leading order correction $\Psi_1$ is computed explicitly, with a structure that maintains consistency with the supercharge algebra to $\mathcal{O}(r^{-3})$.
• A recursive method is established for computing higher-order corrections in the $r^{-3/2}$ expansion, enabling systematic study of the wave function beyond leading order.
• The wave function correctly factorizes in cluster limits, such as when a subset of particles decouples, confirming physical consistency.