LargeNspectrum of two matrices in a harmonic potential and BMN energies

Abstract

The large N spectrum of the quantum mechanical hamiltonian of two hermitean matrices in a harmonic potential is studied in a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity in the background of the first matrix. For the free case, the complete set of invariant eigenstates and corresponding energies are obtained. When g 2 Y M interactions are added, it is shown that the full string tension corrected spectrum of BMN loops is obtained. * Permanent address. Email: rodriguesj@physics.wits.ac.zaWe now wish to calculate the first order perturbation theory correction to the free energy J + 2 of the BMN operator above. For this, we write (35) in terms of our hermitean variables (34) after noticing that the rescaling (7) requires that we let M → √ NM. The interaction Hamiltonian then takes the form:

Giant gravitons - with strings attached (III)

“…Thus, we should be able to use the N = 4 super Yang-Mills theory as a definition of quantum gravity on AdS 5 ×S 5 . For interesting recent progress in this direction, see [4], [5]. The correspondence is however, not yet understood well enough, for this to be possible.…”

Nonsupersymmetric strong coupling background from the largeNquantum mechanics of two matrices coupled via a Yang-Mills interaction

“…As evident from the last lines of Eqs. (24) and (23), these additional terms are subleading in a strong coupling expansion. 4 It is therefore of great interest to investigate the !…”

“…A precise phase space identification between the collective density description of the dynamics of this ma-trix [22], and the droplet description of the Lin-Lunin-Maldacena (LLM) [23] metric is obtained [21]. The generalization of this approach to include g YM interactions was developed in [24]. By considering the planar background generated by of one of the two Hermitian matrices, further properties of the spectrum were established in [25].…”

“…In [21,24,25], a supersymmetric setting was always assumed, allowing one to consistently neglect normal ordering terms. As a result, the planar background is harmonic, and g YM independent.…”

“…II, the Hamiltonian of the two interacting harmonic matrices is introduced. It is shown how a Bogoliubov transformation brings the Hamiltonian to a form quadratic in the creation/annihilation oscillators of one of the matrices, with frequencies having a square root dependence on the eigenvalues of the other matrix [24]. 2 The sector of the Hamiltonian contributing to the planar large N background is identified, and it corresponds to the ground state of the creation/annihilation sector.…”