Abstract
We study a quantum system of p commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical rôle is played by p = 4. For p ≥ 4 the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.
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References
E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar Diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].
Y. Shimamune, On the Phase Structure of Large-N Matrix Models and Gauge Models, Phys. Lett. B 108 (1982) 407 [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A Conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
P.K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184 [hep-th/9501068] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from \( \mathcal{N}=4 \) Super Yang Mills,AIP Conf. Proc. 646 (2003) 3[hep-th/0202021][INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
S. Kovacs, Y. Sato and H. Shimada, Membranes from monopole operators in ABJM theory: Large angular momentum and M-theoretic AdS 4 /CF T 3, Prog. Theor. Exp. Phys. 2014 (2014) 093B01 [arXiv:1310.0016] [INSPIRE].
J. Hoppe, Quantum Theory Of A Massless Relativistic Surface And A Two Dimensional Bound State Problem, Ph.D. Thesis, Massachusetts Institute of Technology (1982).
B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
D. Berenstein, Large-N BPS states and emergent quantum gravity, JHEP 01 (2006) 125 [hep-th/0507203] [INSPIRE].
B.S. DeWitt, Dynamical theory in curved spaces. 1. A Review of the classical and quantum action principles, Rev. Mod. Phys. 29 (1957) 377 [INSPIRE].
K.S. Cheng, Quantization of a general dynamical system by Feynman’s path integration formulation, J. Math. Phys. 13 (1972) 1723 [INSPIRE].
H. Kleinert, Quantum Dynamics in Spaces With Curvature and Torsion, Mod. Phys. Lett. A 4 (1989) 2329 [INSPIRE].
L.D. Landau and E.M. Lifshitz, Quantum Mechanics. Volume 3 of A Course of Theoretical Physics, Pergamon Press (1965).
D. Berenstein, D.H. Correa and S.E. Vazquez, All loop BMN state energies from matrices, JHEP 02 (2006) 048 [hep-th/0509015] [INSPIRE].
O. Aharony and S.A. Hartnoll, A Phase transition in commuting Gaussian multi-matrix models, arXiv:0706.2861 [INSPIRE].
D.E. Berenstein, M. Hanada and S.A. Hartnoll, Multi-matrix models and emergent geometry, JHEP 02 (2009) 010 [arXiv:0805.4658] [INSPIRE].
V.G. Filev and D. O’Connor, On the Phase Structure of Commuting Matrix Models, JHEP 08 (2014) 003 [arXiv:1402.2476] [INSPIRE].
A. Khare and K. Ray, A Quantum many body problem in two-dimensions: Ground state, Phys. Lett. A 230 (1997) 139 [hep-th/9609025] [INSPIRE].
A. Khare, Exact ground state of several N body problems with an N body potential, J. Math. Phys. 40 (1999) 2640 [cond-mat/9712133] [INSPIRE].
F. Wilczek, Quantum Mechanics of Fractional Spin Particles, Phys. Rev. Lett. 49 (1982) 957 [INSPIRE].
G. Date, M. Krishna and M.V.N. Murthy, Asymptotic analysis and spectrum of three anyons, Int. J. Mod. Phys. A 9 (1994) 2545 [cond-mat/9301021] [INSPIRE].
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ArXiv ePrint: 1408.1388
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Filev, V.G., O’Connor, D. Commuting quantum matrix models. J. High Energ. Phys. 2015, 24 (2015). https://doi.org/10.1007/JHEP03(2015)024
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DOI: https://doi.org/10.1007/JHEP03(2015)024