Grassmann matrix quantum mechanics

Anninos, Dionysios; Denef, Frederik; Monten, Ruben

doi:10.1007/JHEP04(2016)138

Title: Grassmann matrix quantum mechanics

Abstract

We explore quantum mechanical theories whose fundamental degrees of freedom are rectangular matrices with Grassmann valued matrix elements. We study particular models where the low energy sector can be described in terms of a bosonic Hermitian matrix quantum mechanics. We describe the classical curved phase space that emerges in the low energy sector. The phase space lives on a compact Kähler manifold parameterized by a complex matrix, of the type discovered some time ago by Berezin. The emergence of a semiclassical bosonic matrix quantum mechanics at low energies requires that the original Grassmann matrices be in the long rectangular limit. In conclusion, we discuss possible holographic interpretations of such matrix models which, by construction, are endowed with a finite dimensional Hilbert space.

Authors:

Anninos, Dionysios ^[1]; Denef, Frederik ^[2]; Monten, Ruben ^[2]

Institute for Advanced Study, Princeton, NJ (United States)
Columbia Univ., New York, NY (United States); Instituut voor Theoretische Fysica, Leuven (Belgium)

Publication Date:: Thu Apr 21 00:00:00 EDT 2016

Research Org.:: Columbia Univ., New York, NY (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1326967

Grant/Contract Number:: SC0011941

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 4; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 1/N expansion; Matrix Models; Gauge-gravity correspondence

Citation Formats


                    Anninos, Dionysios, Denef, Frederik, and Monten, Ruben. Grassmann matrix quantum mechanics.  United States: N. p., 2016. 
Web.  doi:10.1007/JHEP04(2016)138.

Copy to clipboard


                    Anninos, Dionysios, Denef, Frederik, & Monten, Ruben. Grassmann matrix quantum mechanics.  United States.  https://doi.org/10.1007/JHEP04(2016)138

Copy to clipboard


                    Anninos, Dionysios, Denef, Frederik, and Monten, Ruben. Thu .  
"Grassmann matrix quantum mechanics".  United States.  https://doi.org/10.1007/JHEP04(2016)138.  https://www.osti.gov/servlets/purl/1326967.

Copy to clipboard


                    
@article{osti_1326967,

  title        = {Grassmann matrix quantum mechanics},

  author       = {Anninos, Dionysios and Denef, Frederik and Monten, Ruben},

  abstractNote = {We explore quantum mechanical theories whose fundamental degrees of freedom are rectangular matrices with Grassmann valued matrix elements. We study particular models where the low energy sector can be described in terms of a bosonic Hermitian matrix quantum mechanics. We describe the classical curved phase space that emerges in the low energy sector. The phase space lives on a compact Kähler manifold parameterized by a complex matrix, of the type discovered some time ago by Berezin. The emergence of a semiclassical bosonic matrix quantum mechanics at low energies requires that the original Grassmann matrices be in the long rectangular limit. In conclusion, we discuss possible holographic interpretations of such matrix models which, by construction, are endowed with a finite dimensional Hilbert space.},

  doi          = {10.1007/JHEP04(2016)138},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 4,

  volume       = 2016,

  place        = {United States},

  year         = {Thu Apr 21 00:00:00 EDT 2016},

  month        = {Thu Apr 21 00:00:00 EDT 2016}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1007/JHEP04(2016)138

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 13 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Ising model on a dynamical planar random lattice: Exact solution
journal, December 1986

Kazakov, V. A.
Physics Letters A, Vol. 119, Issue 3
DOI: 10.1016/0375-9601(86)90433-0

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Maldacena, Juan
International Journal of Theoretical Physics, Vol. 38, Issue 4, p. 1113-1133
DOI: 10.1023/A:1026654312961

Strings from tachyons: the c = 1 matrix reloaded
journal, December 2003

McGreevy, John; Verlinde, Herman
Journal of High Energy Physics, Vol. 2003, Issue 12
DOI: 10.1088/1126-6708/2003/12/054

Planar diagrams
journal, February 1978

Brézin, E.; Itzykson, C.; Parisi, G.
Communications in Mathematical Physics, Vol. 59, Issue 1
DOI: 10.1007/BF01614153

The supersymmetric one-dimensional string
journal, April 1990

Marinari, Enzo; Parisi, Giorgio
Physics Letters B, Vol. 240, Issue 3-4
DOI: 10.1016/0370-2693(90)91115-R

Adjoint fermion matrix models
journal, July 1994

Makeenko, Yu.; Zarembo, K.
Nuclear Physics B, Vol. 422, Issue 1-2
DOI: 10.1016/0550-3213(94)00061-1

Fermionic Matrix Models
journal, May 1997

Semenoff, Gordon W.; Szabo, Richard J.
International Journal of Modern Physics A, Vol. 12, Issue 12
DOI: 10.1142/S0217751X97001328

Fermionic quantum gravity
journal, January 2001

Paniak, Lori D.; Szabo, Richard J.
Nuclear Physics B, Vol. 593, Issue 3
DOI: 10.1016/S0550-3213(00)00631-3

Large N matrices from a nonlocal spin system
journal, September 2015

Anninos, Dionysios; Hartnoll, Sean A.; Huijse, Liza
Classical and Quantum Gravity, Vol. 32, Issue 19
DOI: 10.1088/0264-9381/32/19/195009

Towards a quantum theory of de Sitter space
journal, December 2006

Banks, Tom; Fiol, Bartomeu; Morisse, Alexander
Journal of High Energy Physics, Vol. 2006, Issue 12
DOI: 10.1088/1126-6708/2006/12/004

De Sitter Holography with a Finite Number of States
journal, January 2005

Parikh, Maulik; Verlinde, Erik
Journal of High Energy Physics, Vol. 2005, Issue 01
DOI: 10.1088/1126-6708/2005/01/054

Micromanaging de Sitter holography
journal, November 2010

Dong, Xi; Horn, Bart; Silverstein, Eva
Classical and Quantum Gravity, Vol. 27, Issue 24
DOI: 10.1088/0264-9381/27/24/245020

Matrix model for de Sitter
journal, April 2002

Li, Miao
Journal of High Energy Physics, Vol. 2002, Issue 04
DOI: 10.1088/1126-6708/2002/04/005

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model
journal, February 1983

Haldane, F. D. M.
Physics Letters A, Vol. 93, Issue 9
DOI: 10.1016/0375-9601(83)90631-X

Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State
journal, April 1983

Haldane, F. D. M.
Physical Review Letters, Vol. 50, Issue 15
DOI: 10.1103/PhysRevLett.50.1153

Models of Gross-Neveu type are quantization of a classical mechanics with nonlinear phase space
journal, June 1978

Berezin, F. A.
Communications in Mathematical Physics, Vol. 63, Issue 2
DOI: 10.1007/BF01220849

Bi-local construction of Sp(2N)/dS higher spin correspondence
journal, January 2013

Das, Diptarka; Das, Sumit R.; Jevicki, Antal
Journal of High Energy Physics, Vol. 2013, Issue 1
DOI: 10.1007/JHEP01(2013)107

Static patch solipsism: conformal symmetry of the de Sitter worldline
journal, March 2012

Anninos, Dionysios; Hartnoll, Sean A.; Hofman, Diego M.
Classical and Quantum Gravity, Vol. 29, Issue 7
DOI: 10.1088/0264-9381/29/7/075002

De Sitter Musings
journal, May 2012

Anninos, Dionysios
International Journal of Modern Physics A, Vol. 27, Issue 13
DOI: 10.1142/S0217751X1230013X

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories
journal, January 2004

Aharony, Ofer; Marsano, Joseph; Minwalla, Shiraz
Advances in Theoretical and Mathematical Physics, Vol. 8, Issue 4
DOI: 10.4310/ATMP.2004.v8.n4.a1

The importance of the Selberg integral
journal, January 2008

Forrester, Peter J.; Warnaar, S. Ole
Bulletin of the American Mathematical Society, Vol. 45, Issue 4
DOI: 10.1090/S0273-0979-08-01221-4

Works referencing / citing this record:

Infrared realization of dS 2 in AdS 2
journal, March 2018

Anninos, Dionysios; Hofman, Diego M.
Classical and Quantum Gravity, Vol. 35, Issue 8
DOI: 10.1088/1361-6382/aab143

Infrared Realization of dS$_2$ in AdS$_2$
text, January 2017

Anninos, Dionysios; Hofman, Diego M.
arXiv
DOI: 10.48550/arxiv.1703.04622

Similar Records in DOE PAGES and OSTI.GOV collections:

Deep Quantum Geometry of Matrices

Journal Article Han, Xizhi ; Hartnoll, Sean A. - Physical Review. X

We employ machine learning techniques to provide accurate variational wave functions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. The variational quantum Monte Carlo method is implemented with deep generative flows to search for gauge-invariant low-energy states. The ground state (and also long-lived metastable states) of an SU(N) matrix quantum mechanics with three bosonic matrices, and also its supersymmetric “mini-BMN” extension, are studied as a function of coupling and N. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wave function. We then describemore »« less
Cited by 22
https://doi.org/10.1103/PhysRevX.10.011069
Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

Journal Article Znojil, Miloslav - Annals of Physics (New York)

It is known that the practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians H≠H{sup †} requires an efficient reconstruction of an ad hoc Hilbert-space metric Θ=Θ(H) which would render the time-evolution unitary. Once one considers just the N-dimensional matrix toy models H=H{sup (N)}, the matrix elements of Θ(H) may be defined via a coupled set of N{sup 2} polynomial equations. Their solution is a typical task for computer-assisted symbolic manipulations. The feasibility of such a model-completion construction is illustrated here via a discrete square well model H=p{sup 2}+V endowed with a k-parametric close-to-the-boundary interaction V. The model ismore »« less
https://doi.org/10.1016/J.AOP.2013.05.016
Grassmann phase space theory and the Jaynes–Cummings model

Journal Article Dalton, B.J., E-mail: bdalton@swin.edu.au ; Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Melbourne, Victoria 3122 ; Garraway, B. M. ; ... - Annals of Physics (New York)

The Jaynes–Cummings model of a two-level atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an n-photon Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherentmore »« less
https://doi.org/10.1016/J.AOP.2013.03.010
Grassmann phase space methods for fermions. II. Field theory

Journal Article Dalton, B.J., E-mail: bdalton@swin.edu.au ; Jeffers, J. ; Barnett, S. M. - Annals of Physics

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggests the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum-atom optics to treat fermionic systems is rather rare, thoughmore »« less
https://doi.org/10.1016/J.AOP.2016.12.026
Q-deformed Grassmann fields and the two-dimensional Ising model

Journal Article Bugrij, A I ; Shadura, V N - Theoretical and Mathematical Physics

We construct an exact representation of the Ising partition function in the form of the SL{sub q} (2, R)-invariant functional integral for the lattice-free q-fermion field theory (q = -1). It is shown that the q-fermionization allows one to rewrite the partition function of the eight-vertex model in an external field through a functional integral with four-fermion interaction. To construct these representations, we define a lattice (l, q, s) -deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At q = -1, l = s = 1, we obtain the lattice q-fermion field which allows usmore »« less
https://doi.org/10.1007/BF02065864

Similar Records

Title: Grassmann matrix quantum mechanics

Abstract

Citation Formats

Ising model on a dynamical planar random lattice: Exact solution journal, December 1986

The Large-N Limit of Superconformal Field Theories and Supergravity journal, January 1999

Strings from tachyons: the c = 1 matrix reloaded journal, December 2003

Planar diagrams journal, February 1978

The supersymmetric one-dimensional string journal, April 1990

Adjoint fermion matrix models journal, July 1994

Fermionic Matrix Models journal, May 1997

Fermionic quantum gravity journal, January 2001

Large N matrices from a nonlocal spin system journal, September 2015

Towards a quantum theory of de Sitter space journal, December 2006

De Sitter Holography with a Finite Number of States journal, January 2005

Micromanaging de Sitter holography journal, November 2010

Matrix model for de Sitter journal, April 2002

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model journal, February 1983

Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State journal, April 1983

Models of Gross-Neveu type are quantization of a classical mechanics with nonlinear phase space journal, June 1978

Bi-local construction of Sp(2N)/dS higher spin correspondence journal, January 2013

Static patch solipsism: conformal symmetry of the de Sitter worldline journal, March 2012

De Sitter Musings journal, May 2012

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories journal, January 2004

The importance of the Selberg integral journal, January 2008

Infrared realization of dS 2 in AdS 2 journal, March 2018

Infrared Realization of dS$_2$ in AdS$_2$ text, January 2017

Ising model on a dynamical planar random lattice: Exact solution
journal, December 1986

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Strings from tachyons: the c = 1 matrix reloaded
journal, December 2003

Planar diagrams
journal, February 1978

The supersymmetric one-dimensional string
journal, April 1990

Adjoint fermion matrix models
journal, July 1994

Fermionic Matrix Models
journal, May 1997

Fermionic quantum gravity
journal, January 2001

Large N matrices from a nonlocal spin system
journal, September 2015

Towards a quantum theory of de Sitter space
journal, December 2006

De Sitter Holography with a Finite Number of States
journal, January 2005

Micromanaging de Sitter holography
journal, November 2010

Matrix model for de Sitter
journal, April 2002

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model
journal, February 1983

Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State
journal, April 1983

Models of Gross-Neveu type are quantization of a classical mechanics with nonlinear phase space
journal, June 1978

Bi-local construction of Sp(2N)/dS higher spin correspondence
journal, January 2013

Static patch solipsism: conformal symmetry of the de Sitter worldline
journal, March 2012

De Sitter Musings
journal, May 2012

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories
journal, January 2004

The importance of the Selberg integral
journal, January 2008

Infrared realization of dS 2 in AdS 2
journal, March 2018

Infrared Realization of dS$_2$ in AdS$_2$
text, January 2017