QSM Systems Associated to Riemann Surfaces Mark Greenfield Introduction and Overview QSM Systems Spectral Triples Riemann Sfcs Previous Results QSM Construction Generalization Conclusions Quantum Statistical Mechanical Systems Associated to Riemann Surfaces Mark Greenfield Mentor: Prof. Matilde Marcolli October 20, 2012
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QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Quantum Statistical Mechanical SystemsAssociated to Riemann Surfaces
Mark GreenfieldMentor: Prof. Matilde Marcolli
October 20, 2012
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
1 Introduction and Overview
2 Quantum Statistical Mechanical Systems
3 Spectral Triples
4 Riemann Surfaces and Uniformization
5 Previous Results
6 Construction of the QSM System
7 Generalization of Construction
8 Conclusions and Further Study
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Introduction
Noncommutative geometry and mathematical physics
• Construct a QSM system holding conformal isomorphism(shape) of a Riemann surface
• Using spectral triple construction of Cornelissen andMarcolli (2008)
• Generalize for larger class of spectral triples
Riemann Surface ! Spectral Triple - (known)
Spectral Triple ! QSM System - (my project)
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Quantum Statistical MechanicalSystems: C ⇤-Dynamical Systems
We use a purely mathematical notion of a QSM system knownas a C ⇤-dynamical system:
(A,�)
• A is a C ⇤-Algebra of observables operating on states
• Operate on state, obtain information about the system
• � time-evolves operators, acting as an automorphismgroup on A parameterized by time
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
C ⇤-Dynamical Systems
• Time evolution � can be defined in terms of Hamiltonianoperator H. At time t on operator a 2 A:
�t(a) = e itHae�itH
• Equilibrium states that do not change in time take form,at inverse temperature � > 0 (a 2 A):
��(a) =tr(ae��H)
tr(e��H)
• Partition function has form, with inverse temperature �:
Z (�) = tr(e��H)
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Spectral Triples
Collection of geometric data in algebraic structure:
• C ⇤-algebra of operators, AR
• Hilbert space H on which AR acts as bounded operators
• Dirac operator D that also acts on H
(A,H,D)
We look at ”zeta functions” of (AR ,H,D):⇣a(s) = tr(aDs), s 2 C,Re(s) negative.
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Riemann Surfaces
Representation of complex-valued functions as manifolds
Figure: The torus is (up to homeomorphism) the only genus 1 Riemann surface.Image credit: http://en.wikipedia.org/wiki/Riemann surface
• Manifold: generalized smooth space
• One complex dimension, 2 real dimensions (”surface”)
• Genus: the number of ”handles” on the surface
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
UniformizationEncodes Riemann surface into a group structure.
• Group of discrete isometries (jump point-to-point,preserving distances)
• Points partitioned into sets reachable from each other
• Each set glued together to get Riemann surface
• Schottky Uniformization gives similar group �
Figure: Isometries define a lattice onthe hyperbolic disk. This is arepresentation of the Fuchsianuniformization of a genus 2 surface.Image credit:http://www.calvin.edu/ ven-ema/courses/m100/F11/escher.html
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
More on Uniformization. . .Schottky Groups �:
• Isomorphic to free group Fg
• Infinite sequence of actions from � lead to ”limit points,”defining the limit set ⇤
Free groups Fg :
• g generating elements, e.g. {G1, . . . ,Gg}• Each string of generators (e.g. GiGj . . .Gk) gives unique”word”
Figure: Graph representing the”embedding” of Fg into theRiemann sphere. Image credit:http://en.wikipedia.org/wiki/Cayley graph
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Spectral Triple Construction ofCornelissen and Marcolli
Construction from: Cornelissen, Gunther and Matilde Marcolli.Zeta Functions that hear the shape of a Riemann surface.Journal of Geometry and Physics, Vol. 58 (2008) N.1 57-69.
Spectral triple (AR ,H,D) constructed from uniformizingSchottky group � and limit set ⇤.
Key Idea: (finite) Words in Fg define subsets of ⇤. The set�!w ⇢ ⇤ contains all infinite words starting with w .
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
(AR ,H ,D) for � and ⇤
Define characteristic functions on ⇤ by:
�w (�) =
⇢1 : � 2 �!w0 : � /2 �!w
We let C ⇤-algebra AR be the closure of the span of thecharacteristic functions. That is, AR = C (⇤).
Hilbert space H is isomorphic to AR , with inner product:< �v |�w >= ”size” (Patterson-Sullivan measure) of �!w \ �!v .