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 .
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Dirac Operator
Define:
• Hn: subspace of H with all �v having len(v) n.
• �n = dim(Hn)3
• Pn: projection operator onto Hn. Pn ”chops o↵” lettersafter nth.
Dirac operator:
D = �0P0 +X
n>0
�n(Pn � Pn�1)
Eigenvectors: composed of single-length words, eigenvalues �k .
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Zeta Functions
We look at Zeta functions for (AR ,HR ,DR), for a 2 AR :
⇣a(s) = tr(aDs)
• Each Riemann surface has a set of zeta functions
• If ⇣1 equal for di↵erent surfaces, algebras AR areisomorphic and other zeta functions can be compared
• If all ⇣a equal for di↵erent surfaces: surfaces areconformally isomorphic
• Want to extract equivalent set of functions from QSMsystem (A,�)
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Construction of the QSM System
• Want to construct (A,�) from which we can get the ⇣afunctions
• Need algebra of observables A and Hamiltonian H
• Will start with Hamiltonian H; implicitly defines�t(a) = e itHae�itH
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Hamiltonian and Time Evolution
Recall: Z (�) = tr(e��H) and ⇣1(s) = tr(Ds)
• Define eH = D ) H = log(D), with �� for s
• Need each �n > 0
• �0 = 1 (spanned by �⇤), and �n = dim(Hn)3
• Define Hamiltonian for (A,�):
H =X
n>0
log(�n)(Pn � Pn�1)
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Algebra of Observables
Define the minimal algebra extending AR :
• A = {e itHae�itH |a 2 AR , t 2 R}• Contains all possible time-evolved operators from AR
• Noncommutative for operators with di↵erent timeparameters
• Hilbert space on which this acts will be H, well-definedsince based on components of spectral triple
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Extracting the Zeta Functionsfrom the QSM System
• Already have: Z (�) = tr(e��H) ⇠ tr(Ds) = ⇣1(s)
• Recall equilibrium states: ��(a) =tr(ae��H)tr(e��H)
• ⇣a(s) = tr(aDs) ⇠ tr(ae��H)
• If Z and all � are equal for two Riemann surfaces, we have:
tr(ae��H1)
tr(e��H1)=
tr(ae��H2)
tr(e��H2)) ⇣a,1(s) = ⇣a,2(s)
• This QSM system encodes the conformal isomorphismclass of a Riemann surface.
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Generalizing to Dirac Operatorswith Nonpositive Eigenvalues
• For complex eigenvalues, use complex logarithm:Log(z) = log(|z |) + iArg(z)
• For a zero eigenvalue, introduce shifting factor:
• Let �k be the zero eigenvalue, Pk be the projectionoperator onto vectors with zero eigenvalue
• Define new D⇤ for some ✏ 2 R:
D⇤ = (�k + ✏)Pk +X
n 6=k
�nPn
• This su�ciently generalizes the construction to a muchlarger class of spectral triples!
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Conclusions and Further Study
• We have a construction of a Quantum StatisticalMechanical System that encodes the conformalisomorphism class of a Riemann surface
• The construction was generalized to be valid for a largeclass of spectral triples
• Part of an ongoing e↵ort to find relationships betweenmathematical structures and QSM Systems
QSM SystemsAssociated to
RiemannSurfaces
MarkGreenfield
Introductionand Overview
QSM Systems
SpectralTriples
Riemann Sfcs
PreviousResults
QSMConstruction
Generalization
Conclusions
Acknowledgements
• SURF Mentor: Professor Matilde Marcolli
• Peers: Adam Jermyn and Aniruddha Bapat
• Caltech SURF O�ce