quantum statistical mechanical systems associated to riemann...

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


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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)


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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)


RiemannSurfaces

MarkGreenfield


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.


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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 .


RiemannSurfaces

MarkGreenfield


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 .


RiemannSurfaces

MarkGreenfield


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 .


RiemannSurfaces

MarkGreenfield


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,�)


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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)


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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.


RiemannSurfaces

MarkGreenfield


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!


RiemannSurfaces

MarkGreenfield


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


RiemannSurfaces

MarkGreenfield


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

quantum statistical mechanical systems associated to riemann...

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