quantum statistical mechanical systems associated to riemann...
TRANSCRIPT
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