Quantum Statistical Mechanical Systems

Transcription

1 Sfcs Quantum Statistical Mechanical Systems Mentor: Prof. Matilde Marcolli October 20, 2012

2 Sfcs 1 2 Quantum Statistical Mechanical Systems 3 4 and Uniformization 5 6 of the System 7 of 8 and Further Study

3 Sfcs Noncommutative geometry and mathematical physics Construct a system holding conformal isomorphism (shape) of a surface Using spectral triple construction of Cornelissen and Marcolli (2008) Generalize for larger class of spectral triples Surface! Triple - (known) Triple! System - (my project)

4 Quantum Statistical Mechanical Systems: C -Dynamical Systems Sfcs We use a purely mathematical notion of a system known as 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 automorphism group on A parameterized by time

5 Sfcs C -Dynamical Systems Time evolution can be defined in terms of Hamiltonian operator H. Attimet on operator a 2 A: t(a) =e ith ae ith Equilibrium states that do not change in time take form, at inverse temperature >0(a2A): (a) = tr(ae H ) tr(e H ) Partition function has form, with inverse temperature : Z( )=tr(e H )

6 Sfcs Collection of geometric data in algebraic structure: C -algebra of operators, A R Hilbert space H on which A R acts as bounded operators Dirac operator D that also acts on H (A, H, D) We look at zeta functions of (A R, H, D): a (s) =tr(ad s ), s 2 C, Re(s) negative.

7 Representation of complex-valued functions as manifolds Sfcs Figure: The torus is (up to homeomorphism) the only genus 1 surface. Image credit: surface Manifold: generalized smooth space One complex dimension, 2 real dimensions ( surface ) Genus: the number of handles on the surface

8 Sfcs Uniformization Encodes 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 surface Schottky Uniformization gives similar group Figure: Isometries define a lattice on the hyperbolic disk. This is a representation of the Fuchsian uniformization of a genus 2 surface. Image credit: venema/courses/m100/f11/escher.html

9 Sfcs Schottky Groups : More on Uniformization... Isomorphic to free group F g Infinite sequence of actions from defining the limit set Free groups F g : g generating elements, e.g. {G 1,...,G g } lead to limit points, Each string of generators (e.g. G i G j...g k ) gives unique word Figure: Graph representing the embedding of F g into the sphere. Image credit: graph

10 Triple of Cornelissen and Marcolli Sfcs from: Cornelissen, Gunther and Matilde Marcolli. Zeta Functions that hear the shape of a surface. Journal of Geometry and Physics, Vol. 58 (2008) N triple (A R, H, D) constructed from uniformizing Schottky group and limit set. Key Idea: (finite) Words in F g define subsets of. The set! w contains all infinite words starting with w.

11 (A R, H, D) for and Sfcs Define characteristic functions on by: 1 : 2! w w ( )= 0 : /2! w We let C -algebra A R be the closure of the span of the characteristic functions. That is, A R = C( ). Hilbert space H is isomorphic to A R,withinnerproduct: < v w >= size (Patterson-Sullivan measure) of! w \! v.

12 Dirac Operator Sfcs Define: H n : subspace of H with all v having len(v) apple n. n = dim(h n ) 3 P n : projection operator onto H n. P n chops o letters after n th. Dirac operator: D = 0 P 0 + X n>0 n(p n P n 1 ) Eigenvectors: composed of single-length words, eigenvalues k.

13 Sfcs Zeta Functions We look at Zeta functions for (A R, H R, D R ), for a 2 A R : a (s) =tr(ad s ) Each surface has a set of zeta functions If 1 equal for di erent surfaces, algebras A R are isomorphic and other zeta functions can be compared If all a equal for di erent surfaces: surfaces are conformally isomorphic Want to extract equivalent set of functions from system (A, )

14 of the System Sfcs Want to construct (A, functions ) from which we can get the a Need algebra of observables A and Hamiltonian H Will start with Hamiltonian H; implicitly defines t(a) =e ith ae ith

15 Hamiltonian and Time Evolution Sfcs Recall: Z( )=tr(e H )and 1 (s) =tr(d s ) Define e H = D ) H = log(d), with for s Need each n > 0 0 =1(spannedby ), and n = dim(h n ) 3 Define Hamiltonian for (A, ): H = X log( n )(P n n>0 P n 1 )

16 Algebra of Observables Sfcs Define the minimal algebra extending A R : A = {e ith ae ith a 2 A R, t 2 R} Contains all possible time-evolved operators from A R Noncommutative for operators with di erent time parameters Hilbert space on which this acts will be H, well-defined since based on components of spectral triple

17 Extracting the Zeta Functions from the System Sfcs Already have: Z( )=tr(e H ) tr(d s )= 1 (s) Recall equilibrium states: (a) = tr(ae H ) tr(e H ) a (s) =tr(ad s ) tr(ae H ) If Z and all are equal for two surfaces, we have: tr(ae H 1 ) = tr(ae H2 ) ) tr(e H 1) tr(e H 2) a,1 (s) = a,2 (s) This system encodes the conformal isomorphism class of a surface.

18 Generalizing to Dirac Operators with Nonpositive Eigenvalues Sfcs 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, P k be the projection operator onto vectors with zero eigenvalue Define new D for some 2 R: D =( k + )P k + X n6=k np n This su ciently generalizes the construction to a much larger class of spectral triples!

19 and Further Study Sfcs We have a construction of a Quantum Statistical Mechanical System that encodes the conformal isomorphism class of a surface The construction was generalized to be valid for a large class of spectral triples Part of an ongoing e ort to find relationships between mathematical structures and

20 Acknowledgements Sfcs SURF Mentor: Professor Matilde Marcolli Peers: Adam Jermyn and Aniruddha Bapat Caltech SURF O ce