In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes Michel L. Lapidus 2000 Mathematics Subject Classification. P rim a ry 1 1 A 4 1 , 1 1 G 20, 1 1 M 06 , 1 1 M 26 , 1 1 M 4 1 , 28 A 8 0, 3 7 N 20, 4 6 L 5 5 , 5 8 B 3 4 , 8 1 T 3 0. F o r a d d itio n a l in fo rm a tio n a n d u p d a tes o n th is b o o k , v isit www.ams.org/bookpages/mbk-51 Library of Congress Cataloging-in-Publication Data Lapidus, Michel L. (Michel Laurent), 1956 In search o f the R iem ann zero s : string s, fractal m em b ranes and no nco m m utativ e spacetim es / Michel L. Lapidus. p. cm . Includes b ib lio g raphical references. IS B N 97 8 -0 -8 2 18 -4 2 2 2 -5 (alk . paper) 1. R iem ann surfaces. 2 . F unctio ns, Z eta. 3 . S tring m o dels. 4 . N um b er theo ry. 5. F ractals. 6. S pace and tim e. 7 . G eo m etry. I. T itle. 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T he A m erican Mathem atical S o ciety retains all rig hts ex cept tho se g ranted to the U nited S tates G o v ernm ent. P rinted in the U nited S tates o f A m erica. ∞ T he paper used in this b o o k is acid-free and falls w ithin the g uidelines estab lished to ensure perm anence and durab ility. V isit the A MS ho m e pag e at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 0 9 0 8 God made the integers, all else is the work of man. L eopold K ronecker, 1 8 8 6 (quoted in [Web], [Bell,p.477] and [Boy,p.617]) S tring theory carries the seeds of a basic change in ou r ideas abou t spacetime and in other fu ndamental notions of p hysics. E dward W itten, 19 9 6 [Wit15 ,p.2 4] Contents Preface xiii A ck n o w led g em en ts xv ii C red its xxiii O v erv iew xxv A b o u t th e C o v er xxix C h ap ter 1 . In tro d u ctio n 1 1 .1 . A rith m etic an d S p acetim e G eo m etry 1 1 .2 . R iem an n ian , Q u an tu m an d N o n co m m u tativ e G eo m etry 2 1 .3 . S trin g T h eo ry an d S p acetim e G eo m etry 3 1 .4 . T h e R iem an n H y p o th esis an d th e G eo m etry o f th e Prim es 6 1 .5 . M o tiv atio n s, O b jectiv es an d O rg an izatio n o f T h is B o o k 9 C h ap ter 2 . S trin g T h eo ry o n a C ircle an d T -D u ality : A n alo g y w ith th e R iem an n Z eta F u n ctio n 2 1 2 .1 . Q u an tu m M ech an ical Po in t-Particle o n a C ircle 2 4 2 .2 . S trin g T h eo ry o n a C ircle: T -D u ality an d th e E xisten ce o f a F u n d am en tal L en g th 2 6 2 .2 .1 . String Theory on a Circle 2 9 2 .2 .2 . Circle D u ality (T-D u ality for Circu lar Spacetim es) 3 4 2 .2 .3 . T-D u ality and the E xistence of a F u nd am ental L ength 4 3 2 .3 . N o n co m m u tativ e S trin g y S p acetim es an d T -D u ality 4 5 2 .3 .1 . Target Space D u ality and N oncom m u tative G eom etry 4 8 2 .3 .2 . N oncom m u tative Stringy Spacetim es : F ock Spaces, V ertex A lgebras and Chiral D irac O perators 5 5 2 .4 . A n alo g y w ith th e R iem an n Z eta F u n ctio n : F u n ctio n al E q u atio n an d T -D u ality 6 6 2 .4 .1 . K ey P roperties of the R iem ann Z eta F u nction: E u ler P rod u ct and F u nctional E qu ation 6 7 2 .4 .2 . The F u nctional E qu ation, T -D u ality and the R iem ann H ypothesis 7 5 2 .5 . N o tes 8 0 C h ap ter 3 . F ractal S trin g s an d F ractal M em b ran es 8 9 ix x CONTENTS 3.1. Fractal Strings: Geometric Zeta Functions, Complex Dimensions and Self-Similarity 91 3.1.1. The Spectrum of a Fractal String 102 3.2. Fractal (and Prime) Membranes: Spectral Partition Functions and Euler Products 103 3.2.1. Prime M embranes 104 3.2.2. Fractal M embranes and Euler Products 109 3.3. Fractal Membranes vs. Self-Similarity: Self-Similar Membranes 123 3.3.1. Partition Functions View ed as Dynamical Zeta Functions 138 3.4. Notes 145 Chapter 4. Noncommutative Models of Fractal Strings: Fractal Membranes and Beyond 155 4.1. Connes Spectral Triple for Fractal Strings 157 4.2. Fractal Membranes and the Second Quantization of Fractal Strings 160 4.2.1. An Alternative Construction of Fractal M embranes 165 4.3. Fractal Membranes and Noncommutative Stringy Spacetimes 170 4.4. Towards a Cohomological and Spectral Interpretation of (Dynamical) Complex Dimensions 174 4.4.1. Fractal M embranes and Q uantum Deformations: A Possible Connection w ith Harans Real and Finite Primes 183 4.5. Notes 183 Chapter 5. Towards an Arithmetic Site: Moduli Spaces of Fractal Strings and Membranes 197 5.1. The Set of Penrose Tilings: Quantum Space as a Quotient Space 200 5.2. The Moduli Space of Fractal Strings: A Natural Receptacle for Zeta Functions 205 5.3. The Moduli Space of Fractal Membranes: A Quantized Moduli Space of Fractal Strings 208 5.4. Arithmetic Site, Frobenius Flow and the Riemann Hypothesis 215 5.4.1. The M oduli Space of Fractal Strings and Deningerss Arithmetic Site 217 5.4.2. The M oduli Space of Fractal M embranes and (Noncommutative) M odular Flow vs.Arithmetic Site and Frobenius Flow 219 5.4.2a. Factors and Their Classifi cation 220 5.4.2b. M odular Theory of von Neumann Algebras 223 5.4.2c. Type III Factors: Discrete vs.Continuous Flow s 227 5.4.2d. M odular Flow s and the Riemann Hypothesis 231 5.4.2e. Tow ards an Extended M oduli Space and Flow 236 5.5. Flows of Zeros and Zeta Functions: A Dynamical Interpretation of the Riemann Hypothesis 241 5.5.1. Introduction 241 5.5.2. Expected Properties of the Flow s of Zeros and Zeta Functions 243 CONTENTS xi 5.5.3. Analogies with Other Geometric, Analytical or Physical Flows 254 5.5.3a. Singular Potentials, Feynman Integrals and Renormalization Flow 255 5.5.3b. KMS-Flow and Deformation of Po´ lya Hilbert Operators 261 5.5.3c.