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ARITHMETIC QUANTUM MECHANICS ON BLACK HOLE HORIZONS

ARITHMETIC QUANTUM MECHANICS ON BLACK HOLE HORIZONS. E.FLORATOS PHYSICS UNIV OF ATHENS CERN ON THE OCCASION OF 60 TH BIRTHDAYS OF COSTAS KOUNNAS 28 SEPTEMBER 2012 NICOSIA-CYPRUS. PLAN OF THE TALK. PRINCIPLES AND CONSTRAINTS OF BH HOLOGRAPHY

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ARITHMETIC QUANTUM MECHANICS ON BLACK HOLE HORIZONS

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  1. ARITHMETIC QUANTUM MECHANICS ON BLACK HOLE HORIZONS E.FLORATOS PHYSICS UNIV OF ATHENS CERN ON THE OCCASION OF 60TH BIRTHDAYS OF COSTAS KOUNNAS 28 SEPTEMBER 2012 NICOSIA-CYPRUS

  2. PLAN OF THE TALK • PRINCIPLES AND CONSTRAINTS OF BH HOLOGRAPHY • ARITHMETIC QUANTUM MECHANICS (AQM) ON THE STRECHED HORIZON • SCRAMPLING TIMES AND FACTORIZATION IN AQM • CONCLUSIONS

  3. 1.PRINCIPLES AND CONSTRAINTSOF BLACK HOLE HOLOGRAPHY • BLACK HOLE INFORMATION PARADOX IS THERE ANY UNITARY CONNECTION OF OBSERVATIONS MADE BY TWO OBSERVERS? A)A FREELY FALLING INTO THE BLACK HOLE HORIZON B) A STATIONARY ONE OUTSIDE THE HORIZON

  4. HAWKING: NO • WE MUST CHANGE QM TO DESCRIBE EVOLUTION BETWEEN PURE AND MIXED STATES INGOING PURE STATE-OUTGOING HAWKING RADIATION • T’HOOFT: YES –THROUGH BH HORIZON HOLOGRAPHY • WE MUST ENCODE ALL THE 3D INFALLING INFORMATION ON THE HORIZON AND THEN IN THE OUTGOING HAWKING RADIATION BY A 2D HOLOGRAPHIC UNITARY SCATTERING MATRIX

  5. PRINCIPLES OF BH INFORMATION PROCESSING SUSSKIND :BLACK HOLE COMPLEMENTARITY PRINCIPLE • BOTH OBSERVERS ,THEY DO NOT SEE VIOLATION OF ANY PHYSICAL LAW • TOTAL HILBERT SPACE H=Ha x Hb • ALL THE INFORMATION OF THE INFALLING OBSERVER IS PROCESSED BY THE INTERIOR OF THE BLACK HOLE AND IT IS HOLOGRAPHICALLY STORED ON THE STRECHED HORIZON 1PL SCALE OUTSIDE THE HORIZON dimH[STRECHED HORIZON]=Exp[A/4] finite • THE OUTGOING HAWKING RADIATION IS ENCODING THE STRECHED HORIZON MEMORY

  6. CONSTRAINTS ON BH HOLOGRAPHY 1) LAWS OF BLACK HOLE THERMODYNAMICS FOR THE MICROSCOPIC DEGREES OF FREEDOM IN THERMAL EQUILIBRIUM ON THE STRECHED HORIZON (SH) S[A,B]<S[A]+S[B] A=BLACK HOLE INTERIOR,B=BLACK HOLE EXTERIOR HAWKING RAD 2) LAWS OF QUANTUM INFORMATION PROCESSING UNITARITY ,ENTANGLEMENT OF A AND B, TRANFER OF QUANTUM INFORMATION THROUGH CHANELS WITH ERASURE NO-CLONING OF INFORMATION BY THE SH 3)FAST HOLOGRAPHIC MIXING OF INFORMATION ON THE SH- PAGE TIME =R/2 BH SCRAMPLING TIME =R Log[R/lp] PRESKIL TIME<<R Log[R/lp]

  7. ARITHMETIC QUANTUM MECHANICS ON THE SH SH DISCRETE AND FINITE LATTICE OF POINTS (k,l)ModN STORAGE AND PROCESSING OF INFORMATION BY AREA PRESERVING MAPS 1BIT PER PLANCK AREA GENERALIZED ARNOLD CAT MAPS A={{a,b},{c,d}} in SL[2,Z[N]] (r,s)->(r,s)A STRONG ARITHMETIC CHAOS ARNOLD,BERRY VOROS,VIVALDI MIXING TIME LOGARITHMIC-HOLOGRAPHY

  8. QUANTIZATION OF CAT MAPS QP=ωPQ, ω=Εxp[2 πi/N], H=2 π/N J[r,s]=ω^(rs/2)P^rQ^s,H-W GROUP A=>U[A],U[AB]=U[A]U[B] U[A]J[r,s]U[A]^(-1)=J[(r,s)A] SCHRONDIGER EVOL |n+1>=U[A]|n>

  9. U[A](k,l)=1/Sqrt[N] ω^φ[Α,k,l] φ[Α,k,l]=-1/2b [ak^2+d l^2-2 k l],k,l=0,N-1 U[A]^n=U[A^n], HARMONIC OSCILLATOR A={{0,-1},{1,0}} U[A](k,l)=1/Sqrt[N] ω^(kl)=F , Q-FOURIER ARNOLD CAT MAP A={{1,1},{2,1}} QUANTUM CHAOS,LOG MIXING

  10. FACTORIZATION OF OBSERVERS-LOCALITY • N=N1xN2 • SL[2,Z[N]]=SL[2,Z[N1]]xSL[2,Z[N2]] • A[N]=A[N1]A[N2] • H[N]=H[N1]xH[N2] • U[A[N]]=U[A[N1]]U[A[N2]] • FAST QUANTUM MAPS N^2->N LogN • S[N]=S[N1]+S[N2]

  11. CONCLUSIONS • BH COMPLEMENTARITY IS BASED ON THE • ASSUMPTION OF THE SCRAMBLING • MIXING TIME BOUND IN ORDER TO HAVE • A CAUSAL ENCODING OFINFORMATION ON • THE EMMITED HAWKING RADIATION • ModN DISCRETIZATIONS AND ARNOLD CAT MAP DYNAMICS ON THE STRECHED HORIZON • SATURATE THE SCRAMBLING TIME BOUND

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