ARITHMETIC QUANTUM MECHANICS ON BLACK HOLE HORIZONS

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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
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
1.PRINCIPLES AND CONSTRAINTS
OF 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
• 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
•
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
• 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 SHPAGE TIME =R/2
BH SCRAMPLING TIME
=R Log[R/lp]
PRESKIL TIME<<R Log[R/lp]
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
QUANTIZATION OF CAT MAPS
QP=ωPQ, ω=Εxp[2 π i/N],
H=2 π/N
J[r,s]=ω^(rs/2)P^r Q^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>
• 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
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]
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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