Ad S 2CFT 1 Whittaker vector WheelerDe Witt

Ad. S 2/CFT 1, Whittaker vector & Wheeler-De. Witt equation Tadashi Okazaki (NTU) ar. Xiv: 1510. 04759

Conclusion Result 1 GKP-Witten relation in Ad. S 2/CFT 1 Result 2 #(conformal dimention) ↔ #(prime number)

Conformal Symmetry in Quantum Mechanics Conf(R) = SL(2, R)

Works so far on CQM DFF model ’ 76 de. Alfaro, Fubini, Furlan Calogero model ’ 69 Calogero • w/ classical Lagrangian • discrete spectrum My work • w/o classical Lagrangian • continuous spectrum

Irreducible Unitary rep of SL(2, R) most analysis so far this talk ’ 46 Gelfand & Namark ’ 47 Bargmann ’ 52 Harish. Chandra Ex) DFF model mock discrete series principal series complementary series discrete series

Whittaker vector principal series rep of SL(n+1, R) Chevalley basis irrep w/ weight Whittaker vector energy eigenstate in CQM

Whittaker function plays a key role in physics & math ! (sorry, I cannot explain today…) CQM Black Hole (Ad. S 2 gravity) Number Theory (Eisenstein series) Whittaker fcn • • Combinatorics Macdonald polynomial Kashiwara crystal • • Geometry (Gromov-Witten of P 1) Integrable model Toda Hamiltonian 6 vertex model pfn

Ad. Sd+1/CFTd GKP-Witten relation generating fcn of corr fcn in bdy CFTd pfn of bulk Ad. Sd+1

Ad. S 2/CFT 1 is special ! Ad. S 2 side I. II. all the extremal BPS BH contain Ad. S 2 ’ 07 Kunduri etal. excited states on bdy CQM side I. II. ’ 08 Figueras etal. ’ 10 Balasubramanian etal. ’ 01 Zamolodchikov etal. not Fock space but Hilbert space (no radial quantization & no CFT method) ’ 08 Sen ’ 11 Chamon etal spatially disconnected bdy but entangled ? ’ 07 Azeyanagi

Result 1 GKP-Witten relation in Ad. S 2/CFT 1 generating fcn of corr fcn in bdy CQM excited states ! pfn of bulk Ad. S 2 = = generating fcn of expectation value of dilatation operator wavefcn of LFT in minisuperspace approx.

CQM 2 d gravity pfn w/ BC wavefc n ’ 06 Kashani-Poor

Wheeler-De. Witt equation !

Dictionary Ad. S 2 radius cf) Ad. S 3 radius ’ 86 Brown Henneaux energy of bdy ground state _ ? flat space

Q. What is I. 　conformal dimension in CQM (physics) II. 　prime numer in Number Theory (math) Encounter of building blocks in physics & math !

Result 2 #(conformal dimention) ↔ #(prime number) asymptotic behavior of counting fcn of Riemann zeros

Hilbert 8 th Problem & Millennium Problem RH (Riemann Hypothesis) All the non-trivial zeros of zeta are complex numbers with real part ½. zeta knows prime number !

1859 Riemann zeros of zeta = Riemann zeros # (prime number) _ # (zeros of zeta)

Hilbert-Pólya conjecture Imaginary part of non-trivial Riemann zeros correspond to eigenvalues of some self-adjoint operator. ’ 99 Berry, Keating ’ 99 Connes asymptotic behavior of counting fcn of Riemann zeros

This is not Hamiltonian in CQM ! In DFF model This is rather Dilatation operator !

Q. What is asymptotic behavior of counting fcn of Riemann zeros !

Conclusion Result 1 GKP-Witten relation in Ad. S 2/CFT 1 Result 2 #(conformal dimention) ↔ #(prime number)

Conformal Quantum Mechanics is unopened treasure box in physics & math!