Download Slide

AdS2/CFT1,
Whittaker vector
& Wheeler-DeWitt equation
Tadashi Okazaki
(NTU)
arXiv:1510.04759
Conclusion
Result 1 GKP-Witten relation in AdS2/CFT1
Result 2
#(conformal dimention) ↔ #(prime number)
Conformal Symmetry
in Quantum Mechanics
Conf(R) = SL(2,R)
Works so far on CQM
DFF model
’76 deAlfaro, 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 HarishChandra
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
(AdS2 gravity)
Number Theory
(Eisenstein series)
Whittaker fcn
•
•
Combinatorics
Macdonald polynomial
Kashiwara crystal
•
•
Geometry
(Gromov-Witten of P1)
Integrable model
Toda Hamiltonian
6 vertex model pfn
AdSd+1/CFTd
GKP-Witten relation
generating fcn of
corr fcn in
bdy CFTd
pfn of
bulk AdSd+1
AdS2/CFT1 is special !
AdS2 side
I.
II.
all the extremal BPS BH contain AdS2
’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 AdS2/CFT1
generating fcn of
corr fcn in
bdy CQM
excited states !
pfn of
bulk AdS2
=
=
generating fcn of
expectation value
of dilatation operator
wavefcn of
LFT in
minisuperspace
approx.
CQM
pfn w/ BC
2d gravity
wavefc
n
’06 Kashani-Poor
Wheeler-DeWitt equation !
Dictionary
AdS2 radius
cf) AdS3
radius
’86 Brown Henneaux
energy of bdy
ground
state
_
?
flat
space
Q. What is
I.
II.
conformal dimension in CQM
(physics)
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 8th 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 AdS2/CFT1
Result 2
#(conformal dimention) ↔ #(prime number)
Conformal Quantum Mechanics
is unopened treasure box in
physics & math!