Black holes, quantum chaos, and the Riemann hypothesis
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CCTP-2020-5
ITCP-IPP-2020/5
Black holes, quantum chaos, and the Riemann
hypothesis
arXiv:2004.09523v1 [hep-th] 20 Apr 2020
Panos Betzios∗1 , Nava Gaddam†2 , and Olga Papadoulaki‡ 3
1
Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics, Department
of Physics, University of Crete, 70013 Heraklion, Greece
2
Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht
University, 3508 TD Utrecht, The Netherlands
3
International Centre for Theoretical Physics Strada Costiera 11, 34151 Trieste, Italy
Abstract
Quantum gravity is expected to gauge all global symmetries of effective theories, in the
ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a
quantum boundary condition in phase space. We find that it provides for a natural semiclassical
regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related
to the Dilation operator. We observe that the said spectrum is in correspondence with the
zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating,
this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this
quantum Hamiltonian captures the dynamics of the scattering matrix of a Schwarzschild black
hole, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum
appears to be erratic despite the unitarity of the scattering matrix.
∗ pbetzios@physics.uoc.gr
† gaddam@uu.nl
‡ Papadoulaki@ictp.it
1Contents
1 Introduction 2
2 The Dilation operator and symmetries 3
3 Identifying CPT conjugate states 6
3.1 Global identification in spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 The local identification in phase space and a chaotic spectrum . . . . . . . . . . . . . 6
4 Conclusions 9
1 Introduction
It has long been proposed that quantum gravity has no global symmetries; all global symmetries
of effective theories shall be gauged [MW1, Pol1, BS1]. This expectation also applies to discrete
global spacetime symmetries [OH1, OH2]. In fact, there is an old idea that shows how discrete
global symmetries may remain in the infrared, upon Higgs-ing local gauge symmetries in the ul-
traviolet [KW1]. This article concerns a global discrete spacetime symmetry, namely CPT. Indeed,
in quantum gravity, a definition of global CPT requires the specification of a choice of time, a
choice of gauge. It appears to intrinsically be an effective symmetry. For instance, should there be
a symmetric phase of quantum gravity, a possible symmetry breaking mechanism could lead to a
vacuum expectation value for the metric [Wit1], and consequently an effective global discrete CPT
symmetry in the low energy semiclassical description. The larger gauge symmetry group might
then subsume such a discrete CPT symmetry.
Often the ultraviolet physics is attributed to new degrees of freedom (such as strings). Could
it be that intricate microscopic gauge symmetries and dynamics result in an effective boundary
condition for the low energy description? There is a precedent for such a mechanism in the Callan-
Rubakov effect. A possible relevance to black hole physics was articulated in [CPW1] and a similar
effect could be at play in Euclidean quantum gravity [BGP1]. Lacking an exact understanding
of the microscopic theory, in this article, we will assume as a postulate that CPT is gauged and
analyse its consequences in an example motivated by the study of quantum black holes.
In a seemingly unrelated pursuit, Berry and Keating proposed [BK1, BK2] that a rich quantum
spectrum could be generated by discrete phase space boundary conditions, inspired by a conjecture1
due to Hilbert and Polya. The latter is the statement that the zeros of the Riemann zeta function
arise as the spectrum of a self-adjoint operator with real eigenvalues. The spectrum of the Riemann
zeros is very rich, a smooth approximation of which is known to produce the statistics of the
Gaussian Unitary Ensemble of random matrices [Gut1]. Nevertheless, several ambiguities remain to
date: the appropriate self-adjoint operator, the exact boundary condition to impose that determines
the space of functions upon which the said operator acts, and the physical origin for this choice.
Needless to say, there have been several proposals for such an operator; see [Sie1] and references
therein.
Could it then be that the effect of the proposed gauging of CPT symmetry provides for a
quantisation of semiclassical continuous spectra? Drawing on work towards constructing a unitary
S-matrix for a Schwarzschild black hole [Hoo1, Hoo2, BGP2], we explore the consequence of such
1 The first documented appearance of which is in [Mon1].
2a gauging—as a quantum boundary condition in phase space—to obtain remarkable results. First,
we find that the spectrum of the Dilation operator is naturally discretised and is given by the
zeros of Riemann zeta and Dirichlet beta functions. Following the observation in [BGP2] that
such a Dilation operator captures the near horizon dynamics of the black hole scattering matrix,
we argue that the boundary condition results in a discrete spectrum of black hole microstates.
Despite the seemingly simple classical Hamiltonian, an almost erratic spectrum can arise. This is
in accord with the expectation that black holes have a “fast scrambling”, quantum chaotic nature
[SS1, MS1, ShS1, MSS1, Pol1]. Our proposal shows how gauging CPT synthesises these ideas neatly.
Edmund Landau was famously said to have asked George Polya2, “You know some physics. Do
you know a physical reason that the Riemann hypothesis should be true?” We venture to speculate
that the answer is that CPT is a gauge symmetry in quantum gravity!
2 The Dilation operator and symmetries
In [BGP2], it was shown that two different Hamiltonians with their associated phase space structure,
result in exactly the same scattering matrix. One is a collection of a tower of independent inverted
harmonic oscillators labelled by spherical harmonic indices l, m:
1 X h
in
i 8πG
Ĥ = Ûlm V̂lm + V̂lm Ûlm with V̂lm , Ûlout
′ m′ = i~λl δll′ δmm′ , λl = ,
2 R2 (l2
+ l + 1)
lm
(1)
where the operators Û and V̂ define the light-cone positions of the partial waves. In this description,
an elementary cell in phase space, and the commutation relations, depend on the partial wave
number l. The second description has a common phase space cell size, and therefore a single unit
of time for the entire tower:
1 X R2 (l2 + l + 1) h i
in
Ĥ = Ûlm V̂lm + V̂lm Ûlm + with V̂lm , Ûlout
′ m′ = i~δll′ δmm′ . (2)
2 8πG
lm
The position operator in both descriptions may be expanded in partial waves as follows
X
Û (Ω) = Ûlm Ylm (Ω) with Ûlm |U ; l, mi = Ulm |U ; l, mi . (3)
lm
Both these Hamiltonians yield the same scattering matrix [BGP2]. Moreover, both of them are
clearly diagonal in the partial wave basis. The complete wavefunctions are specified in the position
basis using the two sphere coordinate, say Ω, by
X
hU ; Ω|Ψi =: Ψ (U ; Ω) = Ψlm (U ) Ylm (Ω) and hU ; l m|Ψi =: Ψlm (U ) . (4)
lm
Since there is no spin dependence in the scattered states and their associated dynamics, the index
m represents a degeneracy; nevertheless, this degeneracy will play a crucial role in this work as we
shall see in Section 3.2.
2 See http://www.dtc.umn.edu/~ odlyzko/polya/index.html, for an account.
3Connection with the Schwarzschild black hole The scattering matrix arising from the two
Hamiltonians described above is identical to the one obtained from gravitational back-reaction
[Hoo1, Hoo2], as was shown in [BGP2]. For this comparison, the constant G is identified with
Newton’s constant and R = 2GM , with the radius of a Schwarzschild black hole with mass M .
That the scattering is diagonal in the partial wave basis is owed to the spherical symmetry of the
black hole.
The wavefunctions of eqn. (4) describe partial waves in the Kruskal coordinates of an eternal
Schwarzschild black hole, which carries no charge other than its mass M . The metric takes the
form
32G3 M 3 c2 r
2
2 2 2
ds = − dU dV + r dΩ , U V = 1 − ec r/2GM . (5)
c6 r 2GM
The Hamiltonians (1) and (2), generate the dynamical evolution of the partial waves on this back-
ground. Furthermore, their evolution is according to the clock of a boosted observer that remains
at a distance from the black hole, reaching asymptotic null infinity [BGP2]. The energy then is
measured in such an observer’s natural units as [Haw1]
E ω 4GM ω
= = , (6)
~ κ c2
where M is the mass of the said black hole, κ is the surface gravity, and ω is the energy (in SI units)
that the asymptotic observer registers. Henceforth, we set c = 1 for simplicity. The coordinate
transformation from Kruskal to the boosted observer’s coordinates is
U = −e−u/4GM , V = ev/4GM ; U = e−ū/4GM , V = −ev̄/4GM , (7)
in terms of the Eddington-Finkelstein coordinates (u = t − r∗ , v = t + r∗ ) in region I (for which
U < 0, V > 0). Similarly, the coordinates ū, v̄ describe region II (for which U > 0, V < 0). Using
these coordinate systems and the clock of the boosted observer, ingoing modes depend on (V, Ω),
and outgoing modes on (U, Ω).
The commutation relations in eqns. (1) and (2) are a result of the gravitational backreaction
of modes in the vicinity of the black hole horizon as shown in [Hoo1, Hoo2, BGP2]. This algebra
is intriguing in that it elucidates the quantum gravitational nature of the effect; on the one hand
it becomes exactly zero when G/R2 → 0, and on the other, it reduces to a classical effect as
~ → 0 when the commutators are replaced by Poisson brackets. Another important aspect of this
algebra, is that it indicates an inherent uncertainty in determining the exact lightcone position of
propagating modes, since ∆V ∆U ≥ ~, the spacetime has minimal size causal diamonds for each
harmonic. This phase space structure will be important in Section 3.2.
Symmetries In the rest of this section, we drop the spectator indices l and m for simplicity, and
study the following system:
1 ∂ 1
Ĥ = Û V̂ + V̂ Û = −i~ V + , [V̂ , Û ] = i~ (8)
2 ∂V 2
The classical counterpart of this Hamiltonian, Hcl = U V , has the chaotic property that any two
nearby solutions diverge exponentially quickly at late times. On the quantum Hamiltonian (8),
symmetry under dilations acting as
U′ = κ U , V ′ = V /κ , (9)
4corresponds to time evolutions of the dynamical system, and the SL(2, R) generators in the V -basis
P̂ = −i∂V , D̂ = −i∆ − iV ∂V and K̂ = −i2∆V − iV 2 ∂V (10)
form the following algebra
[D̂, P̂ ] = iP̂ , [D̂, K̂] = −iK̂, [P̂ , K̂] = −2iD̂ , (11)
where ∆ is the conformal weight of the SL(2, R) representation. The quadratic Casimir is given by
1
C2 = −D̂2 + K̂ P̂ + P̂ K̂ , (12)
2
with eigenvalues ∆(1 − ∆). This is also the Laplace-Beltrami operator on the upper half complex
plane, and the eigenfuctions are automorphic functions with respect to discrete subgroups Γ of
SL(2, R). The energy eigenfunctions
ĤΨE = EΨE (13)
are also to be thought of as eigenfunctions of the dilation operator, D̂, with weight ∆ = 1/2 − iE/~.
Their form is
1 1 1 sign (V )
Ψeven
E (V ) = √ 1 , Ψodd
E (V ) = √ 1 (14)
2π~ |V | 2 −iE/~ 2π~ |V | 2 −iE/~
They belong to the unitary principal series representations of SL(2, R). It is convenient to split
them in odd and even parts under reflection V ↔ −V in order to determine which of the two
branches to consider near the origin V = 0. The spectrum is hence doubly degenerate (if no further
boundary conditions are imposed). These eigenfunctions correspond to plane waves in Eddington-
Finkelstein coordinates (7), as in [BGP2].
Demanding complete conformal invariance on the wavefunctions trivialises them. However, we
can also define a set of discrete inversions acting on the space of coordinates as
1 1
R̂ V = − , Iˆ V = (15)
V V
The first is parity odd, while the second is parity even. The energy eigenfunctions are also eigen-
functions of these discrete conformal inversions, acting on the space of functions as
1
R̂ Ψ(V ) = Ψ(−1/V ) , R̂ Ψeven
E (V ) = Ψeven
E (V ) , R̂ Ψodd odd
E (V ) = −ΨE (V ) (16)
|V |2∆
and
1
Iˆ Ψ(V ) = Ψ(1/V ) , Iˆ Ψeven
E (V ) = Ψeven
E (V ) , Iˆ Ψodd odd
E (V ) = ΨE (V ) (17)
|V |2∆
Therefore, the eigenfunctions are also automorphic forms with respect to the discrete SL(2, R)
subgroup generated by conformal inversions.
53 Identifying CPT conjugate states
3.1 Global identification in spacetime
In view of the global Kruskal notion of time 2T = U + V , the CPT transformation acts on the
wavefunctions as
Ψ(CP T ) (U, V, Ω) = Ψ† (−U, −V, ΩP ) . (18)
Using the following relations for the spherical harmonics
Yl∗m (Ω) = (−1)m Yl −m (Ω) , Yl m (ΩP ) = (−1)l Yl m (Ω) , (19)
a global identification of CPT conjugate wavefunctions describing the partial waves results in
Ψlm (U, V ) ≃ (−1)l−m Ψ†l −m (−U, −V ) . (20)
In particular this identification on the classical spacetime background of the black hole given in
eqn.(5) holds for all V and U , resulting in an antipodal identification on the bifurcation sphere
(V, U ) = (0, 0). This is the global spacetime form of the identification proposed in [Hoo2], that
results in a single spacetime exterior.
3.2 The local identification in phase space and a chaotic spectrum
The elementary cell in phase space has area U V = 2π~. Classically the constant energy trajectories
are U V = ±E. In the quantum mechanical phase space, due to the commutation relations of equa-
tion (8), an antipodal identification between all pairs of points on the V -U plane is not possible;
points are not sharply defined due to the uncertainty principle. A quantum phase space analogue
of the identification must be defined. Our guide is the discrete inversion of eqn. (15), that is a
symmetry of the energy eigenfunctions.
In phase space, remaining consistent with the uncertainty principle, the R̂-inversion (in terms
of a general parameter3 h) is enlarged into [Ane1]
h V 2U
T̂1± : V′ =− , U′ = ± , (21)
V h
h V U2
T̂2± : V′ =− , U′ = ∓ . (22)
U h
These are area preserving canonical transformations, since the Jacobian matrix of the transforma-
tions has a determinant ±1; the sign depends on the superscript of the generator in question. They
were also proposed in [BK1, BK2] as a possible means of obtaining a discrete spectrum for the Di-
lation operator; we will argue that this can be physically motivated by treating the wavefunctions
as partial waves in four dimensions. The Iˆ inversion is related to −T̂1,2
±
in phase space, so these
generators provide a complete phase space description of both transformations. We also notice
that T̂1− , T̂2+ leave the operators ±D̂ and the evolution Hamiltonian ±Ĥ invariant, while P̂ and K̂
change sign. For fixed U V = h, the transformations (21), (22) generate a finite group, the Dihedral
group of the square D4 with eight elements. A geometric property of these transformations is that
3 The choice of name for the parameter is intentional. It will turn out to specify Planckian cells.
6they leave the square centered at the origin invariant; we will call this square the causal diamond.
This diamond consists of four elementary phase space cells, each of which has an area |U V |= h, so
its total area is 4h.
Let us now consider this causal diamond near the bifurcation sphere U = 0 = V . (In fact we
can consider a more general parallelogram of sides LU , LV as long as LU LV = h and make a similar
argument, since any such distorted diamond can be obtained using the scaling symmetry of the
Hamiltonian). The causal diamond can be split into eight fundamental triangular chambers that
get interchanged through the action of the elements of D4 .
On the other hand, the spacetime identification, is expressed as the following equivalence relation
for the points of the V, U plane
(V, U ) ≃ (−V, −U ) , (23)
as can be seen from (20). In phase space, the analogue of this is imposing an identification under
the action of the generator4 T2+ . Since we work in phase space, we can only make this identification
at the edges of the causal diamond, where it corresponds precisely to the lightcone part of the CPT
identification. This endows the causal diamond with an RP2 topology, which is an unorientable
manifold5 .
At the quantum level, we must impose the quantum mechanical version of this identification
on the eigenfunctions of the Hamiltonian or Dilation operator. This is a legitimate quotient as it
corresponds to a symmetry of the quantum Hamiltonian. We begin with the Fourier transform
which describes the scattering matrix arising from the gravitational back-reaction, that relates the
U, V bases of functions Z ∞
dV − i UV in
Ψout (U ) = √ e ~ Ψ (V ) . (24)
−∞ 2π~
Using the eigenfunctions in (14) as ingoing modes, the outgoing wavefunction can be split into even
and odd parts
Γ 14 + i 2~
E
even 1 1 +i E
ΨE,out (U ) = √ E (2~)
~ (25)
E
Γ 41 − i 2~
1
2π~ |U | 2 +i ~
Γ 34 + i 2~
E
1 sign (U ) +i E
Ψodd
E,out (U ) = √ (2~) ~ . (26)
E
Γ 43 − i 2~
1 E
2π~ |U | 2 +i ~
In order to impose the CPT identification, we reinstate the partial wave l, m indices; the energy
index for each harmonic is henceforth E = E(l). The action of CPT on the complete wavefunctions
is then given by the combined action of the operator Tˆ2+ in (22), together with complex conjugation
and parity on the two sphere:
V U2
l−m even†
Ψeven
Elm,out (U ) → (−1) Ψ E l−m,out − , (27)
h
V U2
l−m odd†
Ψodd
Elm,out (U ) → (−1) Ψ E l−m,out − , (28)
h
4 With h = U V , the generator T̂2+ corresponds to the part of CPT that induces V ′ = −V and U ′ = −U .
5A mathematical explanation of this based on the relations between Coxeter elements of the dihedral group, can
be found in [Ane1]. Closed paths fall into two homotopy classes depending on the odd or even-ness of the number of
reflections along the path. This is because π1 (RP2 ) = Z2 ; therefore, to each path, there is an associated number ±1.
7
l−m h
Ψeven
Elm,in (V ) → (−1) Ψeven†
− ,
E l−m,in (29)
U
l−m odd† h
Ψodd
Elm,in (V ) → (−1) ΨE l−m,in − . (30)
U
From the first of the above equations, we have
V U2
l−m
Ψeven
Elm,out (U ) → (−1) Ψeven†
E l−m,out −
h
iE 2iE
1 E
l−m 1 1 (2~) ~ |V | ~ Γ 4+ i 2~
= (−1) √ 1 iE 1 iE 1 E
2π~ |V | 2 + ~ U2 2 − ~ Γ 4− i 2~
h
iE 2iE
1 E
l−m (2~) ~
|V | ~ Γ + i 2~
= (−1) Ψeven†
E l−m,in (V ) 1 iE
4
1 E
. (31)
U2 2+ ~ Γ − i 2~
4
h
The identification of CPT conjugate states on the causal diamond, using the definition of the
fundamental cell h = U V now gives
iE 1 iE
h ~ |V | 2 + ~ Γ 14 + i 2~E
l−m even†
Ψeven
Elm,out (U ) ≃ (−1) Ψ E l−m,in (V ) 1 E
1
− iE
π |U | 2 ~ Γ 4 − i 2~
iE 1
1 ~ |V | 2 Γ 41 + i 2~E
l−m even†
≃ (−1) ΨE l−m,in (V ) 1 1 E
. (32)
π |U | 2 Γ 4 − i 2~
Using the functional relation of the Riemann zeta function
1−s
s−1/2 Γ 2
ζ (s) = π ζ(1 − s) (33)
Γ 2s
with s = 1/2 + iE/~, we find
1 l−m 1
|U | 2 Ψeven
Elm,out (U ) ζ (1/2 + iE/~) ≃ (−1) |V | 2 Ψeven†
E l−m,in (V ) ζ (1/2 − iE/~) . (34)
Using (25) with U = Vh on the left hand side, (14) on the right hand side, and noting that this
equation must hold for all V , we find that it is satisfied provided
ζ (1/2 − iE/~) = ζ (1/2 + iE/~) = 0 , (35)
for all the even modes. This is exactly the condition for the Riemann zeta function ζ(s) to have
non-trivial zeros on the critical line Re(s) = 12 . Notice however, that for all odd waves Ψodd, we
have an additional minus sign arising from the sign (V ) in (14):
1 l−m
|V | 2 Ψodd†
1
|U | 2 Ψodd
Elm,out (U ) β (1/2 + iE/~) ≃ − (−1) E l−m,in (V ) β (1/2 − iE/~) , (36)
where we used the functional relation for the Dirichlet β function
Γ 1 − 2s
β (s) = 2π s−1/2 β(1 − s) . (37)
Γ 21 + 2s
8This implies that the spectrum of odd waves Ψodd is again quantised, but given by the zeros of the
Dirichlet beta function.
In essence, the quantum boundary condition on the phase space generates a discrete spec-
trum, providing a semiclassical regularisation procedure for the Dilation operator that respects the
physical symmetries of our quantum Hamiltonian and its associated dynamics. This is a different
quantisation of the spectrum compared to the cutoff proposed in [BGP2, Hoo3], and takes the phase
space structure in eqns. (1) and (2) into account.
The complete spectrum of the Hamiltonian (2), can be decomposed as follows:
E even X E even
l R2 (l2 + l + 1) 4GM X even
= + = ωl ,
~ ~ 8πG~ c2
l l
E odd X E odd
l R2 (l2 + l + 1) 4GM X odd
= + = ωl , (38)
~ ~ 8πG~ c2
l l
where Eleven /~ is given by the zeros of the Riemann zeta function, and Elodd /~ by those of the
Dirichlet beta function. The two-fold degeneracy between the even and odd modes is broken. There
is still a (2l + 1)-fold degeneracy due to the m index. The variables ωl carry the correct SI units
of energy for each partial wave, as registered by the asymptotic observer. A similar decomposition
can also be written for the Hamiltonian (1).
Apart from the non-trivial zeros, there are also the trivial zeros for which
1 E even 1 E odd
−i l = −2n, −i l = −(2n − 1), with n ∈ Z+ . (39)
2 ~ 2 ~
These zeros, correspond to the poles of the scattering matrix, defined via eqns. (24), (25), (26),
in the axis of negative imaginary energy, and thus to scattering resonances. The interpretation of
such resonances in the context of black holes is that of quasinormal modes.
The lightcone momentum operator We now study the effect of the local phase space iden-
tification on the spectrum of the lightcone momentum operator P̂ of eqn. (10). Its eigenfunctions
are
Φin
P (V ) = c e
in iP V
, P̂ Φin in
P (V ) = P ΦP (V ) . (40)
Reintroducing again the l, m indices, the identification under the action of T̂2+ (for U V = h), results
into
in l−m in† h
ΦP lm (V ) ≃ (−1) ΦP l−m − , for U V = h , (41)
U
leading to c∗lm = (−1)l−m clm . We therefore find that the spectrum of this operator is unaffected
by the identification we propose. This conclusion evidently also holds for the momentum operator
that translates modes along the Kruskal coordinate U , causing a fall into the black hole.
4 Conclusions
Inspired by the expectation that there are no global symmetries in quantum gravity, we observe that
treating CPT as a local gauge symmetry leads to an effective boundary condition on the elementary
diamond of phase space, centered at U = 0 = V . This results in a discretisation of the spectrum
9of the Dilation operator when appended with extra quantum numbers; in the present construction,
these arise from the presence of extra transverse dimensions.
The resulting spectrum is extremely rich, arguably chaotic, and supports several properties
expected of quantum black holes. It goes to show that a simple, unbounded Hamiltonian may
turn out to have an extremely interesting spectrum provided appropriate boundary conditions are
chosen.
Should the approach of Berry and Keating [BK1, BK2] still provide an interesting way forward,
the boundary condition and the Hamiltonian proposed in this article may help the study of the
Riemann hypothesis. An asymptotic analysis of a smooth approximation of the density of zeros of
the Riemann zeta function may be an interesting starting point to compute the partition function of
the theory, and possibly derive the Bekenstein-Hawking entropy of the black hole. This will clarify
whether the present proposal adequately captures all the Schwarzschild black hole microstates.
A study of multi-particle scattering cross sections and the consequence of this boundary condi-
tion on such observables is another future direction of physical interest. The presence of a discrete
spectrum that is unbounded from below remains a puzzle to be understood; perhaps our proposal
of gauging CPT is insufficient to fix the ambiguity in the energy of the ground state in this system.
Acknowledgements
It is a pleasure to acknowledge various helpful discussions with E. Kiritsis, K. Papadodimas, M.M.
Sheikh-Jabbari, T. Tomaras, N. Tsamis, and E. Verlinde, on topics related to the black hole S-
matrix. Special thanks go to Gerard ’t Hooft for pointing out to us that the naive cutoff regular-
isation is in clash with the symmetries of the Dilation operator and the S-matrix, and for several
discussions over the years. We thank David Tong for his talk on gauging discrete symmetries at
the Amsterdam String Workshop 2019. We also thank Michael Berry and Jon Keating for corre-
spondence and feedback on a draft of this article.
P.B is supported by the Advanced ERC grant SM-GRAV, No 669288. N.G is supported by the
Delta-Institute for Theoretical Physics (D-ITP) that is funded by the Dutch Ministry of Education,
Culture and Science (OCW).
References
[MW1] C. W. Misner and J. A. Wheeler, Classical physics as geometry: Gravitation, electromag-
netism, unquantized charge, and mass as properties of curved empty space, Annals Phys. 2
(1957), 525-603 doi:10.1016/0003-4916(57)90049-0
[Pol1] J. Polchinski, Monopoles, duality, and string theory, Int. J. Mod. Phys. A 19S1 (2004),
145-156 doi:10.1142/S0217751X0401866X [arXiv:hep-th/0304042 [hep-th]].
[BS1] T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev.
D 83 (2011), 084019 doi:10.1103/PhysRevD.83.084019 [arXiv:1011.5120 [hep-th]].
[OH1] D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity,
[arXiv:1810.05338 [hep-th]].
[OH2] D. Harlow and H. Ooguri, Constraints on Symmetries from Holography, Phys. Rev. Lett.
122 (2019) no.19, 191601 doi:10.1103/PhysRevLett.122.191601 [arXiv:1810.05337 [hep-th]].
10[KW1] L. M. Krauss and F. Wilczek, Discrete Gauge Symmetry in Continuum Theories, Phys.
Rev. Lett. 62 (1989), 1221 doi:10.1103/PhysRevLett.62.1221
[Wit1] E. Witten, THE SEARCH FOR HIGHER SYMMETRY IN STRING THEORY, Phil.
Trans. Roy. Soc. Lond. A 329 (1989), 349-357 IASSNS-HEP-88/55.
[CPW1] S. R. Coleman, J. Preskill and F. Wilczek, Quantum hair on black holes, Nucl. Phys. B
378 (1992), 175-246 doi:10.1016/0550-3213(92)90008-Y [arXiv:hep-th/9201059 [hep-th]].
[BGP1] P. Betzios, N. Gaddam and O. Papadoulaki, Antipodal correlation on the meron
wormhole and a bang-crunch universe, Phys. Rev. D 97 (2018) no.12, 126006
doi:10.1103/PhysRevD.97.126006 [arXiv:1711.03469 [hep-th]].
[BK1] M. V. Berry and J. P. Keating, H=xp AND THE RIEMANN ZEROS, Supersymmetry and
Trace Formulae: Chaos and Disorder (1999) pp. 355-367.
[BK2] M. V. Berry and J. P. Keating, The Riemann Zeros and Eigenvalue Asymptotics, SIAM
Review Vol. 41. No.2 pp. 236-266 (1999).
[Mon1] H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number
Theory, American Mathematical Society, Proc. Sympos. Pure Math., XXIV (1973), 181 -
193, MRNumber: 0337821.
[Gut1] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag New York
1 (1990), XIV 432 pages, doi:10.1007/978-1-4612-0983-6
[Sie1] G. Sierra, The Riemann zeros as spectrum and the Riemann hypothesis, Symmetry 11 (2019)
no.4, 494 doi:10.3390/sym11040494 [arXiv:1601.01797 [math-ph]].
[BGP2] P. Betzios, N. Gaddam and O. Papadoulaki, The Black Hole S-Matrix from Quantum
Mechanics, JHEP 11 (2016), 131 doi:10.1007/JHEP11(2016)131 [arXiv:1607.07885 [hep-th]].
[Hoo1] G. ’t Hooft, Diagonalizing the Black Hole Information Retrieval Process, [arXiv:1509.01695
[gr-qc]].
[Hoo2] G. ’t Hooft, Black hole unitarity and antipodal entanglement, Found. Phys. 46 (2016) no.9,
1185-1198 doi:10.1007/s10701-016-0014-y [arXiv:1601.03447 [gr-qc]].
[Hoo3] G. ‘t. Hooft, Discreteness of Black Hole Microstates, arXiv:1809.05367 [gr-qc].
[SS1] Y. Sekino and L. Susskind, F ast Scramblers, JHEP 0810 (2008) 065 doi:10.1088/1126-
6708/2008/10/065 [arXiv:0808.2096 [hep-th]].
[ShS1] S. H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 1403 (2014) 067
doi:10.1007/JHEP03(2014)067 [arXiv:1306.0622 [hep-th]].
[MS1] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61
(2013) 781 doi:10.1002/prop.201300020 [arXiv:1306.0533 [hep-th]].
[MSS1] J. Maldacena, S. H. Shenker and D. Stanford, A bound on chaos, JHEP 1608 (2016) 106
[arXiv:1503.01409 [hep-th]].
11[Pol1] J. Polchinski, Chaos in the black hole S-matrix, arXiv:1505.08108 [hep-th].
[Haw1] S. Hawking, Particle Creation by Black Holes,’ Commun. Math. Phys. 43 (1975), 199-220
doi:10.1007/BF02345020
[Ane1] B. Aneva, Symmetry of the Riemann operator, Phys. Lett. B 450 (1999), 388-396
doi:10.1016/S0370-2693(99)00172-0 [arXiv:0804.1618 [nlin.CD]].
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