Download Holographic dual of a time machine

Andrey Bagrov
with Irina Arefeva, Petter Saterskog, and Koenraad Schalm
to appear soon
Holographic duality for condensed matter physics
Beijing, China, July 06-31, 2015
My collaborators
Irina Arefeva
(MI RAS)
Petter Saterskog
(Leiden U.)
Koenraad Schalm
(Leiden U.)
Outline
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Motivation
The dual bulk set up
Prescription for the boundary Green's function:
the geodesic approximation
Results: negative time retarded propagation
Outline





Motivation
The dual bulk set up
Prescription for the boundary Green's function:
the geodesic approximation
Results: negative time retarded propagation
Speculations on possible connections with real life
physics
Physics of time traveling
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Is there any time machine solution in GR that can
be stable?
Is there a physical way to create a time machine?
What dynamical behavior would a physical system
experience when evolving in a time machine
background?
Physics of time traveling
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Is there any time machine solution in GR that can
be stable?
Is there a physical way to create a time machine?
What dynamical behavior would a physical system
experience when evolving in a time machine
background?
Stability of causality violating backgrounds
Hawking’s chronology protection conjecture:
Emergence of macroscopic non-causal structures should be
prevented by strong quantum fluctuations of the stressenergy tensor.
(S. Hawking, Phys. Rev. D 46, 603)
Can be tested within the context of string theory.
Stability of causality violating backgrounds
O-plane orbifold background:
closed timelike curves → the Hagedorn transition
(M. Costa, C. Herdeiro, J. Penedos, N. Sousa; hep-th/0504102)
Goedel type background:
closed timelike curves are hidden behind the
holographic screens
(E. Boyda, S. Ganguli, P. Horava, U. Varadarajan; hepth/0405019)
Null energy condition
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The weakest among energy conditions:
Has to be violated in order to create a time machine
(S. Carrol, E. Farhi, A. Guth, PRL 68, 263 (1992)
In some models it does not hold:
Certain higher derivative theories (V. Rubakov, hep-th/1401.4024)
Galileon inflation (P. Creminelli, M. Luty, A. Nicolis, L. Senatore,
hep-th/1007.0027)
String field theory inspired cosmological models (I. Arefeva, I.
Volovich, hep-th/0612098)
What happens to a physical system in a time
machine?
Classical scalar fields: can evolve self-consistently on globally nonhyperbolic spaces
(O. Groshev, N. Gusev, E. Kuryanovich, I. Volovich, math-ph/0903.0741)
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What happens to a physical system in a time
machine?
Classical scalar fields: can evolve self-consistently on globally nonhyperbolic spaces
(O. Groshev, N. Gusev, E. Kuryanovich, I. Volovich, math-ph/0903.0741)
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Classical billiard ball: a non-zero set of self-consistent initial conditions
(F. Echeverria, G. Klinkhammer, K. Thorne, Phys.Rev. D 44, 1077)
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What happens to a physical system in a time
machine?
Classical scalar fields: can evolve self-consistently on globally nonhyperbolic spaces
(O. Groshev, N. Gusev, E. Kuryanovich, I. Volovich, math-ph/0903.0741)
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Classical billiard ball: a non-zero set of self-consistent initial conditions
(F. Echeverria, G. Klinkhammer, K. Thorne, Phys.Rev. D 44, 1077)
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Quantum mechanics:
multivalued path integrals (D. Deutsch, Phys.Rev. D 44, 3197)
post-selection (S. Lloyd et al., quant-ph/1005.2219)
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What happens to a physical system in a time
machine?
Classical scalar fields: can evolve self-consistently on globally nonhyperbolic spaces
(O. Groshev, N. Gusev, E. Kuryanovich, I. Volovich, math-ph/0903.0741)

Classical billiard ball: a non-zero set of self-consistent initial conditions
(F. Echeverria, G. Klinkhammer, K. Thorne, Phys.Rev. D 44, 1077)
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Quantum mechanics:
multivalued path integrals (D. Deutsch, Phys.Rev. D 44, 3197)
post-selection (S. Lloyd et al., quant-ph/1005.2219)
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Quantum field theories: failure of unitarity
(D. Boulware, hep-th/9207054;
J. Friedman, N. Papastamatiou, J. Simon, Phys.Rev. D46 (1992) 44564469)
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The goal
We would like to address the “grandfather paradox” in a constructive
manner, - to find an explicit solution to a QFT in a time machine.
The goal
We would like to address the “grandfather paradox” in a constructive
manner, - to find an explicit solution to a QFT in a time machine.
Why AdS/CFT?
1. Reduces analysis of non-causal quantum dynamics just to
study of a classical pseudo-Riemannian geometry.
2. No need to impose additional “consistency” boundary
conditions by hands.
Time machines in AdS/CFT
H. Lin, O. Lunin, J. Maldacena prescription (hep-th/0409174):
↔ free fermionic configurations
M. Caldarelli, D. Klemm, P. Silva (hep-th/0411203):
CTC ↔ violation of the Pauli principle!
Conical defect in AdS3
Embedding space representation
The AdS is a hyperboloid embedded into a flat spacetime:
Relation to the global coordinates:
Location of the leading and trailing faces:
Embedding space representation
Consider two conical defects. We can boost each of them independently:
Then they are moving along circular orbits:
The DeDeo-Gott time machine
S. DeDeo, J. Gott, “An eternal time machine in (2+1)-dimensional anti-de
Sitter space”, gr-qc/0212118
Closed timelike curves
Time and angular jumps:
Existence of CTC:
Geodesic approximation in holography
 Conformal dimension of boundary operator ↔ mass of the bulk field
Thus for a large conformal dimension we deal with a heavy (semi)classical
bulk particle
Prescription for the two-point function:
Note that the length should be renormalized near the boundary.
In the case of pure AdS it reproduces the exact answer:
Lensing of geodesics
A number of contributing geodesics
Picture from V. Balasubramanian, B. Chowdhury, B. Czech, J. de Boer, hep-th/1406.5859
Geodesics in the time machine
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Each of the two cones can be bypassed either clockwise or
counterclockwise:
A typical geodesic:
Contributions to the Green function:
Geodesics in the time machine
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The geodesic consists of a number of pure AdS geodesic
segments:
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All complementary point are
algebraically related:
Geodesics in the time machine
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Intersection points can easily be found in the corresponding rest
frame:
Causal propagation
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The easiest way to see how the closed timelike curves change the
physics of the system is to look at an object that would be manifestly
causal otherwise, i.e. the retarded Green function.
How to define “retardness” in the time machine?
In a normal case:
For a non-trivial entwinement structure:
Geodesic approximation for
timelike intervals
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But there are no timelike geodesics connecting AdS boundary points…
Actually, a massive particle Lagrangian is
Introduce kinetic invariants:
Then we get:
Geodesic approximation for
timelike intervals
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No global notion of time → no way to define the Wick
rotation and transform to the Euclidean signature.
For the Poincare patch a solution has been proposed by B.
Craps et al. in hep-th/1212.6060:
Connect timelike separated points by a discontinuous
spacelike geodesic!
Geodesic approximation for
timelike intervals
The corresponding particle Lagrangian is
And the invariants:
Now there are two cases:
Geodesic approximation for
timelike intervals
Geodesics of the second type reach the Poincare horizon r=0 at λ=0.
We can treat this point in a similar way as the complex infinity in the
theory of complex functions:
Two disconnected geodesics possessing the same invariants E and J but
emerging from two different boundary points A and B should be
considered as two branches of a single geodesic reconnected at the
infinity.
This leads to a correct results:
Geodesic approximation for
timelike intervals
Poincare horizon in global coordinates:
Section of a quadric by a plane:
plane
Geodesic approximation for
timelike intervals
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The two-point function of a CFT on a cylinder has a reflection
symmetry:
We can use it to substitute timelike separated points by their spacelike
images:
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Geodesic approximation for
timelike intervals
Winding quasigeodesics
Perturbative expansion
 Number of windings as a control parameter
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
- Generate all possible sequences of windings {Wj} for a given N
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
- Generate all possible sequences of windings {Wj} for a given N
- Impose the “retardness” condition
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
- Generate all possible sequences of windings {Wj} for a given N
- Impose the “retardness” condition
- Solve for the intersection points
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
- Generate all possible sequences of windings {Wj} for a given N
- Impose the “retardness” condition
- Solve for the intersection points
- Check that the segments of geodesic intersect the faces in a
correct order
Perturbative expansion
 Number of windings as a control parameter
 The algorithm:
- Pick up two boundary points
- Generate all possible sequences of windings {Wj} for a given N
- Impose the “retardness” condition
- Solve for the intersection points
- Check that the segments of geodesic intersect the faces in a
correct order
- Calculate the renormalized length
Perturbative expansion
 As we increase the number of windings, lengths of geodesics grow
linearly, and thus the corresponding contributions to the correlator
decrease exponentially:
 On the other hand the number of different winding configurations
increases exponentially:
 So, why do we expect the perturbative expansion to converge?
Perturbative expansion
 Three reasons for the series not to diverge:
- The conformal dimension is a knob. Making it very large we can
make the converging exponent dominate over the diverging (actually, it
is the limit where the geodesic approximation is at work)
- Not all of the topologically different windings satisfy the
“retardness” condition.
- Even if the “retardness” condition is met, the more complicated
the topology of a quasigeodesic becomes (for large N), the less the
chance the quasigeodesic can be located within the physical part of the
double-cone spacetime
Perturbative expansion
Physical
Unphysical
Retarded Green’s function
 We are going to focus on a one-dimensional timelike (nearly lightlike)
slice of the Green function:
 In the co-rotating frame:
Retarded Green’s function:
negative times
Retarded Green’s function:
negative times
Retarded Green’s function:
positive times
 So far we have done simulations up to N=2:
Origin of the peaks
A bit suspicious that for negative times N=4 contribution dominates over
N=2 and N=3…
The reason is that the renormalized length can be negative. Otherwise we
could not reproduce the lightcone singularity in the correlator:
The total length:
The contribution is not exponentially small in the large Δ limit if
Away from these special points the perturbative expansion is convergent.
Convergence of the winding expansion series
For each complementary point
C1 there is a unique final point
B.
The contribution is divergent if
and only if C1 is located on the
generatrix of the light cone.
Retarded Green’s function:
negative times
2D Green’s function for N=2
Boundary vs. bulk causality violation
Time and angular jumps:
Boundary vs. bulk causality violation
“Phase diagram”
 Behavior of the N=2 peak upon
changing the TM parameters:
Red – no causality violation (at
the leading order)
Green – mild causality violation
(suppressed leakage)
Yellow – singular revivals
Blue – non-existence of the bulk
spacetime
How to think about non-unitarity
 Opportunistic: If we really hope to obtain any insight into physics of
paradoxical systems, we should not rely on the “common sense”
intuition and the corresponding fundamental principles. Everything is
fine as long as we can find a way to solve the theory.
 Orthodoxal: Non-unitary evolution is completely physical in the case of
open quantum systems (the Lindblad equation). So we might conjecture
that the AdS time machine is an effective dual to a certain state of that
kind (with some reserve).
What is the dual state?
The total angular deficit is enough to form a black hole.
Classical and quantum stability of the considered solution has not been
analyzed yet, but it is natural to expect that it is less stable than a fully
thermalized BTZ. Stable while driven.
Conical defects: excited states generated by non-local operators in the
boundary theory (V. Balasubramanian et al. hep-th/1406.5859).
“Entanglement = spacetime” (see papers by B. Swingle, X.L. Qi and
others) → states with deformed structure of entanglement links.
A kind of external observer?..
Timelike entanglement
 Since we were interested in the causality violation, we have been
considering “timelike” quasigeodesics.
 But due to the lensing timelike separated points can be also
connected by continuous spacelike geodesics. The latter are directly
related to the entanglement.
 So, do we have a state with a high rate of timelike entanglement on
the boundary?..
(see also S. Olson, T. Ralph, “Entanglement between the future and the past
in the quantum vacuum”; Phys.Rev.Lett.106:110404,2011)
Conclusions
 Using the geodesic approximation we have calculated the two point
Green function of a theory dual to the AdS time machine.
 Leakage of a signal to negative times preceding the moment of
creating excitation is manifest.
 Numerical resummation of non-trivial windings is convergent away
from the singular points that form a subset of zero measure.
 Increasing the “strength” of the bulk time machine somehow makes
the boundary causality violation milder.
Outlook
 Understand the interplay of two non-local operators generating the
conical defects.
 Find an exact solution to the boundary theory using the local
isometry of the spacetime to the pure AdS geometry.
 Suggest an experimental setup for preparing a timelike entangled
state and study its physical properties.