Reduction and Emergence in Holographic Scenarios
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Transcript of Reduction and Emergence in Holographic Scenarios
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Holography and the Emergence
of Gravity
Dennis Dieks, Jeroen van Dongen,
Sebastian de Haro
Reduction and Emergence, MCMP,
Munich, 2013
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“Starting from first principles and general assumptions, we present a heuristic argument that shows that Newton’s law of gravitation naturally arises in a theory in which space emerges through a holographic scenario. Gravity is identified with an entropic force caused by changes in the information associated with the position of material bodies.”
Erik Verlinde, 2010
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Philosophical concerns regarding quantum
gravity holography:
• Can one point to a fundamental ontology
with holographically related theories?
• Is one facing emergence of space, time
and/or gravity?
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Overview
• Introduction to holography (Jeroen)
• AdS/CFT and emergence (Sebastian)
• Emergence and Verlinde’s holographic
scenario (Dennis)
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Black hole thermodynamics and
quantum gravity degrees of freedom
• Bekenstein entropy: 𝑆 = 𝐴/4𝐺 (1972)
• Hawking radiation: 𝑇 = 1/8𝜋𝐺𝑀 (1974)
• Information paradox (1985)
• Holographic hypothesis (1993)
• “Maldacena” conjecture: AdS/CFT (1997)
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Holographic hypothesis of ’t Hooft
• The total number of degrees of freedom, 𝑛, in a region of
spacetime containing a black hole is:
𝑛 =𝑆
log 2=
𝐴
4𝐺log 2
• Hence, “we can represent all that happens inside [a volume] by
degrees of freedom on the surface”
• “This suggests that quantum gravity should be described
entirely by a topological quantum field theory, in which all
degrees of freedom can be projected on to the boundary”
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Holographic hypothesis of ’t Hooft
• “We suspect that there simply are no more
degrees of freedom to talk about than the ones
one can draw on a surface [in bit/Planck
length2]. The situation can be compared with a
hologram of a three dimensional image on a
two dimensional surface”
• Fundamental ontology, emergence?
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AdS/CFT
• 𝐷-dim. anti-de Sitter space
• In local coordinates:
d𝑠2 =ℓ2
𝑟2d𝑟2 − d𝑡2 + d𝐱2
• Fields 𝜙 𝑟, 𝑥
• CFT on ℝ𝐷−1
• Operators 𝒪 𝑥
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Duality Statement
• String theory in AdS space = CFT on boundary
• Fields 𝜙 𝑟, 𝑥 ↔ Operators 𝒪 𝑥
• Partition function 𝑑 = 𝐷 − 1 :
𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥 𝑟=0
= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
• One-to-one map of observables.
• Physical equivalence, mathematical structure
different
• Large distance ↔ high energy divergences
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Renormalization Group
• Radial integration: • Wilsonian
renormalization:
Λ 𝑏Λ 0
𝑘
integrate out
New cutoff 𝑏Λ
rescale 𝑏Λ → Λ until 𝑏 → 0
AdS𝑟
𝜕AdS𝑟 𝜕AdS𝜖
new boundary condition
integrate out
IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT
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Philosophical Questions
• Is one side of the duality more
fundamental?
– If QFT more fundamental, space-time could be
‘emergent’
– If duality not exact: room for emergence (e.g.
thermodynamics vs. atomic theory)
• Exact duality: one-to-one relation between
the values of physical quantities
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Remarks
• External view: meaning of observables
externally fixed, map relates different
physical quantities
– No empirical equivalence, numbers correspond
to different physical quantities
• No external point of view:
– How to decide which description to choose?
– Equivalence of descriptions
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• Holography and emergence of gravity?
• Erik Verlinde’s proposal
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Motivating Thoughts
• Hints from string theory, the holographic
conjecture/principle: there are solid indications
from quantum gravity research that gravitational
theories within a volume correspond to a theory
without gravity on the boundary of the volume
• AdS/CFT duality gives a concrete and detailed
example of the idea. Renormalization steps in the
CFT on the surface correspond to different sizes of
the bulk
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Could it be that gravity is “just
the bulk-description” of a world
without gravity?
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Further Motivating Thoughts
• Gravity is special: it is universal. It applies to all
matter and energy, regardless of specific
interactions; it seems to relate to space itself.
• This universality reminds one of the universal
character of thermodynamical behavior, which is
independent of microscopic details
• Gravity distinguishes itself from other forces
because it is difficult to quantize; is it
fundamentally different?
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Program of Research
• Start with a theory without gravity on a two-
dimensional screen, e.g. the surface of a sphere
• Holography: this theory codifies information about
matter in an additional spatial dimension (“in the
bulk”)
• The microscopic details of this gravitation-less
theory remain unspecified: it is a mere information
processing device, a theory of holographic “bits”
• Make gravity appear as a macroscopic
thermodynamic phenomenon
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Guiding idea about force as a
thermodynamic phenomenon
• Entropic processes: as a result of random motion of
its microscopic constituents a physical system will
end up in a state of greater entropy, i.e. higher
probability: the system seems to be directed
• Although there are no forces on the microscopic
level, on the thermodynamic level the system
appears driven, and this can be described by a
“macroscopic force”
• Like an ink droplet in water, or stretched polymer
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Working this out
• Imagine a sphere, whose area is divided into small
cells with each one “bit”. This information suffices
to describe the inside (holography)
• On the sphere an entropic process takes place: the
distribution of 0-s and 1-s tends to an equilibrium
• This process will correspond to gravitational
motion inside the sphere
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Appearance of Space
• In the surface theory, there are no spatial dimensions
other than those within the surface itself
• Consider several spheres, namely different surface
theories that relate to each other via
“renormalization” (or “coarse-graining”)
• “Coarse-grained” theories encode less information,
i.e. describe less space
• Thus, a spatial dimension x appears as a
bookkeeping device that records the level of coarse
graining on the sphere
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The appearance of gravity
• Number of bits on the sphere:
N ~ A = 4πR2
• Equipartition: E = Mc2 ~ N. T
• F = T ∆S/∆x
• ∆S ~ m.∆x
• From which we get: F ~ M.m/R2
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Microscopic theory
on sphere
Microscopic theory
in bulk
Macroscopic theory
in bulk: Gravity
Macroscopic theory
on sphere: no gravity
holography
Thermodynamic
limit
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• The holographic duality relation may well be a
bijective mapping
• There is no reason in this case to think that one side
is more fundamental than the other (left-right)
• But the thermodynamic limit introduces the
emergence of gravity in an uncontroversial sense
(top-bottom)
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Conclusions
• In the holographic scenario, the microscopic surface
theory is not necessarily more fundamental than the
microscopic bulk theory
• However, the appearance of gravity in the
thermodynamic limit makes it a clear case of
emergence, connected with robustness and novelty
of behavior. This robustness explains the
universality of gravitation
• That gravity is emergent could give rise to new
predictions: the law of gravity is not exact but
subject to fluctuations