Transcript Emergent Physics and Quantum Gravity

Hyun Seok Yang
Institute for the Early Universe
Ewha Womans University
Based on constructive conversations with
Prof. Jungjai Lee (Daejin University).
This talk will be very speculative.
So please be patient for some
tough ideas.
Ether and Special Relativity
Remember that the 19th century physicists introduced
the Ether due to a deep trust of Newtonian mechanics
when Maxwell theory was completed.
Maxwell’s electromagnetism predicts the existence of a
universal velocity 𝑐 which contradicts the Galilean relativity
𝑣 ′ = 𝑣 + 𝑢. Newtonian space and time needs a single universal
frame of reference, Ether, to account the universal velocity.
For Einstein, the Maxwell theory implies a conceptual
change about space and time instead of Ether: The
concept of space or time is not absolute, but can differ
depending on the observer's velocity and only the single
entity, spacetime, has an invariant meaning.
Multiverse and Emergent Spacetime
I want to remark that the 20th century physicists
introduced the Multiverse picture due to a deep trust of
general relativity when string theory was formulated.
There are basically two approaches to formulate a quantum
theory of gravity. The first treats gravity as a fundamental
interaction which it attempts to quantise. In the second
approach gravity is not fundamental but an emergent
phenomenon. String theory falls into the second category.
It has the gratifying feature that not only gravity but also the
gauge interactions which are mediated by a spin one gauge
boson are emergent. String theory thus provides a unifying
framework of all elementary particles and their interactions:
………
M. Blau and S. Theisen, GRG 41 (2009) 743
Emergent Gravity
Basic idea
Gravity is not a fundamental force but a collective or emergent
phenomenon from NC or large N matrices:
1⊗1
𝑆
⇄ 2 ⊕ 0 or ⊂ ⊗ ⊃ ⇄ ○
In a market
(→): AdS/CFT correspondence, Matrix models,
open-closed string duality, KLT relation, etc.
(←): Kaluza-Klein theory, String compactifications
I will be more radical: Generate spacetime as well as matter fields
from NC or large N matrices.
Everything Is Emergent !
Gravity is emergent ⇒ Spacetime should be emergent too.
If spacetime is emergent, every structure supported
on spacetime should be emergent too !
Matter fields and non-Abelian gauge fields (weak and
strong forces) are emergent together with spacetime.
We need extra dimensions where a matter field is a
stable localized geometry (i.e., a topological object in
K-theory) in extra dimensions.
Main sources of multiverse idea
1: Cosmological constant problem with the anthropic principle
2: Chaotic and eternal inflation
3: String landscape
I want to emphasize that the multiverse picture has been invited
to cure some problems arising from the conventional (old)
spacetime picture or derived from an effective theory based on
the general relativity and QFT defined in a preexisting spacetime.
But I will assume that the birth of our Universe was a
cosmological event happened at the Planck epoch simultaneously
generating spacetime as well as all other matter fields.
Big-Bang: Arena of Quantum Gravity
Now I will borrow Hawking’s words (The Nature of Space and Time,
Princeton Univ. Press, 1996):
Unlike the black hole pair creation, one couldn’t say that the de
Sitter universe was created out of field energy in a preexisting
space. Instead, it would quite literally be created out of nothing:
not just out of the vacuum, but out of absolutely nothing at all,
because there is nothing outside the universe.
If so, in order to pose the origin of our Universe,
it is necessary to have a background-independent theory
for quantum gravity in which any spacetime structure
is not a priori assumed but defined by the theory itself.
Therefore, I want to test the multiverse idea with the assumption
that spacetime as well as matter fields are emergent.
Basic Slogan
(H. Elvang and J. Polchinski: hep-th/0209104)
The emergence of gravity requires the emergence
of spacetime itself.
Spacetime is not given a priori but defined by
more fundamental ingredients of the underlying theory.
Emergent gravity should be BACKGROUND INDEPENDENT.
Background Independent Quantum Gravity
0-dimensional IKKT Matrix model: No a priori spacetime
structure
1
𝑆=−
𝑇𝑟 𝑋𝑎 , 𝑋𝑏
8𝜋𝑔𝑠
2
Algebraic relations:
𝑋𝑎 , 𝑋𝑏 , 𝑋𝑐
+ cyclic = 0,
𝑋 𝑎 , 𝑋𝑎 , 𝑋𝑏
= 0.
Specify a vacuum: 𝑋𝑎𝑣𝑎𝑐 ≡ 𝐵𝑎𝑏 𝑦 𝑏 satisfying 𝑦 𝑎 , 𝑦 𝑏 = 𝑖 𝜃 𝑎𝑏
⇔ 𝐵𝑎𝑏 𝑣𝑎𝑐 = 𝜃 −1 𝑎𝑏 where 𝑎, 𝑏 = 1, ⋯ , 𝑑.
Consider fluctuations around the vacuum
𝑋𝑎 = 𝐵𝑎𝑏 𝑦 𝑏 + 𝐴𝑎 (𝑦)
Noncommutative Spacetime as Quantum Geometry
IKKT Matrix model = NC U(1) gauge theory on ℝ𝑑𝑁𝐶
1
2
𝑑
𝑆𝑁𝐶 =
∫
𝑑
𝑦
𝐹
−
𝐵
𝑎𝑏
𝑎𝑏
2
4𝑔𝑌𝑀
where
𝐹𝑎𝑏 = 𝜕𝑎 𝐴𝑏 − 𝜕𝑏 𝐴𝑎 − 𝑖 𝐴𝑎 , 𝐴𝑏 .
NC spacetime: Quantize a symplectic (or Poisson) manifold (𝑀, 𝐵)
à la Dirac
𝑓 𝑦 ,𝑔 𝑦
𝜃
→ −𝑖 𝑓, 𝑔 = 𝑓 𝑦 , 𝑔 𝑦
𝜃
+ 𝑂 𝜃3
∃ isomorphism between noncommutative ∗-algebra 𝐴𝜃 and
generalized vector fields Γ(𝑇𝑀):
For 𝑋𝑎 𝑦 = 𝐵𝑎𝑏 𝑦 𝑏 + 𝐴𝑎 𝑦 ∈ 𝐴𝜃 ,
𝑉𝑎 𝑓 ≡ −𝑖 𝑋𝑎 , 𝑓 = 𝑉𝑎 𝑓 + 𝑂 𝜃 3 .
“Lie algebra” homomorphism:
−𝑖 𝑋𝑎 𝑦 , 𝑋𝑏 𝑦
⇔
= −𝐵𝑎𝑏 + 𝐹𝑎𝑏 𝑦 ∈ 𝐴𝜃 ,
where 𝑉𝑎 ≡ 𝑉𝑋𝑎 , 𝑉𝑏 ≡ 𝑉𝑋𝑏 .
𝑉𝐹𝑎𝑏 = 𝑉𝑎 , 𝑉𝑏
𝑋𝑎 𝑦 , [𝑋𝑏 𝑦 , 𝑋𝑐 𝑦 ] = 𝐷𝑎 𝐹𝑏𝑐 𝑦 ∈ 𝐴𝜃 ,
⇔
𝑉𝐷𝑎 𝐹𝑏𝑐 = 𝑉𝑎 , [𝑉𝑏 , 𝑉𝑐 ] .
Einstein gravity arises from the leading order of NC gauge fields:
𝑑𝑠 2 = 𝐸 𝑎 ⊗ 𝐸 𝑎 = 𝜆2 𝛿𝑎𝑏 𝑉𝜇𝑎 𝑉𝜈𝑏 𝑑𝑥 𝜇 𝑑𝑥 𝜈
𝜇
where 𝑉𝑎 = 𝜆 𝐸𝑎 ∈ Γ 𝑇𝑀 , 𝐸 𝑎 = 𝜆𝑉 𝑎 ∈ Γ 𝑇 ∗ 𝑀 , 𝑉𝜇𝑎 𝑉𝑏 = 𝛿𝑏𝑎 and 𝜆2 =
𝑑𝑒𝑡 −1 𝑉𝜇𝑎 .
𝑋𝑎 , 𝑋𝑏 , 𝑋𝑐
𝑋 𝑎 , 𝑋𝑎 , 𝑋𝑏
+ cyclic = 0 ⇔ 𝑅 𝑎𝑏𝑐 𝑑 + 𝑂 𝜃 3 = 0,
=0 ⇔
𝑅𝑎𝑏 −
where
1
2
𝐺ℏ2
𝑐2
𝛿𝑎𝑏 𝑅 + 𝑂 𝜃 3 = 8𝜋𝐺 𝑇𝑎𝑏 + 𝑂 𝜃 3
2
~ 𝑔𝑌𝑀
𝜃.


Space and time on equal footing; if space
emerges, so should time.
Expect:
◦ Time is not fundamental.
◦ Approximate
(classical) notion of
macroscopic time
◦ Time is fuzzy
(ill defined) near
singularities.





We have no example of derived time.
Locality in time is more puzzling because of
the relation to causality.
Physics is about predicting the outcome of
an experiment before it is performed
(causality). What do we do without time?
How can things evolve without time?
How is a timeless theory formulated?
Emergent Time from Noncommutative Space
Time in quantum mechanics: 𝑥 𝑖 , 𝑝𝑗 = 𝑖ℏ 𝛿𝑗𝑖
𝑑𝑓
𝐻, 𝑓 = 𝑖ℏ
↔ 𝑈 𝑡 + 𝑓 0 𝑈(𝑡) = 𝑓 𝑡
𝑑𝑡
𝑖 𝐻𝑡
where 𝑈 𝑡 = exp
.
ℏ
(Particle) Time is an inner automorphism of noncommutative
algebra 𝐴ℏ .
Note that we have introduced a noncommutative space
𝑦𝑎 , 𝑦𝑏
=𝑖
𝜃 𝑎𝑏
→
𝑌𝑎
𝑌𝑏
Θ𝑎𝑏
𝑦 ,
𝑦 =𝑖
𝑑 𝑓(𝑦)
𝐻(𝑦), 𝑓(𝑦) = 𝑖ℏ
↔ 𝑈 𝑦, 𝑡
𝑑𝑡
𝑖 𝐻(𝑦)𝑡
where 𝑈 𝑦, 𝑡 = exp
.
𝑦 = 𝑖 𝜃 𝐵 − 𝐹(𝑦) 𝜃
+𝑓
𝑎𝑏
𝑦, 0 𝑈(𝑦, 𝑡) = 𝑓 𝑦, 𝑡
ℏ
(Space) Time is an inner automorphism of noncommutative
algebra 𝐴𝜃 .
.
Anatomy of Spacetime
If gravity and spacetime are emergent, it says that
𝑋𝑎𝑣𝑎𝑐 = 𝐵𝑎𝑏 𝑦 𝑏 → 𝑉𝑎𝑣𝑎𝑐 = 𝜕𝑎 → 𝑑𝑠 2 = 𝜂𝜇𝜈 𝑑𝑥 𝜇 ⊗ 𝑑𝑥 𝜈 .
−1
Flat spacetime is coming from the vacuum 𝐵𝑎𝑏 = 𝜃𝑎𝑏
and
the vacuum does not gravitate.
Gauge/gravity correspondence:
2
So 𝜌𝑣𝑎𝑐 ~ 𝐵𝑎𝑏
~ 𝑀𝑃4 where 𝑀𝑃2 =
𝐺ℏ2
𝑐2
1
8𝜋𝐺
2
~ 𝑔𝑌𝑀
𝜃.
in 4 dimensions.
Flat spacetime is NOT free gratis BUT a result of Planck energy
condensation in vacuum.
The flat spacetime and Lorentz symmetry must be very robust against
any perturbations and gravitational fields should be very weak:
𝜇
𝑋𝑎 (𝑦) = 𝐵𝑎𝑏 𝑦 𝑏 + 𝐴𝑎 (𝑦) → 𝑉𝑎 = 𝑉𝑎 𝜕𝜇 → 𝑔𝜇𝜈 = 𝜂𝜇𝜈 + ℎ𝜇𝜈 𝑑𝑥 𝜇 ⊗ 𝑑𝑥 𝜈 .
No Cosmological Constant Problem !
Shift symmetry:
→
𝑋𝑎𝑣𝑎𝑐
=
1
2
4𝑔𝑌𝑀
2
𝐵𝑎𝑏
→
𝑏
2
′
=
1
+ 2 Λ = 4𝑔2
𝜇
𝜈
𝑌𝑀
2
𝐵𝑎𝑏
𝜇
𝜇
= 𝐵𝑎𝑏 𝑦 → ds = 𝜂𝜇𝜈 𝑑𝑥 ⊗ 𝑑𝑥 → 𝑉𝑎 = Λ𝑎𝑏 𝑉𝑏
⇔ Global automorphism of 𝐴𝜃 → 𝐴𝜃 = Global Lorentz transformation .
Planck energy consensation in vacuum ⇒ NC space 𝑦 𝑎 , 𝑦 𝑏 = 𝑖𝜃 𝑎𝑏
→ Spacetime uncertainty principle → ∃ UV/IR mixing
⇒ Exclusion principle of spacetime
We cannot further accumulate Planck energy over NC space and so a
vacuum condensation for spacetime creation can happen only once !
Dark Energy
Recall that a vacuum energy itself does not couple to gravity since it is used
to create a flat spacetime in our picture. So let us calculate “vacuum
fluctuation” energy with the largest possible wavelength 𝐿𝐻 coming from
UV/IR mixing.
1
𝜌 = 𝜌𝑣𝑎𝑐 + 𝛿𝜌 = 2 𝐵𝑎𝑏 − 𝐹𝑎𝑏 𝑦
4𝑔𝑌𝑀
~ 𝑀𝑃4 1 +
𝐿2𝑃
𝐿2𝐻
2
1
𝐿2𝑃 𝐿2𝐻
1
𝐹(𝑦)~ 2 due
𝐿𝐻
2
1
2
=
𝐵
𝑎𝑏 1 + 𝜃 𝐹 𝑦
2
4𝑔𝑌𝑀
~ 𝑀𝑃4 +
where we assumed that
to a dimensional analysis.
Note that the vacuum fluctuation energy 𝛿𝜌 ~
1
2
2𝑔𝑌𝑀
𝐵𝑎𝑏 𝐹 𝑎𝑏 𝑦 ~
1
𝐿2𝑃 𝐿2𝐻
is a total derivative term and so a boundary term on a hypersurface of
radius 𝐿𝐻 . If 𝐿𝐻 = the size of cosmic horizon,
𝛿𝜌 = 10−3 𝑒𝑉 4 = dark energy ?
2
Inflation ?
I don’t know how to describe the dynamical process of the vacuum
energy condensate to generate an extremely large spacetime.
But we know that the dynamical scale of the vacuum condensate is
around the Planck energy ~ 𝑀𝑃 and the time scale for the
condensate is roughly 10−44 ≤ 𝛿𝑡𝐼 ≤ 10−33 .
A simple energy consideration for the condensate leads to the
expansion rate
1 2
𝑣
2 𝐼
−
4𝜋𝐺 𝜌𝑣𝑎𝑐
3
𝑅2 = 0 ⇔ 𝑣𝐼 = 𝐻𝐼 𝑅 where 𝐻𝐼 =
8𝜋𝐺𝜌𝑣𝑎𝑐
3
and 𝜌𝑣𝑎𝑐 = 𝑀𝑃4 .
The dynamical process of the vacuum condensate may correspond
to a cosmic inflation to generate an extremely large spacetime
~ 1030 𝐿𝐻 ~ 𝑒 60 𝐿𝐻 .
Personal Conclusion ?
The emergent spacetime picture seems to show a completely new
perspective about the cosmological constant problem.
A background independent quantum gravity, if any, may cripple all
the rationale to introduce the multiverse and there seems to be no
reason to rely on the picture to explain physical laws.
Nevertheless a background independent theory for emergent
gravity seems to imply an extremely large homogeneous and
isotropic space but the observable (our Universe) is a tiny part
(10−30 ~ 10−26 ) of entire spacetime.
(Maybe Level I multiverse according to M. Tegmark)