From Boiling Water to Quantum Gravity
Miguel F. Paulos - CERN
Theory Division
Critical Phenomena and their Exponents
o Critical phenomena abundant in nature:
o
o
o
o
o
Opalescence (boiling water)
Ferromagnetic transitions (Curie point)
Superfluidity
Quantum Hall plateau transition
…
o Nice observables are critical exponents.
How to determine them?
o Different critical systems have same critical
exponents! Why?
Scale invariance
o
At criticality scale invariance emerges.
o
Explains opalescence of water: air bubbles of all
sizes, including optical wavelengths.
o
No preferred length scale: the correlation length
which sets size of typical bubble, becomes infinite.
In magnets, bubbles => magnetic domains/ spins.
Ordered
Critical
Temperature
"Great fleas have little fleas upon
their backs to bite 'em,
And little fleas have lesser fleas,
and so ad infinitum.”
A. de Morgan, Siphonaptera
Disordered
Renormalization Group
o
o
We average what’s going on at small scales to obtain a coarse-grained perspective at
larger scales.
This amounts to ``zooming out’’ in the figures below – also called renormalizing.
Ordered
Critical
Disordered
Temperature
o
o
When we do this, ordered phases look more ordered and vice-versa; but critical phase
remains the same. This is the meaning of scale invariance.
After zooming out, microscopic details are (mostly) irrelevant! Explains universality of
critical phenomena.
Landau-Ginzburg Theory
o
o
Simple model of phase transitions.
Describes systems by order parameters – e.g. magnetization.
Energy
Magnetisation
Ordered
Critical
Temperature
Disordered
Landau-Ginzburg Theory
o
o
Simple model of phase transitions.
Describes systems by order parameters – e.g. magnetization.
Energy
Magnetisation
o
o
o
Makes predictions for critical exponents.
Model can be complicated, improved, etc.
(Un)fortunately, these don’t always match experiment! Need more sophisticated theory.
Conformal Field Theories
and their Bootstrap
Conformal Invariance
o Transformations that preserve angles but not lengths.
Conformal Field Theory
o A field theory that sees angles but not
lengths.
o Fields means we have a continuous
description – justified by the
renormalization group.
o Critical exponents encoded in the energy
spectrum.
o Any given CFT describes many different
critical systems (universality!).
o Conformal invariance buys a theorist a lot
more than simple scale invariance!
The Conformal Bootstrap:
Use the enhanced symmetries of CFTs to:
1.
2.
Map out their parameter space.
Determine the spectrum (i.e. critical exponents) of
specific models.
Ultimate goal: classify the set of consistent theories
completely.
O(2) model (superfluid He4)
3d Ising model
(boiling water)
CFT Space
Banks-Zaks fixed point
(conformal window of QCD)
Maps of CFT space
Ruled out!
Ising model
CFT Space
Critical
exp
Allowed
Critical
exp
o Bounds above hold for any 3d conformal field theory – 3d critical
phenomena!
o Critical Ising model saturates a bound, and lies on a kink.
Conformal Spectroscopy
Quantum Gravity
Holography
o How much information can we
cram into a given volume?
o Naively, amount is proportional to
volume. Not so in a gravity theory!
o When density is too large, create a
black hole. The entropy
(information measure) of a black
hole grows like its area
(Bekenstein-Hawking).
Crunch time!
Holography
o Holographic principle
(‘t Hooft, Susskind): a
gravitational theory in
D dimensions is
equivalent to a nongravitational theory in
D-1 dimensions.
o Description of any
given volume lies on
the surface of that
volume.
The AdS/CFT correspondence
o Thanks to J. Maldacena ‘97 we
have a concrete example of
holography.
o Conjectured equivalence between
a Conformal Field Theory in 4
dimensions (no gravity) and
Superstring theory in 10
dimensions (gravity and other
things).
o Thanks to this correspondence,
understanding conformal field
theories directly teaches us about
string theory (and quantum
gravity)!
Summary
o Many physical systems display critical phenomena.
o At criticality there is often a description in terms of conformal field
theory.
o These theories have been intensely studied in the last few decades, and
especially in the last eight years.
o This has greatly increased our understanding of critical systems.
o Somewhat miraculously the same theories can describe quantum gravity
via the holographic principle. Much research today is focused on how
the hologram (CFT) can be translated into properties of quantum
gravity.
Thank you!