The holographic approach to critical points Johannes Oberreuter (University of Amsterdam)
Scale invariance power spectrum of CMB P s (k) / k n s 1 Lambda archive WMAP We need to understand critical points!
Examples of critical points
Examples of critical points First order phase transition
Examples of critical points Second order phase transition First order phase transition
Examples of critical points ferromagnet Second order phase transition First order phase transition
Examples of critical points ferromagnet Second order phase transition First order superconductor phase transition
ferromagnet Examples of critical points superfluid Second order phase transition First order superconductor phase transition
ferromagnet Examples of critical points superfluid Second order phase transition First order superconductor quark-gluon plasma phase transition
ferromagnet Examples of critical points superfluid Second order superconductor phase plasma? transition First order quark-gluon phase transition
What is special about a phase transition? Landau-Theory of phase transitions: F = a 2 + b 4 + H
What is special about a phase transition? Landau-Theory of phase transitions: F = a 2 + b 4 + H order parameter: magnetization
What is special about a phase transition? Landau-Theory of phase transitions: F = a 2 + b 4 + H a = a 0 (T T c ) order parameter: magnetization
What is special about a phase transition? Landau-Theory of phase transitions: F = a 2 + b 4 + H a = a 0 (T T c ) order parameter: magnetization = ± r a0 (T T c ) 2b below Tc
What is special about a phase transition? order parameter: magnetization critical temperature: Curie temperature divergence of correlation length power-law behavior: T T c scale invariance universality
What is special about scale invariance? x! x, t! t! scaling dimension classically, it s easy: no natural length scale dimensional analysis L =(@ ) 2 g 4 = D 2 2 (e.g. classical electromagnetism)
What is special about scale invariance? quantum mechanics: neeed renormalization scale invariant theories are very special fixed points necessary for QFT to be well defined can derive scaling behavior from them Conformal Field Theory (CFT) IR UV energy
Scaling behavior in a CFT ho(x)o(y)i ho(x)o(y)i 1 x y 2 1 2g 2 A log( x y +... 0 1 x y 2, = 0 + g 2 A +... Problem: Very often, CFT strongly coupled Series expansion breaks down Curious observation: CFTs are equivalent to gravity
gravitational dual from point of view of gravity, not entirely unexpected A box of gas lowered into a BH carries black hole m entropy What happens to the entropy, when gas vanishes in BH?
The second law of thermodynamics entropy must always increase: S + S 0 direction of time: microscopically reversible macroscopically irreversible
Black hole thermodynamics Things go into a black hole but not out. black hole m black hole A + A 0
Black hole thermodynamics S + S 0 A + A 0 Horizon area plays the role of entropy S BH = k BA 4l 2 P Surprise: entropy of a CFT scales with the volume, not with area
The holographic principle S BH = k BA 4l 2 P Gravity cannot be described by a field theory!
The holographic principle S BH = k BA 4l 2 P Gravity cannot be described by a field theory!... within the same space-time
The holographic principle S BH = k BA 4l 2 P Gravity cannot be described by a field theory!... within the same space-time
The holographic principle S BH = k BA 4l 2 P Gravity cannot be described by a field theory!... within the same space-time...on the boundary!
The holographic principle S BH = k BA 4l 2 P Gravity cannot be described by a field theory!... within the same space-time...on the boundary! The holographic principle
The AdS/CFT correspondence (an explicit example)
holographic renormalization group
specifics of the field theory 2 N N 0 ho 1...i (g 2 N)+(g 2 N) 2 +... 1+ 1 N + 1 N 2 +...
specifics of the field theory 2 N N 0 ho 1...i (g 2 N)+(g 2 N) 2 +... 1+ 1 N + 1 N 2 +...
duality relations g 2 YMN $ Rcurvature l string 4 1 N $ g string 2 opportunities:
duality relations g 2 YMN $ Rcurvature l string 4 1 N $ g string 2 opportunities: describe strongly coupled QFT
duality relations g 2 YMN $ Rcurvature l string 4 1 N $ g string 2 opportunities: describe strongly describe coupled QFT strongly coupled gravity
Use gravity to describe strongly coupled CFT:
Use field theory to describe gravity big bang: gravity strongly coupled quantum gravity caveat: our universe is de Sitter space (positive cosmological constant) has four dimensions
our model of the big bang gravity CFT there is no ground state field theory also singular need to renormalize field theory
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2 f = 3 2 16 2 2 ln µ 2 +finite
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2 f = 3 2 16 2 2 ln µ 2 +finite apple µ @ @µ + ( ) @ @ + n ( ) G(n) (x i : µ, )=0
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2 f = 3 2 16 2 2 ln µ 2 +finite apple µ @ @µ + ( ) @ @ + n ( ) G(n) (x i : µ, )=0 f = 3f 2 8 2 1-loop exact!
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2 f = 3 2 16 2 2 ln µ 2 +finite apple µ @ @µ + ( ) @ @ + n ( ) G(n) (x i : µ, )=0 f = 3f 2 8 2 1-loop exact! V ( )= 4 2 4 9ln( 2 M 2 )
renormalization of the boundary field theory deformed N =4 Super-Yang-Mills theory S = Z d 4 xtr 1 4 F µ F µ 1 2 D µ D µ + 1 4 [ i, j ][ i, j] + fermions f 2N 2 ( tr " ( 1 ) 1 1 5 #) 2 6X ( i ) 2 i=2 f = 3 2 16 2 2 ln µ 2 +finite apple µ @ @µ + ( ) @ @ + n ( ) G(n) (x i : µ, )=0 f = 3f 2 8 2 1-loop exact! V ( )= 4 2 4 9ln( 2 M 2 )
Need to include string-loop effects 1 N $ g string
Need to include string-loop effects 1 N $ g string
Summary: Critical points are very interesting and very important in nature The duality between field theory and gravity provides novel ways to deal with one from the point of view of the other The big bang is a point, where quantum gravity is important With the holographic duality, we can understand the big bang, albeit with difficulties