Introduction to Renormalization

Vincent Rivasseau

LPT Orsay

Cetraro, Summer 2010

Cetraro, Summer 2010, Cetraro, Summer 2010 Vincent Rivasseau, LPT Orsay

What is quantum field theory? Renormalization Constructive tools Constructive theory in zero dimension

Quantum Field Theory

What is quantum field theory?

Is it putting together quantum mechanics and special relativity?

Is it functional integrals?

Is it Feynman diagrams?

Is it axioms (Wightman... C ⋆ algebras...)?

Remark/statement/suggestion: quantum field theory has a soul which isrenormalization.

Trees, Forests, Jungles...

At the core of quantum field theory and renormalization lies thecomputation of connected quantities, hence trees, which are the simplestconnected structures. In fact trees, forests and jungles (ie layered forests)appear in almost endless ways in quantum field theory:

In the parametric representation of Feynman amplitudes(Kirchoff-Symanzik polynomials)

In renormalization theory (Zimmermann’s forests, Gallavotti-Nicolotrees, Connes-Kreimer Hopf algebra...)

In constructive field theory (Brydges-Kennedy-Abdesselam-R. forestformula, cluster expansions, Mayer expansions ...)

In more exotic renormalization group settings (Fermions in condensedmatter, non-commutative field theory, quantum gravity...)

It has even been proposed by Gurau, Magnen and myself that anyQFT should be considered as a scalar product on the vector spacegenerated by all possible trees (Tree Quantum Field Theory).

Quantum Field Theories as weighted species

A quantum field theory F should be the generating function for a certainweighted species, in the sense of combinatorists. It should involve only afew parameters: the space-time dimension, masses, coupling constants.The weights are usually called amplitudes.

Some of these weights may be naively given by divergent integrals, and theseries defining the generating function itself might have zero radius ofconvergence.

Renormalization theory adresses the first problem. → Couplings movewith scale.

Constructive field theory adresses the second problem. Species ofGraphs → Species of Trees.

Renormalization

Renormalization in physics is a very general framework to study how asystem changes under change of the observation scale.

Renormalization

Mathematically it reshuffles the initial theory with divergent weights into aninfinite iteration of a certain ‘renormalization group map” which involvesonly convergent weights.

Renormalization

In combinatoric terms roughly speaking it writes F = limGoGo...G , whereG corresponds to a single scale.

Renormalization

Single scale constructive theory reshuffles G itself. It relabels the “large”species of graphs in terms of the smaller species of trees. The result istypically only defined at small couplings, where it is the Borel sum of theinitial divergent series.

Renormalization

Single scale constructive theory reshuffles G itself. It relabels the “large”species of graphs in terms of the smaller species of trees. The result istypically only defined at small couplings, where it is the Borel sum of theinitial divergent series.

Multiscale constructive theory tries to combine the two previous steps in aconsistent way, but...5

The snag

Even after the first two problems have been correctly tackled, the flow ofthe composition map delivered by the renormalization group may after awhile wander out of the convergence (Borel) radius delivered byconstructive theory.

The snag

Even after the first two problems have been correctly tackled, the flow ofthe composition map delivered by the renormalization group may after awhile wander out of the convergence (Borel) radius delivered byconstructive theory.

This phenomenon always occur (either at the “infrared” or at the‘ultraviolet” end of the renormalization group) in field theory on ordinaryfour dimensional space time (except possibly for extremely special models).This is somewhat frustrating.

Ordinary φ44

It is defined on R4 through its Schwinger functions which are the momentsof the formal functional measure:

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

λ is the coupling constant, positive in order for the theory to be stable;

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

m is the mass, which fixes some scale;

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

a is called the ”wave function constant”, in general fixed to 1;

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

Z is the normalization, so that this measure is a probability measure;

Ordinary φ44

dν =1

Ze−(λ/4!)

φ4−(m2/2)R

φ2−(a/2)R

(∂µφ∂µφ)Dφ, (2.1)

Z is the normalization, so that this measure is a probability measure;

Dφ is a formal product∏

x∈Rd

dφ(x) of Lebesgue measures at each

point of R4.

The ordinary φ44 propagator

An infinite product of Lebesgue measures is ill-defined. So it is better todefine first the Gaussian part of the measure

dµ(φ) =1

Z0e−(m2/2)

φ2−(a/2)R

(∂µφ∂µφ)Dφ. (2.2)

where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.

dµ(φ) =1

Z0e−(m2/2)

φ2−(a/2)R

(∂µφ∂µφ)Dφ. (2.2)

where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.The covariance of dµ is called the (free) propagator

C (p) =1

(2π)21

p2 + m2, C (x , y) =

∫ ∞

0dαe−αm2 e−|x−y |2/4α

α2, (2.3)

dµ(φ) =1

Z0e−(m2/2)

φ2−(a/2)R

(∂µφ∂µφ)Dφ. (2.2)

where Z0 is again the normalization factor which makes (2.2) a probabilitymeasure.The covariance of dµ is called the (free) propagator

C (p) =1

(2π)21

p2 + m2, C (x , y) =

∫ ∞

0dαe−αm2 e−|x−y |2/4α

α2, (2.3)

where we recognize the heat kernel.

Feynman Rules

The full interacting measure may now be written as the multiplication ofthe Gaussian measure dµ(φ) by the interaction factor:

dν =1

Ze−(λ/4!)

φ4(x)dxdµ(φ) (2.4)

Feynman Rules

dν =1

Ze−(λ/4!)

φ4(x)dxdµ(φ) (2.4)

and the Schwinger functions are the normalized moments of this measure:

SN(z1, ..., zN) =

φ(z1)...φ(zN)dν(φ). (2.5)

Feynman Rules

dν =1

Ze−(λ/4!)

φ4(x)dxdµ(φ) (2.4)

and the Schwinger functions are the normalized moments of this measure:

SN(z1, ..., zN) =

φ(z1)...φ(zN)dν(φ). (2.5)

Expanding the exponential as a formal power series in the coupling constantλ we get perturbative field theory:

SN(z1, ..., zN) =1

∞∑

(−λ)n

φ4(x)dx

]nφ(z1)...φ(zN)dµ(φ) (2.6)

Feynman RulesBy Wick theorem, SN is a sum over “Wick contractions schemes”, i.e. waysof pairing together 4n + N fields into 2n + N/2 pairs. There are exactly(4n + N − 1)(4n + N − 3)...5.3.1 = (4n + N)!! such contraction schemes.

Sources

Vertices A Wick contraction

Figure: A contraction scheme

Feynman Rules

Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.

Feynman Rules

Each internal position x1, ..., xn is associated to a vertex of degree 4 andeach external position zi to a vertex of degree 1.The Feynman amplitude is in position space

AG (z1, ..., zN) =

∫ n∏

C (xℓ, x′ℓ) (2.7)

Feynman Rules

AG (z1, ..., zN) =

∫ n∏

C (xℓ, x′ℓ) (2.7)

There is an interesting combinatoric factor in front, which counts howmany ”Wick schemes” lead to that graph and multiplies by 1

n!(−λ4! )n.

Feynman Rules

AG (z1, ..., zN) =

∫ n∏

C (xℓ, x′ℓ) (2.7)

There is an interesting combinatoric factor in front, which counts howmany ”Wick schemes” lead to that graph and multiplies by 1

n!(−λ4! )n.

Feynman amplitudes are functions (in fact distributions) of the externalpositions z1, ..., zN . They may diverge either because of integration over allof R4 or because of the singularity in the propagator C (x , y) at x = y .

A Recipe for Renormalization

In ordinary field theory renormalization relies on the combination of threeingredients:

A scale decomposition

A locality principle

A power counting

The first two elements are quite universal. The third depends on the detailsof the model.

Scale decomposition

It is convenient to perform it using the parametric representation of thepropagator:

Scale decomposition

C =∑

C i , (2.8)

C i (x , y) =

∫ M−2(i+1)

M−2i

dαe−αm2 e−‖x−y‖2/4α

α2(2.9)

6 KM2ie−cM i‖x−y‖ (2.10)

where M is a fixed integer.

Scale decomposition

C =∑

C i , (2.8)

C i (x , y) =

∫ M−2(i+1)

M−2i

dαe−αm2 e−‖x−y‖2/4α

α2(2.9)

6 KM2ie−cM i‖x−y‖ (2.10)

where M is a fixed integer.

Higher and higher values of the scale index i probe shorter and shorterdistances.

Scale attributions, high subgraphs

Decomposing the propagator means that for any graph we have to sumover an independent scale index for each line of the graph.

Scale attributions, high subgraphs

Decomposing the propagator means that for any graph we have to sumover an independent scale index for each line of the graph.

At fixed scale attribution, some subgraphs play an essential role. They arethe connected subgraphs whose internal lines all have higher scale indexthan all the external lines of the subgraph. Let’s call them the ”high”subgraphs. They form a forest for the inclusion relation.

Locality principle

The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.

Locality principle

The locality principle is independent of the dimension: it simply remarksthat every ”high” subgraph looks more and more local as the gap betweenthe smallest internal and the largest external scale grows.Let’s visualize an example

Locality principle

Power counting

Power counting depends on the dimension d .

Power counting

Power counting depends on the dimension d .The key question is whether after spatial integration of the internal verticesof a high subgraph, save one, the sum over the gap between the lowestinternal and highest external scale converges or diverges.

Power counting

Power counting depends on the dimension d .The key question is whether after spatial integration of the internal verticesof a high subgraph, save one, the sum over the gap between the lowestinternal and highest external scale converges or diverges.

In four dimension by the previous estimates of a single scale propagator C ,power counting delivers a factor M2i per line and M−4i per vertexintegration

d4x . There are n − 1 ”internal” integrations to perform tocompare a high connected subgraph to a local vertex.

Perturbative renormalisability of φ44

For a connected φ44 graph, the net factor is 2l(G )−4(n(G )−1) = 4−N(G )

(because 4n = 2l + N). When this factor is strictly negative, the sum isgeometrically convergent, otherwise it diverges.

For instance for this graph the sum over the red scale i at fixed blue scalediverges (logarithmically) because there are two line factors M2i and asingle internal integration M−4i .

High subgraphs with N = 2 and 4 diverge when inserted into ordinary ones.

However this divergence can be absorbed into a change of the threeparameters (coupling constant, mass and wave function) which appeared inthe initial model.

This means physically that the parameters of the model do change with theobservation scale but not the structure of the model itself. This is a kind ofsophisticated self-similarity.

Such models are called (perturbatively) renormalizable. But...

The flow

Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.

The flow

Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4

4 theory the evolution of the coupling constant λ underchange of scale is mainly due to the first non trivial one-particle irreduciblegraph

The flow

Every ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

The flowEvery ”high” subgraph looks more and more local as the gap between thesmallest internal and the largest external scale grows.In the case of the φ4

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

Effective versus renormalized series

Expressing the perturbation theory in terms of the last (renormalized)coupling leads to the renormalized expansion.

The renormalized expansion requires some complicated book-keeping(Zimmermann’s forests, Connes-Kreimer Hopf algebra).

It subtracts local pieces of divergent subgraphs irrespective of whetherthey are high or not.

There is a price to pay, called renormalons.

Renormalons

Let us write the bubble graph in momentum space

AG (k) =

(p2 + m2)(p + k)2 + m2)

Renormalons

AG (k) =

(p2 + m2)(p + k)2 + m2)

AeffG (k) =

(p2 + m2)(p + k)2 + m2)−

|p|≥|k|

(p2 + m2)2

is bounded: |AeffG (k)| ≤ const.

Renormalons

AG (k) =

(p2 + m2)(p + k)2 + m2)

AeffG (k) =

(p2 + m2)(p + k)2 + m2)−

|p|≥|k|

(p2 + m2)2

is bounded: |AeffG (k)| ≤ const.

ArenG (k) =

(p2 + m2)(p + k)2 + m2)−

(p2 + m2)2

is finite but unbounded: |AeffG (k)| ∼|k|→∞ c log |k/m|.

Renormalons, II

A chain of n such graphs as above behaves as [log |q|]n. Inserting them in aconvergent loop leads to a total amplitude of Pn

[log |q|]nd4q

[q2 + m2]3≃n→∞ cnn!

which cannot be summed over n.

Expressing the theory in terms of all the running couplings leads to theeffective expansion (which is not a power series in a single coupling).

Because there is a single forest subtracted there is no book-keeping,(no need for Zimmermann’s forests nor Connes-Kreimer Hopf algebras)

There are also no renormalons, so the effective expansion is good forconstructive purpose.

The unavoidable Landau ghost?

In the case of the φ44 theory the evolution of the coupling constant λ under

change of scale is mainly due to the first non trivial one-particle irreduciblegraph we already saw.

It gives the flow equation

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (2.11)

whose sign cannot be changed without losing stability.This flow diverges ina finite time!

It gives the flow equation

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (2.11)

whose sign cannot be changed without losing stability.This flow diverges ina finite time! In the 60’s all known field theories sufferered from thisLandau ghost.

Asymptotic Freedom

In fact field theory and renormalization made in the early 70’s a spectacularcomeback:

Asymptotic Freedom

Weinberg and Salam unified the weak and electromagnetic interactionsinto the formalism of Yang and Mills of non-Abelian gauge theories,which are based on an internal non commutative symmetry.

Asymptotic Freedom

’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. They used a new technical tool calleddimensionalrenormalization.

Asymptotic Freedom

’t Hooft (1972, unpublished), Politzer, Gross and Wilczek (1973)discovered that these theories did not suffer from the Landau ghost.Gross and Wilczek then developed a theory of this type, QCD todescribe strong interactions (nuclear forces).

Asymptotic Freedom

Around the same time K. Wilson enlarged considerably the realm ofrenormalization, under the name of the renormalization group.

Asymptotic Freedom

Around the same time K. Wilson enlarged considerably the realm ofrenormalization, under the name of the renormalization group.

Happy end, all the people in red in this page got the Nobel prize...25

Constructive Theory

Functional integration looks good for global stability bounds(|

e−λφ4dµ(φ)| ≤ 1)

Constructive Theory

e−λφ4dµ(φ)| ≤ 1) but bad to compute connected functions

Constructive Theory

Feynman graphs have the opposite properties. Connectivity is readeasily on Feynman graphs, but these graphs proliferate too fast atlarge order for convergence

Constructive Theory

Constructive theory is a search for a good compromise between functionalintegral and Feynman graphs.

Constructive Theory

Constructive theory is a search for a good compromise between functionalintegral and Feynman graphs.

Trees are at the core of this compromise because they are not too manyand they ensure connectedness. Cycles/loops should be kept in functionalintegral form. So constructive theory is about finding trees and resummingloops.

A partial history of constructive field theory

70’s: superrenormalizable theories, eg φ42, φ4

3... with non-canonicalcluster and Mayer expansions (lattices of cubes...)

80’s: tools generalized into multiscale expansions to treatrenormalizable theories such as GN2.

Many applications developed in statistical mechanics

90’s: cluster expansions were realized unnecessary for Fermions. Thissimplification was important to constructive analysis of interactingFermions in condensed matter, with many results in the 00’s.

in the late 00’s new canonical methods have been found to dispense ofcluster/Mayer expansions for Bosons as well (loop vertex expansions,tree QFT...) and new domains opened (NCQFT, Quantum Gravity...)

Still no full-fledged constructive four dimensional field theorycompleted. φ4

4 has a Landau ghost, non-Abelian gauge theory hasGribov ambiguities and infrared confinement (1M$ problem...). →Grosse-Wulkenhaar model

Constructive QFT in Single Slid(c)e

Perturbative QFTS =

Problem: it does not converge:

|AG | = +∞

Perturbative QFTS =

|AG | = +∞

Constructive QFT = clever rewriting

Perturbative QFTS =

|AG | = +∞

S =∑

Perturbative QFTS =

|AG | = +∞

S =∑

Solution: it converges!∑

|AT | < +∞

Constructive rewriting

The forest formula provides canonical barycentric weights w(G ,T ) for thevarious trees T in a connected graph G .

w(G ,T ) =

ℓ∈T

ℓ 6∈T

xTℓ (w)

w(G ,T ) =

ℓ∈T

ℓ 6∈T

xTℓ (w)

xTℓ (w) = inf

ℓ′wℓ′ , ℓ′ ∈ unique path in T joining ends of ℓ

w(G ,T ) =

ℓ∈T

ℓ 6∈T

xTℓ (w)

xTℓ (w) = inf

ℓ′wℓ′ , ℓ′ ∈ unique path in T joining ends of ℓ

Barycentric weights means

w(G ,T ) = 1 ∀G .

An Example

w(G , T12) =

0dw2[inf(w1,w2)]

2 = 1/6

An Example

w(G , T12) =

0dw2[inf(w1,w2)]

2 = 1/6

w(G ,T13) = w(G ,T14) = w(G , T23) = w(G , T24)

inf(w1, w3)]

w3 = 5/24

An Example

w(G , T12) =

0dw2[inf(w1,w2)]

2 = 1/6

w(G ,T13) = w(G ,T14) = w(G , T23) = w(G , T24)

inf(w1, w3)]

w3 = 5/24

6+ 4 · 5

Fermions/Bosons

The constructive rewriting is nothing but

S =∑

w(G ,T )

AG =∑

AT , AT =∑

w(G ,T )AG

Fermions/Bosons

S =∑

w(G ,T )

AG =∑

AT , AT =∑

w(G ,T )AG

For Fermions one can apply this method directly to the graphs of theordinary perturbative expansion.

Fermions/Bosons

S =∑

w(G ,T )

AG =∑

AT , AT =∑

w(G ,T )AG

For Fermions one can apply this method directly to the graphs of theordinary perturbative expansion.

For Bosons one should apply this method to the graphs of the intermediatefield expansion (Loop Vertex Expansion).

The Forest Formula or “constructive swiss knife”

Let F be a smooth function of n(n − 1)/2 line variables xℓ, ℓ = (i , j),1 ≤ i < j ≤ n. The forest formula states

F (1, ..., 1) =∑

ℓ∈F

0dwℓ

ℓ∈F

∂xℓF

xF (w)]

, where

F (1, ..., 1) =∑

ℓ∈F

0dwℓ

ℓ∈F

∂xℓF

xF (w)]

, where

the sum over F is over all forests over n vertices,

F (1, ..., 1) =∑

ℓ∈F

0dwℓ

ℓ∈F

∂xℓF

xF (w)]

, where

the ‘weakening parameter” xFℓ (w) is 0 if ℓ = (i , j) with i and j in

different connected components with respect to F ; otherwise it is theinfimum of the wℓ′ for ℓ′ running over the unique path from i to j in F .

F (1, ..., 1) =∑

ℓ∈F

0dwℓ

ℓ∈F

∂xℓF

xF (w)]

, where

Furthermore the real symmetric matrix xFi ,j(w) (completed by 1 on

the diagonal i = j) is positive.

F (1, ..., 1) =∑

ℓ∈F

0dwℓ

ℓ∈F

∂xℓF

xF (w)]

, where

Furthermore the real symmetric matrix xFi ,j(w) (completed by 1 on

the diagonal i = j) is positive.

Five different ways to compute a log

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

is Borel summable. How to compute G (λ) = log F (λ) (and prove it is alsoBorel summable)?

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

Composition of series (XIXth century)

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

A la Feynman (1950)

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

A la Feynman (1950)

‘Classical Constructive”, a la Glimm-Jaffe-Spencer (1970...)

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

A la Feynman (1950)

With loop vertices (2007)...

F (λ) =

∫ +∞

−∞e−λx4−x2/2 dx√

A la Feynman (1950)

With loop vertices (2007)...

Tree QFT (2008)...

Borel Summability

Borel summability of a series an means existence of a function f with twoproperties

Borel Summability

Analyticity in a disk tangent at the origin to the imaginary axis

Borel Summability

plus uniform remainder estimates:

|f (λ) −N

anλn| ≤ KN |λ|N+1N! (4.12)

Borel Summability

plus uniform remainder estimates:

|f (λ) −N

anλn| ≤ KN |λ|N+1N! (4.12)

Given any series an, there is at most one such function f . When there isone, it is called the Borel sum, and it can be computed from the series toarbitrary accuracy.

Composition of series

F = 1 + H, H =∑

ap(−λ)p, ap =(4p)!!

log(1 + x) =∞

(−1)n+1 xn

∞∑

(−1)n+1 H(λ)n

bk(−λ)k ,

(−1)n+1

p1,..,pn≥1p1+...+pn=k

(4pj)!!

F = 1 + H, H =∑

ap(−λ)p, ap =(4p)!!

log(1 + x) =∞

(−1)n+1 xn

∞∑

(−1)n+1 H(λ)n

bk(−λ)k ,

(−1)n+1

p1,..,pn≥1p1+...+pn=k

(4pj)!!

Borel summability is unclear. Even the sign of bk is unclear.

A la Feynman

F = 1 + H, H =∑

ap(−λ)p, ap =1

p!#vacuum graphs on p vertices

A la Feynman

F = 1 + H, H =∑

ap(−λ)p, ap =1

∞∑

(−λ)kbk , bk =1

k!#vacuum connected graphs on k vertices

A la Feynman

F = 1 + H, H =∑

ap(−λ)p, ap =1

∞∑

(−λ)kbk , bk =1

b1 = 3, b2 = 48, b3 = 1584...

A la Feynman

F = 1 + H, H =∑

ap(−λ)p, ap =1

∞∑

(−λ)kbk , bk =1

b1 = 3, b2 = 48, b3 = 1584...

Borel summability unclear. bk ≥ 0 clear.

Classical Constructive

Cluster expansion = Taylor-Lagrange expansion of the functional integral:

F = 1 + H, H = −λ

∫ +∞

−∞x4e−λtx4−x2/2 dx√

Cluster expansion = Taylor-Lagrange expansion of the functional integral:

F = 1 + H, H = −λ

∫ +∞

−∞x4e−λtx4−x2/2 dx√

Mayer expansion: define Hi = −λ∫ 10 dt

∫ +∞−∞ x4

i e−λtx4i −x2

i /2 dxi√2π

= H ∀i ,

εij = 0 ∀i , j and write

F = 1 + H =∞

Hi (λ)∏

1≤i<j≤n

Defining ηij = −1, εij = 1 + ηij = 1 + xijηij |xij=1 and apply swiss knife.

Classical Constructive Field Theory, II

F =∞

Hi (λ)

ℓ∈F

0dwℓ

ℓ6∈F

1 + ηℓxFℓ (w)

F =∞

Hi (λ)

ℓ∈F

0dwℓ

ℓ6∈F

1 + ηℓxFℓ (w)

∞∑

Hi (λ)

ℓ∈T

0dwℓ

ℓ6∈T

1 + ηℓxTℓ (w)

F =∞

Hi (λ)

ℓ∈F

0dwℓ

ℓ6∈F

1 + ηℓxFℓ (w)

∞∑

Hi (λ)

ℓ∈T

0dwℓ

ℓ6∈T

1 + ηℓxTℓ (w)

where the second sum runs over trees!

F =∞

Hi (λ)

ℓ∈F

0dwℓ

ℓ6∈F

1 + ηℓxFℓ (w)

∞∑

Hi (λ)

ℓ∈T

0dwℓ

ℓ6∈T

1 + ηℓxTℓ (w)

Convergence easy because each Hi contains a different ”copy”∫

dxi offunctional integration.

F =∞

Hi (λ)

ℓ∈F

0dwℓ

ℓ6∈F

1 + ηℓxFℓ (w)

∞∑

Hi (λ)

ℓ∈T

0dwℓ

ℓ6∈T

1 + ηℓxTℓ (w)

Convergence easy because each Hi contains a different ”copy”∫

dxi offunctional integration.

Borel summability now easy from the Borel summability of H. But thismethod does not extend to noncommutative theory.

Loop Vertices

Intermediate field representation

Loop Vertices

∫ +∞

−∞e−λx4−x2/2 dx√

∫ +∞

−∞

∫ +∞

−∞e−i

√2λσx2−x2/2−σ2/2 dx√

dσ√2π

∫ +∞

−∞e−

log[1+i√

8λσ]−σ2/2 dσ√2π

∫ +∞

−∞

∞∑

n!dµ(σ) (4.13)

Loop Vertices

∫ +∞

−∞e−λx4−x2/2 dx√

∫ +∞

−∞

∫ +∞

−∞e−i

√2λσx2−x2/2−σ2/2 dx√

dσ√2π

∫ +∞

−∞e−

log[1+i√

8λσ]−σ2/2 dσ√2π

∫ +∞

−∞

∞∑

n!dµ(σ) (4.13)

Apply swiss knife by making copies: V n(σ) → ∏ni=1 Vi (σi ),

dµ(σ) → dµC (σi), Cij = 1 = xij |xij=1.

Loop Vertices, II

∞∑

ℓ∈F

0dwℓ

ℓ∈F

∂σi(ℓ)

∂σj(ℓ)

V (σi )

where CFij = xF

ℓ (w) if i < j , CFii = 1.

Loop Vertices, II

∞∑

ℓ∈F

0dwℓ

ℓ∈F

∂σi(ℓ)

∂σj(ℓ)

V (σi )

where CFij = xF

ℓ (w) if i < j , CFii = 1.

∞∑

ℓ∈T

0dwℓ

ℓ∈T

∂σi(ℓ)

∂σj(ℓ)

V (σi )

Loop Vertices, II

∞∑

ℓ∈F

0dwℓ

ℓ∈F

∂σi(ℓ)

∂σj(ℓ)

V (σi )

where CFij = xF

ℓ (w) if i < j , CFii = 1.

∞∑

ℓ∈T

0dwℓ

ℓ∈T

∂σi(ℓ)

∂σj(ℓ)

V (σi )

Advantages

One can picture the result as a sum over trees on loops, or ”cacti”. Since

∂σklog[1 + i

√8λσ] = −(k − 1)!(−i

√8λ)k [1 + i

√8λσ]−k ,

Advantages

∂σklog[1 + i

√8λσ] = −(k − 1)!(−i

√8λ)k [1 + i

√8λσ]−k ,

Convergence is easy because |[1 + i√

8λσ]−k | ≤ 1.

Advantages

∂σklog[1 + i

√8λσ] = −(k − 1)!(−i

√8λ)k [1 + i

√8λσ]−k ,

8λσ]−k | ≤ 1.

Borel summability is easy.

Advantages

∂σklog[1 + i

√8λσ] = −(k − 1)!(−i

√8λ)k [1 + i

√8λσ]−k ,

8λσ]−k | ≤ 1.

Borel summability is easy.

This method extends to non commutative field theory.

Tree Quantum Field Theory

This new axiomatic approach, initiated by Gurau-Magnen-R., is based oncanonical objects independent on any space-time background.

the universal space E algebraic span of marked trees,

the canonical (BKAR) forest formula,

a positive operator H, which glues one more branch on any tree:

x x* =

Each QFT corresponds to a scalar product on E through the tree formula.Correlation functions are expressed in terms of the resolvent 1

x x* =

Each QFT corresponds to a scalar product on E through the tree formula.Correlation functions are expressed in terms of the resolvent 1

This formalism still in development could eg allow continuous interpolationbetween QFT’s (even in different space-time dimensions).

Renormalization and Condensed Matter

Vincent Rivasseau

LPT Orsay

Cetraro, Summer 2010, Lecture II

Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture II Vincent Rivasseau, LPT Orsay

Grassmann Variables Many Fermions in Condensed Matter Single Scale Toy Model 2 Dimensional Jellium Model

Grassmann variables

Independent Grassmann variables ψ1, ..., ψn satisfy completeanticommutation relations

Grassmann variables

ψiψj = −ψjψi ∀i , j

Grassmann variables

plus a rule called Grassmann integration which is

dψi = 0,

ψidψi = 1.

and the rule that dψ symbols also anticommute between themselves andwith all ψ variables.

Grassmann variables

dψi = 0,

ψidψi = 1.

Any function of Grassmann variables is a polynomial with highestdegree one in each variable.

Grassmann variables

dψi = 0,

ψidψi = 1.

Any function of Grassmann variables is a polynomial with highestdegree one in each variable.

Pfaffians and determinants can be nicely written as Grassmannintegrals.

Determinants

The main important fact is that the determinant of any n by n matrix canbe expressed as a Grassmann Gaussian integral over 2n independentGrassmann variables.

Determinants

It is convenient to name these variables as ψ1, . . . , ψn, ψ1, . . . , ψn, althoughthe bars have nothing to do with complex conjugation.

Determinants

The formula is

detM =

dψidψie−

ij ψiMijψj .

Determinants

The formula is

detM =

dψidψie−

ij ψiMijψj .

Remember that for ordinary commuting variables and a positive n by n

Hermitian matrix M

Determinants

The formula is

detM =

dψidψie−

ij ψiMijψj .

Remember that for ordinary commuting variables and a positive n by n

Hermitian matrix M

∫ +∞

−∞

d φidφie−

ij φiMijφj = det−1M.

Grassmann Gaussian Integrals

Normalized Grassmann Gaussian measures may be written formally as

dµM =

dψidψie−

ij ψiM−1ijψj

∫∏

dψidψie−

ij ψiM−1ijψj

dµM =

dψidψie−

ij ψiM−1ijψj

∫∏

dψidψie−

ij ψiM−1ijψj

and again are characterized by their two point function or covariance

ψiψjdµM = Mij .

dµM =

dψidψie−

ij ψiM−1ijψj

∫∏

dψidψie−

ij ψiM−1ijψj

and again are characterized by their two point function or covariance

ψiψjdµM = Mij .

plus the Grassmann-Wick rule that n-point functions are expressed as sumover Wick contractions with signs.

Determinants

In short Grassmann Gaussian measures are simpler than ordinary Gaussianmeasures for two main reasons:

Determinants

Grassmann Gaussian measures are associated to any matrix M, there isno positivity requirement for M like for ordinary Gaussian measures.

Determinants

Grassmann Gaussian measures are associated to any matrix M, there isno positivity requirement for M like for ordinary Gaussian measures.

their normalization directly computes the determinant of M, not theinverse (square-root of) the determinant of M. This is essential inmany areas where factoring out this determinant is desirable; it explainsin particular the success of Grassmann and supersymmetric functionalintegrals in the study of disordered systems → Tom Spencer’s lectures.

Pfaffians

PfaffiansThe Pfaffian Pf(A) of an antisymmetric matrix A is defined by

detA = [Pf(A)]2.

We can express the Pfaffian as:

Pf(A) =

dχ1...dχne−

i<j χiAijχj =

dχ1...dχne− 1

i,j χiAijχj .

detA = [Pf(A)]2.

Pf(A) =

dχ1...dχne−

i<j χiAijχj =

dχ1...dχne− 1

i,j χiAijχj .

Indeed we write

detA =

dψidψie−

ij ψiAijψj .

detA = [Pf(A)]2.

Pf(A) =

dχ1...dχne−

i<j χiAijχj =

dχ1...dχne− 1

i,j χiAijχj .

Indeed we write

detA =

dψidψie−

ij ψiAijψj .

Performing the change of variables (which a posteriori justifies the complexnotation)

ψi =1√2(χi − iωi ), ψi =

1√2(χi + iωi ),

detA = [Pf(A)]2.

Pf(A) =

dχ1...dχne−

i<j χiAijχj =

dχ1...dχne− 1

i,j χiAijχj .

Indeed we write

detA =

dψidψie−

ij ψiAijψj .

ψi =1√2(χi − iωi ), ψi =

1√2(χi + iωi ),

shows why detA is a perfect square and proves the red formula.

detA = [Pf(A)]2.

Pf(A) =

dχ1...dχne−

i<j χiAijχj =

dχ1...dχne− 1

i,j χiAijχj .

Indeed we write

detA =

dψidψie−

ij ψiAijψj .

ψi =1√2(χi − iωi ), ψi =

1√2(χi + iωi ),

shows why detA is a perfect square and proves the red formula.Reference on Grassmann calculus in QFT: J. Feldman, RenormalizationGroup and Fermionic Functional Integrals.6

Fermionic Fields

The Hamiltonian Fock space formalism for electrons in condensed matter isbetter reexpressed in terms of Grasmannian fields.

Fermionic Fields

We consider independent Grasmmann valued fields ψ(ξ), ψ(ξ) with

Fermionic Fields

ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~

Fermionic Fields

ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~

The propagator is

Cab(k) = δab1

ik0 − [ε(~k) − µ]

where a, b are the spin indices.

Fermionic Fields

ξ = (~x , t), ~x ∈ Rd , t ∈ [− ~

The propagator is

Cab(k) = δab1

ik0 − [ε(~k) − µ]

where a, b are the spin indices.

The momentum vector ~k has d spatial dimensions and ε(~k) is the energyfor a single electron of momentum ~k. The parameter µ corresponds to thechemical potential. The (spatial) Fermi surface is the manifold ε(~k) = µ.

The Jellium Model

For a jellium isotropic model the energy function is invariant under spatialrotations

ε(~k) =~k2

The Jellium Model

ε(~k) =~k2

where m is some effective or “dressed” electron mass. In this case theFermi surface is simply a sphere. This jellium isotropic model is realistic inthe limit of weak electron densities, where the Fermi surface becomesapproximately spherical.

The Jellium Model

ε(~k) =~k2

where m is some effective or “dressed” electron mass. In this case theFermi surface is simply a sphere. This jellium isotropic model is realistic inthe limit of weak electron densities, where the Fermi surface becomesapproximately spherical.

In general a propagator with a more complicated energy function ε(~k) hasto be considered to take into account eg the effect of the lattice of ions in asolid.

The Hubbard model in d = 2, H2

( ( ( ((

The position variable x lives on the lattice Z2, hence

ε(k) = cos k1 + cos k2

( ( ( ((

(! (" (# ($ % $ # " !(!

At µ = 0 the Fermi surface is a square of side size√

2π, joining the points(π, 0), (0, π) in the first Brillouin zone. The particle-hole symmetry makesthe Fermi surface invariant under RG flow.

Matsubara Frequencies

The Matsubara frequencies are:

k0 = ±2n + 1

so the integral over k0 is really a discrete sum over n.

k0 = ±2n + 1

so the integral over k0 is really a discrete sum over n.

For any n we have k0 6= 0, so that the denominator in C (k) can never be 0.This is why the temperature provides a natural infrared cut-off. But whenT → 0, k0 becomes a continuous variable and the propagator diverges onthe “space-time” Fermi surface, defined by k0 = 0 and ε(~k) = µ.

The Interaction

The electron-electron interaction is complicated; in solids it is mostly due tothe underlying lattice (phonons). To study the long range behavior of thesystem it can be efficiently modeled again by a local term:

SΛ = λ

Λdd+1ξ (

a∈↑,↓

ψaψa)2(ξ) .

The Interaction

The electron-electron interaction is complicated; in solids it is mostly due tothe underlying lattice (phonons). To study the long range behavior of thesystem it can be efficiently modeled again by a local term:

SΛ = λ

Λdd+1ξ (

a∈↑,↓

ψaψa)2(ξ) .

Remark that this term is quite unique, as a, the spin index, takes only twovalues.

Renormalization in Condensed Matter

The general renormalization ideas (slice decomposition, locality, powercounting) and even constructive techniques have to be subtly adapted tothe case the electrons of condensed matter.

Scale decomposition

The scales are again defined by slicing gently the propagator according toits spectrum. For instance:

C =∞

Cj ; Cj(k) =fj(k)

ik0 − (ε(~k) − µ)

Scale decomposition

The scales are again defined by slicing gently the propagator according toits spectrum. For instance:

C =∞

Cj ; Cj(k) =fj(k)

ik0 − (ε(~k) − µ)

where the slice function fj(k) effectively forces |ik0 − (ε(~k) − µ)| ∼ M−j ,

for some fixed parameter M > 1.

Slices around J2 Fermi Surface

These slices pinch more and more the Fermi surface as j → ∞.

The multiscale analysis now relies on finding the subgraphs which haveinternal scales j lower than external scales (analog of an infrared QFTproblem), to compute an effective theory for degrees of freedom closer andcloser to the Fermi surface.

Locality Principle

It is surprising and subtle that the locality principle still holds!

Locality Principle

Moving propagators around indeed leads to a non trivial phase factor:

Locality Principle

Clow (y , ..) = Clow (x , ...)+ [ ~(x − y) ·~pF + ~(x − y) · (~∇−~pF )]Clow (x , ...)+ · · ·

Locality Principle

However the most divergent part of eg the bubble graph

Locality Principle

occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,

Locality Principle

occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,in which case the two factors, ~pF · ~(x − y) and −~pF · ~(x − y),approximately cancel each other!

Locality Principle

occurs when the two pairs of entering momenta at both ends approximatelyadds to zero,in which case the two factors, ~pF · ~(x − y) and −~pF · ~(x − y),approximately cancel each other!

This is why the theory is still renormalizable.

Locality Principle, II

More precisely, in momentum space

AG (p1, p2, p3, p4) =

dk0d~k1

ik0 +~k2

2m− µ

iη(k0 + q0) + (~k+~q)2

2m− µ

AG (p1, p2, p3, p4) =

dk0d~k1

ik0 +~k2

2m− µ

iη(k0 + q0) + (~k+~q)2

2m− µ

where q = p1 + p2 = −(p3 + p4) and η = ±1, depending on the arrows forthe lines (from ψ to ψ).

AG (p1, p2, p3, p4) =

dk0d~k1

ik0 +~k2

2m− µ

iη(k0 + q0) + (~k+~q)2

2m− µ

The maximal value occurs for q = 0 and arrows in the same direction(η = −1). This algebraic combination corresponds to the so-called Cooperpairs.

AG (p1, p2, p3, p4) =

dk0d~k1

ik0 +~k2

2m− µ

iη(k0 + q0) + (~k+~q)2

2m− µ

The maximal value occurs for q = 0 and arrows in the same direction(η = −1). This algebraic combination corresponds to the so-called Cooperpairs.

In the case of an attractive BCS interaction the main effect is growth of thecoupling constant with j and a small change in the Fermi radius.

Power Counting is Independent of Dimension

In fact power counting gives a just renormalizable theory in any dimension!

Since the slices are insensitive to the directions tangent to the Fermisphere, the theory has the power counting of a two dimensional model witha 1/|p| kind of propagator and a φ4 kind of interaction, hence is justrenormalizable.

More precisely in momentum representation a vacuum graph with n verticeshas n + 1 loop lines, hence n + 1 loop integrals, each over a M−2j volume;and it has l = 2n propagators, each of which of size M j . The result isneutral in n, which indicates just renormalizability.

The BCS phase transition

It occurs when the coupling constant become of order 1. The RG flow isgoverned by the bubble B(p) and leads to an effective coupling

λeff = λbare/(1 − λbareB(p)).

When λB(0) = 1 we get divergence. Expanding

λB(p) = 1 + p2λB”(0)...

we get the Cooper pair boson propagator in 1/λB”(0)p2, which has apower counting depending of the dimension.

RG and Phase Transition

Expanding a theory around a given vacuum is a saddle pointapproximation. The quadratic part, or Hessian approximation, definesthe propagators of the particles; the rest is their interactions.

A phase transition can occur when the RG flow has moved the theoryout of the convergent region near the initial vacuum.

This means that another saddle point (or orbit) has to be computedwith a different Hessian hence usually completely different particlesand physics.

Some lessons to remember

Nature seems to love renormalizable theories!

Particles and even the nature of their associated renormalization groupcan change completely across a phase transition!

The Fermi Liquid Problem

( ⁄y ( ⁄y

’’ F

Physics Textbooks Definition Says:

( ⁄y ( ⁄y

’’ F

A Fermi liquid is an interacting system of Fermions whose density of statesat zero temperature, like in the free case, is discontinuous at a certainvalue. This value is called the interacting Fermi radius.

( ⁄y ( ⁄y

’’ F

( ⁄y ( ⁄y

’’ F

( ⁄y ( ⁄y

’’ F

Problem: this discontinuity never really occurs for parity-invariant modelsbecause BCS or similar phase transition are generic at low temperatures(Kohn-Luttinger).

( ⁄y ( ⁄y

’’ F

Problem: this discontinuity never really occurs for parity-invariant modelsbecause BCS or similar phase transition are generic at low temperatures(Kohn-Luttinger).

“An interacting Fermi liquid maybe a figment of the imagination” (P.W.Anderson).21

Salmhofer’s CriterionSalmhofer proposed a useful mathematical criterion of Fermi liquidbehavior. Perhaps this is what the standard textbooks “had in mind”.

( ⁄y ( ⁄y

’’ F

(>?⁄y%@A& (>?⁄

⁄* <

’’ F

(>?⁄y%@A& (>?⁄

⁄* <

’’ F

Salmhofer’s Criterion:

(>?⁄y%@A& (>?⁄

⁄* <

’’ F

The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,

(>?⁄y%@A& (>?⁄

⁄* <

’’ F

The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,The self-energy as function of the momentum is bounded uniformlytogether with its first and second derivatives in that domain.

(>?⁄y%@A& (>?⁄

⁄* <

’’ F

The Schwinger functions are analytic in λ in a domain|λ| ≤ K/| log T |,The self-energy as function of the momentum is bounded uniformlytogether with its first and second derivatives in that domain.

A key question after discovery of high Tc supraconductors was theproperties of 2 − d interacting Fermi liquids.22

Toy (Single Scale) Fermionic Model

There is no need to introduce a lattice of cubes to compute connectedfunctions of a Fermionic theory.

Consider eg a Fermionic d-dimensional QFT in an infrared slice with N

colors. Suppose the propagator is diagonal in color space and satisfies thebound

|Cj ,ab(x , y)| ≤ δabM−dj/2

e−M−j |x−y |

We say that the interaction is of the vector type (or Gross-Neveu type) if itis of the form

V = λ

ψa(x)ψa(x))(

ψb(x)ψb(x))

e−M−j |x−y |

V = λ

ψa(x)ψa(x))(

ψb(x)ψb(x))

where λ is the coupling constant.23

Toy Fermionic Model, II

We claim that

The perturbation theory for the connected functions of this single slicemodel has a radius of convergence in λ which is uniform in j and N.

We claim that

The J2 model is roughly similar to that model with the role of colorspayed by N = M j angular sectors around the Fermi surface.

Fermionic Tree Expansion

Expanding the pressure

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

|Λ| log Z (Λ)

through the forest formula leads to

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

dµC (ψ, ψ)eSΛ(ψa,ψa))

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

(λn/n!)N

a1,...,an,b1,...,bn=1

ε(T ,Ω)(

ℓ∈T

0dwℓ

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

(λn/n!)N

a1,...,an,b1,...,bn=1

ε(T ,Ω)(

ℓ∈T

0dwℓ

dx1...dxnδ(x1 = 0)∏

ℓ∈T

Cj ,ab(xℓ, yℓ))

× det[Cj ,ab()]remaining

Bounds on the Remaining Determinant

Bounds on the Remaining Determinant[Cj ,ab()]remaining is the matrix with entries

xTkm(w) · Cj ,ab(xk , ym)

corresponding to the fields ψ(xm) and ψ(yn) which have not beenWick-contracted into the tree T .

Suppose we have written

Cj(xk , ym) < fj ,k , gj ,m >L2

(this is realized through fj ,k = fj(xk , ·) and gj ,m = gj(·, ym) if

fj(p).gj(p) = Cj(p)).

fj(p).gj(p) = Cj(p)). Then we have the Gram inequality:

|det[Cj ,ab()]remaining| ≤∏

anti−fields k

||fj ,k ||∏

fields m

||gj ,m||

anti−fields k

||fj ,k ||∏

fields m

||gj ,m||

Proof: The w dependence disappears by extracting the symmetric squareroot v of the positive matrix xT so that xT

km =∑n

k=1 vTknv

Uniform Radius in j for the Toy Model

There is a factor M−dj/2 per line, or M−dj/4 per field ie entry of theloop determinant. This gives a factor M−dj per vertex

There is a factor M+dj per vertex spatial integration (save one)

Hence the λ radius of convergence is uniform in j .

Uniform Radius in N for the Toy Model

There is a factor N−1/2 per line, or N−1/4 per field ie entry of the loopdeterminant. This gives a factor N−1 per vertex

There is a factor N per vertex (plus one)

The last item is not obvious to prove, because we don’t know all the graph,but only a tree To prove it we organize the sum over the colors from leavesto root of the tree. In this way the pay a factor N at each leaf to know thecolor index which does not go towards the root, then prune the leaf anditerate. The last vertex (the root) is the only special one as it costs two N

factors.

Hence the λ radius of convergence is uniform in N.

J2 Model in a RG Slice

We claim that this model is roughly similar to the Toy Model, withdimension d = 3.

The naive estimate on the slice propagator is (using integration by parts)

|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2

(using Gevrey cutoffs fj to get fractional exponential decay).

|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2

This is much worse than the factor M−3j/2 that would be needed.

|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2

But the situation improves if we cut the Fermi slice into smaller pieces(called sectors).

J2 Sectors

Suppose we divide the j-th slice into M j sectors, each of size roughly M−j

in all three directions.

J2 Sectors

in all three directions.A sector propagator C j ,a has now prefactor M−2j and

|Cj ,ab(x , y)| ≤ δabM−2je−[M−j |x−y |]1/2

J2 Sectors

(using again Gevrey cutoffs fja for fractional power decay).

J2 Sectors

(using again Gevrey cutoffs fja for fractional power decay).But since N = M j

M−2j =M−3j/2

J2 Sectors

M−2j =M−3j/2

so that the bound is identical to that of the toy model.

Momentum Conservation

In two dimensions a rhombus (i.e; a closed quadrilateral whose four sideshave equal lengths) is a parallelogram.

Hence an approximate rhombus should be an approximate parallelogram.

Momentum conservation δ(p1 + p2 + p3 + p4) at each vertex follows fromtranslation invariance of J2. Hence p1, p2, p3, p4 form a quadrilateral. For j

large we have |pk | ≃√

2Mµ hence the quadrilateral is an approximaterhombus. Hence the four sectors to which p1, p2, p3 and p4 should beroughly equal two by two (parallelogram condition).

It means that the interaction is roughly of the color (or Gross-Neveu) typewith respect to these angular sectors:

ψaψa

ψbψb

Anisotropic Sectors

The rhombus rule is not fully correct for almost degenerate rhombuses.

Anisotropic Sectors

This is the source of painful technical complications (anisotropic angularsectors) which were developed by Feldman, Magnen, Trubowitz and myself.

Anisotropic Sectors

One should use in fact M j/2 longer sectors in the tangential direction (oflength M−j/2). The corresponding propagators have dual decay because thesectors are still aprroximately flat.

Anisotropic Sectors

Ultimately the conclusion is unchanged: the radius of convergence of J2 ina slice is independent of the slice index j .

Results on 2d Interacting Fermi Liquid

This plus a lot of work to fill in the “technical details” lead to theproof of Salmhofer’s criterion for J2. This was the first mathematicalconstruction of an interacting Fermi liquid (Disertori-R., 2000).

In 2002 Benfatto, Giuliani and Mastropietro extended our analysis tothe case of non-rotation invariant curves close to the jellium case.

Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.

Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.Following this road, Feldman, Knorrer and Trubowitz built in greatdetail a 2D Fermi liquids with non-parity invariant surfaces in animpressive series of papers completed around 2003-2004.

Next Models

There are not many sectors for H2; it is not a Fermi liquid in the sense ofSalmhofer but a “Luttinger liquid” with logarithmic corrections.(Afchain-Magnen-R., 2004).

Next Models

As the Hubbard filling factor moves from zero to half-filling, there is acrossover between Fermi and Luttinger behavior (Benfatto, Giuliani,Mastropietro)

Next Models

There is no rhombus rule for J3; although it is a Fermi liquid, new methodshave to be developed to treat it constructively (work in progress).

Introduction to Group Field Theory

Vincent Rivasseau

LPT Orsay

Cetraro, Summer 2010, Lecture III

Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture III Vincent Rivasseau, LPT Orsay

GFT Group Field Theory

In quantum field theory (QFT) we said renormalization group relies on

The locality principle

Power counting

However a quantum theory of space-time and gravity may have to startwithout any fixed space time background metric. So what should be thecorresponding background-invariant notion of scale?

Power counting

We propose

Power counting

We propose

Definition: A scale is a slice of eigenvalues of a propagator (slicing gentlyaccording to a geometric progression).

Power counting

We propose

Definition: A scale is a slice of eigenvalues of a propagator (slicing gentlyaccording to a geometric progression).

So we turned the problem into that of finding the right propagator withnon-trivial spectrum for quantum gravity.

Non Commutative Quantum Field Theory

The Grosse-Wulkenhaar φ⋆4 model on the Moyal space R4 with1/(p2 + Ωx2) propagator is just renormalizable. But itsrenormalization relies on

A new scale decomposition (which mixes ultraviolet and infrared),

A new scale decomposition (which mixes ultraviolet and infrared),A new locality principle (Moyality),

A new scale decomposition (which mixes ultraviolet and infrared),A new locality principle (Moyality),A new power counting (under which only regular graphs diverge).

A big unexpected bonus is the existence of a non trivial fixed pointwhich makes the GW model essentially the prime candidate for a fullysolvable 4D field theory.

The new multiscale analysis

The propagator 1/[p2 + Ω2(x)2] has parametric representation in x space

C (x , y) =θ

∫ ∞

0dα e−

µ20θ

(sinhα)2exp

θ sinhα‖x − y‖2 −

θtanh

2(‖x‖2+‖y‖2)

C (x , y) =θ

∫ ∞

0dα e−

µ20θ

(sinhα)2exp

θtanh

2(‖x‖2+‖y‖2)

involving the Mehler kernel rather than the heat kernel.

C (x , y) =θ

∫ ∞

0dα e−

µ20θ

(sinhα)2exp

θtanh

2(‖x‖2+‖y‖2)

We slice it according to the same method

C i (x , y) =

∫ M−2(i−1)

M−2i

dα · · · 6 KM2ie−c1M2i‖x−y‖2−c2M

−2i (‖x‖2+‖y‖2).

C (x , y) =θ

∫ ∞

0dα e−

µ20θ

(sinhα)2exp

θtanh

2(‖x‖2+‖y‖2)

We slice it according to the same method

C i (x , y) =

∫ M−2(i−1)

M−2i

dα · · · 6 KM2ie−c1M2i‖x−y‖2−c2M

−2i (‖x‖2+‖y‖2).

The corresponding scales correspond to a mixture of the previous ultraviolet

and infrared notions.

The Moyal vertex

The Moyal vertex in direct space is proportional to

∫ 4∏

d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

The Moyal vertex

∫ 4∏

d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

This vertex is non-local and oscillates. It has a parallelogram shape, and itsoscillation is proportional to its area:

The Moyal vertex

∫ 4∏

d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

The Moyal vertex

∫ 4∏

d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

The Moyal vertex

∫ 4∏

d4x iφ(x i ) δ(x1 − x2 + x3 − x4) exp(

2ıθ−1(x1 ∧ x2 + x3 ∧ x4))

The Moyality principle

It replaces locality:

This principle applies only to regular, high subgraphs.

Noncommutative Renormalization

Renormalizability of the GW model combines the three elements:

Power counting tells us that only regular graphs with two and four externallegs must be renormalized.

The Moyality principle says that such “high” regular graphs look like Moyalproducts. The corresponding counterterms are therefore of the form of theinitial theory.

Again this means that the theory is renormalizable.

Group Field Theory

Group field theory (GFT) lies at the crossroads between

Group Field Theory

The discretization of Cartan’s 1rst order formalism by loop quantumgravity

Group Field Theory

The higher dimensional tensorial extension of the matrix modelsrelevant for 2D quantum gravity

Group Field Theory

The higher dimensional tensorial extension of the matrix modelsrelevant for 2D quantum gravity

The “Regge” or simplicial approach to quantum gravity

Advantages of Group Field Theory

There are several enticing factors about GFT:

GFT is an attempt to quantize completely space-time, summing at thesame time over metrics and topologies.

GFT could provide a scenario for emergent large, manifold-like classicalspace-time as a condensed phase of more elementary space-timequanta.

If that is the case, there is no reason that some GFT’s cannothopefully be renormalized through a suitably adapted multiscale RG.

The spin foams of loop gravity are exactly Feynman amplitudes ofgroup field theory.

Objections to GFTThe main objections to GFT’s that I heard during the last two years are

1) The GFT’s are tensor generalizations of matrix models; but whereasmatrix models really triangulate 2D Riemann surfaces, GFT’striangulate much more singular objects (not even pseudo-manifoldswith local singularities).

2) The natural GFT’s action does not look positive. Hence thenon-perturbative meaning of the theory is unclear.

3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time.

4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process;

5) GFT’s dont predict space-time dimension, nor the standard model,hence are not a TOE.

Answering the Objections

1 and 2) Colored Field theory could be an answer.

3) It seems difficult to identify ordinary space-time and physicalobservables in GFT’s. In particular the ultraspin limit j → ∞ can beinterpreted either as ultraviolet limit (on the group) or as an infraredlimit for a large ”quantum” of space-time. I believe quantum gravitymight be a theory with ultraviolet/infrared mixing. Experience withNCQFT may help.

4) GFT’s have infinities and it is not clear whether these infinitiesshould or can be absorbed in a renormalization process. We havestarted in Orsay a systematic study of this point.

5) GFT’s dont predict space-time dimension, nor the standard model,hence are not a TOE. Here I have nothing to say, except it is not clearto me that competition (string theory) is faring really so much betteron this point.

Colored GFT

A colored GFT (Gurau, arXiv:0907.2582). is a model with two types ofvertices, one with only incoming and the other with only outgoing fields.

Colored GFT

The D + 1 fields of the vertex each have a distinct color and the propagatorjoins only identical colors.

Colored GFT

The D + 1 fields of the vertex each have a distinct color and the propagatorjoins only identical colors.

Faces are 2-D objects which are the connected bicolor components of thegraphs; their length is therefore even. Higher p-dimensional ”bubbles” arethe connected components with p colors. Therefore colored graphs are trueD-dimensional complexes, with a canonical D dimensional homology.

Colored GFT may solve the first two objections

Colored GFT’s amplitudes are dual to manifolds with only localisolated singularities.arXiv:1006.0714 Lost in Translation: Topological Singularities in Group Field Theory, Razvan Gurau

The Fermionic version of colored GFT seems more natural. This couldessentially also solve the stability problem.

A complete study of the power counting and renormalization is easierin colored GFT’s.

Fermionic D-dimensional colored GFT has a natural but puzzlingSU(D) symmetry. Hence it may be a more promising starting point toinclude standard model matter fields.

GFT as theories of Holonomies

Consider a manifold M with a connection Γ (eg Levi-Civita of a metric). Allinformation is encoded in the holonomies along closed curves γ in M.If X is the vector field solution of the parallel transport equation

dt∂νX

µ + Γµ

νσXνdγσ

dt= 0 , X(0) = X0

dt∂νX

µ + Γµ

νσXνdγσ

dt= 0 , X(0) = X0

then X(T ) = gX0 for some g ∈ GL(TMγ(0)). g (independent of X0) is theholonomy along the curve γ.

dt∂νX

µ + Γµ

νσXνdγσ

dt= 0 , X(0) = X0

then X(T ) = gX0 for some g ∈ GL(TMγ(0)). g (independent of X0) is theholonomy along the curve γ.

Suppose we discretize M with flat n dimensional simplices. Their boundary(n − 1 dimensional) is also flat. The curvature is located at the “joints” ofthese blocks, that is at n − 2 dimensional cells.

Hence holonomies h are associated to blocks of codimension 2.

Discretized Surfaces and Matrix Models

Consider a two dimensional surface M , and fix a triangulation of M.

The holonomy group of a surface is G = U(1).

The holonomy group of a surface is G = U(1). To all vertices (points) inour triangulation we associate a holonomy g , namely parallel transportalong any small curve encircling the vertex.

g’’

..........

g’’

..........

The surface and its metric are specified by the gluing of the triangles andby the holonomies g . The associated weight is a function F (g , g ′, . . . )

g’’

..........

The surface and its metric are specified by the gluing of the triangles andby the holonomies g . The associated weight is a function F (g , g ′, . . . )

Quantization of geometry should sum both over metrics compatible with atriangulation and over all triangulations.

The Dual Graph

The Dual GraphThe discrete information about the triangulation of the surface can beencoded in a ribbon (2-stranded) graph.

The ribbon vertices of the graph are dual to triangles and the ribbon linesof the graph are dual to edges.

In the dual graph the group elements g are associatedto the sides of the ribbons, also called strands. Wedistribute them on all ribbon vertices sharing the samestrand.

Suppose that the weight function F factors into contributions of dualvertices and dual lines:

F =∏

V (g1, g2, g3)∏

K (g1, g2)

Suppose that the weight function F factors into contributions of dualvertices and dual lines:

F =∏

V (g1, g2, g3)∏

K (g1, g2)

Then a 2-stranded graph is a ribbon Feynman graph and its weight is theintegrand of the associated Feynman amplitude.16

2D Group Field Theory

The associated matrix model action is (φ(g1, g2) = φ∗(g2, g1))

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

and the complete correlation function is

< φ(g1, g2) . . . φ(g2n−1, g2n) >=

[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

< φ(g1, g2) . . . φ(g2n−1, g2n) >=

[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .

The insertions φ(g1, g2) fix a boundary triangulation and metric.

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

< φ(g1, g2) . . . φ(g2n−1, g2n) >=

[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .

Each Feynman graph fixes a bulk triangulation.

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

< φ(g1, g2) . . . φ(g2n−1, g2n) >=

[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .

Each Feynman graph fixes a bulk triangulation. The amplitude of a graph isthe sum over bulk metrics compatible with the triangulation.

φ(g1, g2)K−1(g1, g2)φ

∗(g1, g2)

G×G×G

V (g1, g2, g3)φ(g1, g2)φ(g2, g3)φ(g3, g1) ,

< φ(g1, g2) . . . φ(g2n−1, g2n) >=

[dφ]e−Sφ(g1, g2) . . . φ(g2n−1, g2n) .

Each Feynman graph fixes a bulk triangulation. The amplitude of a graph isthe sum over bulk metrics compatible with the triangulation. Thecorrelation function automatically performs both sums!

Matrix Models

Take K ,V = 1 and develop φ in Fourier series

φ(g1, g2) =∑

eımg1e−ıng2φmn .

Matrix Models

φ(g1, g2) =∑

The action takes the more familiar form in Fourier space

φmnφ∗mn + λ

φmnφnkφkm .

Matrix Models

φ(g1, g2) =∑

The action takes the more familiar form in Fourier space

φmnφ∗mn + λ

φmnφnkφkm .

Summing all Feynman graphs generated by the matrix model action sumsover different topologies. This is 2D quantum gravity (David, Ginzparg).

3D GFT

In three dimensions we must use the holonomy group G = SU(2).

3D GFT

In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2).

3D GFT

In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments.

3D GFT

In three dimensions we must use the holonomy group G = SU(2). Groupelements are associated to edges in the triangulation (codimension 2). Eachfield φ is associated to a triangular face of a tetrahedron, therefore it hasthree arguments. The propagator K is the inverse of the quadratic part

φ(ga, gb, gc)K−1(ga, gb, gc)φ

∗(ga, gb, gc)

3D GFT

∗(ga, gb, gc)

The vertex is dual to a tetrahedron. A tetrahedron is bounded by fourtriangles therefore the vertex is a φ4 term

3D GFT

∗(ga, gb, gc)

The vertex is dual to a tetrahedron. A tetrahedron is bounded by fourtriangles therefore the vertex is a φ4 term

V (g , . . . , g)φ(g03, g02, g01)φ(g01, g13, g12)φ(g12, g02, g23)φ(g23, g13, g03)

3D GFT Vertex

3D GFT VertexWhat we call the vertex in QFT usually obeys some kind of locality property.

Proposed Definition A vertex joining 2p strands is called simple if it hasfor kernel in direct group space a product of p delta functions matchingstrands two by two in different half-lines.

Then the natural 3D GFT tetrahedron vertex in 3 dimensions is simple(with p = 6) as it writes

V [φ] = λ

φ(g1, g2, g3)φ(g4, g5, g6)

φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),

with a kernel

V [φ] = λ

φ(g1, g2, g3)φ(g4, g5, g6)

φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),

with a kernel

Q(g1, ..g12) = δ(g3g−14 )δ(g2g

−18 )δ(g6g

−17 )δ(g9g

−110 )δ(g5g

−111 )δ(g1g

−112 )

V [φ] = λ

φ(g1, g2, g3)φ(g4, g5, g6)

φ(g7, g8, g9)φ(g10, g11, g12)Q(g1, ..g12),

with a kernel

Q(g1, ..g12) = δ(g3g−14 )δ(g2g

−18 )δ(g6g

−17 )δ(g9g

−110 )δ(g5g

−111 )δ(g1g

−112 )

satisfying to our definition.20

3D GFT Vertex is dual of a tetrahedron

The tetrahedron is dual to a vertex, its triangles are dual to halflines and itsedges are dual to strands. Hence we could label the 3D GFT vertex as

(23) (02) (03)

(01)(12)

(23) (02) (03)

(01)(12)

The GFT lines connect two vertices,thus are formed of three strands.

(23) (02) (03)

(01)(12)

The GFT lines connect two vertices,thus are formed of three strands.The graph built with such verticesand lines is a 3-stranded graph.

(23) (02) (03)

(01)(12)

(23) (02) (03)

(01)(12)

This corresponds to the kernel written before

Q(g1, ..g12) = δ(g3g−14 )δ(g2g

−18 )δ(g6g

−17 )δ(g9g

−110 )δ(g5g

−111 )δ(g1g

−112 )

How to choose the 3D GFT propagator?

How to choose the 3D GFT propagator?In Cartan’s formalism, 3D quantum gravity is expressed in terms of adreibein e

e i (x) = e ia(x)dxa

and a spin connection ω with values in the so(3) Lie algebra

ωi (x) = ωia(x)dxa

How to choose the 3D GFT propagator?In Cartan’s formalism, 3D quantum gravity is expressed in terms of adreibein e

e i (x) = e ia(x)dxa

and a spin connection ω with values in the so(3) Lie algebra

ωi (x) = ωia(x)dxa

The action is

S(e, ω) =

e i ∧ F (ω)i

where F is the curvature of ω :

F (ω) = dω + ω ∧ ω

3d Gravity

Varying ω gives Cartan’s equation De = 0. Varying e gives F = 0, hence aflat space-time.

3d Gravity

Hence 3D gravity is a topological theory with only global observables, nopropagating degrees of freedom. This is confirmed by perturbation aroundflat space, which leads to the Chern-Simons theory, also topological.

3d Gravity

Nevertheless the theory is physically interesting. Matter can be added;point particles do not curve space but induce an angular deficit proportionalto their mass. Therefore in 3D gravity there is a limit (2π) to anypoint-particle’s mass.

3d Gravity

Nevertheless the theory is physically interesting. Matter can be added;point particles do not curve space but induce an angular deficit proportionalto their mass. Therefore in 3D gravity there is a limit (2π) to anypoint-particle’s mass.

The 3D GFT propagator should just implement this flatness condition.

3D GFT Propagator

3D GFT PropagatorThe following propagator

[Cφ](g1, g2, g3) =

dhφ(hg1, hg2, hg3)

[Cφ](g1, g2, g3) =

dhφ(hg1, hg2, hg3)

is a projector onto gauge invariant fields φ(hg , hg ′, hg”) = φ(g , g ′, g”). Ithas kernel

dh δ(g1hg ′−11 ) δ(g2hg ′−1

2 ) δ(g3hg ′−13 )

[Cφ](g1, g2, g3) =

dhφ(hg1, hg2, hg3)

dh δ(g1hg ′−11 ) δ(g2hg ′−1

2 ) δ(g3hg ′−13 )

Performing the integrations over all group elements of the vertices andkeeping the h elements unintegrated gives for each triangulation, or3-stranded graph G a Feynman amplitude

[Cφ](g1, g2, g3) =

dhφ(hg1, hg2, hg3)

dh δ(g1hg ′−11 ) δ(g2hg ′−1

2 ) δ(g3hg ′−13 )

δ(gf )

[Cφ](g1, g2, g3) =

dhφ(hg1, hg2, hg3)

dh δ(g1hg ′−11 ) δ(g2hg ′−1

2 ) δ(g3hg ′−13 )

δ(gf )

where gf = ~∏e∈f he is the holonomy along the face f .

3D GFT = Boulatov Model

Therefore 3D GFT with the projector C as propagator implements exactlythe correct flatness conditions of 3D gravity!

This was the discovery of Boulatov (1992).

Fourier transforming the model, one gets Ponzano-Regge amplitudes.

Properties of 3D GFT

This 3D GFT is topological and the amplitude of a graph changesthrough a global multiplicative factor under (1-4) or (2-3) Pachnermoves.

Amplitudes may be infinite, for instance the tetraedron graph orcomplete graph K4 diverges as Λ3 at zero external data, if Λ is theultraviolet cutoff on the size of j ’s.

Regularization by going to a quantum group at a root of unity leads towell-defined topological invariants of the triangulation, namely theTuraev-Viro invariants.

However the theory is unsuited for a RG analysis, as the propagatorhas spectrum limited to 0 and 1. How to make slices with such aspectrum?

The 4 Dimensional Case

In the first order Cartan formalism, the action is∫

⋆[e ∧ e] ∧ F where againthe vierbein e and the spin connection ω are considered independentvariables.

Not all two-forms B in four dimenions are of the form e ∧ e. So BF theoryis not gravity in 4 dimension. The difference is expressed through a set ofconstraints classically called the Plebanski constraints.

These constraints render 4D gravity much more complicated and interestingsince they are responsible for the local propagating degrees of freedom, thegravitational waves. (Constraints on B allow richer set of F ’s than justF = 0).

4D GFT Vertex

The vertex is the easy part as it should be again given by gluing rules forthe five tetraedra which join into pentachores:

4D GFT Vertex

The pentachore

4D GFT

The corresponding graphs are made of vertices dual to the pentachores, offour-stranded edges dual to tetraedra, and of faces or closed thread circuitsdual to triangles.

4D GFT

If we keep the same propagator than in the Boulatov theory

4D GFT

[Cφ](g1, g2, g3, g4) =

dhφ(hg1, hg2, hg3, hg4) (2.1)

4D GFT

[Cφ](g1, g2, g3, g4) =

dhφ(hg1, hg2, hg3, hg4) (2.1)

we get a φ5 GFT with 4-stranded graphs called the Ooguri group fieldtheory. It is not 4D gravity, but a discretization of the 4D BF theory, ie thePlebanski constraints are missing.

4D GFT

[Cφ](g1, g2, g3, g4) =

dhφ(hg1, hg2, hg3, hg4) (2.1)

we get a φ5 GFT with 4-stranded graphs called the Ooguri group fieldtheory. It is not 4D gravity, but a discretization of the 4D BF theory, ie thePlebanski constraints are missing.

Again such a topological version of gravity is unsuited for RG analysis.

The Barrett-Crane proposal

A first attempt to implement the Plebanski constraints in this language isdue to Barrett and Crane. They suggested one should restrict to simplerepresentations of SU(2) × SU(2), namely those satisfying j+ = j−.

However this does not seem to work because the Barrett-Crane proposalimplements the Plebanski constraints too strongly. It does not seem inparticular to include correctly the angular degrees of freedom of thegraviton.

The Holst Action

Still another action classically equivalent to the Einstein-Hilbert action isthe Holst action:

The Holst Action

S = −1

[⋆(e ∧ e) +1

γ(e ∧ e)] ∧ F

The Holst Action

S = −1

[⋆(e ∧ e) +1

γ(e ∧ e)] ∧ F

where the first term is the Palatini action, and the second one, the Holstterm, is often called topological since it does not affect the equations ofmotion. The parameter γ is called the (Barbero)-Immirzi parameter, andplays an essential role in Loop quantum gravity.

The Holst Action

S = −1

[⋆(e ∧ e) +1

γ(e ∧ e)] ∧ F

where the first term is the Palatini action, and the second one, the Holstterm, is often called topological since it does not affect the equations ofmotion. The parameter γ is called the (Barbero)-Immirzi parameter, andplays an essential role in Loop quantum gravity.

We can then rewrite this action a la Plebanski and get thePalatini-Holst-Plebanski functional integral, loosely written as:

dν =1

Ze− 1

[⋆B+ 1γB]∧F

δ(B = e ∧ e) DB Dω

The EPR(LS)/FK theory

The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.

The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).

The EPR(LS)/FK theoryParallel works by Engle-Pereira-Rovelli and by Freidel-Krasnov, withcontributions of Livine and Speziale, lead in 2007 to an improved spin foammodel which better implements the Plebanski constraints.It seems to incoprorate better the angular degrees of freedom of thegraviton. It also reproduces the correct Einstein-Hilbert action in the limitof large spins (Barrett et al, Conrady-Freidel, 2009).These spin foams translate into an new proposal for a 4D GFT with a newimproved propagator which incorporates

a SU(2) × SU(2) averaging C at both ends like in Ooguri theory

a simplicity constraint which depends on the value of the Immirziparameter. For instance for 0 < γ < 1 it restricts representations to:

j+ =1 + γ

2j ; j− =

1 − γ

2j (2.2)

a simplicity constraint which depends on the value of the Immirziparameter. For instance for 0 < γ < 1 it restricts representations to:

j+ =1 + γ

2j ; j− =

1 − γ

2j (2.2)

a new projector S averaging on a single SU(2) residual gaugeinvariance in the middle of the propagator.

Renormalization group, at last?

The main new feature is that the full propagator can therefore be written asK = CSC with C 2 = C and S2 = S , hence C and S are two projectors, butthey do not commute, hence K has non-trivial spectrum!

This means that a RG analysis become possible! That’s our program.

In 2008 Perini, Rovelli and Speziale found in some natural normalization ofthe theory a Λ6 divergence for the graph G2

and a tantalizing logarithmic divergence for a radiative correction to thecoupling constant.

What we have done so far

This section is based on joint work with collaborators J. Ben Geloun, T.Krajewski, Karim Noui, Jacques Magnen, Matteo Smerlak, A. Tanasa andP. Vitale.

Our program aims at renormalizing group field theories, eg of theEPR(LS)/FK type; we have performed only some preliminary steps yet

We have investigated the power counting of the Boulatov model, andfound the first uniform bounds for its Freidel-Louapre constructiveregularization, using the loop vertex expansion technique.arXiv:0906.5477, Scaling behaviour of three-dimensional group field theory (MNRS, Class. Quant. Gravity .26

(2009)185012)

What we have done so far, II

We have computed in the Abelian case the power counting of graphsfor BF theory, and found how to relate them to the number of bubblesof the graphs in the colored case.

The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.

The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.arXiv:0911.1719 Bosonic Colored Group Field Theory (BMR)

arXiv:1002.3592, Linearized Group Field Theory and Power Counting Theorems (BKMR, Class. Quant. Grav. 27 (2010)

155012)

The Abelian amplitudes lead to new class of topological polynomialsfor stranded graphs.arXiv:0911.1719 Bosonic Colored Group Field Theory (BMR)

arXiv:1002.3592, Linearized Group Field Theory and Power Counting Theorems (BKMR, Class. Quant. Grav. 27 (2010)

155012)

See also the related work of Bonzom and Smerlak

What we have done so far, III

We have started the study of the general superficial degree ofdivergence of EPR(LS)/FK graphs through saddle point analysis usingcoherent states techniques. We recover in many cases the Abeliancounting, and find also corrections to it in general.

But saddle points may come in many varieties, degenerate or not. Thisrichness is interesting but a major challenge.

We recovered the Λ6 divergence of the EPR(LS)/FK graph G2 in thecase of non-degenerate saddle point configurations, but also a Λ9

divergence for the maximally degenerate saddle points. Our methodalso works at non zero external spins.

We recovered the Λ6 divergence of the EPR(LS)/FK graph G2 in thecase of non-degenerate saddle point configurations, but also a Λ9

divergence for the maximally degenerate saddle points. Our methodalso works at non zero external spins.

arXiv:1007. 3150 Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models (KMRTV)

What we have done so far, IV

We have written still a more compact representation recently for theEPR/FK propagator in terms of traces

We hope that raising eventually the EPR/FK propagator to the rightpower should lead to a just renormalizable model of 4D colored groupfield theory.

EPRL/FK Group Field Theory, arXiv:1008.0354 (BGR)

What remains to be done in our program

Essentially everything!

We need to investigate better the dominance of type 1 or ball-likegraphs, perhaps in the simpler colored BF models and establish at leastrough bounds on all other graphs showing they are smaller.

We need to cut propagators with non trivial spectra like theEPR(LS)/FK into slices according to their sepctrum. This requires twocutoffs. This is required to identify the ”high” subgraphs, and performa true multiscale RG analysis. We have to find the new localityprinciple and the correct power counting. Only then can we find ifsome of these models are renormalizable or not.

We need to find out whether symmetries such as the tantalizingtopological BF symmetry recovered at γ = 1 create fixed points of theRG analysis.

We need to get a better glimpse of whether a phase transition to acondensed phase can lead to emergence of a large smooth space-time.

Conclusion

EPR(LS)/FK group field theory is an interesting 4D group field theory.It has the right gluing rules plus a non-trivial propagator spectrum.

Conclusion

I feel the discovery of the noncommutative RG associated to the GWmodel, which is nonlocal and mixes ordinary scales, is an encouragingstep towards finding a similar renormalization group for group fieldtheories.

Conclusion

I hope black holes on one side, and possibly some mean fieldtransplanckian RG fixed point might bring constraints on GFT- basedscenarios for geometrogenesis and cosmology.

Conclusion

I hope black holes on one side, and possibly some mean fieldtransplanckian RG fixed point might bring constraints on GFT- basedscenarios for geometrogenesis and cosmology.

However ”geometrogenesis” phase transition may be of a higher tensornature, hence harder to analyze than confinement, which is still notwell analytically understood. So computer studies may becomenecessary at some point.

Hubbard Model, Three Dimensions

Vincent Rivasseau

LPT Orsay

Cetraro, Summer 2010, Lecture IV

Cetraro, Summer 2010, Cetraro, Summer 2010, Lecture IV Vincent Rivasseau, LPT Orsay

Single Scale Toy Model 2 Dimensional Jellium Model Hubbard Model 3d Jellium Model

e−M−j |x−y |

V = λ

ψa(x)ψa(x))(

ψb(x)ψb(x))

e−M−j |x−y |

V = λ

ψa(x)ψa(x))(

ψb(x)ψb(x))

where λ is the coupling constant.2

We claim that

The J2 model is roughly similar to that model with the role of colorspayed by N = M j angular sectors around the Fermi surface.

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

(λn/n!)N

a1,...,an,b1,...,bn=1

ε(T ,Ω)(

ℓ∈T

0dwℓ

p = limΛ→∞

|Λ| log Z (Λ)

p = limΛ→∞

(λn/n!)N

a1,...,an,b1,...,bn=1

ε(T ,Ω)(

ℓ∈T

0dwℓ

dx1...dxnδ(x1 = 0)∏

ℓ∈T

Cj ,ab(xℓ, yℓ))

× det[Cj ,ab()]remaining

[Cj ,ab()]remaining is the matrix with entries

fj(p).gj(p) = Cj(p)).

Bounds on the Remaining Determinant, II

Then we have the Gram inequality:

anti−fields k

||fj ,k ||∏

fields m

||gj ,m||

anti−fields k

||fj ,k ||∏

fields m

||gj ,m||

Proof: The w dependence disappears by extracting the symmetric squareroot v of the positive matrix xT so that xT

km =∑n

k=1 vTknv

Hence the λ radius of convergence is uniform in j .

factors.

Hence the λ radius of convergence is uniform in N.

|Cj(x , y)| ≤ M−je−[M−j |x−y |]1/2

But the situation improves if we cut the Fermi slice into smaller pieces(called sectors).

J2 Sectors

in all three directions.

J2 Sectors

(using again Gevrey cutoffs fja for fractional power decay).

J2 Sectors

M−2j =M−3j/2

J2 Sectors

M−2j =M−3j/2

so that the bound is identical to that of the toy model.

It means that the interaction is roughly of the color (or Gross-Neveu) typewith respect to these angular sectors:

ψaψa

ψbψb

Anisotropic Sectors

Ultimately the conclusion is unchanged: the radius of convergence of J2 ina slice is independent of the slice index j .

Anisotropic Sectors

In order for sectors defined in momentum space to correspond topropagators with dual decay in direct space, it is essential that their lengthin the tangential direction is not too big, otherwise the curvature is toostrong for the stationary phase method to apply.

Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.

Another way to build a Fermi liquid is to introduce a magneticregulator instead of a temperature, to get rid of the BCS phasetransition. The magnetic field typically blocks the Cooper pair channelby breaking the parity invariance of the Fermi surface.Following this road, Feldman, Knorrer and Trubowitz built in greatdetail a 2D Fermi liquids with non-parity invariant surfaces in animpressive series of papers completed around 2003-2004.

The Hubbard model at half-filling in 2 dimensions

H2 Sectors

Consider the curvature radius of the curve (cos k1 + cos k2)2 = M−2j , which

R =(sin2 k1 + sin2 k2)

| cos k1 sin2 k2 + cos k2 sin2 k1|.

H2 Sectors

Consider also the distance d(k1) to the Fermi curve cos k1 + cos k2 = 0,and the width w(k1) of the band M−j ≤ | cos k1 + cos k2| ≤

√2M.M−j .

H2 Sectors

√2M.M−j .

One finds

d(k1) ≃ w(k1) ≃M−j

M−j/2 + k1,

H2 Sectors

√2M.M−j .

One finds

d(k1) ≃ w(k1) ≃M−j

M−j/2 + k1,

R(k1) ≃k31 + M−3j/2

M−j,

H2 Sectors, II

( ( ( ((

H2 Sectors, IIRemember the condition that the sector length should not be bigger thanthat anisotropic length.

( ( ( ((

(! (" (# ($ % $ # " !(!

This leads to slice |k1| or |k2| according to a geometric progression from 1to M−j/2 to form the angular sectors in this model.

(! (" (# ($ % $ # " !(!

Sincecos k1 + cos k2 = 2 cos(πk+/2) cos(πk−/2) .

(! (" (# ($ % $ # " !(!

Sincecos k1 + cos k2 = 2 cos(πk+/2) cos(πk−/2) .

it is better to slice cos(πk±/2) with indices (s+, s−)

| cos(πk±/2)| ≃ M−s±

Momentum Conservation Rule, Power Counting

The two smallest indices among sj ,+ for j = 1, 2, 3, 4 differ by at most oneunit, and the two smallest indices among sj ,− for j = 1, 2, 3, 4 differ by atmost one unit.

Because the total number of sectors is growing logarithmically, notpower-like, the scaling of the model resembles more the one-dimensionalmodel (Luttinger liqud) than to the jellium models in 2 or 3 dimensions.

The jellium model in 3 dimensions

Fix m ∈ Zd+1 . The number of 4-tuples S1, · · · S4 of sectors for whichthere exist ki ∈ R

d , i = 1, · · · , 4 satisfying

k ′i ∈ Si , |ki − k ′

i | ≤ const M−j , i = 1, · · · , 4

d , i = 1, · · · , 4 satisfying

k ′i ∈ Si , |ki − k ′

i | ≤ const M−j , i = 1, · · · , 4

and|k1 + · · · + k4| ≤ const (1 + |m|) M−j

d , i = 1, · · · , 4 satisfying

k ′i ∈ Si , |ki − k ′

i | ≤ const M−j , i = 1, · · · , 4

and|k1 + · · · + k4| ≤ const (1 + |m|) M−j

is bounded byconst(1 + |m|)dM(3d−4)j 1 + jδd ,2 .

d , i = 1, · · · , 4 satisfying

k ′i ∈ Si , |ki − k ′

i | ≤ const M−j , i = 1, · · · , 4

and|k1 + · · · + k4| ≤ const (1 + |m|) M−j

is bounded byconst(1 + |m|)dM(3d−4)j 1 + jδd ,2 .

Here, k ′ = k|k| denotes the projection of k onto the Fermi surface.

So for d = 3 we find M5j 4-tuples which correspond to the choice of twosectors (M2j × M2j) and of one angular twist M j .

Therefore the jellium model interaction is not of the vector type.

The power counting corresponds to M−3j per sector propagator. Twopropagators pay for one vertex integration (M4j) and one sector choice(M2j) but there is nothing to pay for the angular twist.

Scaling in direct space

In direct space the problem is different.

The propagator in x-space satisfies a naive bound

Cj(x , y) ≤ M−jeM−j |x−y |

But integrating over angles leads to an additional 1/|x − y | decay, because

∫ π

0sin θdθdφe i cos θ|x−y | = sin |x − y |/|x − y |

∫ π

Hence for typical distances |x − y | ≃ M j the propagator obeys an improvedestimate

Cj(x , y) ≤ M−2jeM−j |x−y |

∫ π

Hence for typical distances |x − y | ≃ M j the propagator obeys an improvedestimate

Cj(x , y) ≤ M−2jeM−j |x−y |

The problem is that this bound is wrong at small distances.21

Solution: Hadamard bound.

Hadamard bound for a 2n by 2n determinant yields

|detaij | ≤∏

|aij |2 ≤ (2n)n| supij

|aij |2n

|detaij | ≤∏

|aij |2n

This bound used after a cluster expansion between cubes solves theproblem for the main part of the propagator.

|detaij | ≤∏

|aij |2n

Correct just renormalizable power counting is recovered for the mainpart of the theory

|detaij | ≤∏

|aij |2n

A factor n! is lost in the bound at order n but this is what is allowedby the 1/n! symmetry factor.

|detaij | ≤∏

|aij |2n

A factor n! is lost in the bound at order n but this is what is allowedby the 1/n! symmetry factor.

An auxiliary (superrenormalizable) expansion is needed to treat the smalldistance part.

Introduction to Renormalization -...

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