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Why RenormalizableNoncommutative

Quantum Field Theories

Vincent Rivasseau

Laboratoire de Physique Theorique

Universite Paris-Sud XI, Orsay

Vienne, November 30 2007

Why Renormalizable Noncommutative Quantum Field Theories, Vienne, November 25 2007 ,

Introduction Noncommutative Field Theory Covariant theories, self-dual theories Ghost Hunting Can we get back the REAL world? Constructive Field Theory Conclusions

Introduction

Noncommutative Field Theory

Covariant theories, self-dual theories

Ghost Hunting

Can we get back the REAL world?

Constructive Field Theory

Conclusions

Scales in the universe

Physical phenomena occur over a wide range of scales from the Planck scale`P =

√~G/c3 ' 1.6 10−35 m to the radius of the observable universe which in

practice in comoving coordinates corresponds to about 45 billion light-years, hencearound 4.4 1026 m or better, 2.7 1061`P .

The universe therefore is made of roughly 61 powers of 10 or 140 powers of e.

Scales in the universePhysical phenomena occur over a wide range of scales from the Planck scale`P =

The universe therefore is made of roughly 61 powers of 10 or 140 powers of e.2

Terra Incognita

We have a fair knowledge of the 45 biggest scales of the universe.

But the fifteen to sixteen scales between `P and 2 10−19 meters (about 1 Tev),make up the last true terra incognita of physics.

Terra Incognita

The “last” big collider?

Next year, the Large Hadron Collider at Cern, Geneva should open up a new powerof 10 to direct observation.

After that treat no new spectacular advance of this type is planned on terraincognita, hence we have some time to deepen our theoretical and mathematicalunderstanding.

The “last” big collider?Next year, the Large Hadron Collider at Cern, Geneva should open up a new powerof 10 to direct observation.

After that treat no new spectacular advance of this type is planned on terraincognita, hence we have some time to deepen our theoretical and mathematicalunderstanding.4

Why R, NC, QFT?

I R because renormalizable models are the ones who survive renormalizationgroup flows. They are the generic building blocks of physics.

I NC because one cannot measure length below the Planck scale: the energy ofthe necessary particles would create a black hole whose horizon would hide themeasurement. Fundamental uncertainties of this type mean noncommutativityof space time itself at that scale.

I QFT because that’s what particle physics is about at the frontier of terraincognita.

RNCQFT is probably not the ultimate theory but it seems a logical and necessarystep on the road...

Why R, NC, QFT?

Renormalization

The first computations in quantum field theory always ended in infinite results.Computations that could be compared to actual experiments were developed onlyafter finding a cure for these infinities, called renormalization.

Renormalization (Dyson, Feynman, Schwinger, Tomonaga...) suppressed the QFTinfinities by reformulating physical laws in terms of parameters that can beobserved at low energy, also called renormalized parameters. Infinities were pushedinto unobservable ”bare” parameters. In spite of its intial success (explanation ofthe Lamb shift and of the anomalous magnetic factor g − 2 for the electron...),renormalization was not easily accepted:

I F. Dyson, one of its inventors, thought “this was just a trick we wanted to usefor a few months until finding something better...”

I Mathematicians (until Alain...) considered renormalization as a kind ofill-defined recipe, which “pulled infinities under the rug...”

I In the 50’s, L. Landau in the Soviet Union discovered an unexpected difficulty:the infinities thrown out through the door sort of reappeared through thewindow. This phenomenon has been called the “Landau ghost”.

Renormalization

The Landau ghost

Landau discovered that the electrodynamic coupling became infinite at a large butfinite energy scale, just like a particle in the flow of a quadratic vector field goes toinfinity in a finite time. Quantum Field Theory therefore did not seem consistentat high energy!

The Landau ghost

Ordinary φ44 QFT

The simplest renormalizable quantum field theory, φ44, is defined on Euclidean

space R4 through its Schwinger functions which are the moments of the formalfunctional measure:

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

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

I m is the mass, which fixes the energy scale of the renormalized theory;

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

I Z is a normalization so that this measure should be a probability measure;

I Dφ is a formal product∏

x∈Rd

dφ(x) of Lebesgue measures at each point of R4.

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

Ordinary φ44 QFT

dν =1

Ze−(λ/4!)

Rφ4−(m2/2)

Rφ2−(a/2)

R(∂µφ∂

µφ)Dφ, (1.1)

x∈Rd

The ordinary φ44 propagator

The covariance of the Gaussian part of the measure dν is called the propagator

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

dαe−αm2 e−|x−y |2/4α

α2, (1.2)

where we recognize the heat kernel.

Schwinger functions can be expanded as a sum of Feynman amplitudes(perturbation theory).The best method to study and understand renormalization in QFT is to cut thepropagator into a sequence of slices a la Wilson for a multi-scale analysis:

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

dα · · · 6 KM2ie−cM i‖x−y‖

This leads to a factor M2i per line and M−4i per vertex integration∫

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

α2, (1.2)

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

α2, (1.2)

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

α2, (1.2)

Schwinger functions can be expanded as a sum of Feynman amplitudes(perturbation theory).

The best method to study and understand renormalization in QFT is to cut thepropagator into a sequence of slices a la Wilson for a multi-scale analysis:

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

α2, (1.2)

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

C (p) =1

(2π)2

p2 + m2, C (x , y) =

∫ ∞0

α2, (1.2)

C =∑

C i , C i (x , y) =

∫ M−2(i+1)

M−2i

The renormalization of ordinary φ44

The renormalization relies on the combination of two arguments:

I The locality principle, independent of the dimension: every subgraph made ofinternal propagators of higher energy than its external lines looks local as thegap between the internal and external energy grows.

I The power counting, dependent on the dimension d : when such a subgraph iscompared to a local vertex, the corresponding weight depends on thedimension and of the type of that subgraph. The sum over the gap betweenthe internal and external energy may therefore either converge or diverge.

For instance this bubble diverges logarithmically when d = 4 because there are twoline factors M2i and a single internal integration M−4i .

For instance this bubble diverges logarithmically when d = 4 because there are twoline factors M2i and a single internal integration M−4i .10

The unavoidable ghost?

In the case of the φ44 theory, power counting tells us that only two and four point

subgraphs do diverge (in the sense explained above). Four point functions divergelogarithmically and govern the renormalization of the coupling constant λ.

The only one-loop graph which is one-particle irreducible is the bubble:

It governs the flow equation

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

whose sign cannot be changed without losing stability. It corresponds to aquadratic one dimensional flow whose solution is well known to diverge in a finitetime! In the 60’s all known field theories sufferered from this Landau ghost.

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

whose sign cannot be changed without losing stability. It corresponds to aquadratic one dimensional flow whose solution is well known to diverge in a finitetime!

In the 60’s all known field theories sufferered from this Landau ghost.

−λi−1 = −λi + β(−λi )2,

di= +β(λi )

2, (1.3)

Asymptotic Freedom

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

I Weinberg and Salam unified the weak and electromagnetic interactions intothe formalism of Yang and Mills of non-Abelian gauge theories, which arebased on an internal non commutative symmetry.

I ’tHooft and Veltmann succeeded to show that these theories are stillrenormalisable. Their technical tour de force used a new technical tool,namely dimensional renormalization.

I ’t Hooft in an unpublished work, then Politzer, Gross and Wilczek discoveredin 1973 that these theories did not suffer from the Landau ghost. Gross andWilczek then developed and advertized a theory of this type, QCD to describestrong interactions, hence nuclear forces.

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

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

Asymptotic Freedom

Noncommutative Geometry

Noncommutative geometry as you all know generalizes ordinary geometry.Ordinary smooth functions or observables form a commutative algebra underordinary multiplication. For instance in classical mechanics observables are smoothfunctions on phase space. Quantum mechanics replaces this commutative algebraby a noncommutative algebra of operators, where Poisson brackets becomecommutators. This is the first physical example of noncommutative geometry.

But direct space-time itself could be of this type; for instance at a certain scalenew uncertainty relations could appear between length and width which wouldgeneralize Heisenberg’s relations.

The Moyal space R4θ

One of the simplest example of a noncommutative geometry is the flat (Euclidean)vector space R4 but with a constant commutator between coordinates, proportionalto a new kind of “Planck constant”, θ, which would be measured in square meters.

[xµ, xν ] = ıΘµν ,

where Θµν is an antisymmetric constant tensor which in the simplest case can bewritten as:

Θµν = θ

0 1−1 0

(0)0 1−1 0

The unique product (associative, but noncommutative) generated by these

relations on Schwarz class functions is called the Moyal-Weyl, product and writes:

(f ? g) (x) =

∫ddk

(2π)dddy f (x +

2Θ · k)g(x + y)eık·y

eıkx ? eık′x = e−

ı2 Θijkikj eı(k+k′)x

Θµν = θ

0 1−1 0

(0)0 1−1 0

(f ? g) (x) =

∫ddk

(2π)dddy f (x +

Θµν = θ

0 1−1 0

(0)0 1−1 0

(f ? g) (x) =

∫ddk

(2π)dddy f (x +

Θµν = θ

0 1−1 0

(0)0 1−1 0

The unique product (associative, but noncommutative) generated by theserelations on Schwarz class functions is called the Moyal-Weyl, product and writes:

(f ? g) (x) =

∫ddk

(2π)dddy f (x +

Θµν = θ

0 1−1 0

(0)0 1−1 0

(f ? g) (x) =

∫ddk

(2π)dddy f (x +

Quantum Field Theory on the Moyal space

To explore quantum field theory on noncommutative geometries (or NCQFT), inparticular on the simple Moyal space is suggested by string theory. It is also a quitenatural extension of the Connes-Chamseddine interpretation of the standard model(in which noncommutativity is restricted to some internal simple space).

Another deep motivation for noncommutative field theory comes from standardphysics but in strong background field. It can be either particle physics orcondensed matter physics.

The most obvious example is the quantum Hall effect. Following the work ofSusskind and Polychronakos, this system can be considered a noncommutativefluid, in need of a better understanding through a noncommutative generalizationof condensed matter renormalization group.

But one can probably also study fruitfuilly other strong field problems (such asquark confinement or the growth of charged polymers under strong magneticfields) with NCQFT techniques. This may help to tackle problems which lookuntractable in the ordinary geometry language simply because they correspond inthat geometry to non-perturbative and non-local effects.

The φ4 theory on Moyal R4θ

I The (naive) version of φ44 on Moyal space has for Lagrangian

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φ

I The Moyal vertex can be computed explicitly. It is proportional to∫ 4∏i=1

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

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

I This vertex is non-local and oscillates. It has a parallelogram shape in eachsymplectic pair, and its oscillation is proportional to its area:

2∂µφ∂µφ+

2φ2 +

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φI The Moyal vertex can be computed explicitly. It is proportional to∫ 4∏

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

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φ

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

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φ

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

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φ

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

2ıθ−1 (x1 ∧ x2 + x3 ∧ x4))

2∂µφ ? ∂µφ+

2φ ? φ +

4φ ? φ ? φ ? φ

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

2ıθ−1(x1 ∧ x2 + x3 ∧ x4))

Ultraviolet-infrared mixing

I Amplitudes of planar graphs remain unchanged.

I Nonplanar amplitudes create a new type of divergence, of the infrared type.

For instance,

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

I These divergences increase with perturbation order.

I All correlation functions are affected and diverge.

Ù The theory cannot be renormalized.

I Nonplanar amplitudes create a new type of divergence, of the infrared type.

For instance,

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

I Nonplanar amplitudes create a new type of divergence, of the infrared type.For instance,

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

p∝ λ

∫d4k

e ipµkνθµν

k2 + m2

∝ λ

p2K1(√

m2p2) ∼p→0 p−2

The Grosse-Wulkenhaar breakthrough:

The Euclidean theory with an additional harmonic potential

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

where xµ = 2Θ−1µν xν , is covariant under a symmetry pµ ↔ xµ called

Langmann-Szabo symmetry and is renormalizable at every order in λ!

This result has now been proved through many independent methods

I matrix base or ”coherent states”;

I direct space

I parametric representation and dimensional renormalization

Related models were explored in many papers by various subsets of an informal”MOV” group (Munster-Orsay-Vienna): Disertori, Gayral, de Goursac, Gurau,Grosse, Magnen, Malbouisson, R, Steinacker, Tanasa, Vignes-Tourneret, Wallet,Wohlgennant, Wulkenhaar...

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

∫d4x

2∂µφ ? ∂

µφ+Ω2

2(xµφ) ? (xµφ) +

2φ ? φ +

4!φ ? φ ? φ ? φ

I direct space

The new propagator

The propagator for this theory is best understood through its parametricrepresentation

G (x , y) =θ

) ∫ ∞0

dα e−µ2

4Ω α (2.4)

(sinhα)2exp

(− Ω

θ sinhα‖x − y‖2 − Ω

θtanh

2(‖x‖2+‖y‖2)

). (2.5)

and involves the Mehler kernel rather than the heat kernel.Multiscale analysis relies again on a slicing of that propagator according to ageometric sequence:

G i (x , y) =

∫ M−2(i−1)

M−2i

dα · · · 6 KM2ie−c1M2i‖x−y‖2−c2M

−2i (‖x‖2+‖y‖2)

The corresponding new renormalization group corresponds to a completely newmixture of the previous ultraviolet and infrared notions. Furthermore there existsonly a half direction which is infinite for this RG.

The new propagator

G (x , y) =θ

) ∫ ∞0

dα e−µ2

4Ω α (2.4)

(sinhα)2exp

(− Ω

θtanh

2(‖x‖2+‖y‖2)

). (2.5)

G i (x , y) =

∫ M−2(i−1)

M−2i

−2i (‖x‖2+‖y‖2)

The new propagator

G (x , y) =θ

) ∫ ∞0

dα e−µ2

4Ω α (2.4)

(sinhα)2exp

(− Ω

θtanh

2(‖x‖2+‖y‖2)

). (2.5)

and involves the Mehler kernel rather than the heat kernel.

Multiscale analysis relies again on a slicing of that propagator according to ageometric sequence:

G i (x , y) =

∫ M−2(i−1)

M−2i

−2i (‖x‖2+‖y‖2)

The new propagator

G (x , y) =θ

) ∫ ∞0

dα e−µ2

4Ω α (2.4)

(sinhα)2exp

(− Ω

θtanh

2(‖x‖2+‖y‖2)

). (2.5)

G i (x , y) =

∫ M−2(i−1)

M−2i

−2i (‖x‖2+‖y‖2)

The new propagator

G (x , y) =θ

) ∫ ∞0

dα e−µ2

4Ω α (2.4)

(sinhα)2exp

(− Ω

θtanh

2(‖x‖2+‖y‖2)

). (2.5)

G i (x , y) =

∫ M−2(i−1)

M−2i

−2i (‖x‖2+‖y‖2)

Renormalisability

It follows again from the combination of two arguments. A new Moyality principlereplaces locality:

This principle applies only to planar graphs with a single external face.

Renormalisability

Planarity and Power Counting

There is also a new power counting. One must take into account cyclicity of thevertex hence graphs have ribbon lines.

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

I V is the numbers of vertices,

I L is the number of lines or propagators,

I F is the number of external faces (following the ribbons borders),

I E is the number of external lines,

I g is the genus defined by Euler’s relation χ = 2− 2g = V − L + F ,

I EF is the number of external faces (i.e. containing arriving external lines).

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

2(F − EF )− L =

(2− E

)− 4g − 2 (EF − 1) si d = 4, ou

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

Ù //ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Examples of graphs

g = 1− (V − L + F )/2 , ω = 2− E/2− 4g − 2(EF − 1)

//V=3L=3F=2E=6

Ù g = 0, ω = −3

//ooMMQQ

//oo oo

//Ù //

ooMMQQ

V=2L=3F=1E=2

Ù g = 1, ω = −3

Noncommutative Renormalization

Renormalization again follows from the combination of two arguments.

Power counting tells us that only planar graphs with two and four external legsarriving on a single external face must be renormalized.

The Moyality priniciple tells us that when the gap grows between internal andexternal lines in the sense of the new renormalization group slicing, these termslook like Moyal products. The corresponding counterterms are therefore of theform of the initial Lagrangian!

Translation invariance

Some critics were addressed to the Grosse-Wulkenhaar model:

I The harmonic potential is just an infrared cutoff. No wonder that it cures themixing!

I There is a preferred origin in this model which cannot therefore be related totrue physics which is translation-invariant.

These critics are not fully justified:

I The harmonic potential is not a simple infrared cutoff among others, it is theone which makes the theory just renormalizable.

I The initial GW model indeed breaks translation invariance. However theremay be scenarios to connect it to translation invariant models at lower energy.

I There exists other classes of models, which we now call covariant, whosepropagator is similar to the one of GW, can be analyzed by the same methodsand is indeed invariant under magnetic translations. Physical quantities(which are gauge-invariant) are then translation-invariant.

Covariant Models

Their kernel is a slight generalization of the Mehler kernel. Defining Ω = 2Ωθ :

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

sinh(2Ωα)(x − y)2

− Ω

cosh(2Ωα)− cosh(2Bα)

sinh(2Ωα)(x2 + y2) +2ıΩ

sinh(2Bα)

sinh(2Ωα)x ∧ y

)The covariant Laplacian in a fixed external field corresponds to the case B = Ω.

Ù The decay away from the origin is replaced by an oscillation:

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

2coth(2Ωα)(x − y)2 + 2ıΩx ∧ y

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

The covariant Laplacian in a fixed external field corresponds to the case B = Ω.

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

Covariant Models

H−1 =(p2 + Ω2x2 − 2ıB

(x0p1 − x1p0

))−1,

H−1(x , y) =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

cosh(2Bα)

− Ω

sinh(2Bα)

sinh(2Ωα)x ∧ y

Q−1 = H−1 =Ω

∫ ∞0

sinh(2Ωα)exp

(− Ω

The Gross-Neveu Model

The noncommutative Gross-Neveu model on R2θ is defined by

2ψ(/p + Ω/x + m

∑a,b

λ1ψa ? ψa ? ψb ? ψb

+ λ2ψa ? ψb ? ψb ? ψa + λ3ψa ? ψb ? ψa ? ψb + λ4ψa ? ψb ? ψb ? ψa

where x = 2(Θ−1x), Θ = θ

(0 1−1 0

)and a, b are color or spin indices.

Ù This covariant model, more difficult to treat, was proved renormalizable byVignes-Tourneret in the so-called orientable case (λ3 = λ4 = 0) and physicalobservables are indeed translation-invariant.

Ù There exists also similar Bosonic models, which generalize the so-calledLangmann-Szabo-Zarembo model.

2ψ(/p + Ω/x + m

∑a,b

(0 1−1 0

2ψ(/p + Ω/x + m

∑a,b

(0 1−1 0

2ψ(/p + Ω/x + m

∑a,b

(0 1−1 0

2ψ(/p + Ω/x + m

∑a,b

(0 1−1 0

2ψ(/p + Ω/x + m

∑a,b

(0 1−1 0

Victory on the Landau ghost!

Contrary to the initial expectations (Snyders, 1947...) noncommutativity ofspace-time does not prevent infinities. φ4

4 remains just renormalisable, with stillinfinitely many primitively divergent graphs. However there is a big improvement:

in the Moyal plane, the Landau ghost has disappeared!

Ghost hunting

This result was obtained in three steps

I One loop: H. Grosse and R. Wulkenhaar,The beta-function in duality-covariant noncommutative φ4-theory,Eur. Phys. J. C35 (2004) 277–282, hep-th/0402093

I Two and three loops: M. Disertori and V. Rivasseau,Two and Three Loops Beta Function of Non Commutative Φ4

4 Theory,Eur. Phys. J. C (2007) hep-th/0610224.

I Any loops: M. Disertori, R. Gurau, J. Magnen and V. Rivasseau,Vanishing of Beta Function of Non Commutative Φ4

4 to all orders,Phys. Lett. B, (2007) hep-th/0612251

Ghost hunting

The ghost killing gun

We must now follow two main parameters under renormalization group flow,namely λ and Ω. At first order one finds

di' a(1− Ωi )λ

di' b(1− Ωi )λi ,

whose solution is:

0 100 200 300 400 500

At Ω = 1 (selfdual point) the field strength renormalization compensates thecoupling constant renormalization so that λφ4 remains invariant.

di' a(1− Ωi )λ

whose solution is:

0 100 200 300 400 500

di' a(1− Ωi )λ

whose solution is:

0 100 200 300 400 500

di' a(1− Ωi )λ

whose solution is:

0 100 200 300 400 500

di' a(1− Ωi )λ

whose solution is:

0 100 200 300 400 500

The Ward identities at all loops

Z (η, η) =

∫dφd φ e−(φXφ+φX φ+Aφφ+λ

2 φφφφ)+φη+ηφ (4.6)

Let U = eıM . One performs the ”left” change of variables:

φ→ φU = φU φ→ φU = Uφ (4.7)

which leads to

∂η∂ηδ ln Z

δMba= 0 (4.8)

and one obtains the Ward identities:

bν bν aν

aµ aµµ b

=(a−b)

b ν νν

aφφφ

φφφ

µ µ µ

Z (η, η) =

φ→ φU = φU φ→ φU = Uφ (4.7)

which leads to

∂η∂ηδ ln Z

δMba= 0 (4.8)

bν bν aν

aµ aµµ b

=(a−b)

b ν νν

aφφφ

φφφ

µ µ µ

Z (η, η) =

φ→ φU = φU φ→ φU = Uφ (4.7)

which leads to

∂η∂ηδ ln Z

δMba= 0 (4.8)

bν bν aν

aµ aµµ b

=(a−b)

b ν νν

aφφφ

φφφ

µ µ µ

Dyson’s equations

φ φ =

G4(1) G

4(2) G

φ φφ

I This is a classification of graphs (no combinatoric to check)!

I The second term has one ”left tadpole insertion”. It vanishes after massrenormalization.

The first term

G 4(1)(0,m, 0,m) = λC0mG 2(0,m)

([G 2(0,m)]2 + G 2

ins(0, 0; m))

The Ward identity gives:

G 2ins(0, 0; m) = lim

ζ→0G 2

ins(ζ, 0; m) = limζ→0

G 2(0,m)− G 2(ζ,m)

= −∂LG2(0,m)→

G 4(1)(0,m, 0,m) = λ[G 2(0,m)]4 C0m

G 2(0,m)[1− ∂LΣ(0,m)] (4.10)

The first term

G 4(1)(0,m, 0,m) = λC0mG 2(0,m)

([G 2(0,m)]2 + G 2

ins(0, 0; m))

The Ward identity gives:

G 2ins(0, 0; m) = lim

ζ→0G 2

ins(ζ, 0; m) = limζ→0

G 2(0,m)− G 2(ζ,m)

= −∂LG2(0,m)→

G 4(1)(0,m, 0,m) = λ[G 2(0,m)]4 C0m

G 2(0,m)[1− ∂LΣ(0,m)] (4.10)

The third term

1PI 1PI CTlost

is obtained by opening the face p of G 4,bare(3) = C0m

∑p G 4,bare

ins (p, 0; m, 0,m)

But in

the renormalized theory we must add the missing mass counterterm.

G 4(3) = C0m

G 4ins(0, p; m, 0,m)− C0m(CTmissing )G 4(0,m, 0,m) (4.11)

But one has CTmissing = ΣR(0, 0) = Σ(0, 0)− T L, hence one concludes:

G 4(3)(0,m, 0,m) = −C0mG 4(0,m, 0,m)

G 2(0, 0)

∂Σ(0, 0)

1− ∂Σ(0, 0)

The third term

1PI 1PI CTlost

∑p G 4,bare

ins (p, 0; m, 0,m) But in

G 4(3) = C0m

G 4(3)(0,m, 0,m) = −C0mG 4(0,m, 0,m)

G 2(0, 0)

∂Σ(0, 0)

1− ∂Σ(0, 0)

The third term

1PI 1PI CTlost

∑p G 4,bare

ins (p, 0; m, 0,m) But in

G 4(3) = C0m

G 4(3)(0,m, 0,m) = −C0mG 4(0,m, 0,m)

G 2(0, 0)

∂Σ(0, 0)

1− ∂Σ(0, 0)

Death of the ghost

One puts G 43 on the left side of Dyson’s equation:

G 4(1 + C0m1

G 2(0, 0)

∂Σ(0, 0)

1− ∂Σ(0, 0))

= λ[G 2(0,m)]4 C0m

G 2(0,m)[1− ∂LΣ(0,m)] (4.12)

and using C0m = 1/(m + Aren); G 2(0,m) = 1/[m(1− ∂Σ) + Aren] one gets:

G 4(1− ∂Σ +Aren

m + Aren∂Σ) = λ[G 2]4(1− ∂Σ)2(1− m

m + Aren∂Σ) (4.13)

hence by simplifying, since red terms are equal, amputating, Γ4 = λZ 2 henceβ = 0!

A beautiful mathematical toy?

The Grosse-Wulkenhaar model seems far from the ordinary world...

It is Euclidean, it has parameters θ and Ω which are unobserved at ordinaryenergies...

However its main advantage is that it is mathematically consistent. May be naturemakes use of it in a certain way?

The Grosse-Wulkenhaar model seems far from the ordinary world...It is Euclidean, it has parameters θ and Ω which are unobserved at ordinaryenergies...

A possible scenario

As a first step towards possible scenarios beyond the standard model that couldmake use of the GW model, we suggest that there could be a whole bunch ofnoncommutative worldlets each with a θi parameter. They would each contain aGWi model with its own Ωi harmonic confining potential, all glued by acommutative space:

A possible scenarioAs a first step towards possible scenarios beyond the standard model that couldmake use of the GW model, we suggest that there could be a whole bunch ofnoncommutative worldlets each with a θi parameter. They would each contain aGWi model with its own Ωi harmonic confining potential, all glued by acommutative space:

A Lattice of wordlets?At low energy only the ordinary world would remain visible... The zero modes ofthe worldlets would create an effective LOCAL interaction in the commutativeworld.

That theory is somewhat similar to a lattice-regularized ordinarycommutative theory, but it has no ultraviolet cutoff.As energy increases, bigger and bigger parts of the worldlets would appear.

We currently try to elaborate this idea in collaboration with R. Gurau and A.Tanasa

A Lattice of wordlets?At low energy only the ordinary world would remain visible... The zero modes ofthe worldlets would create an effective LOCAL interaction in the commutativeworld. That theory is somewhat similar to a lattice-regularized ordinarycommutative theory, but it has no ultraviolet cutoff.As energy increases, bigger and bigger parts of the worldlets would appear.

We currently try to elaborate this idea in collaboration with R. Gurau and A.Tanasa37

Noncommutativity a possible alternative tosupersymmetry?

One of the strongest arguments for supersymmetry is that it tames uv divergences.This was desired to explain the mass hierarchy problem. Also it helps the couplingconstants of the U(1), SU(2) and SU(3) gauge groups of the standard model toconverge at a single scale:

But the ghost killing mechanism is a new mechanism to tame ultravioletdivergences which is completely different from SUSY and deserves some carefulstudy...

Constructive Field Theory = Resummation of Perturbative Field Theory. But isnot obvious:

I Functional integral is good for global existence (|∫

e−λφ4

dµ(φ)| ≤ 1) but badto compute connected functions

I Connectivity is read easily on graphs, but they are too many (about n! atorder n)

To my surprise, NCQFT φ?44 appears to be doable at the constructive level and to

throw some new light on the constructive problems of ordinary commutative fieldtheory

e−λφ4

dµ(φ)| ≤ 1)

but badto compute connected functions

e−λφ4

Open Problems

There are of course many:

I Covariant non-orientable Models

I Covariant self-dual Models

I Gauge Theories (Yang-Mills, Chern-Simons)

I Noncommutative Minkovski Space

I Condensed Matter Applications (Quantum Hall effect, polymer growth instrong magnetic field...)

I Non commutative ”curved” geometries

I Constructive analysis of the GW model to be completed ... Constructive dimReg and dim Ren

I Possible link to between ordinary field theory and quantum gravity

Open Problems

Some good news:

There is more thancompatibility

between non-commutative geometry and quantum field theory.

I Quantum field theory on non-commutative space can be renormalised.

I Quantum field theory is better behaved on non-commutative space than oncommutative space (no Landau ghost).

I It seems it can be fully built at the non-perturbative level.

I Renormalization group flows are modified when there is non commutativity ofspace-time.

Some good news:

There is more than

compatibilitybetween non-commutative geometry and quantum field theory.

Some good news:

Thanks

for yourattention!

Thanks

for yourattention!

why renormalizable noncommutative quantum field theories › vienna.seminar › 2007 › talks ›...

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