renormalizability of noncommutative quantum field...
TRANSCRIPT
![Page 1: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/1.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Renormalizability of noncommutative quantumfield theories
Axel de Goursac
Universite Catholique de Louvain
Quantum Theory and Gravitation, ETH Zurich, June 2011
![Page 2: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/2.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Plan
1 Introduction: Noncommutative Geometry and Physics
2 Noncommutative scalar theory
3 Noncommutative gauge theory
4 Review and perspectives
![Page 3: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/3.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Plan
1 Introduction: Noncommutative Geometry and Physics
2 Noncommutative scalar theory
3 Noncommutative gauge theory
4 Review and perspectives
![Page 4: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/4.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Plan
1 Introduction: Noncommutative Geometry and Physics
2 Noncommutative scalar theory
3 Noncommutative gauge theory
4 Review and perspectives
![Page 5: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/5.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Plan
1 Introduction: Noncommutative Geometry and Physics
2 Noncommutative scalar theory
3 Noncommutative gauge theory
4 Review and perspectives
![Page 6: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/6.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Essence of NCG
Correspondence between spaces and commutative algebras(fields)
Gelfand-Naimark thm: topological spaces X commutativeC*-algebras C (X ) = X → C continuousNoncommutative algebras: viewed as associated to a“noncommutative space”
NCG: generalize geometrical constructions and properties tononcommutative algebras
NCG could be adapted to the unification of quantum physicswith gravitation as an extension of differential geometry(classical QFT, general relativity) within an operator algebras’framework (quantum physics)
![Page 7: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/7.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Essence of NCG
Correspondence between spaces and commutative algebras(fields)
Gelfand-Naimark thm: topological spaces X commutativeC*-algebras C (X ) = X → C continuousNoncommutative algebras: viewed as associated to a“noncommutative space”
NCG: generalize geometrical constructions and properties tononcommutative algebras
NCG could be adapted to the unification of quantum physicswith gravitation as an extension of differential geometry(classical QFT, general relativity) within an operator algebras’framework (quantum physics)
![Page 8: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/8.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Essence of NCG
Correspondence between spaces and commutative algebras(fields)
Gelfand-Naimark thm: topological spaces X commutativeC*-algebras C (X ) = X → C continuousNoncommutative algebras: viewed as associated to a“noncommutative space”
NCG: generalize geometrical constructions and properties tononcommutative algebras
NCG could be adapted to the unification of quantum physicswith gravitation as an extension of differential geometry(classical QFT, general relativity) within an operator algebras’framework (quantum physics)
![Page 9: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/9.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Essence of NCG
Correspondence between spaces and commutative algebras(fields)
Gelfand-Naimark thm: topological spaces X commutativeC*-algebras C (X ) = X → C continuousNoncommutative algebras: viewed as associated to a“noncommutative space”
NCG: generalize geometrical constructions and properties tononcommutative algebras
NCG could be adapted to the unification of quantum physicswith gravitation as an extension of differential geometry(classical QFT, general relativity) within an operator algebras’framework (quantum physics)
![Page 10: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/10.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Essence of NCG
Correspondence between spaces and commutative algebras(fields)
Gelfand-Naimark thm: topological spaces X commutativeC*-algebras C (X ) = X → C continuousNoncommutative algebras: viewed as associated to a“noncommutative space”
NCG: generalize geometrical constructions and properties tononcommutative algebras
NCG could be adapted to the unification of quantum physicswith gravitation as an extension of differential geometry(classical QFT, general relativity) within an operator algebras’framework (quantum physics)
![Page 11: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/11.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
NCG in Physics
classical Standard Model coupled with gravitation: spectralaction principle on C∞(M)⊗ (C⊕H⊕M3(C))(Chamseddine Connes ’07) (non-abelianity)
Quantum space-time at Planck scale (Doplicher et. al. ’94)
(fuzzy structure)
Emergent gravity from nc gauge theory (Steinacker ’07, Yang ’09)
Relationship with string theory (Seiberg Witten ’99) and quantum loopgravity (Freidel Livine ’06, Noui ’08)
other approaches are possible...
![Page 12: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/12.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
NCG in Physics
classical Standard Model coupled with gravitation: spectralaction principle on C∞(M)⊗ (C⊕H⊕M3(C))(Chamseddine Connes ’07) (non-abelianity)
Quantum space-time at Planck scale (Doplicher et. al. ’94)
(fuzzy structure)
Emergent gravity from nc gauge theory (Steinacker ’07, Yang ’09)
Relationship with string theory (Seiberg Witten ’99) and quantum loopgravity (Freidel Livine ’06, Noui ’08)
other approaches are possible...
![Page 13: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/13.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
NCG in Physics
classical Standard Model coupled with gravitation: spectralaction principle on C∞(M)⊗ (C⊕H⊕M3(C))(Chamseddine Connes ’07) (non-abelianity)
Quantum space-time at Planck scale (Doplicher et. al. ’94)
(fuzzy structure)
Emergent gravity from nc gauge theory (Steinacker ’07, Yang ’09)
Relationship with string theory (Seiberg Witten ’99) and quantum loopgravity (Freidel Livine ’06, Noui ’08)
other approaches are possible...
![Page 14: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/14.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
NCG in Physics
classical Standard Model coupled with gravitation: spectralaction principle on C∞(M)⊗ (C⊕H⊕M3(C))(Chamseddine Connes ’07) (non-abelianity)
Quantum space-time at Planck scale (Doplicher et. al. ’94)
(fuzzy structure)
Emergent gravity from nc gauge theory (Steinacker ’07, Yang ’09)
Relationship with string theory (Seiberg Witten ’99) and quantum loopgravity (Freidel Livine ’06, Noui ’08)
other approaches are possible...
![Page 15: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/15.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
NCG in Physics
classical Standard Model coupled with gravitation: spectralaction principle on C∞(M)⊗ (C⊕H⊕M3(C))(Chamseddine Connes ’07) (non-abelianity)
Quantum space-time at Planck scale (Doplicher et. al. ’94)
(fuzzy structure)
Emergent gravity from nc gauge theory (Steinacker ’07, Yang ’09)
Relationship with string theory (Seiberg Witten ’99) and quantum loopgravity (Freidel Livine ’06, Noui ’08)
other approaches are possible...
![Page 16: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/16.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 17: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/17.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 18: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/18.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 19: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/19.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 20: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/20.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 21: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/21.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Quantum spaces
Important class of nc algebras: deformations of (Poisson)manifolds M: (C∞(M), ?θ)such that f ?θ g → f ·g and 1
θ [f , g ]? → f , gPBQFT: construct an action by replacing commutative productwith ?θ
Renormalizability of QFT: consistency of the theory for achange of scale
various examples of lorentzian and riemannian deformedspaces
In this talk: Moyal space in the euclidean framework,deformation quantization of R4
→ Existence of renormalizable gauge theories on this nc space isa crucial question
![Page 22: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/22.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Presentation of the Moyal space
Algebra of functions f : R4 → C (fields)
Deformed Moyal product:
(f ?g)(x) =1
π4θ4
∫d4yd4z f (x + y)g(x + z)e−2iyΘ−1z
Θ = θΣ, Σ =
0 −1 0 01 0 0 00 0 0 −10 0 1 0
Limit θ = 0: (f ? g)(x) = f (x)·g(x)
The Moyal space is the nc space associated to this algebra.Scalar fields are elements of this deformed algebra and will bemultiplied by using ?.
Tracial property:∫
d4x (f ? g)(x) =∫
d4x f (x)g(x)
![Page 23: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/23.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Presentation of the Moyal space
Algebra of functions f : R4 → C (fields)
Deformed Moyal product:
(f ?g)(x) =1
π4θ4
∫d4yd4z f (x + y)g(x + z)e−2iyΘ−1z
Θ = θΣ, Σ =
0 −1 0 01 0 0 00 0 0 −10 0 1 0
Limit θ = 0: (f ? g)(x) = f (x)·g(x)
The Moyal space is the nc space associated to this algebra.Scalar fields are elements of this deformed algebra and will bemultiplied by using ?.
Tracial property:∫
d4x (f ? g)(x) =∫
d4x f (x)g(x)
![Page 24: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/24.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Presentation of the Moyal space
Algebra of functions f : R4 → C (fields)
Deformed Moyal product:
(f ?g)(x) =1
π4θ4
∫d4yd4z f (x + y)g(x + z)e−2iyΘ−1z
Θ = θΣ, Σ =
0 −1 0 01 0 0 00 0 0 −10 0 1 0
Limit θ = 0: (f ? g)(x) = f (x)·g(x)
The Moyal space is the nc space associated to this algebra.Scalar fields are elements of this deformed algebra and will bemultiplied by using ?.
Tracial property:∫
d4x (f ? g)(x) =∫
d4x f (x)g(x)
![Page 25: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/25.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Presentation of the Moyal space
Algebra of functions f : R4 → C (fields)
Deformed Moyal product:
(f ?g)(x) =1
π4θ4
∫d4yd4z f (x + y)g(x + z)e−2iyΘ−1z
Θ = θΣ, Σ =
0 −1 0 01 0 0 00 0 0 −10 0 1 0
Limit θ = 0: (f ? g)(x) = f (x)·g(x)
The Moyal space is the nc space associated to this algebra.Scalar fields are elements of this deformed algebra and will bemultiplied by using ?.
Tracial property:∫
d4x (f ? g)(x) =∫
d4x f (x)g(x)
![Page 26: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/26.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Presentation of the Moyal space
Algebra of functions f : R4 → C (fields)
Deformed Moyal product:
(f ?g)(x) =1
π4θ4
∫d4yd4z f (x + y)g(x + z)e−2iyΘ−1z
Θ = θΣ, Σ =
0 −1 0 01 0 0 00 0 0 −10 0 1 0
Limit θ = 0: (f ? g)(x) = f (x)·g(x)
The Moyal space is the nc space associated to this algebra.Scalar fields are elements of this deformed algebra and will bemultiplied by using ?.
Tracial property:∫
d4x (f ? g)(x) =∫
d4x f (x)g(x)
![Page 27: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/27.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc scalar theory
Real scalar field theory on the euclidean Moyal space:
S [φ] =
∫dDx
(1
2(∂µφ)2 +
m2
2φ2 + λφ?φ?φ?φ
)Feynman rules in impulsions (Filk ’96):
λδ( 4∑
i=1
pi
)7→ λe i
θ2
2(p1Θ−1p2+p1Θ−1p3+p2Θ−1p3)δ
( 4∑i=1
pi
)Vertex symmetries: only cyclic permutations
⇒ Ribbon Feynman graphs (genus: 2− 2g = N2 − V + F )
g = 0 g = 1
![Page 28: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/28.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc scalar theory
Real scalar field theory on the euclidean Moyal space:
S [φ] =
∫dDx
(1
2(∂µφ)2 +
m2
2φ2 + λφ?φ?φ?φ
)Feynman rules in impulsions (Filk ’96):
λδ( 4∑
i=1
pi
)7→ λe i
θ2
2(p1Θ−1p2+p1Θ−1p3+p2Θ−1p3)δ
( 4∑i=1
pi
)Vertex symmetries: only cyclic permutations
⇒ Ribbon Feynman graphs (genus: 2− 2g = N2 − V + F )
g = 0 g = 1
![Page 29: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/29.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc scalar theory
Real scalar field theory on the euclidean Moyal space:
S [φ] =
∫dDx
(1
2(∂µφ)2 +
m2
2φ2 + λφ?φ?φ?φ
)Feynman rules in impulsions (Filk ’96):
λδ( 4∑
i=1
pi
)7→ λe i
θ2
2(p1Θ−1p2+p1Θ−1p3+p2Θ−1p3)δ
( 4∑i=1
pi
)Vertex symmetries: only cyclic permutations
⇒ Ribbon Feynman graphs (genus: 2− 2g = N2 − V + F )
g = 0 g = 1
![Page 30: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/30.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc scalar theory
Real scalar field theory on the euclidean Moyal space:
S [φ] =
∫dDx
(1
2(∂µφ)2 +
m2
2φ2 + λφ?φ?φ?φ
)Feynman rules in impulsions (Filk ’96):
λδ( 4∑
i=1
pi
)7→ λe i
θ2
2(p1Θ−1p2+p1Θ−1p3+p2Θ−1p3)δ
( 4∑i=1
pi
)Vertex symmetries: only cyclic permutations
⇒ Ribbon Feynman graphs (genus: 2− 2g = N2 − V + F )
g = 0 g = 1
![Page 31: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/31.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 32: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/32.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 33: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/33.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 34: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/34.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 35: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/35.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 36: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/36.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
UV-IR mixing
(Minwalla et al. ’00)
Tadpole (F = B = 2, g = 0): 2 broken faces
λ
∫d4k
e ikΘp
k2 + m2∝|p|→0
1
θ2p2
Amplitude is finite but singular for p → 0
Generic behavior for graphs with B ≥ 2
When inserted many times in higher order graphs, p is aninternal impulsion ⇒ IR divergence in p (coming from the UVsector of k)
→ Non-renormalizability of the theory
![Page 37: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/37.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 38: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/38.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 39: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/39.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 40: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/40.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 41: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/41.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 42: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/42.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Addition of a harmonic term to the action:
S [φ] =
∫d4x
(1
2(∂µφ)2 +
Ω2
2x2φ2 +
m2
2φ2 + λφ ? φ ? φ ? φ
)where x = 2Θ−1x
Propagator: Mehler kernel (Gurau Rivasseau Vignes ’06)
Power counting
ω = 4− N − 4(2g + B − 1)
Primitively divergent graphs: planar, B = 1, N = 2, 4
⇒ Renormalizability of the theory to all orders (Grosse Wulkenhaar ’05)
Loss of translation-invariance: essential here for the removingof the UV-IR mixing
![Page 43: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/43.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Properties
Covariant under Langmann-Szabo duality (Langmann Szabo ’02):
S [φ,m, λ,Ω] = Ω2 S[φ,
m
Ω,λ
Ω2,
1
Ω
]where φ is a symplectic Fourier transformation
New properties of the flow at the fixed point Ω = 1: βλ = 0(Grosse Wulkenhaar ’04, Disertori Gurau Magnen Rivasseau ’06)
→ Towards a constructive version of the nc φ4 harmonic model(Grosse Wulkenhaar ’09, Rivasseau Wang ’11)
Vacua of the theory (A.G. Tanasa Wallet ’08)
Rotational invariance at the quantum level if Θ is a tensor(A.G. Wallet ’11)
![Page 44: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/44.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Properties
Covariant under Langmann-Szabo duality (Langmann Szabo ’02):
S [φ,m, λ,Ω] = Ω2 S[φ,
m
Ω,λ
Ω2,
1
Ω
]where φ is a symplectic Fourier transformation
New properties of the flow at the fixed point Ω = 1: βλ = 0(Grosse Wulkenhaar ’04, Disertori Gurau Magnen Rivasseau ’06)
→ Towards a constructive version of the nc φ4 harmonic model(Grosse Wulkenhaar ’09, Rivasseau Wang ’11)
Vacua of the theory (A.G. Tanasa Wallet ’08)
Rotational invariance at the quantum level if Θ is a tensor(A.G. Wallet ’11)
![Page 45: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/45.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Properties
Covariant under Langmann-Szabo duality (Langmann Szabo ’02):
S [φ,m, λ,Ω] = Ω2 S[φ,
m
Ω,λ
Ω2,
1
Ω
]where φ is a symplectic Fourier transformation
New properties of the flow at the fixed point Ω = 1: βλ = 0(Grosse Wulkenhaar ’04, Disertori Gurau Magnen Rivasseau ’06)
→ Towards a constructive version of the nc φ4 harmonic model(Grosse Wulkenhaar ’09, Rivasseau Wang ’11)
Vacua of the theory (A.G. Tanasa Wallet ’08)
Rotational invariance at the quantum level if Θ is a tensor(A.G. Wallet ’11)
![Page 46: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/46.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Properties
Covariant under Langmann-Szabo duality (Langmann Szabo ’02):
S [φ,m, λ,Ω] = Ω2 S[φ,
m
Ω,λ
Ω2,
1
Ω
]where φ is a symplectic Fourier transformation
New properties of the flow at the fixed point Ω = 1: βλ = 0(Grosse Wulkenhaar ’04, Disertori Gurau Magnen Rivasseau ’06)
→ Towards a constructive version of the nc φ4 harmonic model(Grosse Wulkenhaar ’09, Rivasseau Wang ’11)
Vacua of the theory (A.G. Tanasa Wallet ’08)
Rotational invariance at the quantum level if Θ is a tensor(A.G. Wallet ’11)
![Page 47: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/47.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Properties
Covariant under Langmann-Szabo duality (Langmann Szabo ’02):
S [φ,m, λ,Ω] = Ω2 S[φ,
m
Ω,λ
Ω2,
1
Ω
]where φ is a symplectic Fourier transformation
New properties of the flow at the fixed point Ω = 1: βλ = 0(Grosse Wulkenhaar ’04, Disertori Gurau Magnen Rivasseau ’06)
→ Towards a constructive version of the nc φ4 harmonic model(Grosse Wulkenhaar ’09, Rivasseau Wang ’11)
Vacua of the theory (A.G. Tanasa Wallet ’08)
Rotational invariance at the quantum level if Θ is a tensor(A.G. Wallet ’11)
![Page 48: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/48.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 49: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/49.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 50: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/50.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 51: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/51.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 52: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/52.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 53: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/53.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 54: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/54.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(Bieliavsky A.G. Tuynman ’10)
Non-formal deformation of Rp|q: Moyal ⊗ Clifford Cl(q,C)
Product on R4|1: if Φ(x , ξ) = φ0(x) + φ1(x)ξ,
(Φ?Ψ)(x , ξ) =(φ0 ?ψ0 +
iθ
4φ1 ?ψ1 + (φ0 ?ψ1 +φ1 ?ψ0)ξ
)(x)
Trace: Tr(Φ) =∫
d4x Φ(x , 0)
Reduction: Φ(x , ξ) = (1 + bξ)φ(x) with b ∈ RStandard action on the deformed R4|1:
tr(1
2|[− i
2xµ(1 + bξ),Φ]?|2 +
m2
2Φ?2 + λΦ?4
)=
∫d4x
(1
2(∂µφ)2+
b4
8x2φ2+
m2
2φ2+λ(1+
b4θ
16)φ?φ?φ?φ
)→ Philosophy: change the space not the action.
LS duality: grading exchange (A.G. ’10)
![Page 55: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/55.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 56: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/56.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 57: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/57.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 58: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/58.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 59: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/59.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 60: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/60.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Construction of the nc gauge theory
Nc gauge potential: Aµ (real)
Gauge transformation: g † ? g = g ? g † = 1
Aµ 7→ g ? Aµ ? g† + ig ? ∂µg
†
Field strength: Fµν = ∂µAν − ∂νAµ − i [Aµ,Aν ]?
Covariant coordinate: Aµ = Aµ+ 12 xµ
Yang-Mills action:
S [A] =
∫d4x
1
4Fµν ? Fµν
UV-IR mixing (Matusis et al. ’00)
Polarization tensor (with B = 2):
ΠIRµν(p) ∝|p|→0
pµpνp4
![Page 61: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/61.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 62: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/62.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 63: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/63.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 64: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/64.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 65: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/65.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 66: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/66.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 67: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/67.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Harmonic theory
Gauge-invariant harmonic action: (Aµ = Aµ + 12xµ)
S [A] =
∫d4x(1
4Fµν ? Fµν +
β
4Aµ,Aν2
? +κ
2Aµ ?Aµ
)
Coupled to the Grosse-Wulkenhaar model at the 1 loop order(A.G. Wallet Wulkenhaar ’07, Grosse Wohlgenannt ’07)
Quadratic part:∫ (
12 (∂µAν)2 + β
2 x2AνAν + κ
2AνAν)
.
⇒ Good candidate to renormalizability
Mass term for gauge fields without Higgs mechanism
Not Langmann-Szabo covariant
Problem: no trivial vacuum
![Page 68: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/68.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 69: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/69.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 70: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/70.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 71: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/71.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 72: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/72.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 73: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/73.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 74: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/74.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Vacua of the theory
Equation of motion: (Aµ = Aµ + 12xµ)
[[Aµ,Aν ]?,Aν ]? + βAµ,Aν?,Aν? + κAµ = 0
→ No trivial vacuum Aµ = 0
Solution Aµ = −12 xµ: matrix model without dynamic
Computation of the other solutions (product of Bessel,gaussian and hypergeometric functions) (A.G. Wallet Wulkenhaar ’08)
→ Too complicated for an analytic expansion of the theory
A particular gauge fixing permits to give a trivial vacuum tothe theory (in progress)
Ghost sector remains to be fully understood (Blaschke Grosse Kronberger
Schweda Wohlgenannt ’09)
![Page 75: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/75.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(A.G. Masson Wallet ’08)
Reduced gauge potential on the deformed R4|1: (1 + bξ)Aµ(x)
Same gauge transformations
Graded curvature:
F (1)µν = Fµν(x), F (2)
µν = 2bFµν(x)ξ, F (3)µν =
b2θ
4Aµ,Aν?(x)
+ other part
Standard Yang-Mills action:
Tr(∑
i
(F (i)µν )2
)=
∫d4x(1
4(1 +
b2θ
2)(Fµν)2
+b4θ2
16Aµ,Aν2
? + (80
θ+ 20b2 +
θ
2)(Aµ)2
)Symmetry: grading exchange
![Page 76: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/76.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(A.G. Masson Wallet ’08)
Reduced gauge potential on the deformed R4|1: (1 + bξ)Aµ(x)
Same gauge transformations
Graded curvature:
F (1)µν = Fµν(x), F (2)
µν = 2bFµν(x)ξ, F (3)µν =
b2θ
4Aµ,Aν?(x)
+ other part
Standard Yang-Mills action:
Tr(∑
i
(F (i)µν )2
)=
∫d4x(1
4(1 +
b2θ
2)(Fµν)2
+b4θ2
16Aµ,Aν2
? + (80
θ+ 20b2 +
θ
2)(Aµ)2
)Symmetry: grading exchange
![Page 77: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/77.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(A.G. Masson Wallet ’08)
Reduced gauge potential on the deformed R4|1: (1 + bξ)Aµ(x)
Same gauge transformations
Graded curvature:
F (1)µν = Fµν(x), F (2)
µν = 2bFµν(x)ξ, F (3)µν =
b2θ
4Aµ,Aν?(x)
+ other part
Standard Yang-Mills action:
Tr(∑
i
(F (i)µν )2
)=
∫d4x(1
4(1 +
b2θ
2)(Fµν)2
+b4θ2
16Aµ,Aν2
? + (80
θ+ 20b2 +
θ
2)(Aµ)2
)Symmetry: grading exchange
![Page 78: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/78.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(A.G. Masson Wallet ’08)
Reduced gauge potential on the deformed R4|1: (1 + bξ)Aµ(x)
Same gauge transformations
Graded curvature:
F (1)µν = Fµν(x), F (2)
µν = 2bFµν(x)ξ, F (3)µν =
b2θ
4Aµ,Aν?(x)
+ other part
Standard Yang-Mills action:
Tr(∑
i
(F (i)µν )2
)=
∫d4x(1
4(1 +
b2θ
2)(Fµν)2
+b4θ2
16Aµ,Aν2
? + (80
θ+ 20b2 +
θ
2)(Aµ)2
)Symmetry: grading exchange
![Page 79: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/79.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Supergeometrical interpretation
(A.G. Masson Wallet ’08)
Reduced gauge potential on the deformed R4|1: (1 + bξ)Aµ(x)
Same gauge transformations
Graded curvature:
F (1)µν = Fµν(x), F (2)
µν = 2bFµν(x)ξ, F (3)µν =
b2θ
4Aµ,Aν?(x)
+ other part
Standard Yang-Mills action:
Tr(∑
i
(F (i)µν )2
)=
∫d4x(1
4(1 +
b2θ
2)(Fµν)2
+b4θ2
16Aµ,Aν2
? + (80
θ+ 20b2 +
θ
2)(Aµ)2
)Symmetry: grading exchange
![Page 80: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/80.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 81: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/81.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 82: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/82.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 83: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/83.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 84: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/84.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 85: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/85.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 86: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/86.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Another renormalizable scalar theory
Amplitude of the tadpole: singular in 1p2
Addition to the standard φ4 action the term
a
2θ2
∫d4p
1
p2φ(−p)φ(p)
Renormalizable to all orders (Gurau Magnen Rivasseau Tanasa ’09)
Translation invariant
Same flow as in the commutative theory, finiterenormalization for a
Study of the commutative limit (Magnen Rivasseau Tanasa ’09)
Associated gauge models (Blaschke Gieres Kronberger Schweda Wohlgenannt ’08)
![Page 87: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/87.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Other theories
Another interpretation of the harmonic term:noncommutative scalar curvature (Buric Wohlgenannt ’10)
Complex φ4-orientable scalar theory: renormalizable
S [φ] =
∫dDx
(1
2|∂µφ|2 +
m2
2|φ|2 + λφ† ? φ ? φ† ? φ
)
φ3 scalar theories in 2,4,6 dimensions (Grosse Steinacker ’06)
Fermionic theories of type Gross-Neveu in 2 dimensions(Vignes-Tourneret ’07)
UV-IR mixing generic for isospectral deformations (Gayral ’05)
![Page 88: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/88.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Other theories
Another interpretation of the harmonic term:noncommutative scalar curvature (Buric Wohlgenannt ’10)
Complex φ4-orientable scalar theory: renormalizable
S [φ] =
∫dDx
(1
2|∂µφ|2 +
m2
2|φ|2 + λφ† ? φ ? φ† ? φ
)
φ3 scalar theories in 2,4,6 dimensions (Grosse Steinacker ’06)
Fermionic theories of type Gross-Neveu in 2 dimensions(Vignes-Tourneret ’07)
UV-IR mixing generic for isospectral deformations (Gayral ’05)
![Page 89: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/89.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Other theories
Another interpretation of the harmonic term:noncommutative scalar curvature (Buric Wohlgenannt ’10)
Complex φ4-orientable scalar theory: renormalizable
S [φ] =
∫dDx
(1
2|∂µφ|2 +
m2
2|φ|2 + λφ† ? φ ? φ† ? φ
)
φ3 scalar theories in 2,4,6 dimensions (Grosse Steinacker ’06)
Fermionic theories of type Gross-Neveu in 2 dimensions(Vignes-Tourneret ’07)
UV-IR mixing generic for isospectral deformations (Gayral ’05)
![Page 90: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/90.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Other theories
Another interpretation of the harmonic term:noncommutative scalar curvature (Buric Wohlgenannt ’10)
Complex φ4-orientable scalar theory: renormalizable
S [φ] =
∫dDx
(1
2|∂µφ|2 +
m2
2|φ|2 + λφ† ? φ ? φ† ? φ
)
φ3 scalar theories in 2,4,6 dimensions (Grosse Steinacker ’06)
Fermionic theories of type Gross-Neveu in 2 dimensions(Vignes-Tourneret ’07)
UV-IR mixing generic for isospectral deformations (Gayral ’05)
![Page 91: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/91.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Other theories
Another interpretation of the harmonic term:noncommutative scalar curvature (Buric Wohlgenannt ’10)
Complex φ4-orientable scalar theory: renormalizable
S [φ] =
∫dDx
(1
2|∂µφ|2 +
m2
2|φ|2 + λφ† ? φ ? φ† ? φ
)
φ3 scalar theories in 2,4,6 dimensions (Grosse Steinacker ’06)
Fermionic theories of type Gross-Neveu in 2 dimensions(Vignes-Tourneret ’07)
UV-IR mixing generic for isospectral deformations (Gayral ’05)
![Page 92: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/92.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 93: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/93.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 94: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/94.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 95: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/95.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 96: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/96.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 97: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/97.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces
![Page 98: Renormalizability of noncommutative quantum field theoriesconf.itp.phys.ethz.ch/qg11/talks/goursac-zurich2011.pdf · Introduction Scalar theory Gauge theory Review and perspectives](https://reader034.vdocuments.us/reader034/viewer/2022042400/5f0eb5b07e708231d4408e39/html5/thumbnails/98.jpg)
Introduction Scalar theory Gauge theory Review and perspectives
Conclusion
About the theory with harmonic term
Renormalizable theory
New properties of the flow
Interpretation in terms of a nc supergeometry
Good candidate for the gauge theory
Understand BRST formalism
Relation between renormalizability and nc supergeometry
Generalization to other nc spaces