Your SlideShare is downloading. ×
0
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
M. Dimitrijević, Noncommutative models of gauge and gravity theories
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

M. Dimitrijević, Noncommutative models of gauge and gravity theories

290

Published on

Balkan Workshop BW2013 …

Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia

Published in: Education, Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
290
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Balkan Workshop BW2013-Beyond the Standard Models-,Vrnjaˇcka Banja, April 25th-29th, 2013Noncommutative gauge and gravity theoriesMarija Dimitrijevi´cUniversity of Belgrade, Faculty of Physics,Belgrade, SerbiaM. Dimitrijevi´c, V. Radovanovi´c and H. ˇStefanˇci´c, arXiv: 1207.4675[hep-th]P. Aschieri, L. Castellani and M. Dimitrijevi´c, arXiv: 1207.4346[hep-th]
  • 2. OverviewNC geometryMotivationMain results-product approachNC gauge theoryNC gravityGravity as a gauge theoryAdS inspired NC actionConclusions & Outlook
  • 3. NC geometry: motivationWhy Noncommutativity? Physics at small distances (highenergies) not very well understood.Original motivation: to solve the problem of divergences in QFT.Recent motivation: appears in string theory, playground forQuantum theory of gravity, new effects in QFT, . . .[Douglas, Nekrasov ’01; Szabo ’01, ’06,; Castellani ’00;. . . ]Why deformation of gravity?: No renormalizable gravity theory(yet), modified gravity theory could explain the problems such asDark matter and Dark energy,. . .Why deformation of gauge theories?: Standard model is not thefinal theory, renormalizability, SUSY, cosmological constantproblem, DM particles,. . .Different approaches: Connes, Wess (Munich group), Grosse,. . .
  • 4. NC geometry: main resultsMathematics: C∗-algebras, deformation quantizaton, Hopfalgebras and quantum groups,. . .Physics:-renormalizability of NC field theories: UV/IR mixing,renormalizable models [Minwalla, Raamsdonk ’99; Grosse, Wulkenhaar ’05;Buri´c, Radovanovi´c ’04]-NC Standard Model: predictions of new processes (new vertices)[Wess et al. ’01; Trampeti´c et al. ’04,’05]. New vertices:γγγ, γγZ, γZZ, ZZZ, . . .ff γγ, ff γZ, ffZZ, udγW , . . .
  • 5. -NC SUSY theories: models, renormalization [Seiberg ’03; Britto, Feng,Rey ’03; Dimitrijevi´c, Radovanovic´c ’09].-NC gravity: twist approach [Wess et al. ’05, ’06; Ohl, Schenckel ’09],gauge theory of Lorentz/Poincar´e group [Chamseddine ’01,’04, Cardela,Zanon ’03, Aschieri, Castellani ’09,’12]: new terms in the action appear(Rµνρσ)n, . . . .
  • 6. NC geometry: -product approachNoncommutative space ˆAˆx , generated by ˆxµ coordinatesµ = 0, 1, . . . n such that:[ˆxµ, ˆxν] = Θµν(ˆx). (1)It is an associative free algebra generated by ˆxµ and divided by theideal generated by relations (1). Differential calculus, integral,symmetries [Wess et al. ∼1990s] can be discussed, but. . .-product geometry: represent ˆAˆx on the space of commutingcoordinates, but keep track of the deformationˆAˆx → Axˆf (ˆx) → f (x) and ˆf (ˆx)ˆg(ˆx) → f g(x).
  • 7. Example: θ-constant (canonical) deformation leads to Moyal-Weyl-product:f g (x) =∞n=0i2n 1n!θρ1σ1. . . θρnσn∂ρ1 . . . ∂ρn f (x) ∂σ1 . . . ∂σn g(x)= f · g +i2θρσ(∂ρf ) · (∂σg) + O(θ2). (2)Associative, noncommutative; c. conjugation: (f g)∗ = g∗ f ∗.Derivatives, integral are well definedd4x f g = d4x g f = d4x fg. (3)The -product (2) enabled: construction of quantum field theoriesand analysis of their renormalizability properties, construction ofNC Standard Model and the analysis of its phenomenologicalconsequences,. . .
  • 8. NC gauge theoryWe work with θ-constant NC space:f · g → f g = f · g + i2 θαβ(∂αf )(∂βg)−18 θαβθκλ(∂α∂κf )(∂β∂λg) + . . .α, Φ, Aµ, Fµν → ˆα, ˆΦ, ˆAµ,ˆFµν = ∂µˆAν − ∂νˆAµ − i[ˆAµ , ˆAν]δαΨ = iαΨ → δαˆΨ = i ˆα ˆΨδαΦ = i[α, Φ] → δαˆΦ = i[ˆα , ˆΦ]δαAµ = ∂µ + i[α, Aµ] → δαˆAµ = ∂µ ˆα + i[ˆα , ˆAµ]δαFµν = i[α, Fµν] → δαˆFµν = i[ˆα , ˆFµν]
  • 9. Idea of the Seiberg-Witten map: NC gauge transformations areinduced by the commutative gauge transformations, δα → δα.Thenˆα = ˆα(α, Aµ), ˆAµ = ˆAµ(Aµ), ˆΦ = ˆΦ(Φ, Aµ). (4)NC gauge transformations have to close(δαδβ − δβδα)ˆΦ = δ−i[α,β]ˆΦ.This conditions enables to solve for ˆα(α, Aµ), [Ulker, Yapiskan ’08,Aschieri, Castellani ’11]:ˆα(n+1)= −14(n + 1)θκλ{ˆAκ , ∂λ ˆα}(n), (5)with(A B)(n)= A(n)B(0)+ A(n−1)B(1)+ . . .+A(0) (1)B(n−1)+ A(1) (1)B(n−2)+ . . .
  • 10. The SW-map solution for the NC gauge field ˆAµ follows fromδαˆAµ = ∂µ ˆα + i[ˆα , ˆAµ]:A(n+1)µ = −14(n + 1)θκλ{ˆAκ , ∂λˆAµ + ˆFλµ}(n), (6)with the NC field-strength tensor ˆFλµ.Similarly, one calculates solutions of the SW-map for matter fields,field-strength tensor,. . .NC U(1) action, expanded up to first order in the NC parameterθαβS = −14d4x ˆFµνˆFµν= d4x −14FµνFµν+ θαβ(14FαβFµνFµν− FαµFβνFµν) .(7)The commutative abelian gauge theory becomes a NC nonabeliangauge theory ⇒ Photon self-interactions appear!
  • 11. The main result of the SW-map: No new degrees of freedom, NCgauge theory and the corresponding commutative gauge theoryhave the same number of degrees of freedom! This enablesconstrustion and analysis of NC gauge theories.NC Standard Model [Wess et al. ’01,’02]:-phenomenology: [Trampeti´c et al. ’03,’05, . . . ],-renormalization: [Buri´c, Radovanovi´c ’03,. . . , 11; Martin, Tamarit ’05,’07,. . . ]
  • 12. NC gravity: gravity as a gauge theoryLocalization of space-time symmetry ⇒ gravity.Start from the MacDowell-Mansouri action [MacDowell, Mansouri ’77]:S =116πGNd4xl216µνρσabcd R abµν R cdρσ +√−gR − 2√−gΛ (8)and Λ = −3/l2,√−g = det e aµ . Comments:-variables in (8) are spin connection ωµ and vielbeins eµ. They areindependent, 1st order formalsim.-(8) is invariant under SO(1, 3) gauge symmetry, while thediffeomorphism symmetry appears as a consequence of SSB, see[Stelle, West ’80].-varying (8) with respect to ωµ and vielbeins eµ gives equations ofmotion for these fields.-(8) written in the 2nd order formalism has three terms:Gauss-Bonnet topological term, Einstein-Hilbert term and thecosmological constant term.
  • 13. NC gravity: AdS inspired NC actionThe NC generalization of the action (8) isS = −il264πGNd4x µνρσTr (ˆRµνˆRρσγ5) (9)−2il2Tr (ˆRµνˆEρˆEσγ5) +1l4Tr (ˆEµˆEνˆEρˆEσγ5) ,with:-NC SO(1, 3) gauge potential: ˆωµ = ωµ + ˆω(1)µ + ˆω(2)µ + . . .-NC vielbeins: ˆEµ = eµ + ˆE(1)µ + ˆE(2)µ + . . .-NC curvature tensor:ˆRµν = ∂µ ˆων − ∂ν ˆωµ − i[ˆωµ , ˆων]= Rµν + ˆR(1)µν + ˆR(2)µν + . . . .
  • 14. Solutions of the SW map we insert into the action (9) andcalculate corrections. We find:S(0)= commutative action (8)S(1)= 0 unfortunately!!!S(2)= very complicated, not manifestly gauge invariant,no explicit results existed until June 2012.Shortcut: SW map for composite fields [Aschieri, Castellani, Dimitrijevi´c,’12].An example:(ˆEµˆEν)(1)= ˆE(1)µ eν + eµˆE(1)ν +i2θαβ∂αeµ∂βeν (10)= −14θαβ{ωα, ∂β(eµeν) + Dβ(eµeν)} +i2θαβ(Dαeµ)(Dβeν).
  • 15. Inserting these expressions and calculating traces explicitly, weobtainS(2)GB = −l21024πGNθκλθρσ µναβabcd d4x R cdαβ R abµκ R mnνρ Rλσmn−12R cdαβ R abµκ R mnνλ Rρσmn + R mnαβ RµκmnR abνρ R cdλσ+R mnµρ RνσmnR abακ R cdβλ −12R abακ R cdβλ R mnρσ Rµνmn , (11)S(2)CC = −1512πGN l2θκλθαβd4xe 6R abκα Rλβab − 3R abαβ Rκλab+4R γδαβ (Dκeµ)a(Dλeγ)b(eµa eδb + eµb eδa) (12)−4R γδαβ (Dκeγ)a(Dλeδ)a + 4(DκDαeµ)a(DλDβeν )b(eµa eνb − eµb eνa )−8R γδκα (Dβeγ)a(Dλeδ)a + 8R γδκα (Dβeγ)a(Dλeµ)b(eµa eδb + eµb eδa) ,
  • 16. S(2)EH = −1512πGNθαβθκλd4xe12R µνκλ R γδαβ Rµνγδ+Rκλρσ(12R µναβ R ρσµν − 2R µσαβ R ρµ +12RR ρσαβ+4R ρσβν R να + 4R ρα R σβ − 4R νραµ R µσβν )−2R µνκλ R γδαµ Rβνγδ − RR γδκα Rλβγδ−Rµνρσ(2R µνκα R ρσλβ + 4R µσκα R νρλβ )−4R να R γδκβ Rλνγδ + 2Rαµρσ(2R µνκβ R ρσλν−4R ρλ R µσκβ + 4R νρκβ R µσλν + 2R ρσκβ R µλ )+4eµb eνc ((DκRαβ)mb(DλRµν )cm − 2(DκRαµ)mb(DλRβν )cm)+(Dκeρ)m(Dλeσ)m(2R µναβ R ρσµν + 8R µραβ R σµ + 2RR ρσαβ−8R ρσαµ R µβ − 8R ρβ R σα − 8R νραµ R µσβν )−2(DκDαeρ)c(DλDβeσ)d(R(eρc eσd − eσc eρd ) − 3R ρν eνc eσd+3R σν eνc eρd + R ρν eσc eνd − R σν eρc eνd + 2R ρσµν eµc eνd )+4(Dλeσ)aR bmµν (Dαeρ)m(R µνκβ (eρa eσb − eσa eρb ) (13)+2R µσκβ (eνa eρb − eρa eνb ) − 2R µρκβ (eνa eσb − eσa eνb ) + 2R ρσκβ eµa eνb ) .
  • 17. Conclusions & OutlookNC gauge theory-NCSM: no new fields, but new interactions appear;renormalizability and phenomenology studied,. . .-NC gravity: MacDowell-Mansouri action studied, expansionup to second order in the NC parameter written in amanifestly gauge covariant way; phenomenologicalconsequences remain to be studied,. . .future investigation-phenomenological consequences of NC gravity-NC SO(2, 3) symmetry and SSB in NC theory-SW map for composite fields and renormalization of gaugetheories

×