Spin Foams from the LQG point of view

Size: px

Start display at page:

Download "Spin Foams from the LQG point of view"

Zoe Nelson
5 years ago
Views:

1 Spi Foams from the LQG poit of view ILleQGalS, 20th October 2009 Jerzy Lewadowski i collaboratio with: Wojciech Kamiński, Marci Kisielowski, Uiwersytet Warszawski p.1

2 Pla The goal: a geeralizatio of the EPRL spi foam model to all the spi-etwork states What for? For the compatibility with LQG: either we geeralize SFs or we restrict/modify LQG to the piecewise liear spaces Strategy: We go o with the EPRL costructio: sufficietly geeral spi-foam: yes characterizatio of vertex: yes characterizatio of spi-foam: yes vertex amplitude: yes the scheme of the SF models of gravity: yes Barrett-Crae vertex: yes EPRL vertex: yes Techical result: ijectivity of SU(2) ivariats EPRL SU(2) SU(2) ivariats The limits p.2

3 Papers classic: Reiseberger 1994, Reiseberger-Rovelli 1997, Barrett-Crae 1998, Yetter 1998, Barrett 1998, Reiseberger 1998, Baez 2000, Perez 2003 ewer: Biachi-Modesto-Rovelli-Speciale 2006, Alesci-Rovelli 2007, Egle-Livie-Pereira-Rovelli 2008, also Freidel-Krasov 2008 (sorry for ot cosiderig that paper here!) our paper: KKL 2009 p.3

4 Diffeomorphism ivariat theories of coectios Give: a 3-maifold Σ, a Lie group G, its Lie algebra g, ad the set A of the g valued differetial oe-forms A (coectios) o Σ. Parallel trasport defied by A A alog a fiite curve e i Σ: A(e) := Pexp e A The space Cyl(A) of the cylidrical fuctios: a cylidrical Ψ : A C, is defied by a fiite set of fiite, orieted curves e 1,..., e i Σ ad by a cotiues fuctio ψ : G C, Ψ(A) := ψ(a(e 1 ),..., A(e )). There is a atural, diffeomorphism ivariat itegral: : Cyl(A) C Defies the scalar product: (Ψ Ψ ) = ΨΨ the kiematical Hilbert space for diffeomorphism ivariat theories of coectios: ( Cyl(A), ( ) ), the completio LQG: G = SU(2), SFM: G =SU(2) SU(2) reduced to SU(2) by the simplicity costraits The cylidrical fuctios ca be costructed from spi-etworks embedded i Σ: spi-etwork states p.4

5 The spi-etwork states A graph embedded i Σ, irreducible represetatios ρ I of a group G i Hilbert spaces H I, itertwiers: ι 1 Iv(ρ 1 ρ 2 ρ 3 ) H 1 H 2 H 3, ι 2 Iv(ρ 1 ρ 2 ρ 3) H 1 H 2 H 3 Ψ(A) := ρ 1 C 1 B 1 (A(e 1 ))ρ 2 C 2 B 2 (A(e 2 ))ρ 3 C 3 B 3 (A(e 3 ))ι 1 B 1 B 2 B 3 ι 2C1 C 2 C 3 spi-etwork trace: Ψ(0) = ι 1 C 1 C 2 C 3 ι 2C1 C 2 C 3 p.5

6 The idea of spi-etwork state evolutio a history of a graph the iitial ad fial graphs p.6

7 Foam: elemets The circle shows the orietatio p.7

8 Foam: glueig Faces are glued with other faces alog the edges. Mathematically, a foam is a liear 2-cell complex with boudary p.8

9 Spi-foam: coloured foam ι : H f... H f... Hermitia adjoit ι : H f... H f... p.9

10 Spi-etwork iduced o the boudary Spi-foam with boudary Iduced spi-etwork p.10

11 Faces meettig at a vertex a) Edges b) Every face meetig v cotais exactly two of the edges p.11

12 The vertex structure Neigbourhood U of v Spi-etwork iduced o U, a(v) := trace of the spi-etwork = Ψ(0) p.12

13 A o vertex spi-foam The spi-foam structure: A geeral spi-foam is glued from vertex eghbourhouds ad o vertex spi-foams. A o vertex spi-foam: p.13

14 The scheme of (Euclidea) SF models G =SU(2) SU(2) the kiematical Hilbert space: spaed by the spi-etwork states, the spi-etworks embedded i a give 3-maifold Σ histories of spi-etwork states: embedded spi-foams i Σ R the Spi Foam aplitude: iteral vertexes v a(v) faces f dim(ρ f ) boudary vertexes v ι v Simplicity costraits imposed o the itertwiers ι: the Barrett-Crae itertwier we kow that this choice was too restrictive, but sice it was the obligig iterwtier utill recetly, it is good to kow it is easily geeralizated to a arbitrary spi-foam the EPRL itertwier likelly to be the right, eve easier to be geeralized Summig the SF amplitudes: with respect to elemets of orthoormal basis i the subspaces of simple (EPRL) itertwiers p.14

15 The Barrett-Crae itertwier a represetatio of SU(2) SU(2) is a pair of SU(2) represetatios (ρ j, ρ j ) i H j H j ( ) a itertwier is ι Iv (ρ j, ρ 1 j )... (ρ j, ρ j ) Iv(ρ j 1... ρ j ) Iv(ρ j 1... ρ j ) Briefly speakig, the BC itertwier is... the idetity map Iv(ρ j... ρ 1 j ) Iv(ρ j... ρ 1 j ) provided j 1 = j 1 =: j 1,..., j = j =: j Exactly, let ι 1,..., ι k Iv(ρ j1... ρ j ) be ay orthoormal basis. = ι BC A 1...A A 1...A where ǫ j Iv(H j H j ). What is ι BC good for? = k i=1 ιa 1...A i ι ib 1...B (O 1) ι BC = ±(1 O) ι BC ǫ B 1 A 1 j 1...ǫ B A j wheever (O 1) ǫ j1... ǫ j = ±(1 O) ǫ j1... ǫ j Give a graph colored by SU(2) represetatios ρ ji, the corespodig Hilbert space of the BC spi-etworks is either 0 or 1 p.15

16 The EPRL itertwier a represetatio of SU(2) SU(2) is a pair of SU(2) represetatios (ρ j, ρ j ) i H j H j ( ) a itertwier is EPRL Iv (ρ j, ρ 1 j )... (ρ j, ρ j ) Iv(ρ j 1... ρ j ) Iv(ρ j 1... ρ j ) = EPRL A 1...A A 1...A (ι) = k i=1 ιa 1...A i where: c i Iv(H j i H j i H ki ), ι ib 1...B c B 1 A 1 D c B A D ι D1...D j i = γ k i, j i = γ k i ι Iv(ρ k 1... ρ k ). p.16

17 The ijectivity of I ι EPRL The result: For every -valet vertex, the EPRL map SU(2) itertwiers EPRL itertwiers is ijective. The proof splits ito two cases: γ 1 case KKL 2009 oly j = j k is used the key observatio: the map H j H k H j k does ot kill simple tesor products 0 < γ < 1 case Kamiski ot oly j j = k the full j ± = 1 2 (1 ± γ)k was used p.17

18 The sketch of the proof for γ < 1 give ι Iv(ρ k1... ρ k ) let P k12,k 1 k 2 ι 0 be the projectio oto the lowest possible k 12, where P k12,k 1 k 2 : H k1 H k2 H k12 we fid ι P j 12,j 1 j 2 such that P j 12,j 1 j 2 Iv(ρ j 1... ρ j ) Iv(ρ j 1 j 12 j 12 = k 12,... ρ j ) ad (EPRL(ι) ι ) 0. p.18

19 The limits γ ±. Exists at each level: the Holst actio coverges to the Palatii actio, j = j, k = 0, the EPRL itertwier coverges to the BC itertwier, all the EPRL derivatio coverges to a fiite limit. The limit that ca ot be exteded to the etire derivatio although the Holst actio does coverge perfectly well to the self dual actio is γ = ±1. (1) However the EPRL itertwier has a limit i that case: j = 0, k = j ±, ad moreover EPRL(ι) = ι, where ι is a arbitrary SU(2) itertwier, ad the amplitude turs ito the SU(2) BF amplitude. So the limit theory is the SU(2) BF theory. Strage: the self dual actio still defies the same Eistei s (Euclidea) gravity. p.19

20 Limits (cotiued) The limit i which the Holst actio is o loger equivalet to the Palatii actio ad (upo the rescalig by γ) defies the SU(2) SU(2) BF theory, is The γ = 0. (2) j = j, k = j j (3) but, quite surprisigly, the EPRL theory does ot resemble the SU(2) SU(2) theory at all. THANK YOU p.20

arxiv: v5 [gr-qc] 23 Sep 2011

arxiv: v5 [gr-qc] 23 Sep 2011 Spi-Foams for All Loop Quatum Gravity Wojciech Kamiński, Marci Kisielowski, Jerzy Lewadowski,2 Istytut Fizyki Teoretyczej, Uiwersytet Warszawski, ul. Hoża 69, 00-68 Warszawa (Warsaw), Polska (Polad) 2