New results: Operator Spin Foams, SL(2,C)

Transcription

1 New results: Operator Spin Foams, SL(2,C) WARSZAWA, NOVEMBER 2, 2010 Jerzy Lewandowski papers: Benjamin Bahr, Frank Hellmann, JL, Wojciech Kamiński, Marcin Kisielowski arxiv: Wojciech Kamiński arxiv: p.1

2 Plan Operator spin foams: a tool using the structure of the spin foams definition the equivalence relation the glueing the contraction at vertices emergence of the face and the boundary edge amplitudes as consistency conditions Operator spin foam models of the maximal symmetry Application to the EPRL intertwiners The door to the SL(2,C) case opened with the finiteness (and understanding) of the evaluation. p.2

3 Operator spin foam: definition G - a given compact Lie group, (κ, ρ, P) - operator spin foam: κ - 2-cell complex (set of polygons, glueings along sites, no self-glueing) ρ - coloring of the faces with the irreducuble representations, f ρ f (1) P - coloring of each in ternal (contained in at least two faces) edge with operators e P e (2) P e : H e H e (3) p.3

4 Operators spin foam: definition P e : H e H e, H e = algebraic dual (4) p.4

5 Equivalence relation: edge reorientation Given an operator spin foam (κ, ρ, P), the following operator spin foam (κ, ρ, P ) is considered equivalent: κ obtained from κ by flipping an edge: e 1 = e 1 1. P e 1 = P e 1, the remaining colorings unchanged: P e = P e for e e 1, ρ = ρ p.5

6 Equivalence relation: face reorientation Given an operator spin foam (κ, ρ, P), the following operator spin foam (κ, ρ, P) is considered equivalent: κ obtained from κ by flipping the orientation of a face f 1, and denoting the result f 1 ρ f 1 = ρ f 1, the remaining colorings unchanged: ρ f = ρ f for f f 1, P = P p.6

7 Equivalence relation: edge splitting Given an operator spin foam (κ, ρ, P), any operator spin foam (κ, ρ, P) defined below is considered equivalent: κ obtained from κ by splitting an edge: e 0 = e 1 e 2 P e 1 and P e 2 constrained by: P e 1 P e 2 = P e 0, (5) the remaining colorings unchanged: P e = P e for e e 1, e 2, ρ = ρ p.7

8 Equivalence relation: face splitting Given an operator spin foam (κ, ρ, P), the following operator spin foam (κ, ρ, P ) is considered equivalent: κ obtained from κ by splitting a face f 0 into f 1 and f 2 oriented as f 0 and glued along a new edge e 0, ρ f 1 = ρ f 2 = ρ f 0, P e 0 = id (6) the remaining colorings unchanged: ρ f = ρ f for f f 1, f 2, P e = P e for e e 0. p.8

9 Equivalence relation: the other cases Given an operator spin foam (κ, ρ, P), any of the following operator spin foams (κ, ρ, P ), (κ, ρ, P ) and (κ, ρ, P ) is considered equivalent: 1. Adding a trivial face: κ obtained from κ by adding a face f 0 colored by the trivial representation, ρ f 0 = trivial, and ρ f = ρ f for f f 0, P = P. 2. Rescalling of the operators: κ = κ, ρ = ρ, P obtained by rescaling, that is P e = a e P e such that e a e = 1 3. Replacing a representation by an equivalent one: κ = κ, for f 0, one of the faces, use a unitary map I : H ρf0 H, to define an equivalent representation ρ f 0 = I ρ f0 I 1, and the corresponding operators P e 1, P e 2, P e 3,... for every edge e 1, e 2, e 3,... contained in f 0. p.9

10 Glueing the operator spin foams Given (κ, ρ, P) and (κ, ρ, P ), glueing the colored 2-cell complexes into (κ#κ, ρ#ρ ) along links of the boundary is accompanied by the composition of the operators coloring each pair e, e of edges connected into a single edge of κ #κ, P e P e = (P #P) e e (7) p.10

11 The contraction Tr(κ,ρ,P) (P e w) A... A... = P e A...B... A...B...w B... B.... (8) indices assigned to vertices. Tr v,f P e P e Tr v,f P e =: Tr(κ, ρ, P) H e He (9) e e v,f e e/e are incoming to/outgoing from κ. p.11

12 Consistency with the equivalence and glueing Tr(κ, ρ, P) is inconsistent with: the face splitting equivalence relation with the glueing However, for every (κ, ρ, P ) obtained by splitting a face of any (κ, ρ, P), ( f A f )Tr(κ, ρ, P) = ( f A f )Tr(κ, ρ, P ) A f = dimρ f. (10) In that way, the face amplitude emerges. For the consistency with the glueing we introduce the boundary edge amplitude, Z(κ, ρ, P) = ( A f )Tr(κ, ρ, P). (11) l κa l)( f For any pair of the operator spin foams glued along a graph γ, Z(κ #κ, ρ #ρ, P #P) = Tr γ Z(κ, ρ, P)Z(κ, ρ, P) A l = 1 dfl. (12) p.12

13 Passage to the spin foams P e = P ι e ι e ι e ι e (13) ι e B e ι e B e in any basis, B e H e and the conjugate B e H e A v = Tr vf ι e ι e, (14) f : f v e incoming e outgoing Z(κ, ρ, P) = ι e P ι e ι e f d f A v v l 1 df l ιẽ ι ẽ (15) ẽ ẽ p.13

14 Middle summary We have introduced operator spin foams (OSF) (κ, ρ, P) The equivalence relations identify pairs of OSF, alow to reorient and refine OSFs The natural contraction at the vertices applied to e P e leads to the operator Z(κ, ρ, P) H κ The operator involves the face amplitude A f = d ρf for the consistency with the equivalence with respect to the face splitting The operator involves the boundary amplitude A l = 1/ d l for the consistency with the glueing of OSF The operator consist of the spin foams, the contractions give the familiar vertex amplitudes (evaluation of vertex spin-networks), the edge operators give conditions of glueing different vertices This is just an algebraic combinatorial framework, no physical postulates have been made thus far, the framework can be used to express formulae of any spin foam model p.14

15 Natural operator spin foam models We define a G operator spin foam model to be an assignment (κ, ρ) (κ, ρ, P), (16) of an operator coloring P to each 2-cell complex κ colored by ρ with the irreducible representations of a compact group G; the assignment is assumed to be consistent with the equivalence relation in the set of the operator spin foams and with the glueing of the operator spin foams. The set of maximal symmetry assumptions defines a natural operator spin foam model: operator coloring e P e depends only on the unordered sequence of labels ρ f coloring the faces f such that e f for every (internal) edge e P e = P e. (17) p.15

16 Natural OSFM: a general solution The consistency with the face splitting (see transparency 8) determines that is PBB AA = 1 d ρf δb AδA B H e = H ρf H ρ f P e = id (18) the edge splitting equivalence (see transparency 7) P e P e = P e, for every e. (19) Hence, each operator P e is the orthogonal projection onto a subspace H s e H e. (20) For every e, the subspace depends only on the unordered sequence of the representations ρ f given by the faces f containing e. p.16

17 Examples of the natural OSFM The spin foam model of the G 3d BF theory: H s e = H e, and P e = id (21) The Barrett-Crane model: G = SU(2) + SU(2), P e = 0 unless ρ + f = ρ f, then P e = ι BC,e ι BC,e. (22) The EPRL or FK solutions to the simplicity constraints: G = SU(2) + SU(2). For every edge e, there is defined the subspace H EPRL/FK e H e (23) of intertwiners which satisfy the suitably defined simplicity constraints. Notation: H ρf = H j H f j +. f He EPRL Inv( f H ρf ) non-empty, provided j ± f = 1±γ 2 k f, k f 1 2 N. One can use H EPRL e to define a natural operator spin foam model. p.17

18 Natural OSFM for the EPRL intertwiners H EPRL e is defined to be the immage of the EPRL map ι EPRL e : Inv SU(2) f e H kf Inv SU(2) SU(2) f e H j f H j + f (24) The map ι EPRL e is one-to-one, but not an isometry Kamiński,Kisielowski,L Maps an orthonormal basis B e Inv SU(2) f e H k f into a non-orthonormal basis of He EPRL. Therefore, P e = η e B e,η e B e P η e η e ιeprl e (η e ) ι EPRL e (η e), (25) η e P η e η e ( ι EPRL e (η e) ι EPRL e (η e ) ) = e δη e η e. (26) p.18

19 Natural OSFM for the EPRL in terms of SFs Coloring η: two independent SU(2) intertwiners η e, η e assigned to each e. Z(κ, ρ, P) = η e l P η e η e (2j + f l (2j + f + 1)(2j f + 1) f v 1 + 1)(2j f l + 1) ẽ A v ι EPRL e (η e ) ẽ ( ι EPRL e (η e )) (27) f - the set of faces, v - the set of the internal vertices, l - the set of the boundary edges (links), and ẽ/ẽ - the set of edges which intersect κ at the end/beginning point. A v is the vertex amplitude and f l is the (unique) face containing l. p.19

20 Asymptotics 1 define σ = EPRL. define ι σ (η) = ι EPRL (η) Compare the originaly proposed state sum that uses P σ = η ι σ (η)ι σ (η) (28) with the natural state sum that uses P σ = η,η ι σ (η)h(η, η )ι σ (η ) (29) with h(η, η ) such that P σ ι σ (η) = ι σ (η). Both operators have the same range: P σ Pσ = P σ p.20

21 Coherent state basis Asymptotics 2 η(j i, n i ) = dg g n i ji i (or for simplicity η(n i )) then has for γ < 1: P σ = dn i ι σ( ) η(n i ) ι σ ( ) η(n i ) (30) and P σ = dn i dn i ι σ( ( ) η(n i )) h η(n i ), η(n i) ι σ ( ) η(n i) (31) Now from P σ ι σ( ) η(n i ) = ι σ( ) η(n i ) we have dn h ( ) i η(n i ), η(n i) ι σ (η(n i)) ι σ (η(n i )) = δ(n i, n i ). (32) p.21

22 Coherent state basis Asymptotics 3 η(n i ) = dg g n i ji i then has for γ < 1: h 1( ) η(n i ), η(n i) = dg ± i n i g + n i (1+γ)j i n i g n i (1 γ)j i (33) Some observations: Goes to zero exponentially unless i j in i = 0 and η(n i ) η (n i ). On the diagonal we have h 1( ) η(j i, n i ), η(j i, n i ) = η((1 + γ)j i, n i ) 2 η((1 γ)j i, n i ) 2. p.22

23 Asymptotics 4 For normalised coherent intertwiners η(j i, n i ), ) h ( η(j 1 i, n i ), η(j i, n i ) is η(n i ) 2. Asymptotically the two proposals are very similar, differ by edge meassure factor. Original proposal is polynomially small for large j. p.23

24 The SL(2,C) case Brief outline of what we have given a graph γ: The map SU(2) spin network a function on SL(2,C) γ, solution to the EPRL constraints, s ψ s The immage defines a space of functions F A natural scalar product in F A rigging map η : F F. Each η(ψ(s)) is an SL(2,C) invariant distribution on F. Evaluation of each η(ψ(s)) at (I,..., I) SL(2,C) γ which is proven to be taking finite values: the long standing conjecture by Baez and Barrett, passed to the EPRL case by Engle and Pereira proved Kamiński Thank you p.24

25 Summary The framework of operator spin foams encodes the structure of the spin networks and spin foams. Emergence of the face amplitude as consistency condition A class of the natural operator spin foam models (NOSFM) defined by maximal symmetry. The EPRL simplicity constraints can be used to define an example of the NOSFM. The result is inequivalent to the original EPRL proposal. We have calculated the asymptotics of that model for large j and compaired with the original EPRL model. The door to the SL(2,C) extensions open due to the finiteness of the evaluation (suitably defined). p.25