New applications for LQG

Transcription

1 ILleQGalS, Oct 28, 2014

2 Plan position / momentum representations L, Sahlmann (coming soon) Application 1: a new operator Application 2: The BF vacuum ˆq ab ˆφ,a ˆφ,b detˆq Symmetric scalar constraint operators Ĉ(N) for all the laps functions L, Sahlmann

3 The Ashtekar-Barbero variables Other fields and the constraints The holonomy-flux variables The kinematical Hilbert space The spin-network states Σ - an underlying 3d manifold, x a = x 1, x 2, x 3 local coordinates The variables: su(2) - the Lie algebra of SU(2), τ i = τ 1, τ 2, τ 3 su(2) ( A i a, Ei a ) a, b = 1, 2, 3, i = 1, 2, 3 - the field variables {A i a(x), Ei b (y)} = δj iδb a δ(x, y) {A, A} = 0 = {E, E} The relation with intrinsic/extrinsic geometry e i a / K ab of Σ Ashtekar, Barbero, Immirzi 2Tr(τ i τ j ) = δ ij, A i a = Γ i a + γk i a, E a i = 1 16πGγ ej b ek c ɛ abc ɛ ijk.

4 The Ashtekar-Barbero variables Other fields and the constraints The holonomy-flux variables The kinematical Hilbert space The spin-network states Other fields {φ α (x), π β (y)} = δ β αδ(x, y) {φ, φ} = 0 = {π, π} The constraints (I - class) Σ d 3 xn(x)c(x) =: C(N) = C gr (N) + C matt (N) Σ d 3 xn a (x)c a (x) =: C( N) = C gr ( N) + C matt ( N) Σ d 3 xλ i G i (x) =: G(Λ) = G gr (Λ) + G ferm (Λ) Other constraints ( G YM (Λ YM ) ) The free functions: N C(Σ) - a laps function N Γ(T (Σ)) - a shift vector field Λ C(Σ, su(2)).

5 The Ashtekar-Barbero variables Other fields and the constraints The holonomy-flux variables The kinematical Hilbert space The spin-network states Rovelli, Smolin 1988

6 The Ashtekar-Barbero variables Other fields and the constraints The holonomy-flux variables The kinematical Hilbert space The spin-network states Ashtekar, L 1992 e : [t 0, t 1 ] Σ h e (A) := Pexp A Ψ(A) = ψ(h e1 (A),..., h en (A)), ψ Poly(SU(2) n ) (1) e Cyl := {Ψ C(A) : (1), {e 1,..., e n } embedded graph in Σ} dµ(a)ψ(a) = dg 1...dg n ψ(g 1,...g n ) H gr = L 2 (Ω (1) (Σ) su(2), µ)

7 The Ashtekar-Barbero variables Other fields and the constraints The holonomy-flux variables The kinematical Hilbert space The spin-network states U g Ψ(A) = Ψ(g 1 Ag + g 1 dg) (2) The invariant elements in H gr Penrose 1970(?), Rovelli, Smolin, Baez 1993: Γ e I j I, Γ v ι v InvV j1... V jk Γ, j, ι > - a spin-network state

8 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum [ˆq, ˆp] = i (3) ˆp p > = p p >, e i p ˆq p > = p + p > (4) ˆq q > = q q >, e i q ˆp q > = q + q > (5) q >: p e i qp, p >: q e i qp (6) < q p > = e i qp

9 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum {φ(x), π(y)} = δ(x, y), [ ˆφ(x), ˆπ(y)] = i δ(x, y) (7) The polymer representation: d 3 x ˆπ(x)ϕ(x) p > = x Σ p x ϕ(x) p > (8) e i p : Σ R, supp p < (9) x Σ p x ˆφ(x) p > = p + p > (10) p > : φ e i x Σ px φ(x) (11) Define < ϕ to be: < ϕ p > := e i x Σ px ϕ(x) (12)

10 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum < ϕ p > := e i x Σ px ϕ(x) (13) < ϕ e i < ϕ e i x Σ px ˆφ(x) d 3 xϕ (x)ˆπ(x) = < ϕ e i x Σ px ϕ(x) (14) = < ϕ + ϕ (15) < ϕ ˆφ(x) = < ϕ ϕ(x). (16)

11 The dual Polymer Representation The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum < ϕ ϕ > = δ ϕ,ϕ = 0 or 1 (17) H φ = Span( ϕ > : ϕ C(Σ)) (18) ˆφ ϕ > = ϕ(x) ϕ >, e i d 3 xϕ (x)ˆπ(x) ϕ > = ϕ + ϕ > (19) Notice that ˆφ,a... (x) ϕ > = ϕ,a... (x) ϕ >. (20)

12 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum L,Sahlmann 2014 (coming) where H φ H gr ϕ > Γ, j, ι > d 3 xn(x) ˆφ,a ˆφ,b Êi a Êi b ϕ > Γ, j, ι > = ( ) = (8πGγ) 2 ji (j I + 1) N dϕ ϕ > Γ, j, ι > I e I e N dϕ = dtn(e(t)) dϕ(t) dt

13 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum Similar results in the LQG literature: the Ma-Ling operator Ma,Ling 2000 defined in H gr ˆQ(ω) Γ, j, ι > = d 3 x ˆω a ˆω b Êi a Êi b Γ, j, ι > = (21) ( ) (8πGγ) 2 ji (j I + 1) ω Γ, j, ι > I e I (22) The Schroedinger equation of the Rovelli-Vidotto QM on graphs Rovelli, Vidotto 2010

14 Toy equations Plan The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum A toy model of the quantum scalar constraint ( ) d 3 x π(x) + a Êi a(x)ê i b (x) φ,a (x) φ,b (x) Ψ = 0. (23) A general (modulo linearity) solution is: Ψ = e ia d 3 x φ Ê a i Êi b φ,a φ,b ϕ ψ 0 (ϕ) < ϕ < Γ, j, ι where ϕ ψ 0 (ϕ), is an arbitrary function which satisfies the following condition d dɛ ψ 0(ϕ + ɛ) = 0. (24)

15 The Quantum Mechanics analogy The scalar field The new operator Application 2: the BF vacuum [ a (x), Ê b (x)] = i δ b a δ(x, y) (25) The LQG momentum representation: d 3 xê a a a e > = a e > e i  e e > = e e > (26) e The dual position representation spanned by the dual states < a : < a e > := e i e a. (27) < a a > = δ a,a In this representation < a Âa(x) = < a a a (x). (28) In particular < a = 0 = BF vacuum proposed by Dittrich, Geiller 2014

16 The issue Plan The issue of Ĉ gr (N) The new Hilbert space Operators in H new Ĉ gr (N) not defined in the kinematical H gr Ĉ(N) : H Diff suitable dual space, N breaks Diffs In principle we can write Ĉ gr (N)Ψ = 0 however: solutions non-normalizable to large space, which Ψ to select? (Ψ Ψ ) phys =? For GR coupled to a scalar field we need the Rovelli-Smolin d 3 x 2 detê(x)ĉgr(x) how to define the Ĉgr (x)? Σ

17 Obstacles Plan The issue of Ĉ gr (N) The new Hilbert space Operators in H new Perhaps, we can extend H Diff H to accomodate Ĉ(N)??? The obstacle: Ψ Diff invariant (Ĉ(N)Ψ Ĉ(N)Ψ) = 0 for every N C 0 (Σ). L, Marolf 1999

18 Partial ways out Plan The issue of Ĉ gr (N) The new Hilbert space Operators in H new Take N = 1, Ĉ(1) : H Diff H Diff Ĉ gr,sym := 1 ) (Ĉgr (1) + Ĉ 2 gr(1). But N = 1 is not enough... Try to define directly either d 3 x det Ê(x)Ĉ(x) or d 3 Ĉ 2 (x) x det Ê(x) the Master constraint program (Thiemann)

19 The issue of Ĉ gr (N) The new Hilbert space Operators in H new L, Sahlmann 2014 H new = {x 1,...,x k } Σ H {x1,...,x k } H {x1,...,x k } is spanned by all the Γ, j, ι > based at {x 1,..., x k }, averaged with respect to Diff(Σ, {x 1,..., x k }).

20 The issue of Ĉ gr (N) The new Hilbert space Operators in H new We define operators in H new by passing by the duality from the kinematical H gr operators of suitable symmetries, or we derive Ĉ gr (N) from scratch ( Thiemann s regularization Thiemann 1997 works in H new very well, for Ricci see Assanioussi, Alesci, L 2013): O(x) = dete(x), dete(x)ric(x), Ĉ gr (x). We obtain operators of the following form d 3 xn(x)ô(x) = x Σ N(x)Ô x Ô x : H {x1,...,x k } H {x1,...,x k } Ô x : H {x1,...,x k } 0 unless x = x 1,..., x k

21 The quantum scalar contraint The issue of Ĉ gr (N) The new Hilbert space Operators in H new With this result we can: Consider Ĉ (N) (it has sufficiently large domain). Define Ĉ(x) sym = 1 2 (Ĉ + Ĉ ) Find a self adjoint extension Ĉs.a. Spectrally expand each H {x1,...,x k } = dc 1...dc k H c 1...c k {x 1,...,x k } promote H constraint {x 1,...,x k } to be solutions to the quantum scalar

22 The issue of Ĉ gr (N) The new Hilbert space Operators in H new Solutions to the matter free LQG The elements of H {x1,...,x k } have to be farther averaged with respect to the vertex non-preserving Diff(Σ) (the vector constraint), H {x1,...,x k } η new (Ψ) 1 η new (U f Ψ) k! [f ] Diff(Σ)/Diff(Σ) x1,...,x k The map passes to the H Hilbert space {x 1,...,x k } H (k) space, and it s image defines a of k-vertex solutions to the scalar AND vector constraint. The full Hilbert space is H phys = k H (k)

23 How to use the new gadgets? We apply it to (set 8πGγ = 1) ˆπ(x) = ± Where, given operators d 3 x Â(x), ˆφ,a ˆφ,b Êi a Êi b 2V ( φ(x)) detê ˆ detê Ĉgr 2 Σ Σ d 3 x ˆB(x),... Using sufficiently fine partitions Σ = k Σ k we define d 3 x Â(x) + ˆB(x) +... ϕ > Γ, j, ι >:= Σ ( 2 ( 2 d 3 x Â(x)) + d 3 x ˆB(x)) +... ϕ > Γ, j, ι > Σ k Σ k k provided the RHS is independent of the refinements.

24 Thank You