Quantum Reduced Loop Gravity: matter fields coupling

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1 Quantum Reduced Loop Gravity: matter fields coupling Jakub Bilski Department of Physics Fudan University December 5 th, 2016 Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

2 Outline 1 Canonical approach to Loop Quantum Gravity Background-independent quantum field theory Gravitational field Scalar field Gauge field 2 Regularization, quantization and reduction Thiemann s method Alesci-Cianfrani s method Illustration of a basic cell state 3 Results Hamiltonian constraint for quantum-gravitational scalar electrodynamics Lattice (constructional and quantum corrections 4 Conclusions and open problems 5 References Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

3 Background-independent quantum field theory Quantum gravity as the natural regulator of matter quantum field theories ˆF 2 1 (x lim d 3 y χ ε(x y ε 0 ε ˆF (x ˆF (y 3 Matter fields in general relativity S (gr + S (φ + S (A +... = 1 κ + 1 2λ d 4 x gr+ M d 4 x g ( g µν ( µφ( νφ V (φ + M 1 d 4Q M 4 x gg µν g ρσ F I 2 µρ F I νσ +... Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

4 Gravitational field: Wheeler-DeWitt equation The ADM decomposition results in the Hamiltonian, which is a constraint: ( Ĥ sc (gr 1 ψ(q = 2 q Ĝabcdˆπ abˆπ cd q (3ˆR ψ(q = 0 where G abcd = q acq bd + q ad q bc q ab q cd. After the quantization, canonical operators read: ˆq ab ψ(q = qab ψ(q, ˆπ ab ψ(q = i κ 2 δ ψ(q, δq ab [ˆqab (t, x, ˆπ cd (t, y ] = i κ 2 δc aδ d b δ (3 ( x y. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

5 Gravitational field: Ashtekar variables Introducing the Ashtekar variables: Ei a = q ei a, A i a = Γ i a + γk i a, { A i a (t, x, Ej b (t, y } = γ κ 2 δb aδjδ i (3 ( x y, one obtains the total Hamiltonian: H (gr = 1 d 3 x N ( A i tg (gr i κ + N a V (gr a + NH sc (gr, where tha scalar constraint density is given by the formula: H sc (gr = 1 d 3 x 1 ( F i ab (γ 2 + 1ɛ ilm K l ak m b ɛ ijk Ej a κ q E b k The Schrödinger representation reads: Â i a ψ = A i a ψ, Ê a i κ δ ψ = i γ 2 δa i a ψ, [ A i a (t, x, E b j (t, y ] = i γ κ 2 δb aδ i jδ (3 ( x y. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

6 Gravitational field: canonical variables and the Hilbert space Holonomies of Ashtekar-Barbero connections and by fluxes of densitized triads: ( h γ = P exp A j a(γ(sτ j γ a (s, E j(s = ɛ pqr dl q dl r E p j (v γ The kinematical Hilbert space is defined as: := H (gr kin Γ H (gr Γ = L 2 ( A, dµal, S l p (v while the states are cylindrical functions of all links l i Γ and they are defined as Ψ Γ,f (A := A Γ, f := f ( h l 1(A, h l 2(A,..., h l L(A for f : SU(2 L C. The basis states, called spin network states and are given by the expression: Ψ Γ,jl,i v (h = h {Γ, j l, i v} = i v D j l (h l v Γ l and the action of the canonical operators reads: ĥ γd j l (h l = h γd j l (h l, Ê id j l (h l = γ κ 2 σ(s, γdj l (h l1 τ id j l (h l2. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

7 Scalar field: classical Hamiltonian and discretization Hamiltonian of the scalar field: [ H (φ = Single-point states: Σ t d 3 x N a π aφ + N ( ] λ q q 2 q π2 + 2λ qab aφ b φ + 2λ V (φ v; U ψ := e iψvφv Action of diffeomorphism: w; U ψ v; U ψ := δ w,vδ ψ,ψ ϕ v; U ψ = ϕ(v; U ψ Canonical Poisson brackets: { } φ(x, Π(y = χε(x, y, Π(v := d 3 uχ ε(v, uπ(u Action of basic operators: e iψw ˆφ w v; U ψ = e iψwφw v; U ψ = v w; U ψ ˆΠ(v v; U ψ = i φ(v v; U ψ = ψ v v; U ψ Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

8 Scalar field: canonical variables and the Hilbert space Kinematical Hilbert space: H (φ kin = { a 1U ψ a nu ψn : a i C, n N, ψ i R } U ψ := e i v Σ ψvφv := U ψ U ψ U ψ := δ ψ,ψ H (φ kin := L2 ( RBohr Σ is obtained from the single-point one L 2 ( RBohr, where the Bohr measure is defined as follows: dµ Bohr (φe iψvφv = δ 0,v R Bohr Basic variables: Û ψ U ψ = U ψ+ψ, ˆΠ(V Uψ = ψ v U ψ v V Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

9 Gauge field: canonical variables and the total Hilbert space Hamiltonian of the gauge field: ( H (A = d 3 x A I t D a Ea I + N a F I ab Eb I + N Q2 Σ t 2 q q ( ab E a I Eb I + Ba I Bb I Natural lattice representation: fluxes and holonomies ( E I (S p ε 2 E a I (v ( δp a, ɛ pqr tr τ I h q r (v Q 2 ε 2 B a I (v V(v, ε δp a, where the expansion h q r = ε2 F qr + O(ε 4 has been applied. The phase space variables: ( h γ = P exp Aa( I γ(s τ I γ a (s γ, E I (S p = ɛ pqr dl q dl r E p I (v S l p (v Total kinematical Hilbert space: H (tot kin = H(gr kin H(φ kin H(A kin Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

10 Thiemann s method: lattice-regularization Thiemann s trick: ( q Ea i n 2 δv n = = 4 { A i n δei a a, V n}, Ka i = δk nγκ δei a = 2 { A i a, K } γκ Example: gravitational part of the Hamiltonian constraint H (gr sc = 1 κ lim ε 0 25 (γ γ 3 κ 3 ( 1 d 3 x Nɛ abc 2 3 ( ε 3 γκ tr { h a b h 1 l V, c hl c} + ( { tr h 1 l K, a hl a } { } { h 1 l K, b hl b h 1 l V, c hl c}, where F ab = F j ab τj, Aa = Ai aτ i, τ j = i 2 σj and K = d 3 xk i ae a i. Canonical quantization: {, } 1 [, ], canonical variables operators i Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

11 Alesci-Cianfrani s method: reduction Canonical variables in the diagonal gauge: E a i = p i δ a i, A i a = c iδ i a Cuboidal lattice: Γ Γ R Reduced group elements: D j m n(h l D j m ±j l (u l l D j l m l m l (h l (D 1 j ±j l n (u l, h l SU(2, where the projected Wigner matrices (SU(2 U(1 read: l D j l m l m l (h l = (D 1 j ±j l m (u l D j m n (h l D j n ±j l (u l = m l, u l D j l (h l ml, u l. The basis element of the Hilbert space after gauge-fixing: j l h l j l j l, m m l, u l m l, u l D j l (h l ml, u l ml, u l jl, m, m l = ±j l. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

12 Alesci-Cianfrani s method: Quantum Reduced Loop Gravity The kinematical Hilbert space: R H (gr kin := Γ R H (gr Γ. Reduced states are given by the formula: R Ψ Γ,ml,i v (h = h {Γ, m l, i v} = j l, i v m l, u l l D j l m l m l (h l, m l = ±j l, v Γ where j l, i v m l, u l are reduced (one-dimensional intertwiners. l The scalar product between reduced intertwiners is given by: Γ, ml, i v Γ, m l, i v = δγ,γ δ ml,m m l, u l j l, i l v j l, i v ml, u l. v Γ l Γ Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

13 Illustration of a basic cell state: a spider state Γ; j l, i v; n l, i v ; U π R = Γ; j l, i v R Γ; n l, i v R Γ; U π R = e iψ x,y,z+1φ x,y,z+1 j (3 x,y,z n (3 x,y,z x direction h (3 x,y,z h (3 x,y,z x direction j (3 x,y,z n (3 x,y,z j (2 j (2 x,y 1,z x,y 1,z h (2 j x,y,z (2 x,y-1,z h (2 j x,y,z (2 x,y,z = e iψ x,y,z+1φ x,y,z+1 e iψ x,y,z+1φ x,y,z+1 e iψ x,y,z+1φ x,y,z+1 n (2 h (2 n (2 x,y 1,z x,y-1,z x,y 1,z n (2 x,y,z h (2 x,y,z n (2 x,y,z j (3 x,y,z 1 n (3 x,y,z 1 Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

14 Hamiltonian constraint for quantum-gravitational scalar electrodynamics Scalar constraint operator: Ĥ sc Γ; j l, i v; n l, i v ; U π R = (Ĥ(gr sc +Ĥ(A E +Ĥ(A B +Ĥ(φ kin +Ĥ(φ der +Ĥ(φ pot Γ; j l, i v; n l, i v ; U π R Action in a form containing gravitational eigenvalues and matter operators: Ĥ sc Γ; j l, i v; n l, i v ; U π R = N v Ĥ v,sc Γ; j l, i v; n l, i v ; U π R Volume operator v ( (8πγl n ˆV n (v x,y,z Γ; j l, i v R = Vv n 2 3 x,y,z Γ; j l, i v R = P Σ (1 v x,y,z Σ (2 v x,y,z Σ (3 2 v x,y,z Γ; j l, i v R, ( where Σ (i v := 1 (i 2 j v + j (i v e i denotes the mean value of the spin along a direction i. [( J (i,n v = 1 1 8γπlP 2 2 ( j (i v n ( j (i v e i 2 ( j (i v n ] 1 + j (i v e i j s denote the quantum number around which semiclassical states are peaked Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

15 Hamiltonian constraint for quantum-gravitational scalar electrodynamics i Ĥ v,sc = 2 3 κγ 2( 8πγlP Q 2 V v + 25 Q 2 Vv 3 ( i=1 3 ( i= λ V 3 v [( V 3 v 3 4 λ 3 i=1 {l i l j l k } J (i, 1 4 v J (i, 1 4 v ɛ ijk tr ( ĥ l i l j ĥ 1 l k ˆV ĥl k + 2 (ÊI ( S i (v ( J (x, 1 4 v J (y, 4 1 v J (z, 1 4 v J (y, 3 8 v J (z, 8 3 v {l j,l k } i {l l,l m } i 2 ˆΠ2 v + + Vv 2λ ˆV ( φ v + + scalar-electrodynamical interactions ɛ ijk ɛ ilm ĥ l j l k(v ĥl l l m(v + e i( ˆφ v ˆφ v ex e i( ˆφ v+ ex ˆφ 2 v + 2i ( x y y z + z x ( x z ] y x + z y Classical-continuum (large-j limit precisely coincides with the classical expression! Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

16 Hamiltonian constraint for quantum-gravitational scalar electrodynamics Picture of the QRLG action from: E. Alesci and F. Cianfrani, Int. J. Mod. Phys. D 25, no. 08, (2016. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

17 Lattice (constructional and quantum corrections Lattice corrections: - gravitational, e.g. h p ( (v = 1 + εap(v + O(ε 2, h q r ( (v = ε2 F qr(v + O(ε 4, -matter field, e.g. pφ(v 1 e i(φ v+ ep φv e i(φv φ v ep, ε 2i Large-j expansion: J (i,n v = 8γπlP 2 n ( (i j v + j (i v e i [ 1 + (n 2(n 1 24 ( j v (i + j (i 2 + O v e i ( ( j (i v 1 + j (i v e i 4 ] Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

18 Conclusions and open problems Conclusions 1 QRLG allows to construct diffeomorphism invariant matter field theories 2 matrix elements of matter field Hamiltonian constraint are analytic 3 large-j limit approach the classical Hamiltonian at the leading order and gives convergent series of next-to-the-leading-order quantum corrections Open problems 1 construction of semiclassical states for matter fields 2 ambiguity in construction of discrete representations for matter fields 3 different powers of gravitational corrections (representation dependent 4 derivation of gravitational action in terms of the original state space (recoupling of intertwiners 5 studies of phenomenological applications (in cosmology 6 fermion field coupling Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

19 References 1 J. Bilski, E. Alesci and F. Cianfrani, Quantum reduced loop gravity: extension to scalar field, Phys. Rev. D 92, no. 12, (2015 doi: /physrevd [arxiv: [gr-qc]]. 2 J. Bilski, E. Alesci and F. Cianfrani, P. Doná, A. Marcianò, Quantum reduced loop gravity: extension to a gauge vector field, (2016 [arxiv: [gr-qc]]. 3 E. Alesci, F. Cianfrani, Loop Quantum Cosmology from Loop Quantum Gravity, Phys. Rev. D 92, no. 12, (2015 doi: /physrevd [arxiv: [gr-qc]]. 4 E. Alesci and F. Cianfrani, Quantum Reduced Loop Gravity and the foundation of Loop Quantum Cosmology, Int. J. Mod. Phys. D 25, no. 08, (2016 doi: /s [arxiv: [gr-qc]]. 5 T. Thiemann, Quantum spin dynamics (QSD, Class. Quant. Grav. 15, 839 (1998 [gr-qc/ ]. 6 T. Thiemann, QSD 5: Quantum gravity as the natural regulator of matter quantum field theories, Class. Quant. Grav. 15, 1281 (1998 [gr-qc/ ]. 7 T. Thiemann, Modern canonical quantum general relativity, Cambridge, UK: Cambridge Univ. Pr. (2007. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, / 19

Symmetry reductions in loop quantum gravity. based on classical gauge fixings

Symmetry reductions in loop quantum gravity based on classical gauge fixings Norbert Bodendorfer University of Warsaw based on work in collaboration with J. Lewandowski, J. Świeżewski, and A. Zipfel International