Quantum Reduced Loop Gravity: matter fields coupling

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, 2016 1 / 19

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, 2016 2 / 19

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, 2016 3 / 19

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, 2016 4 / 19

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, 2016 5 / 19

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, 2016 6 / 19

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, 2016 7 / 19

Scalar field: canonical variables and the Hilbert space Kinematical Hilbert space: H (φ kin = { a 1U ψ1 +... + 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, 2016 8 / 19

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 = 1 + 1 2 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, 2016 9 / 19

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 2 + 1 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, 2016 10 / 19

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, 2016 11 / 19

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, 2016 12 / 19

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, 2016 13 / 19

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 ( 1 1 + + 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, 2016 14 / 19

Hamiltonian constraint for quantum-gravitational scalar electrodynamics i Ĥ v,sc = 2 3 κγ 2( 8πγlP 2 + 2 5 Q 2 V v + 25 Q 2 Vv 3 ( i=1 3 ( i=1 + 2 17 λ V 3 v [( + 215 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 2 + 2 ( 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, 2016 15 / 19

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, 1642005 (2016. Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, 2016 16 / 19

Lattice (constructional and quantum corrections Lattice corrections: - gravitational, e.g. h p ( (v = 1 + εap(v + O(ε 2, h q r ( (v = 1 + 1 2 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, 2016 17 / 19

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, 2016 18 / 19

References 1 J. Bilski, E. Alesci and F. Cianfrani, Quantum reduced loop gravity: extension to scalar field, Phys. Rev. D 92, no. 12, 124029 (2015 doi:10.1103/physrevd.92.124029 [arxiv:1506.08579 [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:1612.00324 [gr-qc]]. 3 E. Alesci, F. Cianfrani, Loop Quantum Cosmology from Loop Quantum Gravity, Phys. Rev. D 92, no. 12, 124029 (2015 doi:10.1103/physrevd.92.124029 [arxiv:1410.4788 [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, 1642005 (2016 doi:10.1142/s0218271816420050 [arxiv:1602.05475 [gr-qc]]. 5 T. Thiemann, Quantum spin dynamics (QSD, Class. Quant. Grav. 15, 839 (1998 [gr-qc/9606089]. 6 T. Thiemann, QSD 5: Quantum gravity as the natural regulator of matter quantum field theories, Class. Quant. Grav. 15, 1281 (1998 [gr-qc/9705019]. 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, 2016 19 / 19