On deparametrized models in LQG

On deparametrized models in LQG Mehdi Assanioussi Faculty of Physics, University of Warsaw ILQGS, November 2015

Plan of the talk 1 Motivations 2 Classical models General setup Examples 3 LQG quantum models Physical Hilbert space Implementation of the physical Hamiltonian 4 Summary 5 What can we do next? 6 Conclusions

Motivations General covariance Dynamics encoded in constraints General relativity impossible to isolate the true dynamical degrees of freedom Quantization of GR: conceptual problems in interpreting quantum theory technical obstacles in solving the constraints Deparametrization: introduces a privileged coordinates system all or part of the constraints of the covariant theory can be solved Imposing the resolved scalar constraint, as an operator restriction on the quantum states, yields a functional Schrödinger equation offers a clean framework and satisfactory approach to tackle the problem of time in quantum gravity provides a complete setup to study generic phenomenology of quantum models of gravity coupled to matter fields 3 / 29

General setup Recipe 5 / 29

General setup Recipe 1 st class & 2 nd class constraints 5 / 29

General setup Recipe 1 st class & 2 nd class constraints solution(s) (of 2 nd cl constr) 5 / 29

General setup Recipe 1st class & 2nd class constraints solution(s) (of 2nd cl constr) Deparametrized model 5 / 29

General setup [K Giesel, T Thiemann, ClassQuantGrav 32 (2015) 135015] S = d 4 x L G + L R + L M L R = 1 2 [ g g µν ( ρ( µt )( ν T ) + A(ρ)(ω j µs j )(ω k ν S k ) + 2B(ρ)( µt )(ω l ν S l ) ] ) + Λ(ρ) (T, P ), (S j, P j ) Reference fields; (ρ, π), (ω j, π j ) Lagrange multipliers; A(ρ), B(ρ), Λ(ρ) Arbitrary functions of the field ρ; 6 / 29

General setup Hamiltonian analysis: (N, Π), (N a, Π a ) Primary constraints Secondary constraints Z = Π Z a = Π a z = π z j = π j ζ j = ζ j (P, P j, W j ) H tot, stability of pc C = C gr + C R + C M C a = C gr a + CR a + CM a s = H tot ρ K = H tot π 3 (K appears if the ζ j 's are not independent) (ζ j, j = 1, 2, 3 or 1, 2 depending on A and B) Solving the second class constraints plus manipulating the first class constraints, C and C a, lead to C = P h(t, C gr, C M, C gr a, CM a ), C a = P T,a + P j S j,a + Cgr a + CM a C is deparametrized 7 / 29

General setup In case h = h(c gr, C M, C gr a, CM a ) we obtain {h(x), h(y)} = 0 Imposing the gauge T = t, defines the physical Hamiltonian of the theory as H = d 3 xh(x) Σ Observables: C a = P T,a + P j S j,a + Cgr a + CM a pull back all tensors and spinors by the diffeomorphism x S j (x) (T, P, q ab, P ab, ) (T, P, q ab, P ab, ) C a = P T,a + C gr a + CM a spatial diff inv quantities are yet to be constructed Any F (T, P, q ab, P ab, ) is spatial diff inv do dt (T ) = {H, O} + O T (T ) 8 / 29

Example 1: Massless KG scalar field model A = B = Λ = 0, ρ = 1, 1 S = d 4 x g R 1 g g µν ( µ T )( ν T ) 2 The constraints are C a = C gr a + P T,a C = C gr 1 + 2 det(q) ) (P 2 + E a l Eb l T,aT,b The Hamiltonian constraint is solved for P (x) using the diff constraint P = ± det(q)c gr ± det(q) Cgr 2 Ea l Eb l Cgr a C gr b (±, ±) select different regions of the phase space 1 [L Smolin 89'], [C Rovelli, L Smolin 93'], [M Domagala, K Giesel, W Kaminski, J Lewandowski 10'] 9 / 29

Example 1: Massless KG scalar field model A = B = Λ = 0, ρ = 1, 1 S = d 4 x g R 1 g g µν ( µ T )( ν T ) 2 The constraints become C a = C gr a + P T,a = P h C The constraint C is equivalent to C in the region (+, +) with h := det(q)c gr + det(q) Cgr 2 Ea l Eb l Cgr a C gr b (+, +) includes the special case of a homogeneous and isotropic geometry coupled to a scalar field 1 [L Smolin 89'], [C Rovelli, L Smolin 93'], [M Domagala, K Giesel, W Kaminski, J Lewandowski 10'] 9 / 29

Example 1: Massless KG scalar field model A = B = Λ = 0, ρ = 1, 1 S = d 4 x g R 1 g g µν ( µ T )( ν T ) 2 The constraints become C a = C gr a + P T,a = P h C Imposing the gauge T = t gives C a = C gr a h = 2 det(q)c gr 1 [L Smolin 89'], [C Rovelli, L Smolin 93'], [M Domagala, K Giesel, W Kaminski, J Lewandowski 10'] 9 / 29

Example 2: Rotational vs irrotational dust models a Rotational dust (Brown-Kuchar timelike dust) A = B = Λ = ρ, 2 S = d 4 x g R 1 [ g ρ g µν ( ( µt )( ν T ) + (ω j µs j )(ω k ν S k ) + 2( µt )(ω l ν S l ) ] ) + 1 2 2 [JD Brown, KV Kuchar 95'] 10 / 29

Example 2: Rotational vs irrotational dust models a Rotational dust (Brown-Kuchar timelike dust) A = B = Λ = ρ, 2 S = d 4 x g R 1 [ g ρ g µν ( ( µt )( ν T ) + (ω j µs j )(ω k ν S k ) + 2( µt )(ω l ν S l ) ] ) + 1 2 The constraints are with C D a := P T,a + P j Sj,a Ca = C gr a + CD a C = C gr P 2 + 2ρ + ρ det(q) (q ab det(q) 2P 2 Ca D CD b + P 2) = P 2 det(q) ρ 2 + (q ab det(q) P 2 C a D CD b + P 2) + other second class constraints We also have sgn(p ) = sgn(ρ) = sgn(c gr ) and P j = P ω j Introducing the inverse of S,a j and rewriting C and Ca we get C j ( = P j + Sa j C a gr ) + h T,a C = P sgn(p ) C gr 2 q ab Ca gr Cgr b =: P h 2 [JD Brown, KV Kuchar 95'] 10 / 29

Example 2: Rotational vs irrotational dust models a Rotational dust (Brown-Kuchar timelike dust) A = B = Λ = ρ, 2 S = d 4 x g R 1 [ g ρ g µν ( ( µt )( ν T ) + (ω j µs j )(ω k ν S k ) + 2( µt )(ω l ν S l ) ] ) + 1 2 The constraints are with C D a := P T,a + P j Sj,a Ca = C gr a + CD a C = C gr P 2 + 2ρ + ρ det(q) (q ab det(q) 2P 2 Ca D CD b + P 2) = P 2 det(q) ρ 2 + (q ab det(q) P 2 C a D CD b + P 2) + other second class constraints We also have sgn(p ) = sgn(ρ) = sgn(c gr ) and P j = P ω j Introducing the inverse of S,a j and rewriting C and Ca we get C j ( = P j + δa j C a gr ) + h T,a imposing the gauge T = t, S j = x j C = P sgn(p ) C gr 2 q ab Ca gr Cgr b =: P h 2 [JD Brown, KV Kuchar 95'] 10 / 29

Example 2: Rotational vs irrotational dust models b Irrotational dust Λ = ρ, A = B = 0, 3 S = d 4 x g R 1 g ρg µν ( ( µt )( ν T ) + 1 ] 2 The constraints are Ca = C gr a + P T,a C = C gr P 2 + 2ρ + det(q) ρ det(q) 2 = P 2 ρ 2 det(q) ( q ab ) T,aT,b + 1 ( + det(q) q ab ) T,aT,b + 1 We also have sgn(p ) = sgn(ρ) = sgn(c gr ) Solving and using Ca we get Ca = C gr a + P T,a C = C gr + P 3 [V Husain, T Pawlowski 11'], [K Giesel, T Thiemann 12'], [J Świeżewski 13'] 11 / 29

Example 2: Rotational vs irrotational dust models b Irrotational dust Λ = ρ, A = B = 0, 3 S = d 4 x g R 1 g ρg µν ( ( µt )( ν T ) + 1 ] 2 The constraints are Ca = C gr a + P T,a C = C gr P 2 + 2ρ + det(q) ρ det(q) 2 = P 2 ρ 2 det(q) ( q ab ) T,aT,b + 1 ( + det(q) q ab ) T,aT,b + 1 We also have sgn(p ) = sgn(ρ) = sgn(c gr ) Solving and using Ca we get Ca = Ca gr + P T,a C = C gr + P imposing the gauge T = t 3 [V Husain, T Pawlowski 11'], [K Giesel, T Thiemann 12'], [J Świeżewski 13'] 11 / 29

Example 3: Gaussian dust model A = 0, B = 1, Λ = ρ, 4 S = d 4 x g R 1 [ g g µν ( ρ( µt )( ν T ) + 2( µt )(ω l ν S l ) ] ) + ρ 2 The constraints are Ca = Ca gr + P T,a + P j Sj,a C = C gr + ϵ P q ab q ab T,a P j S j,b T,aT,b + 1 + q ab T,aT,b + 1 + second class constraints with ϵ = sgn(ṫ Na T,a) Using Ca in C and vice-versa we get C j ( = P j + Sa j Ca gr ) + h T,a C = P + ϵ C q ab T,aT,b + 1 q ab T,aC gr b =: P h 4 [KV Kuchar, CG Torre 91'] 12 / 29

Example 3: Gaussian dust model A = 0, B = 1, Λ = ρ, 4 S = d 4 x g R 1 [ g g µν ( ρ( µt )( ν T ) + 2( µt )(ω l ν S l ) ] ) + ρ 2 The constraints are Ca = Ca gr + P T,a + P j Sj,a C = C gr + ϵ P q ab q ab T,a P j S j,b T,aT,b + 1 + q ab T,aT,b + 1 + second class constraints with ϵ = sgn(ṫ Na T,a) Imposing the gauge conditions T = t, S j = x j gives C j = P j + δa j Cgr a C = P + ϵ C 4 [KV Kuchar, CG Torre 91'] 12 / 29

Spacetime reference models All constraints are solved on the classical level (except of the Gauss constraints) reduced phase space formulation for the model with a physical Hamiltonian Quantum theory kinematical Hilbert space (up to Gauss gauge invariance) = physical Hilbert space In canonical loop quantization context, there are two possible options: Standard LQG gauge invariant Hilbert space H LQG G ; Algebraic quantum gravity H AQG G ; H LQG G H AQG G Infinite number of embedded graphs Non-separable Subgraphs of one infinite algebraic graph Non-separable Topology and differential structure provided Topology and differential structure absent 14 / 29

Time reference models Unsolved constraints: Spatial diffeomorphism constraint and Gauss constraint Those constraints must be implemented and solved in the quantum theory in order to obtain the physical Hilbert space In canonical loop quantization context: Standard LQG gauge invariant Hilbert space H LQG G ; Algebraic quantum gravity H AQG G ; CAUTION: In both cases, the kinematical Hilbert space must be reduced with respect to the diffeomorphism constraint! 15 / 29

Regularization of the gravitational Hamiltonian Option 1: 5 Graph changing VS Option 3: 7 Graph preserving Option 2: 6 Possibilities of implementing the Hamiltonian Spacetime reference models Time reference models Vacuum (No reference fields) LQG X unknown option 1 option 1 & 2 LQG* option 3 all options all options AQG option 2 & 3 X unknown option 2 & 3 5 [EAlesci, MA, JLewandowski, IMäkinen 15'] 6 [T Thiemann 96'] 7 [T Thiemann 03'] 17 / 29

Regularization of the gravitational Hamiltonian Option 1: 5 Graph changing VS Option 3: 7 Graph preserving Option 2: 6 Possibilities of implementing the Hamiltonian Massless KG scalar field B-K dust Irrotational dust Gaussian dust LQG op 1 X op 1 X LQG* op 3 (Master program) op 3 AQG X op 2 & 3 (Master program) X op 2 & 3 5 [EAlesci, MA, JLewandowski, IMäkinen 15'] 6 [T Thiemann 96'] 7 [T Thiemann 03'] 18 / 29

Concrete quantization of the gravitational Hamiltonian C gr = 1 2k Σ = 1 2kβ 2 d 3 x Σ ( Ea i Eb j ϵ ijk Fab k + 2 ( s β 2) ) K[a i Kj det E b] ( d 3 x s ϵ ijkei aeb j F ab k + ( s β 2) ) det E R det E Euclidean part Lorentzian part (version 1) Lorentzian part (version 2) C E = Σ d 3 ϵ ijk E i a Eb j F ab k x det E First Thiemann's trick ϵ ijk E i a Eb j ϵ abc {A k c, V } det E C L = Σ d 3 E i a Eb j Ki [a Kj b] x det E Second Thiemann's trick K i a {Ai a, {CE, V }} C L = Σ d 3 x det E(x) R(x) Regge's approximation & external regularization The curvature operator 8 Graph preserving F ab hα ab The loop α ab is assigned following one of the three prescriptions 8 [EAlesci, MA, JLewandowski 14'] 19 / 29

Concrete quantization of the gravitational Hamiltonian The straightforward quantization of the classical expressions lead to non-symmetric operators Therefore it is necessary to construct symmetric extensions without altering the semi-classical limit of the operator Simple idea: use the adjoint operator of Ĉ to make a symmetric operator Ĉ Sym := Sym(ĈE, ĈE, ĈL, ĈL ) Typical examples: 1 2 (Ĉ + Ĉ ) ĈĈ Ĉ Ĉ Ĉ xĉx x Σ Self-adjoint extensions? admit self-adjoint extensions 20 / 29

Concrete quantization of the gravitational Hamiltonian Special case: h = h(c, Ca gr ), and Ca gr = s kβ Σ d 3 x E b i F i ab 0 Terms such as q ab C gr a C gr b must be implemented as operators acting on the kinematical Hilbert space Master constraint program C φ := Σ d 3 x qab Ca gr C gr b det E quantized similarly to the Euclidean part in C 21 / 29

Issues & possible treatments in defining proper physical Hamiltonian 1 The form of the Hamiltonian in deparametrized models depends on the chosen reference fields In general, the Hamiltonian may come with a classical sign constraint in its definition that needs to be implemented in the quantum theory Two possibilities: Work out the spectral analysis of the Hamiltonian operator as defined in the full ''physical'' Hilbert space, then define the ''reduced physical'' Hilbert space as the completed span of the positive part of the Hamiltonian spectrum; Modify the form of the Hamiltonian: h = Q(C, C a ) or h = Q(C, C a ), h 0 Q(C, C a ) P 0 Q(C, C a ) P = 1 ) (Q(C, C a ) P + Q(C, C a ) 2 P Q (C, C a ) = 1 2 (Q(C, C a) + Q(C, C a ) ) We obtain Q (C, C a ) P = Q(C, C a ) P, Q (C, C a ) P = 0 Hence, in the quantum theory, the constraint of sign in the Hamiltonian can be traded with the square root of a positive operator 22 / 29

Issues & possible treatments in defining proper physical Hamiltonian 2 But, a square root implies that we need to perform the complete spectral analysis of the operator under it! 23 / 29

Issues & possible treatments in defining proper physical Hamiltonian 2 But, a square root implies that we need to perform the complete spectral analysis of the operator under it! Wellit is not that bad ;) We have perturbation theory Example: semi-classical analysis of the Master constraint operator in AQG Another interesting idea on how to apply perturbation theory on Hamiltonian operators with square roots: stay tuned for some details on slide 27 23 / 29

Summary Unified setup to treat various deparametrized models suitable for quantization (and invent new models); Spacetime reference models, while can be implemented in AQG, imply modifications in the standard LQG quantization, Time reference models can be quantized in context of standard LQG, but do not admit a consistent implementation in AQG so far; Physical Hilbert spaces of the quantum models can be derived; The physical Hamiltonian operators can be defined consistently; Relevant technical issues for the quantum evolution can be overcome and a computable framework can be obtained; 25 / 29

What can we do next? 1 Test the quantum dynamics: Approximate the evolution generated by the (chosen) Hamiltonian operator on coherent states Study the possibility of obtaining semi-classical dynamics with particular states 2 Locality: Graph changing VS Graph preserving Option 1: Option 2: Option 3: 1 Ĉ v may indeed seem (be) local, but what about Ĉ = Ĉ v? v 2 H = H( C, ) Ĥ = H( C ˆ Ĉ, ), ( Rq: Ker( C ˆ Ĉ) ) = Ker(Ĉ) 26 / 29

What can we do next? 3 Perturbation theory for the Hamiltonian: Ĉ = 1 (ĈE 2kβ 2 + ( s β 2) ) ˆR = 1 ( ˆR 1 [ĈE 2k β 2 + s ˆR ] ) β dependent perturbation: β >> 1 Ĉ 0 := ˆR, δĉ := 1 [ĈE β 2 + s ˆR ] Ĉ = Ĉ0 + δĉ Why this is advantageous when ĈL = ˆR? ˆR is graph preserving, it acts only in the intertwiners spaces ˆV The spectral analysis is then much more accessible than in the case of a graph changing Lorentzian part (practically impossible so far) The issue of computing evolution and investigating possible semi-classical regimes gets relatively simplified! 27 / 29

What can we do next? 4 Coupling gravity to matter fields: The aim: construct a quantum model where all matter and gravity degrees of freedom are considered Quantize the Hamiltonian including matter fields part Define the quantum dynamics of matter fields on quantum geometry Starting point example: KG scalar field Investigate the possible modifications in the dynamics of matter fields that emerge in highly quantum regimes of geometry relation to standard quantum field theory on fixed background; possibly grasping some understanding concerning the construction of the continuum limit in LQG; 28 / 29

Conclusions Deparametrization is a powerful technical tool that allows to circumvent conceptual and technical issues arising in the fully constrained theory; LQG and AQG provide a clear and (relatively) clean program to how to quantize deparametrized models and define the corresponding physical Hilbert spaces; Perturbative treatments of the quantum dynamics is possible, which suggests a computable framework; The ambiguity in choice of the reference fields is not an obstacle in investigating generic phenomenon and properties that (could or not) manifest in LQG models; Including, additionally, standard matter fields in the models is a promising route in: 1 understanding the interaction between quantum degrees of freedom of gravity and matter; 2 investigating new ideas to construct the continuum limit in LQG; 29 / 29