Some analytical results of the Hamiltonian operator in LQG

Transcription

1 Some analytical results of the Hamiltonian operator in LQG Cong Zhang January 22, / 61

2 Outline Deparametrized model of LQG coupled to scalar field General works about the physical Hamiltonian Restriction on a special case A toy model of LQG cosmology Conclusion and outlook 2 / 61

3 Deparametrized model of LQG coupling to a scalar field(alesci et al., 2015) ntroduce the other fields as reference frame classically. 3 / 61

4 Deparametrized model of LQG coupling to a scalar field(alesci et al., 2015) ntroduce the other fields as reference frame classically. Solve the constraint equation to obtain a physical Hamiltonian. n the model coupling to a scalar field where h(x) = det E C (x) = π(x) ± h(x) ( ) C gr ± (C gr ) 2 q ab Ca gr C gr b 4 / 61

5 Deparametrized model of LQG coupling to a scalar field(alesci et al., 2015) ntroduce the other fields as reference frame classically. Solve the constraint equation to obtain a physical Hamiltonian. n the model coupling to a scalar field where h(x) = det E Quantize this system. C (x) = π(x) ± h(x) ( ) C gr ± (C gr ) 2 q ab Ca gr C gr b 5 / 61

6 Main results of the quantum Theory(Domaga la et al., 2010, Lewandowski et al., 2011) The physical Hilbert space H phy : the Hilbert space space of pure gravity, satisfying Gaussian and vector constraints. 6 / 61

7 Main results of the quantum Theory(Domaga la et al., 2010, Lewandowski et al., 2011) The physical Hilbert space H phy : the Hilbert space space of pure gravity, satisfying Gaussian and vector constraints. The dynamic: i d dt Ψ = ĤΨ where t is a parameter of the transformations φ φ + t. 7 / 61

8 Main results of the quantum Theory(Domaga la et al., 2010, Lewandowski et al., 2011) The physical Hilbert space H phy : the Hilbert space space of pure gravity, satisfying Gaussian and vector constraints. The dynamic: i d dt Ψ = ĤΨ where t is a parameter of the transformations φ φ + t. The quantum Hamiltonian Ĥ = d 3 x 2 det E(x) C gr (x) 8 / 61

9 Main results of the quantum Theory(Domaga la et al., 2010, Lewandowski et al., 2011) The physical Hilbert space H phy : the Hilbert space space of pure gravity, satisfying Gaussian and vector constraints. The dynamic: i d dt Ψ = ĤΨ where t is a parameter of the transformations φ φ + t. The quantum Hamiltonian Ĥ = d 3 x 2 det E(x) C gr (x) What do is to study this (physical) Hamiltonian operator. 9 / 61

10 General works about the physical Hamiltonian p Classical expression of det E (x) C gr (x) p det E (x) C gr (x) 1 k = ijk Eia (x)ejb (x)fab (x) + (1 + β 2 ) det E (x) R(x) 16πβ 2 G 1 =: (H E (x) + H L (x)). 16πβ 2 G 10 / 61

11 General works about the physical Hamiltonian p Classical expression of det E (x) C gr (x) p det E (x) C gr (x) 1 k = ijk Eia (x)ejb (x)fab (x) + (1 + β 2 ) det E (x) R(x) 16πβ 2 G 1 =: (H E (x) + H L (x)). 16πβ 2 G Volume operator is not involved. 11 / 61

12 General works about the physical Hamiltonian p Classical expression of det E (x) C gr (x) p det E (x) C gr (x) 1 k = ijk Eia (x)ejb (x)fab (x) + (1 + β 2 ) det E (x) R(x) 16πβ 2 G 1 =: (H E (x) + H L (x)). 16πβ 2 G Volume operator is not involved. The analysis is simpler. 12 / 61

13 General works about the physical Hamiltonian p Classical expression of det E (x) C gr (x) p det E (x) C gr (x) 1 k = ijk Eia (x)ejb (x)fab (x) + (1 + β 2 ) det E (x) R(x) 16πβ 2 G 1 =: (H E (x) + H L (x)). 16πβ 2 G Volume operator is not involved. The analysis is simpler. t is possible to start from the simplest case of 2-valent graph. 13 / 61

14 General works about the physical Hamiltonian The chosen loop to quantize Fab (Thiemann, 2007, Yang and Ma, 2015) 14 / 61

15 General works about the physical Hamiltonian The chosen loop to quantize Fab (Thiemann, 2007, Yang and Ma, 2015) Expression of the Hamiltonian operator(alesci et al., 2015) H = p 1 E (l) Hv,ee 0 = ijk Tr L Hee 0 r := X s X 16πG β 2 v V (γ) i e,e 0 at v j (e, e 0 )(H E v,ee 0 + (H E v,ee 0 ) ) + (1 + β 2 )H L v,ee 0 k (hα 0 τ )Jv,e Jv,e 0 ee 2π 0 0 j j δii 0 ijk Jv,e J k 0 i 0 j 0 k 0 Jv,e J k 0 α π + arccos q v,e v,e δkl Jvk,e Jvl,e 0 r 0 0 δkk 0 Jvk,e Jvk,e δkk 0 J k 0 J k 0 v,e v,e 15 / 61

16 General works about the physical Hamiltonian The physical Hilbert space: Degenerate vertex: 16 / 61

17 General works about the physical Hamiltonian The physical Hilbert space: Degenerate vertex: Vnd (γ): non-degenerate vertices. DiffVnd :diffeomorphisms preserving Vnd. Diff(γ)Tr : diffeomorphisms acting trivially on γ 17 / 61

18 General works about the physical Hamiltonian The physical Hilbert space: Degenerate vertex: Vnd (γ): non-degenerate vertices. DiffVnd :diffeomorphisms preserving Vnd. Diff(γ)Tr : diffeomorphisms acting trivially on γ Physical state from ψγ i X (ψγ = Nγ Uφ ψγ i := η ψi φ DiffVnd /Diff(γ)Tr 18 / 61

19 Restriction on a special case The graphs: (H E ) n : γ 0 :=. =: γ n 19 / 61

20 Restriction on a special case The graphs: (H E ) n : γ 0 :=. =: γ n The Hilbert space: Spin networks on γ n. 20 / 61

21 Restriction on a special case. Spin networks on γn : γn,~j,~l i 21 / 61

22 Restriction on a special case. Spin networks on γn : γn,~j,~l i Ambiguity of choosing spin l:. (l) ijk Tr (hαee 0 τ i )Jvj,e Jvk,e 0 :. P Cjn+1. jn+1 22 / 61

23 Restriction on a special case. Spin networks on γn : γn,~j,~l i Ambiguity of choosing spin l:. (l) ijk Tr (hαee 0 τ l(jn ) = 1 1/2 i )Jvj,e Jvk,e 0 :. P Cjn+1. jn+1, jn = 1/2,, jn = 6 1/2. 23 / 61

24 Restriction on a special case. Spin networks on γn : γn,~j,~l i Ambiguity of choosing spin l:. (l) ijk Tr (hαee 0 τ 1 1/2 i )Jvj,e Jvk,e 0. : P Cjn+1. jn+1, jn = 1/2,, jn = 6 1/2. l(jn ) = Hphy = span( ([γn ],~j := N P Uφ γn,~j, l(~j ) i) φ Diffv /DiffTr (γn ) 24 / 61

25 Restriction on a special case The graphs:. (H E )n : γ0 := Hilbert space: Hphy = span( ([γn ],~j := N =: γn P Uφ γn,~j, l(~j ) i). φ Diffv /DiffTr (γn ) 25 / 61

26 Restriction on a special case The graphs:. (H E )n : γ0 := Hilbert space: Hphy = span( ([γn ],~j := N =: γn P Uφ γn,~j, l(~j ) i). φ Diffv /DiffTr (γn ) Operator: Hamiltonian operator restricted on the Hilbert space. 26 / 61

27 Restriction on a special case.. 27 / 61

28 Restriction on a special case The graphs:. (H E )n : γ0 := Hilbert space: Hphy = span( ([γn ],~j := N =: γn P Uφ γn,~j, l(~j ) i). φ Diffv /DiffTr (γn ) Operator: Hamiltonian operator restricted on the Hilbert space, H Hphy. 28 / 61

29 Restriction on a special case The graphs:. (H E )n : γ0 := Hilbert space: Hphy = span( ([γn ],~j := N =: γn P Uφ γn,~j, l(~j ) i). φ Diffv /DiffTr (γn ) Operator: Hamiltonian operator restricted on the Hilbert space, H Hphy. Question: Self-adjointness of the restricted H. 29 / 61

30 Self-adjointness of the Hamiltonian operator Theorem Let N be a self-adjoint operator with N 1. Let H be a symmetric operator with domain D which is a core for N. Suppose that: i For some c and all ψ D, ii For some d and all ψ D, Hψ c Nψ. (Hψ, Nψ) (Nψ, Hψ) d N 1/2 ψ 2 Then A is essential self-adjoint on D and its closure is essentially self-adjoint on any core for N. 30 / 61

31 Self-adjointness of the Hamiltonian operator Theorem Let N be a self-adjoint operator with N 1. Let H be a symmetric operator with domain D which is a core for N. Suppose that: i For some c and all ψ D, ii For some d and all ψ D, Hψ c Nψ. (Hψ, Nψ) (Nψ, Hψ) d N 1/2 ψ 2 Then A is essential self-adjoint on D and its closure is essentially self-adjoint on any core for N. ( We can choose ( the operator N, diagonalized under the basis, that is [γ n ], j N = [γ n ], j N(j n+1 ), such that N(j) = j n with n / 61

32 Summation We prove the operator H, restricted on the simplest graph, is self-adjoint. 32 / 61

33 Summation We prove the operator H, restricted on the simplest graph, is self-adjoint. Turn to the graph preserving version. 33 / 61

34 Summation We prove the operator H, restricted on the simplest graph, is self-adjoint. Turn to the graph preserving version. Consider the coherent state peaking at the cosmology phase space. 34 / 61

35 Summation We prove the operator H, restricted on the simplest graph, is self-adjoint. Turn to the graph preserving version. Consider the coherent state peaking at the cosmology phase space. Calculate the semiclassical dynamics with the coherent state as the initial data. 35 / 61

36 The heat kernel coherent state Classical phase space of the cosmology phase space: 2/3 Aia = cvo 1/3 ω ai, Eia = pv0 qo e ia 36 / 61

37 The heat kernel coherent state Classical phase space of the cosmology phase space: 2/3 Aia = cvo 1/3 ω ai, Eia = pv0 qo e ia The graph: dipole graph with two N-valent vertices. 37 / 61

38 The heat kernel coherent state Classical phase space of the cosmology phase space: 2/3 Aia = cvo 1/3 ω ai, Eia = pv0 qo e ia The graph: dipole graph with two N-valent vertices. Coherent state for p 1 (Bahr and Thiemann, 2009, Bianchi et al., 2010): Ψc,p (~ g ) = Y e E (γ) X 1 tje (je +1)+νpje iµcje je Dj,j (ne ge ne ) (2je + 1)e je e e 38 / 61

39 The heat kernel coherent state Classical phase space of the cosmology phase space: 2/3 Aia = cvo 1/3 ω ai, Eia = pv0 The graph: dipole graph with two N-valent vertices. Coherent state for p 1 (Bahr and Thiemann, 2009, Bianchi et al., 2010): Ψc,p (~ g ) = Y e E (γ) qo e ia X 1 tje (je +1)+νpje iµcje je Dj,j (ne ge ne ) (2je + 1)e e e je The coherent states inspires us to consider the Hilbert space O j Hcos = span(h~g ~j i := Djeeje (ne 1 gne )) e γ 39 / 61

40 The heat kernel coherent state Classical phase space of the cosmology phase space: 2/3 Aia = cvo 1/3 ω ai, Eia = pv0 The graph: dipole graph with two N-valent vertices. Coherent state for p 1 (Bahr and Thiemann, 2009, Bianchi et al., 2010): Ψc,p (~ g ) = Y e E (γ) qo e ia X 1 tje (je +1)+νpje iµcje je Dj,j (ne ge ne ) (2je + 1)e e e je The coherent states inspires us to consider the Hilbert space O j Hcos = span(h~g ~j i := Djeeje (ne 1 gne )) e γ The factor e tj(j+1)+νpj iµcj is a Gaussian function on j peaking at j0 = νp 2t, so we focus on large j limit when calculating. 40 / 61

41 Action of holonomy and flux operators on Hcos For the holonomy operator: he1/2 Dnjee ne (g ) =D 1/2 (ne )! j +1/2 p Dnee ne (g ) 0 D 1/2 (ne 1 ) + O(1/ j) je 1/2 0 Dne ne (g ) j with the abbreviation of Dnn (g ) := Djjj (n 1 gn) 41 / 61

42 Action of holonomy and flux operators on Hcos For the holonomy operator: he1/2 Dnjee ne (g ) =D 1/2 (ne )! j +1/2 p Dnee ne (g ) 0 D 1/2 (ne 1 ) + O(1/ j) je 1/2 0 Dne ne (g ) j with the abbreviation of Dnn (g ) := Djjj (n 1 gn) For the Flux operator p ~Jv,e D je (g ) = je ~ne D je (g ) + O( j) ne ne ne ne (1) 42 / 61

43 Action of holonomy and flux operators on Hcos For the holonomy operator: he1/2 Dnjee ne (g ) =D 1/2 (ne )! j +1/2 p Dnee ne (g ) 0 D 1/2 (ne 1 ) + O(1/ j) je 1/2 0 Dne ne (g ) j with the abbreviation of Dnn (g ) := Djjj (n 1 gn) For the Flux operator p ~Jv,e D je (g ) = je ~ne D je (g ) + O( j) ne ne ne ne (1) The Hilbert space Hcos is preserved approximately for large j. 43 / 61

44 The Hamiltonian operator Action of H on Hcos, H = j 0 (E ) j D e (g )Dne 0 n 0 (ge 0 ) v,ee 0 ne ne e e e H =( q P q P ee 0 HvE,ee 0 + HvL,ee 0 v j +1/2 je + 1/2Dnee ne (ge ) q j 1/2 je 1/2Dnee ne (ge )) p je q q q j 0 +1/2 j 0 1/2 ( je 0 + 1/2Dne 0 n 0 (ge 0 ) je 0 1/2Dne 0 n 0 (ge 0 )) je 0 e j e e e j 0 0 j L j Hv,ee 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) =αee 0 sin(θee 0 )je je 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) e e e e 44 / 61

45 The Hamiltonian operator Action of H on Hcos, H = j 0 (E ) j D e (g )Dne 0 n 0 (ge 0 ) v,ee 0 ne ne e e e H =( q P q P ee 0 HvE,ee 0 + HvL,ee 0 v j +1/2 je + 1/2Dnee ne (ge ) q j 1/2 je 1/2Dnee ne (ge )) p je q q q j 0 +1/2 j 0 1/2 ( je 0 + 1/2Dne 0 n 0 (ge 0 ) je 0 1/2Dne 0 n 0 (ge 0 )) je 0 e j e e e j 0 0 j L j Hv,ee 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) =αee 0 sin(θee 0 )je je 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) e e e e HvE,ee and HvL,ee are self-adjoint. 45 / 61

46 The Hamiltonian operator Action of H on Hcos, H = j 0 (E ) j D e (g )Dne 0 n 0 (ge 0 ) v,ee 0 ne ne e e e H =( q P q P ee 0 HvE,ee 0 + HvL,ee 0 v j +1/2 je + 1/2Dnee ne (ge ) q j 1/2 je 1/2Dnee ne (ge )) p je q q q j 0 +1/2 j 0 1/2 ( je 0 + 1/2Dne 0 n 0 (ge 0 ) je 0 1/2Dne 0 n 0 (ge 0 )) je 0 e e e j e j 0 0 j L j Hv,ee 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) =αee 0 sin(θee 0 )je je 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) e e e e HvE,ee and HvL,ee are self-adjoint. HvE,ee can be rewritten as HvE,ee 0 = Hv,e Hv,e 0 with j Hv,e Dne n (ge ) = i e e q j +1/2 (ge ) e e je + 1/2Dne n q p j 1/2 (ge ) je 1/2Dne n je e e 46 / 61

47 The Hamiltonian operator Action of H on Hcos, H = j 0 (E ) j D e (g )Dne 0 n 0 (ge 0 ) v,ee 0 ne ne e e e H =( q P q P ee 0 HvE,ee 0 + HvL,ee 0 v j +1/2 je + 1/2Dnee ne (ge ) q j 1/2 je 1/2Dnee ne (ge )) p je q q q j 0 +1/2 j 0 1/2 ( je 0 + 1/2Dne 0 n 0 (ge 0 ) je 0 1/2Dne 0 n 0 (ge 0 )) je 0 e e e j e j 0 0 j L j Hv,ee 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) =αee 0 sin(θee 0 )je je 0 Dnee ne (ge )Dne 0 n 0 (ge 0 ) e e e e HvE,ee and HvL,ee are self-adjoint. HvE,ee can be rewritten as HvE,ee 0 = Hv,e Hv,e 0 with j Hv,e Dne n (ge ) = i e e q j +1/2 (ge ) e e je + 1/2Dne n q p j 1/2 (ge ) je 1/2Dne n je e e For large j, Hv,e = i p je d p dje c je =: He which is a self-adjoint in the Hilbert space L2 (R+ ) with iω ln(je ) eigenvector ϕω (je ) = e j. e 47 / 61

48 Some issues about the Minkowski condition Minkowski condition: e at v j e n e = 0 48 / 61

49 Some issues about the Minkowski condition Minkowski condition: e at v j e n e = 0 The volume operator under this condition: V j e n e ɛ ee e j e j e j e n e ( n e n e ) j e n e + o( j) coincide with the classical expression. 49 / 61

50 Some issues about the Minkowski condition Minkowski condition: e at v j e n e = 0 The volume operator under this condition: V j e n e ɛ ee e j e j e j e n e ( n e n e ) j e n e + o( j) coincide with the classical expression. The operator ee at v H v,ee doesn t preserve the condition. 50 / 61

51 Some issues about the Minkowski condition Minkowski condition: e at v j e n e = 0 The volume operator under this condition: V j e n e ɛ ee e j e j e j e n e ( n e n e ) j e n e + o( j) coincide with the classical expression. The operator ee at v H v,ee doesn t preserve the condition. Does the evolution operator e iht preserve this condition? 51 / 61

52 A toy model for LQG cosmology Regardless of the Lorentz part of the Hamiltonian operator, 52 / 61

53 A toy model for LQG cosmology Regardless of the Lorentz part of the Hamiltonian operator, Consider the continuous limit of the above model 53 / 61

54 A toy model for LQG cosmology Regardless of the Lorentz part of the Hamiltonian operator, Consider the continuous limit of the above model The Hilbert space Hcos = N span(h~g ~j i := e γ Djjeeje (ne 1 gne )) L2 ((R+ )N, d~x ) 54 / 61

55 A toy model for LQG cosmology Regardless of the Lorentz part of the Hamiltonian operator, Consider the continuous limit of the above model The Hilbert space Hcos = N span(h~g ~j i := e γ Djjeeje (ne 1 gne )) L2 ((R+ )N, d~x ) P qp E The Hamiltonian operator H = v e,e 0 Hv,ee 0 H = q p P P p 2 ee 0 Hec Hec0 = xe xe 0 xe x 0 xe xe 0 ee 0 e 55 / 61

56 Semiclassical analysis: Coherent state Ψ(~x ) = Z ψ(x) := dωe Q e γ (ω ω0 )2 +iξ0 ω 2σ 2 φ ψ(xe ) s ω (x) = σ 2 σ2 (ξ ln(x))2 iω (ξ ln(x)) e x 56 / 61

57 Semiclassical analysis: Coherent state Ψ(~x ) = Z ψ(x) := dωe Q e γ (ω ω0 )2 +iξ0 ω 2σ 2 φ ψ(xe ) s ω (x) = σ 2 σ2 (ξ ln(x))2 iω (ξ ln(x)) e x The semiclassical condition: ξ0 1, σ 1, while σξ0 1 for ln x/ ln x / 61

58 Semiclassical analysis: Coherent state Ψ(~x ) = Z ψ(x) := dωe Q e γ (ω ω0 )2 +iξ0 ω 2σ 2 φ ψ(xe ) s ω (x) = σ 2 σ2 (ξ ln(x))2 iω (ξ ln(x)) e x The semiclassical condition: ξ0 1, σ 1, while σξ0 1 for ln x/ ln x 1. Solving the dynamic: Ψ(~ x, τ ) = e ihτ 1 Ψ(~ x) = pq i Z xi P N d ωe 2 q i (ωi ω0 ) +i P (ξ ln(x ))ω +i P i i i 0 i6=j ωi ωj τ 2σ 2 58 / 61

59 Semiclassical analysis: Coherent state Ψ(~x ) = Z ψ(x) := dωe Q e γ (ω ω0 )2 +iξ0 ω 2σ 2 φ s ω (x) = σ 2 σ2 (ξ ln(x))2 iω (ξ ln(x)) e x The semiclassical condition: ξ0 1, σ 1, while σξ0 1 for ln x/ ln x 1. Solving the dynamic: Ψ(~ x, τ ) = e ihτ 1 Ψ(~ x) = pq i ψ(xe ) Z xi P N d ωe 2 q i (ωi ω0 ) +i P (ξ ln(x ))ω +i P i i i 0 i6=j ωi ωj τ 2σ 2 The trajectory where the state peaks: N 1 ln xi ξ0 = q τ 2 CN2 59 / 61

60 Conclusion and outlook Semiclassically, the quantum dynamic gives us an expanding universe. The peak satisfies the Minkowski condition. Future works: the quantum phenomenon, generalization to general graph, same problem with graph changing Hamiltonian 60 / 61

61 Thanks! 61 / 61