Asymptotics of the EPRL/FK Spin Foam Model and the Flatness Problem

Transcription

1 Asymptotics of the EPRL/FK Spin Foam Model and the Flatness Problem José Ricardo Oliveira University of Nottingham Supervisor: John W. Barrett April 5, 2013 Talk for Quantum Fields, Gravity and Information 2013

2 Outline 1 2 The Asymptotics Problem 3 Past Work and the Flatness Problem 4 A Concrete Example: 3 5 Future Work

3 Regge calculus: a discrete GR Modern versions of LQG attempt the quantization of a discrete version of GR, based on Regge calculus. (d = 2 example) A triangulated d-manifold is a skeleton of d-simplices, where the bones are the (d 2)-simplices f on their boundaries.

4 Regge calculus: a discrete GR The triangulated manifold is everywhere at except on the bones, where there are conical defects with angles Θ f. In the continuum limit, R abcd ρ(x)θ(x), where ρ is the density of bones around x. Regge showed that triangulated manifolds approximate solutions of the Einstein equations, and the discrete version of the Einstein-Hilbert action is given by S R [A f ] = fi A f Θ f + fb A f Θ B f where A f is the (d 2)-volume of the bone f.

5 Regge calculus: a discrete GR The triangulated manifold is everywhere at except on the bones, where there are conical defects with angles Θ f. In the continuum limit, R abcd ρ(x)θ(x), where ρ is the density of bones around x. Regge showed that triangulated manifolds approximate solutions of the Einstein equations, and the discrete version of the Einstein-Hilbert action is given by S R [A f ] = fi A f Θ f + fb A f Θ B f where A f is the (d 2)-volume of the bone f.

6 Regge calculus: a discrete GR The triangulated manifold is everywhere at except on the bones, where there are conical defects with angles Θ f. In the continuum limit, R abcd ρ(x)θ(x), where ρ is the density of bones around x. Regge showed that triangulated manifolds approximate solutions of the Einstein equations, and the discrete version of the Einstein-Hilbert action is given by S R [A f ] = fi A f Θ f + fb A f Θ B f where A f is the (d 2)-volume of the bone f.

7 Spin Foam models aim to describe the dynamics of Loop Quantum Gravity. They are state sum models, interpreting spacetime as a statistical system where the quantum states are the spin network states of LQG. Taking advantage of the duality between the 2-complexes that make up spin networks and simplicial manifolds, Spin Network Simplicial Manifold (dim= d) d = 4 vertex v d-simplex σ v 4-simplex edge e (d 1)-simplex t e tetrahedron face f (d 2)-simplex f triangle the model can be described on a simplicial manifold, making the analogy to Regge calculus more apparent. The most general partition function is Z = W f (c) W e(c) W v (c) e v colorings f

8 Spin Foam models aim to describe the dynamics of Loop Quantum Gravity. They are state sum models, interpreting spacetime as a statistical system where the quantum states are the spin network states of LQG. Taking advantage of the duality between the 2-complexes that make up spin networks and simplicial manifolds, Spin Network Simplicial Manifold (dim= d) d = 4 vertex v d-simplex σ v 4-simplex edge e (d 1)-simplex t e tetrahedron face f (d 2)-simplex f triangle the model can be described on a simplicial manifold, making the analogy to Regge calculus more apparent. The most general partition function is Z = W f (c) W e(c) W v (c) e v colorings f

9 Spin Foam models aim to describe the dynamics of Loop Quantum Gravity. They are state sum models, interpreting spacetime as a statistical system where the quantum states are the spin network states of LQG. Taking advantage of the duality between the 2-complexes that make up spin networks and simplicial manifolds, Spin Network Simplicial Manifold (dim= d) d = 4 vertex v d-simplex σ v 4-simplex edge e (d 1)-simplex t e tetrahedron face f (d 2)-simplex f triangle the model can be described on a simplicial manifold, making the analogy to Regge calculus more apparent. The most general partition function is Z = W f (c) W e(c) W v (c) e v colorings f

10 In the gure: 4-simplex and dual vertex. A 4-simplex is oriented by ordering its points. From that we can induce orientations on the tetrahedra and triangles within it. If a face f is shared by two tetrahedra, its orientations with respect to both of them are opposite.

11 In the gure: 4-simplex and dual vertex. A 4-simplex is oriented by ordering its points. From that we can induce orientations on the tetrahedra and triangles within it. If a face f is shared by two tetrahedra, its orientations with respect to both of them are opposite.

12 In the gure: 4-simplex and dual vertex. A 4-simplex is oriented by ordering its points. From that we can induce orientations on the tetrahedra and triangles within it. If a face f is shared by two tetrahedra, its orientations with respect to both of them are opposite.

13 The EPRL/FK model In the (Euclidean) EPRL/FK model the colorings required are SU(2) quantum numbers (j + f, j f ) in each face SO(4) group elements g ve = (g + ve, g ve) in each tetrahedron on a simplex Livine-Speziale coherent states n ef where n ef S 2 To construct the vertex amplitude, dene the SU(2) coherent intertwiners ˆ î e = dh e h e k ef, n ef SU(2) then construct the SO(4) = SU(2) SU(2) interwiners i e from them: i e = ˆ ( ) dg π j π j + (g) C j j + {j f f î f } k e SO(4) f f ef k f ef f e

14 The EPRL/FK model In the (Euclidean) EPRL/FK model the colorings required are SU(2) quantum numbers (j + f, j f ) in each face SO(4) group elements g ve = (g + ve, g ve) in each tetrahedron on a simplex Livine-Speziale coherent states n ef where n ef S 2 To construct the vertex amplitude, dene the SU(2) coherent intertwiners ˆ î e = dh e h e k ef, n ef SU(2) then construct the SO(4) = SU(2) SU(2) interwiners i e from them: i e = ˆ ( ) dg π j π j + (g) C j j + {j f f î f } k e SO(4) f f ef k f ef f e

15 The EPRL/FK model In the (Euclidean) EPRL/FK model the colorings required are SU(2) quantum numbers (j + f, j f ) in each face SO(4) group elements g ve = (g + ve, g ve) in each tetrahedron on a simplex Livine-Speziale coherent states n ef where n ef S 2 To construct the vertex amplitude, dene the SU(2) coherent intertwiners ˆ î e = dh e h e k ef, n ef SU(2) then construct the SO(4) = SU(2) SU(2) interwiners i e from them: i e = ˆ ( ) dg π j π j + (g) C j j + {j f f î f } k e SO(4) f f ef k f ef f e

16 The EPRL/FK model Tensor the intertwiners corresponding to the 5 tetrahedra in a 4-simplex and close them up using the inner product ɛ : V j V j C. The vertex amplitude has the form W v ˆ ˆ ( ) ( ) dg ve dn ef K vf î e k ve ef f e ef

17 The EPRL/FK model The tetrahedron amplitude is (for γ < 1) W e f e δ jf, j + f +j f that is, it simply projects the k ef rep. into the highest spin component of the j + f j f rep of SU(2). The triangle amplitude is a measure of the quantized triangle area, usually chosen in accordance to LQG but with (so far) no consensus on whether it is the correct choice. So W f j j f (?)

18 The EPRL/FK model The tetrahedron amplitude is (for γ < 1) W e f e δ jf, j + f +j f that is, it simply projects the k ef rep. into the highest spin component of the j + f j f rep of SU(2). The triangle amplitude is a measure of the quantized triangle area, usually chosen in accordance to LQG but with (so far) no consensus on whether it is the correct choice. So W f j j f (?)

19 The EPRL/FK model The tetrahedron amplitude is (for γ < 1) W e f e δ jf, j + f +j f that is, it simply projects the k ef rep. into the highest spin component of the j + f j f rep of SU(2). The triangle amplitude is a measure of the quantized triangle area, usually chosen in accordance to LQG but with (so far) no consensus on whether it is the correct choice. So W f j j f (?)

20 The Asymptotics Problem Stating the Problem For a QG model to be accurate, it must replicate GR in the semiclassical limit ( 0). In spin foam models the limit is usually taken by xing j f =λj f and assuming λ. This corresponds to a limit of large areas, since in LQG the area spectrum is A f = γl 2 p jf (j f + 1) j γj f L 2 p In this limit the average size of each triangle in the manifold is much larger than L 2 p, so that quantum eects at the triangulation scale are negligible.

21 The Asymptotics Problem Stating the Problem For a QG model to be accurate, it must replicate GR in the semiclassical limit ( 0). In spin foam models the limit is usually taken by xing j f =λj f and assuming λ. This corresponds to a limit of large areas, since in LQG the area spectrum is A f = γl 2 p jf (j f + 1) j γj f L 2 p In this limit the average size of each triangle in the manifold is much larger than L 2 p, so that quantum eects at the triangulation scale are negligible.

22 The Asymptotics Problem Stating the Problem For a QG model to be accurate, it must replicate GR in the semiclassical limit ( 0). In spin foam models the limit is usually taken by xing j f =λj f and assuming λ. This corresponds to a limit of large areas, since in LQG the area spectrum is A f = γl 2 p jf (j f + 1) j γj f L 2 p In this limit the average size of each triangle in the manifold is much larger than L 2 p, so that quantum eects at the triangulation scale are negligible.

23 The Asymptotics Problem EPRL/FK Asymptotics The spin foam model's partition function can be cast into a sum-over-histories form: Z = ˆ ˆ µ(j f ) dg ve dn ef e f S f j f f ve ef where S = f S f can be seen as the classical action. For the EPRL/FK model we have S f = log j f, J n ef Y g ev g ve Y j f, n e f v f = 2j ± log J n f ef g ± ev g ± ve n ef v f ± n ef are the Livine-Speziale SU(2) coherent states; g ± ev are the components of g ev according to SO(4) = SU(2) SU(2).

24 The Asymptotics Problem EPRL/FK Asymptotics The spin foam model's partition function can be cast into a sum-over-histories form: Z = ˆ ˆ µ(j f ) dg ve dn ef e f S f j f f ve ef where S = f S f can be seen as the classical action. For the EPRL/FK model we have S f = log j f, J n ef Y g ev g ve Y j f, n e f v f = 2j ± log J n f ef g ± ev g ± ve n ef v f ± n ef are the Livine-Speziale SU(2) coherent states; g ± ev are the components of g ev according to SO(4) = SU(2) SU(2).

25 The Asymptotics Problem EPRL/FK Asymptotics When j f, the exponential integral becomes steeply peaked, and one can apply the stationary phase method: the dominant contributions come from the solutions of the variational problem δs = 0 which additionally satisfy Re S = 0. These are the critical points. For EPRL/FK we get the equations of motion Re S = 0 R(g ve) n ± ef = R(g ± ve ) n e f (gluing condition) δ gve S = 0 2j ± ɛ f ef (v)r(g ve) n ± ef = 0 (closure condition) f e ± δ nef S = 0 automatically satised given the two above The ideal situation is that in the semiclassical limit Z approaches the expression from the path integral formulation of Regge calculus, i.e. S critical is Regge Z(j fb ) µ(j f )e is Regge (j f ) j fi

26 The Asymptotics Problem EPRL/FK Asymptotics When j f, the exponential integral becomes steeply peaked, and one can apply the stationary phase method: the dominant contributions come from the solutions of the variational problem δs = 0 which additionally satisfy Re S = 0. These are the critical points. For EPRL/FK we get the equations of motion Re S = 0 R(g ve) n ± ef = R(g ± ve ) n e f (gluing condition) δ gve S = 0 2j ± ɛ f ef (v)r(g ve) n ± ef = 0 (closure condition) f e ± δ nef S = 0 automatically satised given the two above The ideal situation is that in the semiclassical limit Z approaches the expression from the path integral formulation of Regge calculus, i.e. S critical is Regge Z(j fb ) µ(j f )e is Regge (j f ) j fi

27 The Asymptotics Problem EPRL/FK Asymptotics When j f, the exponential integral becomes steeply peaked, and one can apply the stationary phase method: the dominant contributions come from the solutions of the variational problem δs = 0 which additionally satisfy Re S = 0. These are the critical points. For EPRL/FK we get the equations of motion Re S = 0 R(g ve) n ± ef = R(g ± ve ) n e f (gluing condition) δ gve S = 0 2j ± ɛ f ef (v)r(g ve) n ± ef = 0 (closure condition) f e ± δ nef S = 0 automatically satised given the two above The ideal situation is that in the semiclassical limit Z approaches the expression from the path integral formulation of Regge calculus, i.e. S critical is Regge Z(j fb ) µ(j f )e is Regge (j f ) j fi

28 Past Work and the Flatness Problem The Reconstruction Theorem Proven by Barrett et al (2009) for a single 4-simplex; extended to a general triangulated manifold with boundary by Han, Zhang (2011) Given a set of Regge-like boundary data and a set of non-degenerate interior data that obeys the semiclassical equations of motion, it is possible to construct a classical, non-degenerate discrete geometry matching the spin foam variables. This geometry is unique up to global symmetries. Additionally, the semiclassical action can be rewritten in the form S = ±i f [j f N f π γsign(v 4 )j f Θ f ] where N f is an integer. Since the semiclassical area of f is γj f, second term is identied as the Regge action. Based on this we will rewrite our action as S fi j f Θ qf (g ve, n ef ) where Θ qf λ ±γθ f can be seen as a quantum decit angle.

29 Past Work and the Flatness Problem The Reconstruction Theorem Proven by Barrett et al (2009) for a single 4-simplex; extended to a general triangulated manifold with boundary by Han, Zhang (2011) Given a set of Regge-like boundary data and a set of non-degenerate interior data that obeys the semiclassical equations of motion, it is possible to construct a classical, non-degenerate discrete geometry matching the spin foam variables. This geometry is unique up to global symmetries. Additionally, the semiclassical action can be rewritten in the form S = ±i f [j f N f π γsign(v 4 )j f Θ f ] where N f is an integer. Since the semiclassical area of f is γj f, second term is identied as the Regge action. Based on this we will rewrite our action as S fi j f Θ qf (g ve, n ef ) where Θ qf λ ±γθ f can be seen as a quantum decit angle.

30 Past Work and the Flatness Problem The Reconstruction Theorem Proven by Barrett et al (2009) for a single 4-simplex; extended to a general triangulated manifold with boundary by Han, Zhang (2011) Given a set of Regge-like boundary data and a set of non-degenerate interior data that obeys the semiclassical equations of motion, it is possible to construct a classical, non-degenerate discrete geometry matching the spin foam variables. This geometry is unique up to global symmetries. Additionally, the semiclassical action can be rewritten in the form S = ±i f [j f N f π γsign(v 4 )j f Θ f ] where N f is an integer. Since the semiclassical area of f is γj f, second term is identied as the Regge action. Based on this we will rewrite our action as S fi j f Θ qf (g ve, n ef ) where Θ qf λ ±γθ f can be seen as a quantum decit angle.

31 Past Work and the Flatness Problem The Flatness Problem (Freidel/Conrady, Bonzom) Notice that in the above study, we did not consider an equation of motion for the j f, i.e. δ jf S = 0. It does, at rst sight, seem that one should use it though. The j-equation raises two important problems, though: How to make sense of the variation with respect to a discrete variable? The origin of the atness problem: if one naively calculates δ jf S, the equation obtained is = 0, f Θ f implying that the semiclassical limit of EPRL/FK includes at geometries only. This contradicts GR! However, the stationary phase method will not work when applied to j, since the action is linear in those variables and there are no local extrema where constructive interference can happen. We will present an example on a small manifold to try and shed some light on the subject...

32 Past Work and the Flatness Problem The Flatness Problem (Freidel/Conrady, Bonzom) Notice that in the above study, we did not consider an equation of motion for the j f, i.e. δ jf S = 0. It does, at rst sight, seem that one should use it though. The j-equation raises two important problems, though: How to make sense of the variation with respect to a discrete variable? The origin of the atness problem: if one naively calculates δ jf S, the equation obtained is = 0, f Θ f implying that the semiclassical limit of EPRL/FK includes at geometries only. This contradicts GR! However, the stationary phase method will not work when applied to j, since the action is linear in those variables and there are no local extrema where constructive interference can happen. We will present an example on a small manifold to try and shed some light on the subject...

33 Past Work and the Flatness Problem The Flatness Problem (Freidel/Conrady, Bonzom) Notice that in the above study, we did not consider an equation of motion for the j f, i.e. δ jf S = 0. It does, at rst sight, seem that one should use it though. The j-equation raises two important problems, though: How to make sense of the variation with respect to a discrete variable? The origin of the atness problem: if one naively calculates δ jf S, the equation obtained is = 0, f Θ f implying that the semiclassical limit of EPRL/FK includes at geometries only. This contradicts GR! However, the stationary phase method will not work when applied to j, since the action is linear in those variables and there are no local extrema where constructive interference can happen. We will present an example on a small manifold to try and shed some light on the subject...

34 Past Work and the Flatness Problem The Flatness Problem (Freidel/Conrady, Bonzom) Notice that in the above study, we did not consider an equation of motion for the j f, i.e. δ jf S = 0. It does, at rst sight, seem that one should use it though. The j-equation raises two important problems, though: How to make sense of the variation with respect to a discrete variable? The origin of the atness problem: if one naively calculates δ jf S, the equation obtained is = 0, f Θ f implying that the semiclassical limit of EPRL/FK includes at geometries only. This contradicts GR! However, the stationary phase method will not work when applied to j, since the action is linear in those variables and there are no local extrema where constructive interference can happen. We will present an example on a small manifold to try and shed some light on the subject...

35 A Concrete Example: 3 A Concrete Example: 3

36 A Concrete Example: 3 A Concrete Example: 3 Working out the semiclassical EOM, we get for closure conditions j n CA + j41 C nc 4,41 + j 42 C nc 4,42 + j 43 C nc 4,43 = 0 j = j41 C nc 4,41 + j 42 C nc 4,42 + j C 43 nc 4,43 n CA = j 41 C nc 4,41 +j 42 C nc 4,42 +j 43 C nc 4,43 j41 C nc 4,41 +j 42 C nc 4,42 +j 43 C nc 4,43 and similar for the vertices A and B. These conditions guarantee geometric consistency of all tetrahedra containing f x the j variable in terms of boundary data. Since j is not summed over the j-equation does not exist.

37 A Concrete Example: 3 A Concrete Example: 3 Working out the semiclassical EOM, we get for closure conditions j n CA + j41 C nc 4,41 + j 42 C nc 4,42 + j 43 C nc 4,43 = 0 j = j41 C nc 4,41 + j 42 C nc 4,42 + j C 43 nc 4,43 n CA = j 41 C nc 4,41 +j 42 C nc 4,42 +j 43 C nc 4,43 j41 C nc 4,41 +j 42 C nc 4,42 +j 43 C nc 4,43 and similar for the vertices A and B. These conditions guarantee geometric consistency of all tetrahedra containing f x the j variable in terms of boundary data. Since j is not summed over the j-equation does not exist.

38 A Concrete Example: 3 A Concrete Example: 3 Eliminating all constrained variables, the partition function becomes ˆ Z SC dβ ± 1i dβ± 2i dɛ± i where the quantum Regge angle is i e ij Θ(ɛ ± ) i 2 + ± (1 ± γ)e2i(β± 1i +β± ) 2i 2 + ± (1 ± γ)e 2i(β± 1i +β± 2i +2ɛ± ) i Θ = ±2 ± (1 ± γ) i ɛ ± i Z has several poles, which is a conceptual problem in itself. But ignoring that as a consequence of the approximation, we can determine the most probable conguration by looking for the pole with the highest degree of divergence, i.e. making all the denominator factors zero.

39 A Concrete Example: 3 A Concrete Example: 3 Eliminating all constrained variables, the partition function becomes ˆ Z SC dβ ± 1i dβ± 2i dɛ± i where the quantum Regge angle is i e ij Θ(ɛ ± ) i 2 + ± (1 ± γ)e2i(β± 1i +β± ) 2i 2 + ± (1 ± γ)e 2i(β± 1i +β± 2i +2ɛ± ) i Θ = ±2 ± (1 ± γ) i ɛ ± i Z has several poles, which is a conceptual problem in itself. But ignoring that as a consequence of the approximation, we can determine the most probable conguration by looking for the pole with the highest degree of divergence, i.e. making all the denominator factors zero.

40 A Concrete Example: 3 A Concrete Example: 3 The maximal pole is obtained when { e 2i(β± 1i +β± ) 2i = 1 e 2i(β± 1i +β± 2i +2ɛ± ) i = 1, i {1, 2, 3} which amounts to and therefore ɛ ± i = k ± i Θ = γπ i π 2, k i Z (k + i k i ) which means that the Regge decit angle we predict is either 0 or π. BAD!

41 Future Work Where to go from here? Is there a (re)normalization procedure to get rid of the poles in the above calculation? Repeat the analysis for other simple cases related to Pachner moves ( 4, 5 ) What role do the triangulation, Planck, and experimental length scales play in asymptotics? Possible to work out at least rst order quantum corrections?...

42 References References J.C. Baez, An introduction to spin foam models of quantum gravity and BF theory, Lect. Notes Phys., vol 543, p. 25, 2000 [gr-qc/ ] J. Engle, E. Livine, R. Pereira, C. Rovelli, LQG vertex with nite Immirzi parameter, Nucl. Phys. vol. B799, pp , 2008 [gr-qc/ ] L. Freidel, K. Krasnov, A New Spin Foam Model for 4D Gravity, Class. Quant. Grav. vol. 25, p , 2008 [gr-qc/ ] E. Livine, S. Speziale, A new spinfoam vertex for quantum gravity, Phys. Rev. D vol. 76, p , 2007 [gr-qc/ ] M. Han, M. Zhang, Asymptotics of Spin Foam Amplitude on Simplicial Manifold: Euclidean Theory [gr-qc/ ] M. Han, M. Zhang, Asymptotics of Spin Foam Amplitude on Simplicial Manifold: Lorentzian Theory [gr-qc/ ] C. Rovelli, L. Smolin, Loop space representation of quantum general relativity, Nucl. Phys. B vol. 331, issue 1, pp J.W. Barrett, R. Dowdall, W. Fairbairn, H. Gomes, F. Hellmann, Asymptotic analysis of the EPRL four-simplex amplitude, J. Math. Phys. 50:112504, 2009 [gr-qc/ ]

43 References References L. Kauman, State models and the Jones polynomial, Topology 26 (1987), no. 3, pp J.W. Barrett, R. Dowdall, W. Fairbairn, F. Hellmann, R. Pereira, Lorentzian spin foam amplitudes: graphical calculus and asymptotics [gr-qc/ ] E. Bianchi, D. Regoli, C. Rovelli, Face amplitude of spinfoam quantum gravity, Class. Quant. Grav. 27:185009, 2010 [gr-qc/ ] V. Bonzom, Spin foam models for quantum gravity from lattice path integrals, Phys. Rev. D 80:064028, 2009 [gr-qc/ ] T. Regge, General Relativity Without Coordinates, Il Nuovo Cimento, Vol. 19, N. 3, p. 558 F. Conrady, L. Freidel, Path integral representation of spin foam models of 4D gravity, Class. Quant. Grav. 25, , 2008 [gr-qc/ ] F. Conrady, L. Freidel, On the semiclassical limit of 4D spin foam models, Phys. Rev. D 78, , 2008 [gr-qc/ ] J.W. Barrett, L. Crane, Relativistic Spin Networks and Quantum Gravity, J.Math.Phys. 39, , 1998 [gr-qc/ ]