Metric-affine vs Spectral Theoretic Characterization of the Massless Dirac Operator

Transcription

1 Metric-affine vs Spectral Theoretic Characterization of the Massless Dirac Operator Elvis Barakovic, Vedad Pasic University of Tuzla Department of Mathematics August 26, 2013

2 Structure of presentation Metric affine gravity: The results of previous joint work. Physical interpretation of these results. Future work for my PhD: The spectrum of the massless Dirac operator. Discussion.

3 Metric affine gravity Alternative theory of gravity. Natural generalization of Einstein s GR, which is based on a spacetime with Riemannian metric g of Lorentzian signature. We consider spacetime to be a connected real 4-manifold M equipped with Lorentzian metric g and an affine connection Γ. SPACETIME MAG={M, g, Γ} The 10 independent components of the symmetric metric tensor g µν and 64 connection coefficients Γ λ µν are unknowns of MAG.

4 Metric affine gravity Alternative theory of gravity. Natural generalization of Einstein s GR, which is based on a spacetime with Riemannian metric g of Lorentzian signature. We consider spacetime to be a connected real 4-manifold M equipped with Lorentzian metric g and an affine connection Γ. SPACETIME MAG={M, g, Γ} The 10 independent components of the symmetric metric tensor g µν and 64 connection coefficients Γ λ µν are unknowns of MAG.

5 Metric affine gravity Alternative theory of gravity. Natural generalization of Einstein s GR, which is based on a spacetime with Riemannian metric g of Lorentzian signature. We consider spacetime to be a connected real 4-manifold M equipped with Lorentzian metric g and an affine connection Γ. SPACETIME MAG={M, g, Γ} The 10 independent components of the symmetric metric tensor g µν and 64 connection coefficients Γ λ µν are unknowns of MAG.

6 Metric affine gravity We define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R. The system of Euler Lagrange equations: S g S Γ = 0, (2) = 0. (3) Objective: To study the combined system of field equations (2), (3) which is system of real nonlinear PDEs with real unknowns.

7 Metric affine gravity We define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R. The system of Euler Lagrange equations: S g S Γ = 0, (2) = 0. (3) Objective: To study the combined system of field equations (2), (3) which is system of real nonlinear PDEs with real unknowns.

8 Metric affine gravity We define our action as S := q(r) (1) where q(r) is a quadratic form on curvature R. The system of Euler Lagrange equations: S g S Γ = 0, (2) = 0. (3) Objective: To study the combined system of field equations (2), (3) which is system of real nonlinear PDEs with real unknowns.

9 New representation of the field equations We write down explicitly our field equations (2), (3) under following assumptions: (i) our spacetime is metric compatible, (ii) curvature has symmetries R κλµν = R µνκλ, ε κλµν R κλµν = 0, (iii) scalar curvature is zero.

10 New representation of the field equations We write down explicitly our field equations (2), (3) under following assumptions: (i) our spacetime is metric compatible, (ii) curvature has symmetries R κλµν = R µνκλ, ε κλµν R κλµν = 0, (iii) scalar curvature is zero.

11 Main result so far The main result is Lemma Under the above assumptions (i) (iii), the field equations (2), (3) are 0 = d 1 W κλµν Ric κµ + d 3 ( Ric λκ Ric ν κ 1 4 g λν Ric κµ Ric κµ ) (4)

12 Main result so far The main result is Lemma Under the above assumptions (i) (iii), the field equations (2), (3) are 0 = d 1 W κλµν Ric κµ + d 3 ( Ric λκ Ric ν κ 1 4 g λν Ric κµ Ric κµ ) (4)

13 New representation of the field equations 0 = d 6 λ Ric κµ d 7 κric λµ ( + d 6 Ricκ η ( d 7 Ric η λ (K µηλ K µλη ) g λµw ηζ κξ (K ξ ηζ K ξ ζη ) g µλric η ξ K ξ ηκ +g µλ Ricκ η K ξ ξη K ξ ξλ Ricκµ + 1 ) 2 g µλricκ ξ (K η ξη K η ηξ ) (Kµηκ Kµκη) gκµwκζ λξ (K ξ ηζ K ξ ζη ) gµκric η ξ K ξ ηλ +g κµric η λ K ξ ξη K ξ ξκ Ric λµ + 1 ) 2 gµκric ξ λ (K η ξη K η ηξ ) + b 10 (g µλ W ηζ κξ (K ξ ζη K ξ ηζ ) + gµκwηζ λξ (K ξ ηζ K ξ ζη ) +g µλ Ricκ ξ (K η ηξ K η ξη ) + gµκric ξ λ (K η ξη K η ηξ ) +g κµric η λ K ξ ξη g λµricκ η K ξ ξη + RicµκK η λη Ric µλk η κη ) + 2b 10 (W η µκξ (K ξ ηλ K ξ λη ) + Wη µλξ (K ξ κη K ξ ηκ ) ) K µξη W ηξ κλ K ξ ξη Wη µλκ (5) where d 1, d 3, d 6, d 7, b 10 are some real constants.

14 The first task We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metric and purely axial torsion which are solutions of the field equations (2), (3). Expectations: to prove or disprove conjecture above. to give a physical interpretation of the new solutions and compare them with existing Riemannian solutions.

15 The first task We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metric and purely axial torsion which are solutions of the field equations (2), (3). Expectations: to prove or disprove conjecture above. to give a physical interpretation of the new solutions and compare them with existing Riemannian solutions.

16 The first task We are going to try to generalize pp waves as follows Conjecture There exists a new class of spacetimes with pp metric and purely axial torsion which are solutions of the field equations (2), (3). Expectations: to prove or disprove conjecture above. to give a physical interpretation of the new solutions and compare them with existing Riemannian solutions.

17 Physical interpretation Massless Dirac action: ( ) S neutrino := 2i ξ a σ µ ( µξḃ) ( aḃ µ ξ a )σ µ aḃ ξḃ. In Einstein Weyl theory the action is given by: S EW = S neutrino + k R. We obtain the well known Einstein Weyl field equations S EW g S EW ξ = 0, (6) = 0. (7)

18 Physical interpretation Massless Dirac action: ( ) S neutrino := 2i ξ a σ µ ( µξḃ) ( aḃ µ ξ a )σ µ aḃ ξḃ. In Einstein Weyl theory the action is given by: S EW = S neutrino + k R. We obtain the well known Einstein Weyl field equations S EW g S EW ξ = 0, (6) = 0. (7)

19 Physical interpretation Massless Dirac action: ( ) S neutrino := 2i ξ a σ µ ( µξḃ) ( aḃ µ ξ a )σ µ aḃ ξḃ. In Einstein Weyl theory the action is given by: S EW = S neutrino + k R. We obtain the well known Einstein Weyl field equations S EW g S EW ξ = 0, (6) = 0. (7)

20 The massless Dirac operator The massless Dirac operator is the matrix operator ( W = iσ α x α + 1 ( { )) σ β β 4 σ β }σ x α + γ. (8) αγ The massless Dirac operator (8) describes a single massless neutrino living in 3-dimensional compact universe M. The eigenvalues of the massless Dirac operator are the energy levels.

21 The massless Dirac operator The massless Dirac operator is the matrix operator ( W = iσ α x α + 1 ( { )) σ β β 4 σ β }σ x α + γ. (8) αγ The massless Dirac operator (8) describes a single massless neutrino living in 3-dimensional compact universe M. The eigenvalues of the massless Dirac operator are the energy levels.

22 The massless Dirac operator Let M be a 3-dimensional connected oriented manifold equipped with a Riemnnian metric g αβ and let W be the corresponding massless Dirac operator (8). Two basic examples when the spectrum of W can be calculated explicitly: the unit torus T 3 equipped with Euclidean metric. the unit sphere S 3 equipped with metric induced by the natural embedding of S 3 in Euclidean space R 4. In both examples the spectrum turns out to be symmetric about zero.

23 The massless Dirac operator Let M be a 3-dimensional connected oriented manifold equipped with a Riemnnian metric g αβ and let W be the corresponding massless Dirac operator (8). Two basic examples when the spectrum of W can be calculated explicitly: the unit torus T 3 equipped with Euclidean metric. the unit sphere S 3 equipped with metric induced by the natural embedding of S 3 in Euclidean space R 4. In both examples the spectrum turns out to be symmetric about zero.

24 The massless Dirac operator Physically, this means that in these two examples there is no difference between the properties of the particle (massless neutrino) and antiparticle (massless antineutrino). For a general oriented Riemannian 3-manifold there is no reason for the spectrum of massless Dirac operator W to be symmetric (M. F. Atiyah, V. K. Patodi and I. M. Singer). Producing explicit examples of spectral asymmetry is a very difficult task.

25 The massless Dirac operator Physically, this means that in these two examples there is no difference between the properties of the particle (massless neutrino) and antiparticle (massless antineutrino). For a general oriented Riemannian 3-manifold there is no reason for the spectrum of massless Dirac operator W to be symmetric (M. F. Atiyah, V. K. Patodi and I. M. Singer). Producing explicit examples of spectral asymmetry is a very difficult task.

26 The massless Dirac operator Physically, this means that in these two examples there is no difference between the properties of the particle (massless neutrino) and antiparticle (massless antineutrino). For a general oriented Riemannian 3-manifold there is no reason for the spectrum of massless Dirac operator W to be symmetric (M. F. Atiyah, V. K. Patodi and I. M. Singer). Producing explicit examples of spectral asymmetry is a very difficult task.

27 The massless Dirac operator Pfäffle: the example based on the idea of choosing a 3-manifold with flat metric but highly nontrivial toplogy! Barakovic, Pasic, Vassiliev: the simplest possible topology and construct an explicit example of spectral asymmetry by perturbing the metric! Spectral asymmetry of the massless Dirac operator on the unit torus T 3 is achieved: Spectral asymmetry of the massless Dirac operator on a 3-torus, D. Vassiliev, R.J.Downes and M.Levitin. Available as preprint arxiv:

28 The massless Dirac operator Pfäffle: the example based on the idea of choosing a 3-manifold with flat metric but highly nontrivial toplogy! Barakovic, Pasic, Vassiliev: the simplest possible topology and construct an explicit example of spectral asymmetry by perturbing the metric! Spectral asymmetry of the massless Dirac operator on the unit torus T 3 is achieved: Spectral asymmetry of the massless Dirac operator on a 3-torus, D. Vassiliev, R.J.Downes and M.Levitin. Available as preprint arxiv:

29 The massless Dirac operator Pfäffle: the example based on the idea of choosing a 3-manifold with flat metric but highly nontrivial toplogy! Barakovic, Pasic, Vassiliev: the simplest possible topology and construct an explicit example of spectral asymmetry by perturbing the metric! Spectral asymmetry of the massless Dirac operator on the unit torus T 3 is achieved: Spectral asymmetry of the massless Dirac operator on a 3-torus, D. Vassiliev, R.J.Downes and M.Levitin. Available as preprint arxiv:

30 The plan Try to achieve the spectral asymmetry of the massless Dirac operator (8) on the unit sphere S 3. The ideas: the metric which depends to small parameter ε, Hopf fibration.

31 The plan Try to achieve the spectral asymmetry of the massless Dirac operator (8) on the unit sphere S 3. The ideas: the metric which depends to small parameter ε, Hopf fibration.

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