Curvature operator and gravity coupled to a scalar field: the physical Hamiltonian operator

1. Curvature operator and gravity coupled to a scalar field: the physical Hamiltonian operator (On going work) E. Alesci, M. Assanioussi, Jerzy Lewandowski Faculty of Physics, University of Warsaw FFP14 Conference, Marseilles 2014

2. Plan: • Gravity (minimally) coupled to a massless scalar field: Overview • Quantization of the model: • Hilbert space and Gauss constraint • Hilbert space of gauge & diff. invariant states • Physical Hamiltonian • The curvature operator • Regularization of the Euclidean part • The adjoint operator & the matrix elements • Outlooks & Summary

3. Gravity (minimally) coupled to a massless scalar field: Overview The theory of 3+1 gravity (Lorentzian) minimally coupled to a free massless scalar field Φ(x) is described by the action Where Arbitrary functions [K.V. Kuchar93’], [L. Smolin 89’], [C. Rovelli & L. Smolin 93], [K.V. Kuchar & J.D. Romano 95’], [M. Domagala, K. Giesel, W. Kaminski, J. Lewandowski 10’], [M. Domagala, M. Dziendzikowski, J. Lewandowski 12’]

4. Gravity (minimally) coupled to a massless scalar field: Overview Assuming that The Hamiltonian constraint is solved for π using the diff. constraint Select different regions of the phase space

5. Gravity (minimally) coupled to a massless scalar field: Overview Φbecomes the emergent time! In the region (+,+), an equivalent model could be obtained by keeping the Gauss and Diff. constraints and reformulating the scalar constraints Where Φ no longer occurs in the function h the scalar constraints deparametrized

6. Gravity (minimally) coupled to a massless scalar field: Overview The scalar constraints strongly commute As a consequence For a Dirac observable O [M. Domagala, K. Giesel, W. Kaminski, J. Lewandowski 10’], [M. Domagala, M. Dziendzikowski, J. Lewandowski 12’]

7. Quantization of the model: Hilbert space and Gauss constraint The kinematical Hilbert space is defined as Where its elements are The gauge invariant subspace It is the space of solutions of the Gauss constraint obtained by group averaging with respect to the YM gauge transformations.

8. Quantization of the model: Hilbert space of gauge & diff. invariant states To construct the Hilbert space of gauge & Diff. invariant states, an averaging procedure is performed w.r. to the group This averaging is achieved through the “rigging” map The space of the Gauss & vector constraints is defined as

9. Quantization of the model: Physical Hamiltonian Solving the scalar constraint in the quantum theory is equivalent to finding solutions to Also, given a quantum observable , the dynamics in this quantum theory is generated by where

10. Quantization of the model: construction of the Hamiltonian operator We are interested in constructing the quantum operator corresponding to the classical quantity Consider

11. Quantization of the model: construction of the Hamiltonian operator Lorentzian part: Volume operator [C. Rovelli & L. Smolin95’], [A. Ashtekar & J. Lewandowski 97’] Ricci scalar term is promoted to the curvature Operator Introduced in [E. Alesci, M. A., J. Lewandowski. Phys. Rev. D 124017 (2014)]

12. Quantization of the model: construction of the Hamiltonian operator Regularization of the curvature term: [E. Bianchi 08’] ,[E. Alesci, M. A., J. Lewandowski 14’]

13. Quantization of the model: construction of the Hamiltonian operator

14. Quantization of the model: construction of the Hamiltonian operator Regularized expression: The final quantum operator corresponding to the Lorentzian part:

15. Quantization of the model: construction of the Hamiltonian operator Properties of this operator: • Gauge & Diffeomorphism invariant; • Not cylindrically consistent (however it’s possible to achieve if the averaging used in defining the curvature operator is restricted to only non zero contributions); • Self-adjoint; • Discrete spectrum & compact expression for the matrix elements (expressed explicitly in terms of the coloring) on the spin network basis;

16. Quantization of the model: construction of the Hamiltonian operator Euclidean part: The quadrant of the generated surface which contains the loop α α

17. Quantization of the model: construction of the Hamiltonian operator A fixed regularization angle Link above the surface α α Link below the surface [Contribution of J. Lewandowski in T. Thiemann “Quantum Spin Dynamics (QSD)” 96’]

18. Quantization of the model: construction of the Hamiltonian operator The resulting operator: Coefficient resulting from the averaging over configurations corresponding to pairs of links

19. Quantization of the model: construction of the Hamiltonian operator Action of the (Euclidean part) operator on a node of a spin network

20. Quantization of the model: construction of the Hamiltonian operator Let us denote by the Hilbert space of cylindrical functions defined on a graph which contains a number of loops such the one introduced by the action of the Euclidean part of .Which means that each loop is associated to a pair of links originating from the same node. In such subspaces, the action of the full on a spin network state can be expressed as

21. Quantization of the model: construction of the Hamiltonian operator we introduce the adjoint operator of : Such that for two spin network states we have

22. Quantization of the model: construction of the Hamiltonian operator We can hence define a symmetric operator: Which acts on s-n states as

23. Quantization of the model: construction of the Hamiltonian operator We can finally define the physical Hamiltonian for this deparametrized model inherits automatically properties from . However, to express its explicit action and derive its spectra we will need to make a spectral analysis of

24. Quantization of the model: construction of the Hamiltonian operator Properties of the final operator : • Gauge & Diffeomorphism invariant; • Not cylindrically consistent (however it’s possible to achieve if the averaging used in defining the operator is restricted to only non zero contributions); • Symmetric (Self-adjoint?); • Discrete spectrum & compact expression for the matrix elements (expressed explicitly in terms of the coloring) on the spin network basis (volume operator not involved);

25. Outlook Back to Vacuum theory! In vacuum theory we have the scalar constraint Using the same implementation described above, an interesting candidate for the constraint operator in the vacuum theory appears Which has the property of preserving the subspaces and in the case λ=0, this operator preserves the graph.

26. Summary We presented a way of implementing the Hamiltonian operator in the case of the deparametrized model of gravity with a scalar field using: • The simple and well defined curvature operator; • A new regularization scheme that allows to define an adjoint operator for the regularized expression and hence construct a symmetric Hamiltonian operator; This operator verifies the properties of gauge symmetries and cylindrical consistency could be imposed. The missing points: • Study the self-adjointness property; • Make a spectral analysis in both cases of scalar field and vacuum; • Test the induced evolution in some simple models;

27. Thank you!