Curvature operator and gravity coupled to a scalar field:

the physical Hamiltonian operator(On going work)

E. Alesci, M. Assanioussi, Jerzy LewandowskiFaculty of Physics, University of Warsaw

FFP14 Conference, Marseilles 2014

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

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. Kuchar 93’], [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’]

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

Gravity (minimally) coupled to a massless scalar field: Overview

Φ becomes 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

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’]

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.

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

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

Quantization of the model: construction of the Hamiltonian operator

We are interested in constructing the quantum operator corresponding to the classical quantity

Consider

Quantization of the model: construction of the Hamiltonian operator

Lorentzian part:

Volume operator

Ricci scalar term is promoted to the curvature Operator Introduced in [E. Alesci, M. A., J. Lewandowski. Phys. Rev. D 124017 (2014)]

[C. Rovelli & L. Smolin 95’], [A. Ashtekar & J. Lewandowski 97’]

Quantization of the model: construction of the Hamiltonian operator

Regularization of the curvature term:

[E. Bianchi 08’] ,[E. Alesci, M. A., J. Lewandowski 14’]

Quantization of the model: construction of the Hamiltonian operator

Quantization of the model: construction of the Hamiltonian operator

Regularized expression:

The final quantum operator corresponding to the Lorentzian part:

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;

Quantization of the model: construction of the Hamiltonian operator

Euclidean part:

The quadrant of the generated surfacewhich contains the loop α α

Quantization of the model: construction of the Hamiltonian operator

Link above the surface

Link below the surface

A fixed regularization angle

α α

[Contribution of J. Lewandowski in T. Thiemann “Quantum Spin Dynamics (QSD)” 96’]

Quantization of the model: construction of the Hamiltonian operator

The resulting operator:

Coefficient resulting from the averaging over configurations corresponding to pairs of links

Quantization of the model: construction of the Hamiltonian operator

Action of the (Euclidean part) operator on a node of a spin network

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

Quantization of the model: construction of the Hamiltonian operator

we introduce the adjoint operator of :

Such that for two spin network states we have

Quantization of the model: construction of the Hamiltonian operator

We can hence define a symmetric operator:

Which acts on s-n states as

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

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);

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.

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;

Thank you!