# Curvature operator and gravity coupled to a scalar field: the physical Hamiltonian operator by Alesci, Assanioussi and Lewandowski

This week I been reviewing some conference proceedings the FFP14 Conference, Marseilles 2014. One paper I particularly like is this one by Alesci, Assanioussi and Lewandowski. I have been doing a some collaboration with Alesci and Assanoussi on the Hamiltonian Operator.

Gravity (minimally) coupled to a massless scalar field

The theory of 3+1 gravity (Lorentzian) minimally coupled to a free massless scalar field Φ(x) is described by the action

where,

Assuming that

The Hamiltonian constraint is solved for π using the diff. constraint

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

Quantization of the model:Hilbert space and Gauss constraint

The kinematical Hilbert space is defined as

Where its elements are

The gauge invariant subspace

Quantization of the model: Hilbert space of gauge & diff. invariant states

The space of the Gauss & vector constraints is defined as

Quantization of the model: Physical Hamiltonian

Solving the scalar constraint C‘ in the quantum theory is equivalent to finding solutions to

given a quantum observable  ,  the dynamics in this quantum theory is generated by

where

Quantization of the model: construction of the Hamiltonian operator

Interested in constructing the quantum operator corresponding to the classical quantity

Consider

Lorentzian part:

The final quantum operator corresponding to the Lorentzian part:

where,

Properties of this operator:

• Gauge & Diffeomorphism invariant
• Cylindrically consistent  if the averaging used in defining the curvature operator is restricted to non zero contributions;
• Discrete spectrum & compact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis

Euclidean part:

The resulting operator:

The action of the full H on a spin network state can be expressed as

Introduce the adjoint operator of  H

For two spin network states have

Define a symmetric operator

Which acts on s-n states as

Define the physical Hamiltonian for this deparametrized model

Properties of the final operator   hphys[N]   :

• Gauge and Diffeomorphism invariant
• Cylindrically consistent  if the averaging used in defining the operator is restricted to only non zero contributions
• Discrete spectrum  andcompact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis – volume operator not involved

The paper 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.

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