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;
- Self-adjoint;
- 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
- Symmetric /Self-adjoint
- 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.

**Related articles**

- A curvature operator for LQG by Alesci, Assanioussi and Lewandowski (quantumtetrahedron.wordpress.com)
- Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint by Alesci, Thiemann, and Zipfel (quantumtetrahedron.wordpress.com)
- Gravity and Scalar Fields [CL] (arxiver.wordpress.com)
- Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: II) The Weyl Tensor, Weyl Scalars, the Weyl Eigenvalues and the Problem of the Observables of the Gravitational Field [CL] (arxiver.wordpress.com)
- Matrix Elements of Lorentzian Hamiltonian Constraint in LQG by Alesci,Liegener and Zipfel (quantumtetrahedron.wordpress.com)
- Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: I) The Riemann Tensor and Ricci Scalars [CL] (arxiver.wordpress.com)