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

Picture1

where,

Picture2

Assuming that

Picture3

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

Picture4

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

Picture5

Where

Picture6

Quantization of the model:Hilbert space and Gauss constraint

The kinematical Hilbert space is defined as

Picture11

 

Where its elements are

Picture12

The gauge invariant subspace

Picture13

 

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

The space of the Gauss & vector constraints is defined as

Picture14

Quantization of the model: Physical Hamiltonian

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

Picture15

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

Picture16

where

Picture17

Picture18

Quantization of the model: construction of the Hamiltonian operator

Interested in constructing the quantum operator corresponding to the classical quantity

Picture19

Consider

Picture20

Lorentzian part:

Picture22

The final quantum operator corresponding to the Lorentzian part:

Picture23

where,

Picture24

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:

Picture25

The resulting operator:

Picture27

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

Picture28

Introduce the adjoint operator of  H

Picture29

 

For two spin network states have

Picture30 Picture31

Define a symmetric operator

Picture32

Which acts on s-n states as

Picture33

Define the physical Hamiltonian for this deparametrized model

Picture34

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.

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