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
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
solutions to
in the quantum theory is equivalent to finding
Also, given a quantum observable
generated by
where
, the dynamics in this quantum theory is
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
[C. Rovelli & L. Smolin 95’], [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)]
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 surface
which contains the loop α
α
Quantization of the model: construction of the Hamiltonian operator
A fixed regularization angle
Link above the surface
α
Link below the surface
[Contribution of J. Lewandowski in T. Thiemann “Quantum Spin Dynamics (QSD)” 96’]
α
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
expressed as
on a spin network state can be
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
λ=0, this operator preserves the graph.
and in the case
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!