approximation methods for the dynamics in deparametrized lqg

Mehdi Assanioussi

Faculty of Physics, University of Warsaw

work in collaboration with J. Lewandowski & I. Mäkinen

[arXiv:1702.01688]

Fifth Tux workshop on QG

Tux, Feb 2017

Approximation methods for the

dynamics in Deparametrized LQG

Plan of the talk

I. LQG deparametrized models

II. Physical Hamiltonian operators

III. Approximation methods for the dynamics

i. Expansion: Hamiltonian op. / Evolution op.

ii. Examples

IV. Summary & outlook

LQG deparametrized models


Motivation

1

• Test-ground for LQG quantization methods

• Insights on the continuum limit of LQG

• Insights on the dynamics of Matter fields in presence of QG


• Complete quantum gravity models:

* Physical Hilbert space

* Admissible Hamiltonian operator

• Computable dynamics (time evolution)

Motivation

1

• Test-ground for LQG quantization methods

• Insights on the continuum limit of LQG

• Insights on the dynamics of Matter fields in presence of QG

First steps…


Deparametrization Matter fields as coordinates

2


Massless K.G. scalar field Non-rotational dust

Deparametrization Matter fields as coordinates

e.g: time ref. mod.

2

Deparametrization


Time ref. models

Spacetime ref.

models

3

Deparametrization


Time ref. models

Spacetime ref.

models

Successful

Unsuccessful

3

Deparametrization


Time ref. models

Spacetime ref.

models

Successful

Unsuccessful

Depends…

Successful

3



e.g: time ref. mod.

4



e.g: time ref. mod.

4

Physical Hilbert space:



e.g: time ref. mod.

4

Proper implementation of the functional C

* Self-adjoint Hamiltonian operator

* Spatial diffeomorphism invariant

* Computable dynamics

Physical Hilbert space:

Physical Hamiltonian operators


Scalar constraint functional

5

Immirzi-Barbero

parameter



Lorentzian part

(Curvature operator)

Euclidean part

(Euclidean operator)

5

Immirzi-Barbero

parameter



Lorentzian part

(Curvature operator)

Euclidean part

(Euclidean operator)

Constructing the operator on

5

Immirzi-Barbero

parameter


Curvature operator

(Regge calculus)

6


Euclidean operator

7


Euclidean operator

Regularization

7


Euclidean operator

Regularization

Moving to :

7


8


Properties

• The curvature op. preserves the graphs;

• The Euclidean op. removes loops from the graphs;

• SU(2) gauge inv. & spatial diff. covariant ;

• Anomaly-free;

8


Properties

• The curvature op. preserves the graphs;

• The Euclidean op. removes loops from the graphs;

• SU(2) gauge inv. & spatial diff. covariant ;

• Anomaly-free;

• Existence of symmetric extensions (self-adjoint?);

• Domain decomposes into stable separable sub-spaces;

8

Approximation methods for the dynamics


Dynamics

9


Dynamics

• Transition amplitudes:

• Quantum observables:

9


Dynamics

• Transition amplitudes:

• Quantum observables:

• Expansion of the evolution operator:

9


10

Dynamics


• Perturbation theory for the Hamiltonian op.:

10

Dynamics


• Perturbation theory for the Hamiltonian op.:

10

Dynamics Self-adjointSymmetric


Concretely? For what range of β?

11



Examples:

11

Evolution of volume & curvature expectation values



Examples:

11

• initial states = eigenvectors of volume (4-valent vertex)




Examples:

11


• β-expansion is taken up to 2nd

order:




Examples:

11


• β-expansion is taken up to 2nd

order:

• time-expansion is taken up to 4th

order:



Scalar field model in examples

12

j=2 j=10


Scalar field model in examples

12

Periodic evolution at 0th

order

j=2 j=10


Dust field model in examples

13

j=10j=10


Dust field model in examples

14

j=10j=2

Summary & outlook

Summary & outlook

Deparametrized models:

Symmetric Hamiltonian operators

Approximation method (pert. th.) for the Hamiltonian op.:

* treatment of the square root in the SF model

* explicit computation of dynamics

Self-adjointness proofs

Identification & interpretation of relevant physical states

Include Standard Model

approximation methods for the dynamics in deparametrized lqg

Documents