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Nov 11, 2022 130 likes •445 views

gra raph coherent st states tes in loo oop quantum um gra ravi vity ty Mehdi Assanioussi II. Institute for Th. Physics, University of Hamburg Jurekfest, September 2019 Problem lems to s solve I. Dynamics in LQG II. Coherent

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Mehdi Assanioussi

  • II. Institute for Th. Physics, University of Hamburg

Jurekfest, September 2019

gra raph coherent st states tes in loo

  • op quantum

um gra ravi vity ty

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Problem lems to s solve

  • I. Dynamics in LQG
  • II. Coherent states & semi-classical limit in LQG
  • III. Matter fields coupling in LQG
  • IV. LQG cosmology
  • V. Black holes in LQG
  • VI. ...
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Warsaw L LQG Co Commando On t the m e move.. e...

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… We a e are n e not d done ne yet!

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Pla lan o

  • f t

f the t talk lk

  • I. Motivation
  • II. Graph coherent states
  • i. Construction concept
  • ii. Example 1: closed loops in the U(1) gauge theory
  • iii. Example 2: closed loops in non-Abelian gauge ths.
  • III. Concluding comments
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  • I. M

. Motiv tivati tion

  • n
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Graphs in in L LQG QG

2

Graph change?

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Graph superposition

Graphs in in L LQG QG

Physical solution? Continuum limit?

2

Graph change? Semi-classical limit?

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Coherent states

Coherent ent s states es in in L LQG QG

Intertwiners

e.g: Livine-Speziale coherent intertwiners

Spins

e.g.: complexifier (Area) coherent states

4

Graphs

???

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  • II. G

. Graph c coh

  • herent

erent states tes

  • i. Con
  • nstr

struc uction

  • n con
  • ncep

ept

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1 Define a graph change on a colored graph: exp.: 2 Construct the structure of a harmonic oscillator space: exp.: New coherent states (graph coherent states);

Construction concept

5

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+ Key observation: A (generic) graph change (with a “finite structure”) provides a decomposition of the into separable subspaces , which are stable under the action of the operators inducing such graph change.

  • ancestor graphs: colored graphs with no graph “excitation”

Construction concept

6

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  • II. G

. Graph c coh

  • herent

erent states tes

  • ii. Example 1: closed loops in

the U(1) gauge theory

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Example 1: closed loops in the U(1) gauge theory

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Closed loop graph excitation

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Example 1: closed loops in the U(1) gauge theory

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Canonical structure:

+ Annihilation & creation operators + Wedge coherent states + Graph coherent states

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Example 1: closed loops in the U(1) gauge theory

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Operators correspondence:

+ Closed loop holonomies + fluxes

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Graph coherent states I: Maxwell Field

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Coherence properties:

+ Expectation values: + Relative variances:

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Example 1: closed loops in the U(1) gauge theory

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Example of an observable:

+ Maxwell field Hamiltonian: “The action of the Hamiltonian at a vertex is reformulated as an action in a space of a finite number of Harmonic oscillators”

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  • II. G

. Graph c coh

  • herent

erent states tes

  • ii. Example 2: closed loops in

non-Abelian gauge theories

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Example 2: closed loops in non-Abelian gauge theories

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Hilbert space structure:

+ Decomposition of the Hilbert space: , , . . .

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Example 2: closed loops in non-Abelian gauge theories

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Canonical structure:

+ Generalized annihilation & creation operators Imposing

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Example 2: closed loops in non-Abelian gauge theories

13

Canonical structure:

+ Generalized annihilation & creation operators Imposing

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Example 2: closed loops in non-Abelian gauge theories

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Canonical structure: Abelian case Non-Abelian case

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Example 2: closed loops in non-Abelian gauge theories

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Choice of canonical structure:

+ Non-uniqueness of vacuum states + Non-uniqueness of the mapping between G-tensors

Graph coherent states:

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Example 2: closed loops in non-Abelian gauge theories

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+ Expectation values: + Relative variances:

Example of canonical structure:

  • r

. . .

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III. . Con

  • ncludi

ding ng com

  • mments

ts

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Concluding comments

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Comments on the construction:

+ The graph change Various valid graph changes, even “non-local” ones + Freedom in choice of vacuum states & tensor mappings adapting the choice of CS to the choice of observables + Canonical structure with fluxes

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Summary:

✔ New coherent states which exhibit graph coherence; ✔ The construction method applies to various graph excitations; ✔ Flexibility of the mapping on the intertwiner space;

Outlook:

Explore the dynamics of such states; Explore the possible choices of the intertwiners mapping; Explore different graph excitations for the purposes of investigating semi-classical and continuum limits

Concluding comments

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Than ank you

  • u & Ha

& Happy birt y birthda day!!! y!!!