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Bubble networks: framed discrete geometry for quantum gravity

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Abstract

In the context of canonical quantum gravity in 3 \(+\) 1 dimensions, we introduce a new notion of bubble network that represents discrete 3d space geometries. These are natural extensions of twisted geometries, which represent the geometrical data underlying loop quantum geometry and are defined as networks of \(\mathrm {SU}(2)\) holonomies. In addition to the \(\mathrm {SU}(2)\) representations encoding the geometrical flux, the bubble network links carry a compatible \(\mathrm {SL}(2,{{\mathbb {R}}})\) representation encoding the discretized frame field which composes the flux. In contrast with twisted geometries, this extra data allows to reconstruct the frame compatible with the flux unambiguously. At the classical level this data represents a network of 3d geometrical cells glued together. The \(\mathrm {SL}(2,{{\mathbb {R}}})\) data contains information about the discretized 2d metrics of the interfaces between 3d cells and \(\mathrm {SL}(2,{{\mathbb {R}}})\) local transformations are understood as the group of area-preserving diffeomorphisms. We further show that the natural gluing condition with respect to this extended group structure ensures that the intrinsic 2d geometry of a boundary surface is the same from the viewpoint of the two cells sharing it. At the quantum level this gluing corresponds to a maximal entanglement along the network edges. We emphasize that the nature of this extension of twisted geometries is compatible with the general analysis of gauge theories that predicts edge mode degrees of freedom at the interface of subsystems.

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Notes

  1. We can think of fluxes as discrete analog Lie algebra valued 2-forms, while frames are the discrete analog of Lie algebra valued 1-form.

  2. The explicit exponentiated action of the \(\ell \)’s is easily computed as:

    $$\begin{aligned}&e^{\{\lambda _{0}\ell _{0}+\lambda _{+}\ell _{-}+\lambda _{-}\ell _{+},\,\cdot \,\}} \,\left( \begin{array}{c}\vec {x}\\ \vec {p}\end{array} \right) = \left( \begin{array}{cc}-\,\lambda _{0} &{} \quad -\,\lambda _{-}\\ \lambda _{+} &{} \quad \lambda _{0}\end{array} \right) \,\left( \begin{array}{c}\vec {x}\\ \vec {p}\end{array} \right) = M\,\left( \begin{array}{c}\vec {x}\\ \vec {p}\end{array} \right) , \quad tr M=0 , \quad M^{2}=(\lambda _{0}^{2} -\lambda _{+}\lambda _{-})\,{\mathbb {I}}=\Delta \,{\mathbb {I}}, \\&\Omega =e^{M}=\cosh \sqrt{\Delta }\,{\mathbb {I}}+\frac{\sinh \sqrt{\Delta }}{\sqrt{\Delta }}\,M \quad \text {if}\,\, \Delta >0 \quad \text {or}\quad \cos \sqrt{-\Delta }\,{\mathbb {I}}+\frac{\sin \sqrt{-\Delta }}{\sqrt{-\Delta }}\,M \quad \text {if}\,\, \Delta <0 . \end{aligned}$$

  3. To be more precise, let us sketch a quantization scheme in terms of coherent states. We use the Segal–Bargmann representation for the pair of conjugate vectors \((\vec {x},\vec {p})\in ({{\mathbb {R}}}^3)^{\times 2}\). For i running from 1 to 3, we quantize each vector component \((x_{i},p_{i})\) as a harmonic oscillator and represent them at the quantum level as acting on holomorphic wave-functions \(\phi (z_{i})\), with \(z_{i}\) being the label of the coherent state and the annihilation (resp. creation) operator represented as the multiplication operator \(a_{i}=z_{i}\) (resp. the derivation operator \(a_{i}^\dagger =\partial _{z_{i}}\). Group transformations in \(\mathrm {SO}(3)\) act as 3d rotations on the complex vector \((z_{1},z_{2},z_{3})\), while the \({\mathfrak {sl}}(2,{{\mathbb {R}}})\) algebra is generated by the total energy \(\sum _{i}(a_{i}^\dagger a_{i}+1/2)\) and the squeezing operators \(\sum _{i} a_{i}^2\) and \(\sum _{i} a_{i}^\dagger {}^2\):

    $$\begin{aligned}&\left[ \sum _{i}a_{i}^\dagger a_{i}, \sum _{i} a_{i}^2\right] =-\,2\sum _{i} a_{i}^2 ,\quad \left[ \sum _{i}a_{i}^\dagger a_{i}, \sum _{i} a_{i}^\dagger {}^2\right] =+\,2\sum _{i} a_{i}^\dagger {}^2 , \\&\left[ \sum _{i} a_{i}^\dagger {}^2,\sum _{i} a_{i}^2\right] = -2\sum _{i}\left( a_{i}^\dagger a_{i}+\frac{1}{2}\right) . \end{aligned}$$

    Considering an edge, we have two copies of this structure, one at its source in terms of coherent state label \(z_{i}\) with operators \(a_{i},a_{i}^\dagger \) and one at its target in terms of label \(w_{i}\) with operators \(b_{i},b_{i}^\dagger \). The \({\mathfrak {sl}}(2,{{\mathbb {R}}})\) matching constraints are:

    $$\begin{aligned} \sum _{i}z_{i}\partial _{z_{i}}=\sum _{i}w_{i}\partial _{w_{i}} ,\quad \sum _{i}z_{i}^2=\sum _{i}\partial _{w_{i}}^2 ,\quad \sum _{i}\partial _{z_{i}}^2=\sum _{i}w_{i}^2 . \end{aligned}$$

    It is straightforward to check that a basis of solutions to these constraints is given by the entangled states \(\cosh [z_{i}h_{ij}w_{j}]\) and \(\sinh [z_{i}h_{ij}w_{j}]\) labeled by a group element \(h\in \mathrm {SO}(3)\), which are exactly the even and odd superpositions of all coherent states at the source and target such that the two states differ that the given rotation h. In order to realize the explicit quantization of the bubble network phase space in terms of extended spin networks, we would need to refine this analysis using irreducible representations of the symmetry group \(\mathrm {SU}(2)\times \mathrm {SL}(2,{{\mathbb {R}}})\).

  4. This reconstruction of the position vector is actually very similar to the definition of position Dirac observables for a relativistic particle [33].

  5. More technically, we would normalize \({\hat{e}}_{z}\wedge \vec {J}\) to define \({\hat{v}}_{x}\) and define the third direction of this orthonormal frame as \({\hat{v}}_{y}={\hat{J}}\wedge {\hat{v}}_{x}\), then

    $$\begin{aligned} {\hat{p}}=\cos \theta {\hat{v}}_{x}+\sin \theta {\hat{v}}_{y}. \end{aligned}$$

  6. Considering a triangle made of three edges, \(\vec {v}_{1,2,3}\) satisfying a closure condition \(\vec {v}_{1}+\vec {v}_{3}=\vec {v}_{2}\), with the normal vector defined as \(\vec {N}=\vec {v}_{1}\wedge \vec {v}_{2}=\vec {v}_{1}\wedge \vec {v}_{3}=\vec {v}_{2}\wedge \vec {v}_{3}\), we can define a Poisson bracket:

    $$\begin{aligned} \left\{ v^a_{1},v^b_{2}\right\} =\left\{ v^a_{1},v^b_{3}\right\} =\left\{ v^a_{2},v^b_{3}\right\} =\delta ^{ab}. \end{aligned}$$

    If we choose the pair of vectors \((\vec {v}_{1},\vec {v}_{2})\) as frame fields, then change root vertex and switch to the pair of vectors \((\vec {v}_{1},\vec {v}_{3})\), this is a simple canonical transformation realized as a \(\mathrm {SL}(2,{{\mathbb {R}}})\) transformation.

  7. This algebra can be derived from the following Poisson brackets,

    $$\begin{aligned} \left\{ \frac{1}{2}\langle z|\sigma _a|z\rangle ,\langle z|\sigma _b|z]\right\} \,=\, \frac{-2i}{2}\langle z|\sigma _a\sigma _b|z] \,=\, \frac{-2i}{2}\langle z|\delta _{ab}{\mathbb {I}}+i\epsilon _{abc}\sigma _c|z] \,=\, \epsilon _{abc}\langle z|\sigma _c|z], \end{aligned}$$

    as well as

    $$\begin{aligned} \left\{ \frac{1}{2}\langle z|\sigma _a|z\rangle ,[ z|\sigma _b|z\rangle \right\} \,=\, \epsilon _{abc}[ z|\sigma _c|z\rangle ,\quad \{\langle z|\sigma _a|z],[ z|\sigma _b|z\rangle \} \,=\, 4i\delta _{ab}\langle z|z\rangle -4\epsilon _{abc}\langle z|\sigma _c|z\rangle . \end{aligned}$$

  8. An intriguing remark is that isothermal coordinates for minimal surfaces allow for the Weierstrass–Enneper representation in terms of holomorphic coordinates [36]. This might open the door to a direct link between spinning geometries (whose boundary surfaces are all minimal surfaces) and spinor networks.

References

  1. Freidel, L., Geiller, M., Ziprick, J.: Continuous formulation of the loop quantum gravity phase space. Class. Quantum Gravity 30, 085013 (2013). arXiv:1110.4833

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Freidel, L., Ziprick, J.: Spinning geometry = twisted geometry. Class. Quantum Gravity 31(4), 045007 (2014). arXiv:1308.0040

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Freidel, L., Perez, A.: Quantum gravity at the corner. arXiv:1507.02573

  4. Donnelly, W., Freidel, L.: Local subsystems in gauge theory and gravity. JHEP 09, 102 (2016). arXiv:1601.04744

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Rovelli, C.: Why gauge? Found. Phys. 44(1), 91–104 (2014). arXiv:1308.5599

    Article  ADS  MATH  Google Scholar 

  6. Freidel, L., Speziale, S.: Twisted geometries: a geometric parametrisation of SU(2) phase space. Phys. Rev. D 82, 084040 (2010). arXiv:1001.2748

    Article  ADS  Google Scholar 

  7. Girelli, F., Sellaroli, G.: SO*(2N) coherent states for loop quantum gravity. J. Math. Phys. 58(7), 071708 (2017). arXiv:1701.07519

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Freidel, L., Livine, E.R.: U(N) coherent states for loop quantum gravity. J. Math. Phys. 52, 052502 (2011). arXiv:1005.2090

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Livine, E.R., Tambornino, J.: Spinor representation for loop quantum gravity. J. Math. Phys. 53, 012503 (2012). arXiv:1105.3385

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Livine, E.R., Tambornino, J.: Loop gravity in terms of spinors. J. Phys. Conf. Ser. 360, 012023 (2012). arXiv:1109.3572

    Article  MATH  Google Scholar 

  11. Freidel, L., Speziale, S.: From twistors to twisted geometries. Phys. Rev. D 82, 084041 (2010). arXiv:1006.0199

    Article  ADS  Google Scholar 

  12. Dittrich, B., Ryan, J.P.: Phase space descriptions for simplicial 4d geometries. Class. Quantum Gravity 28, 065006 (2011). arXiv:0807.2806

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Dittrich, B., Speziale, S.: Area-angle variables for general relativity. New J. Phys. 10, 083006 (2008). arXiv:0802.0864

    Article  ADS  Google Scholar 

  14. Dupuis, M., Ryan, J.P., Speziale, S.: Discrete gravity models and loop quantum gravity: a short review. SIGMA 8, 052 (2012). arXiv:1204.5394

    MathSciNet  MATH  Google Scholar 

  15. Speziale, S., Wieland, W.M.: The twistorial structure of loop-gravity transition amplitudes. Phys. Rev. D 86, 124023 (2012). arXiv:1207.6348

    Article  ADS  Google Scholar 

  16. Haggard, H.M., Rovelli, C., Wieland, W., Vidotto, F.: Spin connection of twisted geometry. Phys. Rev. D 87(2), 024038 (2013). arXiv:1211.2166

    Article  ADS  Google Scholar 

  17. Bianchi, E., Dona, P., Speziale, S.: Polyhedra in loop quantum gravity. Phys. Rev. D 83, 044035 (2011). arXiv:1009.3402

    Article  ADS  Google Scholar 

  18. Livine, E.R.: Deformations of polyhedra and polygons by the unitary group. J. Math. Phys. 54, 123504 (2013). arXiv:1307.2719

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Barbieri, A.: Quantum tetrahedra and simplicial spin networks. Nucl. Phys. B 518, 714–728 (1998). arXiv:gr-qc/9707010

    Article  ADS  MathSciNet  Google Scholar 

  20. Baez, J.C., Barrett, J.W.: The Quantum tetrahedron in three-dimensions and four-dimensions. Adv. Theor. Math. Phys. 3, 815–850 (1999). arXiv:gr-qc/9903060

    Article  MathSciNet  MATH  Google Scholar 

  21. Livine, E.R.: Deformation operators of spin networks and coarse-graining. Class. Quantum Gravity 31, 075004 (2014). arXiv:1310.3362

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Dittrich, B., Geiller, M.: A new vacuum for loop quantum gravity. Class. Quantum Gravity 32(11), 112001 (2015). arXiv:1401.6441

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Dittrich, B., Geiller, M.: Flux formulation of loop quantum gravity: classical framework. Class. Quantum Gravity 32(13), 135016 (2015). arXiv:1412.3752

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Charles, C., Livine, E.R.: The fock space of loopy spin networks for quantum gravity. Gen. Relativ. Gravit. 48(8), 113 (2016). arXiv:1603.01117

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Charles, C., Livine, E.R.: The closure constraint for the hyperbolic tetrahedron as a Bianchi identity. Gen. Relativ. Gravit. 49(7), 92 (2017). arXiv:1607.08359

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Freidel, L., Perez, A., Pranzetti, D.: Loop gravity string. Phys. Rev. D 95(10), 106002 (2017). arXiv:1611.03668

    Article  ADS  MathSciNet  Google Scholar 

  27. Borja, E.F., Freidel, L., Garay, I., Livine, E.R.: U(N) tools for loop quantum gravity: the return of the spinor. Class. Quantum Gravity 28, 055005 (2011). arXiv:1010.5451

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Dupuis, M., Speziale, S., Tambornino, J.: Spinors and twistors in loop gravity and spin foams. PoS QGQGS2011, 021 (2011). arXiv:1201.2120

    Google Scholar 

  29. Dupuis, M., Livine, E.R.: Holomorphic simplicity constraints for 4d spinfoam models. Class. Quantum Gravity 28, 215022 (2011). arXiv:1104.3683

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Bonzom, V., Livine, E.R.: Generating functions for coherent intertwiners. Class. Quantum Gravity 30, 055018 (2013). arXiv:1205.5677

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Freidel, L., Hnybida, J.: On the exact evaluation of spin networks. J. Math. Phys. 54, 112301 (2013). arXiv:1201.3613

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Bonzom, V., Costantino, F., Livine, E.R.: Duality between spin networks and the 2D Ising model. Commun. Math. Phys. 344(2), 531–579 (2016). arXiv:1504.02822

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Freidel, L., Girelli, F., Livine, E.R.: The relativistic particle: Dirac observables and Feynman propagator. Phys. Rev. D 75, 105016 (2007). arXiv:hep-th/0701113

    Article  ADS  MathSciNet  Google Scholar 

  34. Dittrich, B., Ryan, J.P.: Simplicity in simplicial phase space. Phys. Rev. D 82, 064026 (2010). arXiv:1006.4295

    Article  ADS  Google Scholar 

  35. Dupuis, M., Freidel, L., Livine, E.R., Speziale, S.: Holomorphic Lorentzian simplicity constraints. J. Math. Phys. 53, 032502 (2012). arXiv:1107.5274

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Kilchrist, M., Packard, D.: The Weierstrass–Enneper representations. Dynamics at the Horsetooth, vol. 4 (2012). http://www.math.colostate.edu/~shipman/47/volume42011/M641_MKilchrist_Packard.pdf

  37. Långvik, M., Speziale, S.: Twisted geometries, twistors and conformal transformations. Phys. Rev. D 94(2), 024050 (2016). arXiv:1602.01861

    Article  ADS  MathSciNet  Google Scholar 

  38. Yang, J., Ma, Y.: Quasi-local energy in loop quantum gravity. Phys. Rev. D 80, 084027 (2009). arXiv:0812.3554

    Article  ADS  MathSciNet  Google Scholar 

  39. Feller, A., Livine, E.R.: Quantum surface and intertwiner dynamics in loop quantum gravity. Phys. Rev. D 95(12), 124038 (2017). arXiv:1703.01156

    Article  ADS  MathSciNet  Google Scholar 

  40. Bonzom, V., Dupuis, M., Girelli, F., Livine, E.R.: Deformed phase space for 3d loop gravity and hyperbolic discrete geometries. arXiv:1402.2323

  41. Bonzom, V., Dupuis, M., Girelli, F.: Towards the Turaev–Viro amplitudes from a Hamiltonian constraint. Phys. Rev. D 90(10), 104038 (2014). arXiv:1403.7121

    Article  ADS  Google Scholar 

  42. Dupuis, M., Girelli, F., Livine, E.R.: Deformed spinor networks for loop gravity: towards hyperbolic twisted geometries. Gen. Relativ. Gravit. 46(11), 1802 (2014). arXiv:1403.7482

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Dittrich, B., Geiller, M.: Quantum gravity kinematics from extended TQFTs. New J. Phys. 19(1), 013003 (2017). arXiv:1604.05195

    Article  ADS  Google Scholar 

  44. Delcamp, C., Dittrich, B., Riello, A.: Fusion basis for lattice gauge theory and loop quantum gravity. JHEP 02, 061 (2017). arXiv:1607.08881

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Dittrich, B.: (3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces. JHEP 05, 123 (2017). arXiv:1701.02037

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Freidel, L., Yokokura, Y.: Non-equilibrium thermodynamics of gravitational screens. Class. Quantum Gravity 32(21), 215002 (2015). arXiv:1405.4881

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

    Laurent Freidel & Etera R. Livine

  2. Laboratoire de Physique, ENS de Lyon, CNRS, UCBL, Université de Lyon, 69342, Lyon, France

    Etera R. Livine

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Correspondence to Etera R. Livine.

Appendix A: Holonomy reconstruction from the canonical pair of vectors

Appendix A: Holonomy reconstruction from the canonical pair of vectors

Lemma A.1

Let us consider a pair of 3-vectors \((\vec {x},\vec {p})\) such that \(|\vec {x}\wedge \vec {p}|\ne 0\). We consider the symplectic generators:

$$\begin{aligned} \ell _{0}=\vec {x}\cdot \vec {p},\quad \ell _{-}=|\vec {x}|^{2},\quad \ell _{+}=|\vec {p}|^{2},\quad {{\mathcal {D}}}=(\ell _{-}\ell _{+}-\ell _{0}^{2})=|\vec {x}\wedge \vec {p}|^{2}\ne 0. \end{aligned}$$

There exists a unique rotation \(h_{\vec {x},\vec {p}}\in \mathrm {SO}(3)\) mapping the reference pair \((|\vec {x}|{\hat{e}}_{1},\vec {v})\) to \((\vec {x},\vec {p})\) with:

$$\begin{aligned} |\vec {x}|{\hat{e}}_{1} = |\vec {x}| \left( \begin{array}{c}1\\ 0\\ 0\end{array} \right) ,\quad \vec {v}= \frac{1}{|\vec {x}|} \left( \begin{array}{c}\vec {x}\cdot \vec {p}\\ |\vec {x}\wedge \vec {p}|\\ 0 \end{array} \right) = \frac{1}{\sqrt{\ell _{-}}} \left( \begin{array}{c}\ell _{0} \\ \sqrt{{{\mathcal {D}}}}\\ 0\end{array} \right) , \end{aligned}$$

which is given by:

$$\begin{aligned} h_{\vec {x},\vec {p}} = \Bigg ( \frac{\vec {x}}{|\vec {x}|} ,\, \frac{(\vec {x}\wedge \vec {p})\wedge \vec {x}}{|\vec {x}|\,|\vec {x}\wedge \vec {p}|} ,\, \frac{\vec {x}\wedge \vec {p}}{|\vec {x}\wedge \vec {p}|} \Bigg ) = \Bigg ( \frac{\vec {x}}{|\vec {x}|} ,\, \frac{|\vec {x}|^{2}\vec {p}-(\vec {x}\cdot \vec {p})\vec {x}}{|\vec {x}|\,|\vec {x}\wedge \vec {p}|} ,\, \frac{\vec {x}\wedge \vec {p}}{|\vec {x}\wedge \vec {p}|} \Bigg ) \nonumber \\ \end{aligned}$$
(40)

Proof

The matrix \(h_{\vec {x},\vec {p}}\) maps the (Oxy) plane to the plane spanned by the two vectors \((\vec {x},\vec {p})\) and sends the direction \({\hat{e}}_{3}\) to the angular momentum \(\vec {J}\). One simply needs to check that the three columns of h form a positive orthonormal basis of \({{\mathbb {R}}}^{3}\) to prove that \(h\in \mathrm {SO}(3)\). \(\square \)

Now we can combine the two rotations \(h_{\vec {x}^{s},\vec {p}^{s}}\) and \(h_{\vec {x}^{t},\vec {p}^{t}}\) to get the \(\mathrm {SO}(3)\) holonomy living along the oriented edge e:

$$\begin{aligned} h = h_{\vec {x}^{t},\vec {p}^{t}}\,\big (h_{\vec {x}^{s},\vec {p}^{s}}\big )^{-1} = h_{\vec {x}^{t},\vec {p}^{t}}\,{}^{t}h_{\vec {x}^{s},\vec {p}^{s}}. \end{aligned}$$
(41)

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Freidel, L., Livine, E.R. Bubble networks: framed discrete geometry for quantum gravity. Gen Relativ Gravit 51, 9 (2019). https://doi.org/10.1007/s10714-018-2493-y

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