At the Frontier of Spacetime pp 247-308 | Cite as

# Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

## Abstract

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic \(\mathbb {R}^{4}\), and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called *smooth quantum gravity* by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard \(\mathbb {R}^{4}\) to the exotic \(R^{4}\) is a quantization of geometry.

## Notes

### Acknowledgments

I have to thank Carl for 20 years of friendship and collaboration as well numerous discussions. Special thanks to Jerzy Król for our work and many discussions about fundamental problems in math and physics. Now I understand the importance of Model theory. Many thanks to Paul Schultz for reading and corrections.

## References

- 1.H. Abchir, TQFT invariants at infinity for the Whitehead manifold, in
*Knots in Hellas ’98*, ed. by C. McA, V.F.R. Gordan, L. Jones, S. Lambropoulou Kauffman, J.H. Przytycki (World Scientific, Singapore, 1998), pp. 1–17Google Scholar - 2.S. Akbulut, R. Kirby, Mazur manifolds. Mich. Math. J.
**26**, 259–284 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 3.A. Ashtekar, J. Engle, D. Sloan, Asymptotics and Hamiltonians in a first order formalism. Class. Quant. Grav.
**25**, 095020 (2008), arXiv:0802.2527 - 4.A. Ashtekar, D. Sloan, Action and Hamiltonians in higher dimensional general relativity: first order framework. Class. Quant. Grav.
**25**, 225025 (2008), arXiv:0808.2069 - 5.T. Asselmeyer, Generation of source terms in general relativity by differential structures. Class. Quant. Grav.
**14**, 749–758 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 6.T. Asselmeyer-Maluga, Exotic smoothness and quantum gravity. Class. Q. Grav.
**27**, 165002 (2010), arXiv:1003.5506 - 7.T. Asselmeyer-Maluga, C.H. Brans, Cosmological anomalies and exotic smoothness structures. Gen. Rel. Grav.
**34**, 1767–1771 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 8.T. Asselmeyer-Maluga, C.H. Brans,
*Exotic Smoothness and Physics*(World Scientific Publishing, Singapore, 2007)CrossRefzbMATHGoogle Scholar - 9.T. Asselmeyer-Maluga, C.H. Brans,
*Smoothly Exotic Black Holes*, Space Science, Exploration and Policies (NOVA publishers, 2012), pp. 139–156Google Scholar - 10.T. Asselmeyer-Maluga, C.H. Brans, How to include fermions into general relativity by exotic smoothness. Gen. Relativ. Grav.
**47**, 30 (2015), doi: 10.1007/s10714-015-1872-x, arXiv:1502.02087 - 11.T. Asselmeyer-Maluga, J. Król, Exotic smooth \({\mathbb{R}}^{4}\), noncommutative algebras and quantization (2010), arXiv:1001.0882
- 12.T. Asselmeyer-Maluga, J. Król, Small exotic smooth \(R^4\) and string theory, in
*International Congress of Mathematicians ICM 2010 Short Communications Abstracts Book*, ed. by R. Bathia (Hindustan Book Agency, 2010), p. 400Google Scholar - 13.T. Asselmeyer-Maluga, J. Król, Constructing a quantum field theory from spacetime (2011), arXiv:1107.3458
- 14.T. Asselmeyer-Maluga, J. Król, Topological quantum d-branes and wild embeddings from exotic smooth \(R^4\). Int. J. Mod. Phys. A
**26**, 3421–3437 (2011), arXiv:1105.1557 - 15.T. Asselmeyer-Maluga, J. Król, On topological restrictions of the spacetime in cosmology. Mod. Phys. Lett. A
**27**, 1250135 (2012), arXiv:1206.4796 - 16.T. Asselmeyer-Maluga, J. Król, Quantum geometry and wild embeddings as quantum states. Int. J. Geom. Methods Modern Phys.
**10**(10) (2013), will be published in Nov. 2013, arXiv:1211.3012 - 17.T. Asselmeyer-Maluga, J. Król, Abelian gerbes, generalized geometries and foliations of small exotic \(R^4\), arXiv:0904.1276v5, subm. to Rev. Math. Phys. (2014)
- 18.T. Asselmeyer-Maluga, J. Król, Inflation and topological phase transition driven by exotic smoothness. Adv. HEP, Article ID 867460, 14p (2014), doi: 10.1155/2014/867460
- 19.T. Asselmeyer-Maluga, R. Mader, Exotic \(R^4\) and quantum field theory, in
*7th International Conference on Quantum Theory and Symmetries (QTS7)*, ed. by C. Burdik et al. (IOP Publishing, Bristol, UK, 2012), p. 012011, arXiv:1112.4885, doi: 10.1088/1742-6596/343/1/012011 - 20.T. Asselmeyer-Maluga, H. Rosé, On the geometrization of matter by exotic smoothness. Gen. Rel. Grav.
**44**, 2825–2856 (2012), doi: 10.1007/s10714-012-1419-3, arXiv:1006.2230 - 21.N. Berline, M. Vergne, E. Getzler,
*Heat kernels and Dirac Operators*(Springer, New York, 1992)CrossRefzbMATHGoogle Scholar - 22.P. Bertozzini, R. Conti, W. Lewkeeratiyutkul, Modular theory, non-commutative geometry and quantum gravity. SIGMA
**6**, 47pp (2010), arXiv:1007.4094 - 23.Z. Bizaca, An explicit family of exotic Casson handles. Proc. AMS
**123**, 1297–1302 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Z̆. Biz̆aca, R Gompf, Elliptic surfaces and some simple exotic \({\mathbb{R}}^4\)’s. J. Diff. Geom.
**43**, 458–504 (1996)Google Scholar - 25.H.J. Borchers, On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys.
**41**, 3604–3673 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 26.R. Bott, L.W. Tu,
*Differential Forms in Algebraic Topology*. Graduate Texts in Mathematics, vol. 82 (Springer, 1995)Google Scholar - 27.B.H. Bowditch, Markoff triples and quasifuchsian groups. Proc. Lond. Math. Soc.
**77**, 697–736 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 28.C. Branciard, How \(\varPsi \)-epistemic models fail at explaining the indistinguishability of quantum states. Phys. Rev. Lett.
**113**, 020409 (2014), doi: 10.1103/PhysRevLett.113.020409 - 29.C. Brans, Absolulte spacetime: the twentieth century ether. Gen. Rel. Grav.
**31**, 597 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 30.C.H. Brans, Exotic smoothness and physics. J. Math. Phys.
**35**, 5494–5506 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 31.C.H. Brans, Localized exotic smoothness. Class. Quant. Grav.
**11**, 1785–1792 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 32.C.H. Brans, D. Randall, Exotic differentiable structures and general relativity. Gen. Rel. Grav.
**25**, 205 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 33.R. Budney, JSJ-decompositions of knot and link complements in the 3-sphere. L’enseignement Mathématique
**52**, 319–359 (2006), arXiv:math/0506523 - 34.D. Bullock, A finite set of generators for the Kauffman bracket skein algebra. Math. Z.
**231**, 91–101 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 35.D. Bullock, C. Frohman, J. Kania-Bartoszyńska, Topological interpretations of lattice gauge field theory. Comm. Math. Phys.
**198**, 47–81 (1998), arXiv:q-alg/9710003 - 36.D. Bullock, C. Frohman, J. Kania-Bartoszyńska, Understanding the Kauffman bracket skein module. J. Knot Theory Ramif.
**8**, 265–277 (1999), arXiv:q-alg/9604013 - 37.D. Bullock, C. Frohman, J. Kania-Bartoszyńska, The Kauffman bracket skein as an algebra of observables. Proc. AMS
**130**, 2479–2485 (2002), arXiv:math/0010330 - 38.D. Bullock, J.H. Przytycki, Multiplicative structure of Kauffman bracket skein module quantization. Proc. AMS
**128**, 923–931 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 39.A. Casson,
*Three Lectures on New Infinite Constructions in 4-Dimensional Manifolds*, vol. 62. Birkhäuser, Progress in Mathematics Edition (1986). Notes by Lucian Guillou, first published 1973Google Scholar - 40.V. Chernov, S. Nemirovski, Smooth cosmic censorship. Comm. Math. Phys.
**320**, 469–473 (2013), arXiv:1201.6070 - 41.A. Connes, A survey of foliations and operator algebras. Proc. Symp. Pure Math.
**38**, 521–628 (1984). See www.alainconnes.org - 42.A. Connes,
*Non-commutative Geometry*(Academic Press, 1994)Google Scholar - 43.A. Connes, C. Rovelli, Von neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories. Class. Quant. Grav.
**11**, 2899–2918 (1994), arXiv:gr-qc/9406019 - 44.M. Culler, P.B. Shalen, Varieties of group representations and splittings of 3-manifolds. Ann. Math.
**117**, 109–146 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 45.G. Daskalopoulos, S. Dostoglou, R. Wentworth, Character varieties and harmonic maps to \({\mathbf{r}}\)-trees. Math. Res. Lett.
**5**, 523–533 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 46.S. DeMichelis, M.H. Freedman, Uncountable many exotic \({R}^4\)’s in standard 4-space. J. Diff. Geom.
**35**, 219–254 (1992)MathSciNetzbMATHGoogle Scholar - 47.S. Donaldson, An application of gauge theory to the topology of 4-manifolds. J. Diff. Geom.
**18**, 279–315 (1983)MathSciNetzbMATHGoogle Scholar - 48.S. Donaldson, Irrationality and the h-cobordism conjecture. J. Diff. Geom.
**26**, 141–168 (1987)MathSciNetzbMATHGoogle Scholar - 49.C. Duston, Exotic smoothness in 4 dimensions and semiclassical Euclidean quantum gravity. Int. J. Geom. Meth. Mod. Phys.
**8**, 459–484 (2010), arXiv:0911.4068 - 50.Fermi GBM/LAT Collaborations, Testing Einstein’s special relativity with Fermi’s short hard gamma-ray burst GRB090510. Nature
**462**, 331–334 (2009), arXiv:0908.1832 - 51.R. Fintushel, R. Stern, Knots, links, and 4-manifolds. Inv. Math,
**134**, 363–400 (1998), arXiv:dg-ga/9612014 - 52.A. Floer, An instanton invariant for 3-manifolds. Comm. Math. Phys.
**118**, 215–240 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 53.M.H. Freedman, The topology of four-dimensional manifolds. J. Diff. Geom.
**17**, 357–454 (1982)MathSciNetzbMATHGoogle Scholar - 54.M.H. Freedman, The disk problem for four-dimensional manifolds. Proc. Internat. Cong. Math. Warzawa
**17**, 647–663 (1983)Google Scholar - 55.M.H. Freedman, There is no room to spare in four-dimensional space. Not. Am. Math. Soc.
**31**, 3–6 (1984)ADSMathSciNetzbMATHGoogle Scholar - 56.D. Friedan, Nonlinear models in \(2+\epsilon \) dimensions. Phys. Rev. Lett.
**45**, 1057–1060 (1980)ADSMathSciNetCrossRefGoogle Scholar - 57.T. Friedrich, On the spinor representation of surfaces in euclidean 3-space. J. Geom. Phys.
**28**, 143–157 (1998), arXiv:dg-ga/9712021v1 - 58.C. Frohman, R. Gelca, Skein modules and the noncommutative torus. Trans. AMS
**352**, 4877–4888 (2000), arXiv:math/9806107 - 59.S. Ganzell, Ends of 4-manifolds. Top. Proc.
**30**, 223–236 (2006), http://faculty.smcm.edu/sganzell/ends.pdf - 60.D. Gabai, Foliations and the Topology of 3-Manifolds. J. Diff. Geom.
**18**, 445–503 (1983)MathSciNetzbMATHGoogle Scholar - 61.W.M. Goldman, The symplectic nature of the fundamental groups of surfaces. Adv. Math.
**54**, 200–225 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 62.M. Golubitsky, V. Guillemin,
*Stable Mappings and their Singularities*. Graduate Texts in Mathematics, vol. 14 (Springer, New York-Heidelberg-Berlin, 1973)Google Scholar - 63.R. Gompf, Infinite families of casson handles and topological disks. Topology
**23**, 395–400 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 64.R. Gompf, An infinite set of exotic \({\mathbb{R}}^4\)’s. J. Diff. Geom.
**21**, 283–300 (1985)MathSciNetzbMATHGoogle Scholar - 65.R. Gompf, Periodic ends and knot concordance. Top. Appl.
**32**, 141–148 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 66.R.E. Gompf, Three exotic \(R^4\)’s and other anomalies. J. Diff. Geom.
**18**, 317–328 (1983)MathSciNetzbMATHGoogle Scholar - 67.R.E. Gompf, S. Singh, On Freedman’s reimbedding theorems, in
*Four-Manifold Theory*, vol. 35, ed. by C. Gordan, R. Kirby (AMS, Providence Rhode Island, 1984), pp. 277–310Google Scholar - 68.R.E. Gompf, A.I. Stipsicz,
*4-manifolds and Kirby Calculus*(American Mathematical Society, 1999)Google Scholar - 69.S.W. Hawking, The path-integral approach to quantum gravity, in
*General Relativity. An Einstein Centenary Survey*, ed. by I. Hawking (Cambridge University Press, Cambridge, 1979), pp. 746–789Google Scholar - 70.M.W. Hirsch,
*Differential Topology*(Springer, New York, 1976)CrossRefzbMATHGoogle Scholar - 71.J. Hubbard, H. Masur, Quadratic differentials and foliations. Acta Math.
**142**, 221–274 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 72.S. Hurder, A. Katok, Secondary classes and trasnverse measure theory of a foliation. Bull. AMS
**11**, 347–349 (1984). Announced results onlyGoogle Scholar - 73.W. Jaco, P. Shalen,
*Seifert Fibered Spaces in 3-manifolds*, Memoirs of the American Mathematical Society, vol. 21 (AMS, 1979)Google Scholar - 74.T. Kato, ASD moduli space over four-manifolds with tree-like ends. Geom. Top.
**8**, 779–830 (2004), arXiv:math.GT/0405443 - 75.M.S. Leifer, Is the quantum state real? an extended review of \(\psi \)-ontology theorems. Quanta
**3**, 67–155 (2014), doi: 10.12743/quanta.v3i1.22 - 76.J. Mather, Stability of \(C^\infty \) mappings. VI: the nice dimensions, in
*Proccedings of the Liverpool Singularities Symposium*, pp. 207–253. Springer Lecture Notes in Mathematyics, vol. 192 (1971)Google Scholar - 77.J. Milnor, A unique decomposition theorem for 3-manifolds. Am. J. Math.
**84**, 1–7 (1962)MathSciNetCrossRefzbMATHGoogle Scholar - 78.G.D. Mostow, Quasi-conformal mappings in \(n\)-space and the rigidity of hyperbolic space forms. Publ. Math. IHÉS
**34**, 53–104 (1968)MathSciNetCrossRefzbMATHGoogle Scholar - 79.R. Penrose, Nonlinear gravitons and curved twistor theory. Gen. Relativ. Grav.
**7**, 31–52 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 80.H. Pfeiffer, Quantum general relativity and the classification of smooth manifolds. Report number: DAMTP 2004-32 (2004)Google Scholar
- 81.V.V. Prasolov, A.B. Sossinisky,
*Knots, Links, Braids and 3-Manifolds*(AMS, Providence, 1997)Google Scholar - 82.B.L. Reinhart, J.W. Wood, A metric formula for the Godbillon-Vey invariant for foliations. Proc. AMS
**38**, 427–430 (1973)MathSciNetzbMATHGoogle Scholar - 83.M. Ringbauer, B. Duffus, C. Branciard, E.G. Cavalcanti, A.G. White, A. Fedrizzi, Measurements on the reality of the wavefunction. Nat. Phys.
**11**, 249–254 (2015), doi: 10.1038/NPHYS3233 - 84.P. Scott, The geometries of 3-manifolds. Bull. Lond. Math. Soc.
**15**, 401–487 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 85.A.R. Skovborg, The Moduli Space of Flat Connections on a Surface Poisson Structures and Quantization. Ph.D. thesis, Universty Aarhus (2006)Google Scholar
- 86.J. Sładkowski, Strongly gravitating empty spaces. Preprint arXiv:gr-qc/9906037 (1999)
- 87.J. Sładkowski, Gravity on exotic \({\mathbb{R}}^{4}\) with few symmetries. Int. J. Mod. Phys. D
**10**, 311–313 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 88.J. Stallings, Piecewise-linear structure of euclidean space. Proc. Camb. Phil. Soc.
**58**, 481 (1962)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 89.A.A. Starobinski, A new type of isotropic cosmological models without singularity. Phys. Lett.
**91B**, 99–102 (1980)ADSCrossRefGoogle Scholar - 90.K. Strebel,
*Quadratic Differentials*. A Series of Modern Surveys in Mathematics, vol. 5 (Springer, Berlin-Heidelberg, 1984)Google Scholar - 91.M. Takesaki,
*Tomita’s Theory of Modular Hilbert Algebras and its Applications*, Lecture Notes in Mathematics, vol. 128 (Springer, Berlin, 1970)Google Scholar - 92.I. Tamura,
*Topology of Foliations: An Introduction*. Translations of Mathematical Monographs, vol. 97 (AMS, Providence, 1992)Google Scholar - 93.C.H. Taubes, Gauge theory on asymptotically periodic 4-manifolds. J. Diff. Geom.
**25**, 363–430 (1987)MathSciNetzbMATHGoogle Scholar - 94.W. Thurston, Noncobordant foliations of \(S^3\). Bull. AMS
**78**, 511–514 (1972)MathSciNetCrossRefzbMATHGoogle Scholar - 95.W. Thurston,
*Three-Dimensional Geometry and Topology*, 1st edn. (Princeton University Press, Princeton, 1997)zbMATHGoogle Scholar - 96.M. Tomita, On canonical forms of von neumann algebras, in
*Fifth Functional Analysis Sympos (Tôhoku Univ., Sendai, 1967)*, pp. 101–102, Sendai (1967). Tôhoku Univ., Math. InstGoogle Scholar - 97.V. Turaev, Algebras of loops on surfaces, algebras of knots, and quantization. Adv. Ser. Math. Phys.
**9**, 59–95 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 98.V.G. Turaev, Skein quantization of poisson algebras of loops on surfaces. Ann. Sci. de l’ENS
**24**, 635–704 (1991)MathSciNetzbMATHGoogle Scholar - 99.J.H.C. Whitehead, A certain open manifold whose group is unity. Quart. J. Math. Oxf.
**6**, 268–279 (1935)ADSCrossRefzbMATHGoogle Scholar - 100.B. Whitt, Fourth order gravity as general relativity plus matter. Phys. Lett.
**145B**, 176–178 (1984)ADSMathSciNetCrossRefGoogle Scholar - 101.D.K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions. SIGMA
**5**, 080 (2009), arXiv:0904.1738 - 102.D.K. Wise, Macdowell-Mansouri gravity and Cartan geometry. Class. Quantum Grav.
**27**, 155010 (2010), arXiv:gr-qc/0611154 - 103.E. Witten, 2\(+\)1 dimensional gravity as an exactly soluble system. Nucl. Phys.
**B311**, 46–78 (1988/89)Google Scholar - 104.E. Witten, Topology-changing amplitudes in 2\(+\)1 dimensional gravity. Nucl. Phys. B
**323**, 113–140 (1989)ADSMathSciNetCrossRefGoogle Scholar - 105.E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group. Comm. Math. Phys.
**137**, 29–66 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 106.M. Wolf, Harmonic maps from surfaces to \({\mathbf{R}}\)-trees. Math. Z.
**218**, 577–593 (1995)MathSciNetCrossRefzbMATHGoogle Scholar