Particulate exotica. (English) Zbl 1493.81007
Summary: Recent advances in differential topology single out four-dimensions as being special, allowing for vast varieties of exotic smoothness (differential) structures, distinguished by their handlebody decompositions, even as the coarser algebraic topology is fixed. Should the spacetime we reside in takes up one of the more exotic choices, and there is no obvious reason why it shouldn’t, apparent pathologies would inevitably plague calculus-based physical theories assuming the standard vanilla structure, due to the non-existence of a diffeomorphism and the consequent lack of a suitable portal through which to transfer the complete information regarding the exotic physical dynamics into the vanilla theories. An obvious plausible consequence of this deficiency would be the uncertainty permeating our attempted description of the microscopic world. We tentatively argue here, that a re-inspection of the key ingredients of the phenomenological particle models, from the perspective of exotica, could possibly yield interesting insights. Our short and rudimentary discussion is qualitative and speculative, because the necessary mathematical tools have only just began to be developed.
MSC:
| 81V25 | Other elementary particle theory in quantum theory |
| 14R10 | Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) |
| 57R55 | Differentiable structures in differential topology |
| 57K40 | General topology of 4-manifolds |
References:
| [1] | Wheeler, JA, Geometrodynamics (1963), New York: Academic Press, New York · Zbl 0124.22207 |
| [2] | Misner, CW; Wheeler, JA, Classical physics as geometry, Ann. Phys., 2, 6, 525 (1957) · Zbl 0078.19106 · doi:10.1016/0003-4916(57)90049-0 |
| [3] | Wheeler, JA, Geons, Phys. Rev., 97, 511 (1955) · Zbl 0064.22404 · doi:10.1103/PhysRev.97.511 |
| [4] | Anderson, E.: Geometrodynamics: Spacetime or Space?, arXiv e-prints gr-qc/0409123 (2004) |
| [5] | Atiyah, M.; Manton, NS; Schroers, BJ, Geometric models of matter, Proc. R. Soc. Lond., A468, 1252 (2012) · Zbl 1364.53046 · doi:10.1098/rspa.2011.0616 |
| [6] | Thomson, W., On vortex atoms, Proc. R. Soc. Edinburgh, 1, 94 (1967) |
| [7] | Candelas, P.; Horowitz, GT; Strominger, A.; Witten, E., Vacuum configurations for superstrings, Nucl. Phys. B, 258, 46 (1985) · doi:10.1016/0550-3213(85)90602-9 |
| [8] | Rubakov, V.; Shaposhnikov, M., Do we live inside a domain wall?, Phys. Lett. B, 125, 2, 136 (1983) · doi:10.1016/0370-2693(83)91253-4 |
| [9] | Daverman, RJ; Venema, GA, Embeddings in Manifolds (2009), Providence: American Mathematical Society, Providence · Zbl 1209.57002 · doi:10.1090/gsm/106 |
| [10] | Gompf, RE; Stipsicz, AI, 4-Manifolds and Kirby calculus (1999), Providence: American Mathematical Society, Providence · Zbl 0933.57020 · doi:10.1090/gsm/020 |
| [11] | Asselmeyer-Maluga, T.; Brans, C., Exotic Smoothness and Physics (2007), Singapore: World Scientific, Singapore · Zbl 1177.83001 · doi:10.1142/4323 |
| [12] | Scorpan, A., The Wild World of 4-Manifolds (2005), Providence, Gainesville: American Mathematical Society/, Rhode Island, University of Florida, Providence, Gainesville · Zbl 1075.57001 |
| [13] | Gompf, RE, An infinite set of exotic \({\mathbf{R}}^4\)’s, J. Differ. Geom., 21, 2, 283 (1985) · Zbl 0562.57009 · doi:10.4310/jdg/1214439566 |
| [14] | Stallings, J., The piecewise-linear structure of euclidean space, Math. Proc. Camb. Philos. Soc., 58, 3, 481-488 (1962) · Zbl 0107.40203 · doi:10.1017/S0305004100036756 |
| [15] | Akhmedov, A., Park, B.D.: Exotic smooth structures on \(S^2\times S^2\), arXiv e-prints arXiv:1005.3346 (2010) |
| [16] | Mazur, B.: A note on some contractible 4-manifolds. Ann. Math. 73(1), 221 (1961). http://www.jstor.org/stable/1970288 · Zbl 0127.13604 |
| [17] | Poénaru, V.: Les décompositions de l’hypercube en produit topologique, Bulletin de la Société Mathématique de France 88, 113 (1960). doi:10.24033/bsmf.1546. http://www.numdam.org/item/BSMF_1960__88__113_0 · Zbl 0135.41704 |
| [18] | Yasui, K.: Nonexistence of stein structures on 4-manifolds and maximal thurston-bennequin numbers, arXiv: Geometric Topology (2015) · Zbl 1369.57030 |
| [19] | Knudsen, B., Kupers, A.: Embedding calculus and smooth structures, arXiv e-prints arXiv:2006.03109 (2020) |
| [20] | Zhang, F., Equiaffine braneworld, Galaxies, 8, 4, 73 (2020) · doi:10.3390/galaxies8040073 |
| [21] | Asselmeyer-Maluga, T.; Król, J., How to obtain a cosmological constant from small exotic \(\text{ R}^4 \), Phys. Dark Univ., 19, 66 (2018) · doi:10.1016/j.dark.2017.12.002 |
| [22] | Duston, C., Exotic smoothness in four dimensions and euclidean quantum gravity, Int. J. Geom. Meth. Mod. Phys., 8, 459 (2011) · Zbl 1219.83082 · doi:10.1142/S0219887811005233 |
| [23] | Etesi, G., Global solvability of the vacuum Einstein equation and the strong cosmic censorship in four dimensions, J. Geom. Phys., 164, 104164 (2021) · Zbl 1464.83015 · doi:10.1016/j.geomphys.2021.104164 |
| [24] | Brans, CH; Randall, D., Exotic differentiable structures and general relativity, Gen. Rel. Grav., 25, 205 (1993) · Zbl 0771.57012 · doi:10.1007/BF00758828 |
| [25] | Brans, CH, Exotic smoothness and physics, J. Math. Phys., 35, 5494 (1994) · Zbl 0829.57010 · doi:10.1063/1.530761 |
| [26] | Asselmeyer-Maluga, T., Exotic smoothness and quantum gravity, Class. Quant. Grav., 27, 165002 (2010) · Zbl 1195.83035 · doi:10.1088/0264-9381/27/16/165002 |
| [27] | Planat, M.; Aschheim, R.; Amaral, MM; Irwin, K., Quantum computation and measurements from an exotic space-time R4, Symmetry, 12, 5, 736 (2020) · doi:10.3390/sym12050736 |
| [28] | Moise, E.E.: Affine structures in 3-manifolds: V. the triangulation theorem and hauptvermutung. Ann. Math. 56(1), 96 (1952). http://www.jstor.org/stable/1969769 · Zbl 0048.17102 |
| [29] | Bing, R.H.: The cartesian product of a certain nonmanifold and a line is e4. Ann. Math. 70(3), 399 (1959). http://www.jstor.org/stable/1970322 · Zbl 0089.39501 |
| [30] | Akbulut, S., 4-Manifolds (2016), Oxford: Oxford University Press, Oxford · Zbl 1377.57002 · doi:10.1093/acprof:oso/9780198784869.001.0001 |
| [31] | Cartan, É.: La théorie des groupes finis et continus et la géométrie différentielle, traitées par la méthode du repere mobile. Leçons professées à la Sorbonne. (1951) · Zbl 0054.01401 |
| [32] | Ivey, T.; Landsberg, J., Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (2003), New York: American Mathematical Society, New York · Zbl 1105.53001 |
| [33] | Hooft, G.t.: Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy, arXiv e-prints arXiv:2103.04335 (2021) |
| [34] | Iwase, Z., Dehn surgery along a torus \(t^2\)-knot ii, Jap. J. Math., 16, 2, 171 (1990) · Zbl 0719.57013 · doi:10.4099/math1924.16.171 |
| [35] | Akbulut, S., Yasui, K.: Corks, Plugs and exotic structures, J. Gökova Geom. Topol. GGT (2), arXiv:0806.3010 (2008) · Zbl 1214.57027 |
| [36] | Melvin, P., Schwartz, H.: Higher order corks, arXiv e-prints arXiv:1902.02840 (2019) · Zbl 1472.57026 |
| [37] | Kirby, R.: Akbulut’s corks and h-cobordisms of smooth simply connected 4-manifolds, arXiv Mathematics e-prints math/9712231 (1997) |
| [38] | Matveyev, R., A decomposition of smooth simply-connected h-cobordant 4-manifolds, J. Differ. Geom., 44, 571 (1996) · Zbl 0885.57016 · doi:10.4310/jdg/1214459222 |
| [39] | Curtis, CL; Freedman, MH; Stong, HWCR, A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds, Invent. Math., 123, 2, 343 (1996) · Zbl 0843.57020 · doi:10.1007/s002220050031 |
| [40] | Akbulut, S.; Matveyev, R., A convex decomposition theorem for 4-manifolds, Int. Math. Res. Notices, 7, 371 (1998) · Zbl 0911.57025 · doi:10.1155/S1073792898000245 |
| [41] | Akbulut, S.: Exotic Structures on smooth 4-manifolds, arXiv e-prints arXiv:0807.4248 (2008) · Zbl 0779.57010 |
| [42] | Donaldson, SK, An application of gauge theory to four-dimensional topology, J. Differ. Geom., 18, 2, 279 (1983) · Zbl 0507.57010 · doi:10.4310/jdg/1214437665 |
| [43] | Seiberg, N.; Witten, E., Electric-magnetic duality, monopole condensation, and confinement in n=2 supersymmetric yang-mills theory, Nucl. Phys. B, 426, 1, 19 (1994) · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [44] | Akbulut, S.: On infinite order corks, arXiv e-prints arXiv:1605.09348 (2016) · Zbl 1378.32007 |
| [45] | Darboux, J.G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal (Gauthier-Villars, Paris, 1915), vol. 2. Livre 4, chapitre 3: L’équation d’Euler et de Poisson · JFM 45.0881.05 |
| [46] | Colombeau, JF, New Generalized Functions and Multiplication of Distributions (2000), Amsterdam, New York, Oxford: North Holland, Amsterdam, New York, Oxford · Zbl 0532.46019 |
| [47] | Zhang, F., Neutrinos propagating in curved spacetimes, Eur. Phys. J. Plus, 132, 446 (2017) · doi:10.1140/epjp/i2017-11721-4 |
| [48] | Penrose, R., On gravity’s role in quantum state reduction, Gen. Relat. Gravit., 28, 5, 581 (1996) · Zbl 0855.53046 · doi:10.1007/BF02105068 |
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