Time Between Real and Imaginary: what Geometries Describe Universe near Big Bang?

Yuri I. Manin

doi:10.31392/2307-4515/2017-9.1

Автор(и)

Yuri I. Manin Національний педагогічний університет імені М.П. Драгоманова, Німеччина

DOI:

https://doi.org/10.31392/2307-4515/2017-9.1

Анотація

For about a century, a great challenge for theoretical physics consisted in understanding the role of quantum mode of description of our Universe ("quantum gravity"). Einstein space–times on the scale of observable Universe do not easily submit to any naive quantization scheme. There are better chances to concoct a satisfying quantum picture of the very early space–time, near the Big Bang, where natural scales of events like inﬂation extrapolated from current observations resist any purely classical description and rather require quantum input.
Many physicists and mathematicians tried to understand the quantum early Universe, sometimes unaware of input of the other community. One of the goals of this article is to contribute to the communication of the two communities. In the main text, I present some ideas and results contained in the recent survey/research papers [Le13] (physicists) and [MaMar14], [MaMar15] (mathematicians).

Біографія автора

Yuri I. Manin, Національний педагогічний університет імені М.П. Драгоманова

директор наукової бібліотеки, доцент кафедри інформаційних систем і технологій

Посилання

E. Artin. Ein mechanisches System mit quasiergodischen Bahnen. Abh. Math. Sem. Univ. Hamburg 3 (1924), no. 1, pp. 170–175.

The Accelerating Universe. Scientific Background on the Nobel Prize in Physics 2011. http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/advanced-physicsprize2011.pdf

M. V. Babich, D. A. Korotkin. Self–dual SU (2)–Invariant Einstein Metrics and Modular Dependence of Theta–Functions. Lett. Math. Phys. 46 (1998), 323–337. arXiv:gr-qc/9810025v2

A. Balbi. Cosmology and time. arXiv:1304.3823 [physics.hist-ph]

O.I. Bogoyavlensky. Methods in the qualitative theory of dynamical systems in astrophysics and gas dynamics. Springer Series in Soviet Mathematics. Springer Verlag, Berlin, 1985. ix+301 pp.

O.I. Bogoyavlenskii, S. P. Novikov. Singularities of the cosmological model of the Bianchi IX type according to the qualitative theory of differential equations. Zh. Eksp. Teor. Fiz. 64 (1973), pp. 1475–1494.

L.S. Cirio, G. Landi, R.J. Szabo. Algebraic deformations of toric varieties II: noncommutative instantons. Adv. Theor. Math. Phys. 15 (2011), no. 6, 1817–1907.

L.S. Cirio, G. Landi, R.J. Szabo. Instantons and vortices on noncommutative toric varieties. arXiv:1212.3469.

L.S. Cirio, G. Landi, R.J. Szabo. Algebraic deformations of toric varieties I. General constructions. Adv. Math. 246 (2013), 33–88.

A. Connes, G. Landi. Noncommutative manifolds, the instanton algebra and isospectral deformations. Comm. Math. Phys. 221 (2001), no. 1, 141–159.

T. Eguchi, A.J. Hanson, Self-dual solutions to Euclidean Gravity, Annals of Physics, 120 (1979), 82–106.

T. Eguchi, A.J. Hanson, Gravitational Instantons, Gen. Relativity Gravitation 11, No 5 (1979) , 315–320.

Ch. Estrada, M. Marcolli. Noncommutative Mixmaster Cosmologies. Int. J. of Geom. Methods in Modern Phys., Vol. 10, No. 1, 2013.

T. Furusawa. Quantum chaos of Mixmaster Universe. Progr. Theor. Phys., Vol. 75, No. 1 (1986), pp. 59–67.

N. J. Hitchin. Twistor spaces, Einstein metrics and isomonodromic deformations. J. Diff. Geo., Vol. 42, No. 1 (1995), 30–112.

I. M. Khalatnikov, E. M. Lifshitz, K. M. Khanin, L. N. Shchur, and Ya. G. Sinai. On the stochasticity in relativistic cosmology. Journ. Stat. Phys., Vol.38, Nos. 1/2 (1985), pp. 97–114.

D. Lambert. The atom of the universe. Copernicus Center Press, Krak´ow, 2015.

´

D. Lambert. L’itin'eraire spirituel de Georges Lemaitre. Editions Lessius, Bruxelles, 2007.

O. M. Lecian. Reflections on the hyperbolic plane. International Journal of Modern Physics D Vol. 22, No. 14 (2013), 1350085 (53 pages) DOI: 10.1142/S0218271813500855. arXiv:1303.6343 [gr-qc]

L.M.Lugo, P.A. Chauvet. BKL method in the Bianchi IX universe model revisited. Appl. Phys. Res., Vol. 5, No. 6 (2013), pp. 107–117.

Yu. Manin. Sixth Painlev´e equation, universal elliptic curve, and mirror of P2 . In: geometry of Differential Equations, ed. by A. Khovanskii, A. Varchenko, V. Vassiliev. Amer. Math. Soc. Transl. (2), vol. 186, pp. 131–151. Preprint alg-geom/9605010.

Yu. Manin, M. Marcolli. Continued fractions, modular symbols, and non–commutative geometry. Selecta math., new ser. 8 (2002), 475–521; arXiv:math.NT/0102006

Yu. Manin, M. Marcolli. Modular shadows. In: Modular forms on Schiermonnikoog, ed. by Bas Edixhoven, G. van der Geer, B. Moonen, Cambridge UP, 2008, pp. 189–238. arXiv:math.NT/0703718

Yu.I. Manin, M. Marcolli. Big Bang, Blow Up, and Modular Curves: Algebraic Geometry in Cosmology. SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 073, 20 pp. Preprint arXiv:1402.2158

Yu.I. Manin, M. Marcolli. Symbolic dynamics, modular curves, and Bianchi IX cosmologies. Preprint arXiv:1504.04005 [gr-qc]

M. Marcolli. Solvmanifolds and noncommutative tori with real multiplication. Commun. Number Theory and Physics, Vol. 2 (2008) No. 2, 421–476.

D. Mayer. Relaxation properties of the Mixmaster Universe. Phys. Lett. A, Vol. 121, Nos. 8–9, 1987, pp. 390–394.

E. T. Newman. A fundamenal solution to the CCC equations. arXiv:1309.7271 [gr-qc]

E. Newman, L. Tamburino, T. Unti, Empty-space generalization of the Schwarzschild metric, Journ. Math. Phys. 4 (1963), 915–923.

K. A. Olive, J. A. Peacock. Big–Bang Cosmology. Online publication: http://pdg.lbl.gov/2005/reviews/bigbangrpp.pdf

R. Penrose. Conformal treatment of infinity. In: Relativity, groups and topology. Ed. by B. deWitt and C. deWitt, Gordon and Breach, 1964, pp. 565–584.

R. Penrose (1979). Singularities and Time-Asymmetry. In "General Relativity: An Einstein Centenary Survey", (S. W. Hawking and W. Israel, Eds.), Cambridge University Press, 1979, pp. 581–638.

R. Penrose. Twistor geometry of conformal infinity. In: The conformal structure of space–time, ed. by J. Frauendiener, H. Friedrich, LN in Phys. 604, 2002, pp. 113–121.

R. Penrose. Cycles of time. An Extraordinary New View of the Universe. Vintage Books, 2010.

P.M. Petropoulos, P. Vanhove, Gravity, strings, modular and quasimodular forms, Ann. Math. Blaise Pascal 19, No. 2 (2012), 379–430.

C. Series. The modular surface and continued fractions. J. London MS, Vol. 2, no. 31, 1985, pp. 69–80.

K. Takasaki. Painlev´e–Calogero correspondence revisited. Journ. Math. Phys., Vol. 42, No. 3 (2001), pp. 1443–1473.

A. H. Taub. Empty space–times admitting a three parameter group of motions. Annals of Math. 53 (1951), pp. 472–490.

K. P. Tod. Self–dual Einstein metrics from the Painlev´e VI equation. Phys. Lett. A 190 (1994), 221–224.

P. Tod. Penrose’s circles in the CMB and a test of inflation.. Gen. Rel. and Gravitation, vol. 44, issue 11, (2013), pp. 2933–2938. arXiv: 1107.1421 [gr-qc]

W. van Suijlekom. The noncommutative Lorentzian cylinder as an isospectral deformation. J. Math. Phys. 45 (2004), no. 1, 537–556.

S. Weinberg. The First Three Minutes. A Modern View of the Origin of the Universe. Bantam Book, 1977.