Mirror Symmetry as a Basis for Constructing a Space-Time Continuum
By mirroring a one-dimensional oriented set in a complex space specially created on the basis of a symmetry, a mirror n-dimensional space with n > 1 has been constructed. The geometry of the resulting space is described by the Clifford algebra. On the basis of the algebra of hyperbolic hypercomplex numbers, a pseudo-Euclidean space has been constructed with the metric of the Minkowski space. The conditions for a function of a hyperbolic hypercomplex argument to be analytic (h-analyticity) are obtained. The conditions implicitly contain the Maxwell equations for the 4-potential in a free space.
Yu.S. Vladimirov, Relational Theory of Space-Time and Interactions, Parts 1 and 2 (Moscow State Univ. Publ. House, Moscow, 1996–1998) (in Russian).
A.P. Efremov, Quaternion Spaces, Frames, and Physical Fields (RUDN Publ. House, Moscow, 2005) (in Russian).
A.V. Berezin, Yu.A. Kurochkin, and E.A. Tolkachev, Quaternions in Relativistic Physics (Editorial URSS, Moscow, 2003) (in Russian).
D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic, New York, 2002).
D. Hestenes, Clifford Algebra to Geometric Calculus (D. Reidel, Dordrecht, 1987).
H. Weyl, Symmetry (Princeton Univ. Press, Princeton, NJ, 1956).
Sh.-T. Yau and S. Nadis, The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions (Basic Books, New York, 2010), p. 151.
A.N. Kapustin and D.O. Orlov, Usp. Matem. Nauk 59, N 5, 101 (2004).
E. Cartan, ´ Le¸cons sur la Th’eorie des Spineurs (Hermann, Paris, 1938).
P.K. Rashevsky, Usp. Matem. Nauk 10, No. 2, 3 (1955).
I.M. Yaglom, Complex Numbers in Geometry (Academic Press, New York, 1968).
E. Hitzer, SICE J. Control Meas. 4, 1 (2011).
D.G. Pavlov and S.S. Kokarev, Giperkompl. Chisla Geom. Fiz. 7, No. 1, 78 (2010).
D.G. Pavlov and S.S. Kokarev, Giperkompl. Chisla Geom. Fiz. 7, No. 2, 11 (2010).
R. Penrose and W. Rindler, Spinors and Space-Time, Vol. 1: Two-Spinor Calculus and Relativistic Fields (Cambridge Univ. Press, Cambridge, 1987).
M.A. Lavrent'ev and B.V. Shabat, Problems in Hydrodynamics and Their Mathematical Models (Nauka, Moscow, 1977) (in Russian).
D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, A Unified Language for Mathematics and Physics (Kluwer, Dordrecht, 1984).
S.V. Terekhov, Vestn. Novgorod. Gos. Univ., No. 26, 56 (2004).
A.L. Glebov, Teor. Mat. Fiz. 48, 340 (1986).
E. Wigner, Phys. Today 17, 34 (1964).
Yu.B. Rumer, Spinor Analysis (ONTI, Moscow, 1936) (in Russian).
V.I. Vysotskii and M.V. Vysotskyy, Eur. Phys. J. A 44, 279 (2010).