Abstract. The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex sp~tce as original spaces. We can abstracta class of the hyperbolic inner product space and the hyperbolic Hilbert space.
Abstract.In this paper, applying the direction strangeness of the hyperbolic space-time, we introduce the double metric space and the multi-topological structure, and in the wave-particle topology, we give one kind of imaginary metric representation for lightquantum hypothesis, uncertainty principle, energy level transition and interference conditions, etc. in quantum mechanics.Key words: the same kind of isotropic element; quasi-metric space, imaginary metric space, directional phase lattice, wave-particle topology, uncertainty principle.
Abstraer.The hyperbolic complex (HC) space is congruent with Minkowski space time. HC is a special kind of non-Euclidean space with continuous odd-points. The Clifford algebraic spinor and the Dirac wave equation can be introduced in the hyperbolic complex space. The Clifford algebraic spinor contains eight independent elements and the Dirac wave equations 64 coeffŸ For Dirac particles 4 x 8 and for antiparticles 4 x 8 variables which ate Hermitian conjugate to each other (on four dimensional space-time).
In the four-dimensional hyperbolic space, we can derive Euler formula and establish the hyperbolic spherical polar coordinate and the hyperbolic spherical harmonics.
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