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Based on the quaternionic approach developed by one of us [Z.D. Zhang, Phil.Mag. 87 (2007) 5309.] for the three-dimensional (3D) Ising model, we study inthis work conformal invariance in three dimensions. We develop a procedure fortreating the 3D conformal field theory. The 2D conformal field theory isgeneralized to be appropriate for three dimensions, within the framework ofquaternionic coordinates with complex weights. The Virasoro algebra stillworks, but for each complex plane of quaternionic coordinates. The quaternionicgeometric phases appear in quaternionic Hilbert space as a result ofdiagonalization procedure which involves the smoothing of knots/crossings inthe 3D many-body interacting spin Ising system. Possibility for application ofconformal invariance in three dimensions on studying the behaviour of the worldvolume of the brane, or the world sheet of the string in 3D or (3+1)D, isbriefly discussed.

Conformal invariance in the three dimensional (3D) Ising model and quaternionic geometric phase in quaternionic Hilbert space