Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2011-10-23
Physics
Condensed Matter
Statistical Mechanics
16 pages, 0 figures
Scientific paper
Based on the quaternionic approach developed by one of us [Z.D. Zhang, Phil. Mag. 88, (2008) 3097.] for the three-dimensional (3D) Ising model, we study in this work conformal invariance in three dimensions. We develop a procedure for treating the 3D conformal field theory. The 2D conformal field theory is generalized to be appropriate for three dimensions, within the framework of quaternionic coordinates with complex weights. The Virasoro algebra still works, but for each complex plane of quaternionic coordinates. The quaternionic geometric phases appear in quaternionic Hilbert space as a result of diagonalization procedure which involves the smoothing of knots/crossings in the 3D many-body interacting spin Ising system. Possibility for application of conformal invariance in three dimensions on studying the behaviour of the world volume of the brane, or the world sheet of the string in 3D or (3+1)D, is briefly discussed.
March Norman H.
Zhang Zhi-dong
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