Jian Dai and Xing-Chang Song 2001 J. Phys. A: Math. Gen. 34 5571 doi:10.1088/0305-4470/34/27/307
Jian Dai and Xing-Chang Song
Show affiliationsOne of the key ingredients of Connes's noncommutative geometry is a generalized Dirac operator which induces a metric (Connes's distance) on the pure state space. We generalize such a Dirac operator devised by Dimakis et al, whose Connes distance recovers the linear distance on an one-dimensional lattice, to the two-dimensional case. This Dirac operator has the local eigenvalue property and induces a Euclidean distance on this two-dimensional lattice, which is referred to as `natural'. This kind of Dirac operator can be easily generalized into any higher-dimensional lattices.
02.40.Gh Noncommutative geometry
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
34L40 Particular operators (Dirac, one-dimensional Schrödinger, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean n-space
58B34 Noncommutative geometry (à la Connes)
81T75 Noncommutative geometry methods (See also 46L85, 46L87, 58B34)
Issue 27 (13 July 2001)
Received 19 January 2001, in final form 10 May 2001
Published 29 June 2001
Jian Dai and Xing-Chang Song 2001 J. Phys. A: Math. Gen. 34 5571
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