Fuzzy Physics and Matrix Geometry

Abstract

There has been a lot of fuzzy mathematics around to which we make no reference whatsoever. We will deal with a more or less precise theory of fuzzy physics following [9, 11, 12, 16, 41, 44, 46, 45, 69, 51, 146, 137, 138, 139, 208, 232, 257, 281, 284, 285, 286, 287, 288, 289, 290, 282, 329, 333, 336, 338, 344, 345, 353, 354, 389, 393, 435, 447, 449, 484, 451, 452, 528, 569, 573, 586, 626, 631, 646, 648, 658, 669](see especially [670] which appeared after much of this chapter was first written - we have now included some material from this however along with more elementary examples in later remarks and sections). There are several approaches possible and we will begin with the fuzzy sphere. It is probably better to start with some early papers (e.g. [284, 290, 451, 452] along with [144]) rather than say [447] which is interlaced with classical ideas from differential geometry and becomes heavier reading. We will assume known many ideas from classical geometry (with suitable reminders) and will concentrate on the fuzzy or q-versions here. Subsequently we will sketch other fuzzy material (especially coherent states) and indicate connections to matrix models and large N behavior in string or M theory. We have tried to maintain a consistent notation but occasionally this changes to fit with other notation; there should be no problem since definitions are often repeated.