Further Aspects of FQM
The words finite quantum mechanics (FQM) are given a precise formulation in  for example (cf. also [238, 251, 252, 632]). One works with a torus phase space and looks at discrete elements of SL(2, R) (i.e. at SL(2, Z) ~ modular group) studied on discretizations of the torus with rational coordinates (q, p) = (n 1/ℓ, n 2/ℓ) ∈ Г. The periodic trajectories mod 1 correspond to periods of A ∈ SL(2, Z) mod ℓ and the action mod 1 becomes mod ℓ on ℓГ.The classical motion of such discrete dynamical sysems is usually maximally disconnected and chaotic. Then FQM is defined as the quantization of these discrete linear maps and the corresponding one time evolution operators UA are ℓ × ℓ unitary matrices called quantum maps. The matter can be connected to quantization of SL(2, F p ) where F p is a finite field of p elements (p prime) via ℓ=p n for example. We will however use the term FQM in a more general sense here to refer to QM based on a discrete background as in Chapters 6–8. In particular this involves a discrete calculus and could be considered as the principal reason for revisiting calculus in this book from a discrete point of view (even if that seems to go backwards historically). Despite the obvious validity of classical mechanics on a macroscopic scale there seem to be convincing arguments for discretization at very small scales. In particular time would be discrete which is in many ways appealing (although not from Zeno’s point of view). We make to attempt to review all the literature but will extract from various sources to give a flavor of some of the ideas. Chapters 7–8 indicated some approaches which are developing into full blown theories and in this chapter we expand upon Chapters 7–8 and deal also with some more fragmentary material. We will omit discussion of the fast Fourier transform however even though this is clearly related to QM and we will not give any sort of complete discussion of lattice gauge theory (cf. ). Thus in addition to the papers already cited in Chapters 6–8 let us begin by mentioning also [26, 28, 71, 72, 168, 169, 170, 191, 193, 196, 197, 198, 199, 226, 227, 228, 238, 251, 252, 263, 339, 374, 375, 419, 453, 456, 457, 510, 532, 559, 587, 632] with apologies for omissions (other references will arise as we go along).
KeywordsHopf Algebra Conformal Block Braid Group Differential Calculus Linear Connection
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