Fuzzy topology, quantization and gauge invariance
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uzzy Topology, Quantization and Gauge Invariance1 S. N. Mayburov Lebedev Institute of Physics, Leninsky Prospect 53, Moscow, RU117924 Russia email: [email protected] Abstract—Quantum spacetime with Dodson–Zeeman topological structure is studied. In its framework the states of massive particle m correspond to elements of fuzzy set called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σx. Quantization formalism is derived from consideration of m evolution in fuzzy phase space with minimal number of additional assumptions. Particle’s interactions on fuzzy manifold are studied and shown to be gauge invariant. DOI: 10.1134/S1063779612050267
1 The article is published in the original.
tute the ordered set. For the elements of partially ordered set (Poset) {di}, beside the standard ordering relation between its elements dk ≤ dl (or vice versa), the incomparability relation dk dl is also permitted; if it’s true, then both dk ≤ dl and dl ≤ dk propositions are false. To illustrate its meaning, consider Poset DT = A ∪ B, which includes the subset of ‘incomparable’ elements B = {bj}, and the ordered subset A = {ai}. Let’s suppose that in A the element’s indexes grow correspondingly to their ordering, so that ∀i, ai ≤ ai + 1. As the example, consider some interval {al, al + n} and suppose that bj ∈ {al, il + n}, i.e. al ≤ bj; bj ≤ al + n and bj ai; iff l + 1 ≤ i ≤ l + n – 1. In this case, bj in some sense is ‘smeared’ over {al, al + n} interval. To introduce the fuzzy rela tions, let’s put in correspondence to each bj, ai pair the ~
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ó1 Structure of spacetime at microscopic (Plank) scale and its relation to axiomatic of Quantum Mechanics (QM) is actively discussed now [1, 2]. In particular, it was proposed that such fundamental properties of spacetime manifold MST as its metrics and topology can differ significantly at Planck scale from standard Riemanian formalism [2, 3]. Recently it was shown that Posets and the fuzzy ordered sets (Fos ets) can be used for the construction of different vari ants of Fuzzy Topology (FT) and corresponding geometry [4, 5], hence it’s instructive to study what kind of physical theory such topologies induces [1, 3]. In our previous works it was shown that in its frame work the quantization procedure by itself can be defined as the transition from classical ordered phase space to fuzzy one. Therefore, the quantum properties of particles and fields can be deduced directly from FT of their phase space and don’t need to be postulated separately of it [1, 3]. As the simple example, the quantization of nonrelativistic particle was regarded; it was shown that FT induces the particle’s dynamics which is equivalent to QM evolution [1, 3]. Yet in its derivation some phenomenological assumptions were used, here the new and simple formalism which per mits to drop them will be described. It will be shown also that the interactions on such fuzzy manifold are gauge invariant and under simple assumptions corre spond to Yang–Mills fields [
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