| [ sciEncE ] in KIDS 글 쓴 이(By): guest (무식한사람) <muon.kaist.ac.kr> 날 짜 (Date): 2000년 9월 18일 월요일 오후 10시 36분 41초 제 목(Title): Re: Question! Physics! Here is a mathematical explanation for you. Read it in conjunction with the previous posting. A principal bundle P over a manifold M with structure group G is a twisted version of M * G. (G = Gauge group) In this setup, there is the group of bundle automorphisms of P, which is called the group of "gauge transformations". A gauge transformation is a fiber preserving map s: P -> P which satisfies s( p g ) = s(p) g for p belonging to P and g belonging to G. Equivalently, s is a cross section of the bundle of groups Aut n = P *_G G, where G acts by conjugation. To give you a bit more idea about Aut n, one notes that Aut n can also be described as the U_{all x belonging to M} Aut n_x, where Aut n_x is the Lie group automorphism of the fiber n_x which lie in G. Now that you know what the gauge transformation is, I can simply say "something is gauge invariant" when that is invariant under the gauge transformation. Since you know something about "Beltrami", I might add: consider the familiar moduli spaces for Riemann surfaces of fixed genus. Here one starts with the obviously infinite dimensional space of all possible surfaces of the same genus. After dividing out by conformal equivalence, the moduli space obtained is a finite dimensional cell complex, the Riemann moduli space. In this process, you've just divided the original moduli space by orbits of the "gauge transformation". |