| [ sciEncE ] in KIDS 글 쓴 이(By): mkjung (BElTRAMi) 날 짜 (Date): 2000년 9월 19일 화요일 오전 01시 38분 27초 제 목(Title): gauge transform You can move each of them by an arbitrary amount (stupid physicists call this "local gauge transformation", unlike we, the mathematicians). Now after this transformation, you notice that the bundle looks exactly the same as before; in other words, it's "gauge-invariant". ============= ha.... this is very neat explanation! but why "dumb" physicists (i think they are smart but let just pretend they are dumb kiki.... ) call it "local" ? is that mean there is also "global" gauge transform? it begs me a question. ^^ (like SU(n), corresponding to a nonabelian gauge theory, or the Yang-Mills theory). This can be used to prove that there are fake R^4s, which are homeomorphic to R^4 but not diffeomorphic to R^4 !!! ============== OK. there is a familar thing I have been hearing all the time but don't really know. What is Yang-Mills equation? where is it used to solve what ? hope to see you next time for the explanation of the "moduli space"... "Anything" can be explained in a simple manner, but typing those things take lots of time. ============== i am just fooling around here but my guess about moduli space might be something like this. correct me if i am wrong. ^^ Suppose you have a certain vector space X, Y, where Y is a subspace of X. Then the moduli space Y/X might be another subspace such that you are classifying the subspace of Y with respect to X in a sense. kiki... hope this makes sense. F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu 가 있죠. (A_\mu는 scalar potential과 vector potential을 합친 4-vector notation입니다. F_{\mu\nu}도 electric field와 magnetic field와 같은 거구요.) 이때 F_{\mu\nu}는 A_\mu를 A_\mu(x) --> A_\mu(x) + \partial_\mu f(x) 로 바꿔도 변하지 않습니다. (f(x)는 임의의 함수) 이럴 때 F_{\mu\nu}가 gauge invariant하다고 합니다. 즉, 위와 같은 변환을 gauge transformatin이라 하고 그러한 transformation에 대해 invariant하다는 뜻입니다. 그런데 전자기학의 모든 것이라 할 수 있는 맥스웰 방정식은 F_{\mu\nu}로만 써지므로 맥스웰 방정식은 gauge invariant하죠. 그래서 U(1) gauge theory 라고 합니다. (U(1)인 이유는 f(x)가 그냥 함수이기 때문입니다.) ================================ i like Mr. soliton's explanation better. so it simply means when you have a certain tensor and this tensor doesn't change under a certain transformation of its components, then it is called gauge invariant! OK. i got that. now why people call it "gauge"? who was the first person to name it? gauge means "measurement" right? gauge theory는 사실은 수학입니다. 환상님이 수학을 많이 공부하셨으므로 제가 추측하기로는 수학 용어와 물리 용어를 서로 연결만 할 수 있으면 될 겁니다. 미분기하의 fiber bundle theory에서 connection이 gauge field에, curvature가 field strength에 해당합니다. 그리고 일반 상대론도 일종의 gauge theory이죠. general coordinate transformation이 gauge transformation에 해당합니다. gauge invariance는 general coordinate invariance이구요. =============== ya this explanation, i do understand! i know that certain intrinsic geometrical properties are invariant under coordinate transform! kiki.... |