| [ sciEncE ] in KIDS 글 쓴 이(By): guest (무식한사람) <adsl-seongbook-2> 날 짜 (Date): 2000년 9월 19일 화요일 오전 12시 45분 53초 제 목(Title): Re: Question! Physics! OK, here is a "high schooler's version" of the same story. Now, pick, as a manifold M, a disk (strictly speaking, this is a manifold with a boundary) lying on the z=0 plane in 3 dimensions. At each interior point of the disk, place a fiber (or a straw) parallel to the z-axis. This is a simple example of fiber bundles (imagine a bundle of straws). Assume, for the sake of arguments even if it's unrealistic, that the length of each straw is infinite. Then, an example of a gauge transformation is to move each fiber (or each straw) upward or downward. 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". In this example, the bundle is the bundle of straws, a base manifold (with a boundary) is the disk, a fiber is a straw placed at each point of the disk interior, the structure group is the translation group along the z-axis, a bundle automorphism is an act of moving each straw up and down (it's an automorphism when the geometry of the whole bundle doesn't change), and the cross section s encodes how much each straw moved up or down. Going to a non-trivial example, imagine a four dimensional space-time we are living in. Now at each point of the space-time, place a four dimensional vector (now we're talking about a vector bundle). Now one decides to "see" only the x and y components of each vector. Upon this agreement, the rotation of each vector along the "circle" (whose center is located at the point where we placed the vector) parallel to the (t,z) plane does not change the whole bundle geometry. Now the gauge transformation in this context is the aforementioned rotation of each vector by an arbitrary amount, or in other words, the gauge transformations form an abelian (or U(1)) group. This construction, in the end, corresponds to an "over-simplified" version of the U(1) gauge theory (or Maxwell's electromagnetism). A generalization of substantial mathematical importance (and physical importance too, according to physicists) is to change the structure group (z-translations or circle rotations in the above) to a nonabelian Lie group (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 !!! p.s. 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. |