不正经的规范场介绍

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Gauge fields are inevitable when we require the nature has some symmetries. For example the symmetry among different frames of reference requires the existence of gravitation. The requirement that all the inertial observers are equivalent is the the principle of special relativity and the requirement that all the possible observers are equivalent is the principle of general relativity. Roughly speaking each observer has his own frames to describe all the possible phenomenons and we believe the nature itself is objective. The observation from different observer must be somehow consistent and covariant. In order to compare the different results from different observers we have to introduce some connections to link different frames. There is still arbitrariness to choose this connection. One of the simplest and most natural quantities which could or should be measured is the distance. It would be better and convenient to demand there is some agreement on distance for different observers. This requirement will pick a choice of the connection and define the geometry of the spacetime. Until here we are using some intuition and common sense but not physics yet. Basically physical laws describe how things change. Gravity is one kind of forces which affect motions or changes. So if we postulate that motion in the spacetime depends on the geometry of the spacetime then we have defined some kind of force. Beautifully this is exactly the gravity.

How about other forces? From the argument of gravity we can see the origin of forces is the requirement of some symmetries. But the symmetry of what? Or what exactly a symmetry is? We can think the existence of a symmetry reflects that we have multiple ways to describe one thing. For gravity the symmetry is that all the possible frames are allowed and on the same footing. The frame is necessary to describe the position and velocity of particles. Particles also have some other important properties needed to describe, say the phase. As we know there is a duality between waves and particles. And phase it important property of wave. So we also need a reference for the phase. Like before different observers can have their own reference. Again we need to introduce a new corresponding connection to link different phase reference. If we postulate that the motion also depends this phase connection then we have defined a another kind of force. It turns out to be the usual electromagnetic force.