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Both concepts are mathematical in character and they ultimately describe the same characteristics or situations. "Invariance" is a more technical word because it says "what has to be equal to what" for us to say that the symmetry exists. In particular, the "invariance under a symmetry transformation" means that an object, like the action $S$, has the same ...

2

Formally, the meaning you assign is just the usual meaning of the derivative. $$\partial_\mu \psi(x^\nu) = \lim_{h \to 0} \frac{\psi(x^\nu + h\delta^\nu_\mu) - \psi(x^\nu)}{h}$$ You can indeed compute it componentwise, because you can subtract two spinors, as in the equation above, just by subtracting their components. The object you get has sixteen ...

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Basically your confusion is caused by a bias about what a vector is. Everyone agrees that you can add two vector and get another. Everyone agrees you can scale a vector and get another vector. Sometimes we square a vector and get a scalar, but some people say that is "merely" an abuse of notation. But that is just a special case of ...

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version1 The laws of physics are the same in all inertial reference frames. This is a correct principle, but it is useless unless you also state which things are the laws of physics. If they the laws include Newton's laws as usually written then you get Galilean Relativity, if the laws include Maxwell's equation then you get special relativity. ...

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Special Relativity has different postulates correspond to several kinds of symmetries. Symmetry means everything will be the same if you do certain operations. Two of the symmetries are the two postulates you stated, that are written at beginning of every textbook that teaches special relativity. The postulate one says a symmetry of uniform velocity in a ...

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That motion was relative was realized by Galileo, so there was a theory of relativity -- Galilean relativity -- long before Einstein. That the speed of light should be the same according to all observers is indeed inconsistent with the Galilean relativity. This is because in Galilean relativity time is absolute. But it is not mathematically inconsistent to ...

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Absolute four-momentum is not an observable. Relative four-momentum is. We cannot find the four-momentum of the lab itself, but we can (and do, regularly) measure the four-momentum of particles relative to a given lab, which then allows us to calculate the four-momentum of said particles relative to any frame you care to name. Whoever said that wasn't ...

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