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In the case of fields, it is clear what a local transformation in the internal space of the field is:

$$\phi(x) \to \phi'(x')= G(x) \phi(x),$$

as opposed to a global transformation, where $G$ would not depend on $x$.

But I don't understand the difference between local and global coordinate transformations.

It is said that general coordinate transformations, $$x^\mu \to x'^\mu =x^\mu+\epsilon^\mu(x)$$ are local, while rotations $$x^\mu \to x'^\mu =x^\mu+\sigma^\mu_\nu x^\nu$$ are global.

How can we distinguish between the two? In both cases, the variation depends on $x$.

Qmechanic
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jinawee
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1 Answers1

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In the vector notation, $x \rightarrow x' = (1+\sigma)x$. The transformation acts identically on each point. So, the rotation is global, as long as $\sigma$ is independent of $x$. Each point is given same the angular shift.

Consider, in the general transformation, if $\epsilon(x) = kx$ and $k$ is a constant scaler, then this isotropic scaling transformation is global as well. Each line segment is stretched by same factor, wherever it is situated.

So, the distinction between global and local is not dependent on variation of coordinates, as such. Instead, its dependent on the variation of the transformation $T,c$ over the manifold $M$ its acting upon. $$x \mapsto Tx + c ; \ x,c \in M$$

Shark
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