I started reading a document by Gerard 't Hooft which can be found here. Right at the start I am puzzled by a simple expression. It is equation 2.2 showing how a scalar function transforms. I repeat the equation here: $$ \phi^\prime(x) = \phi(x^\prime). \tag{2.2} $$ This does not look right to me. I would expect this to read like: $$ \phi^\prime(x^\prime) = \phi(x). $$ Am I missing something trivial or is this a typo?
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There are tons of differing conventions for this kind of thing -- everybody sets it up their own way. It gets worse -- for the transformation of the action there are fifteen possible conventions, which I outlined here. – knzhou Jan 12 '19 at 21:19
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In my opinion the best thing to do is to try to set it all up yourself, in a way that works for you (i.e. so that your calculations get correct results), then basically ignore whatever anybody else does. You can only learn physics by reinventing it, anyway. – knzhou Jan 12 '19 at 21:20
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@knzhou-- thanks for the link and comments. Yes this is a confusing bag of notation. I think I need to study also active versus passive transformations. Actually never heard of that before. – K7PEH Jan 12 '19 at 21:25
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OK, after a brief look at active and passive transformations I guess I do know what these are -- just didn't know they had a name. – K7PEH Jan 12 '19 at 21:27
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Yeah. God knows if that's what t'Hooft had in mind. He could also have just swapped the usual meanings of $x$ and $x'$. There are no universal conventions on this. A more 'casual' document like the one you linked just wouldn't be careful to be consistent on this either. – knzhou Jan 12 '19 at 21:30
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As long as you can see what's going on and do calculations correctly, you're fine! – knzhou Jan 12 '19 at 21:31