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In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter $\omega$ is explicitly $x$-dependent, so that $\omega = \omega(x)$ for a local transformation and then finally at the end considering a global transformation that consequently results in Noether's theorem. The derivation is on pages 39-41.

I understand all of the derivation, however, it has been brought to my attention that the whole derivation seems to rest on an inconsistent starting point that would therefore make the rest of the argument, while mathematically correct, completely useless.

On P.39, Di Francesco writes that the generic infinitesimal transformations of the coordinates and field are, respectively, $$x'^{\mu} = x^{\mu} + \omega_a \frac{\delta x^{\mu}}{\delta \omega_a}$$$$\Phi'(x') = \Phi(x) + \omega_a \frac{\delta F}{\delta \omega_a}(x).$$ This is written under the assumption that the $\omega_a$ are infinitesimal parameters ,not functions of $x$. So when he uses this result in his derivation on P.40, and says that he will make the supposition that the $\omega_a$ are dependent on $x$, how can he do this?

For clarity, in case I missed something, I was having this discussion here: http://www.physicsforums.com/showthread.php?t=760137 and in post 14 is where the subtlety arises. So is it really a flaw? I am just really looking for another opinion on this. I asked one of the professors at my university and he said provided the $x$ dependence is 'small' then it is valid, but I am not quite sure exactly what that means.

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1 Answers1

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I) OP's question (v2) seems to be essentially a question of mathematical precision vs. the way physicists express themselves concisely when speaking of "infinitesimals" without becoming too technical by introducing epsilons and deltas and what not. See also this related Phys.SE post.

Perhaps the easiest and most elementary way to make sense of the "infinitesimal function" $\omega(x)$ in Di Francesco et. al., CFT, is to think of it as a product$^1$

$$ \omega(x)~=~\delta~f(x), $$

where $f(x)$ is a bounded function$^2$, e.g. $|f(x)|\leq 1$; and $\delta$ is an "infinitesimal constant". An "infinitesimal constant" is just physics-jargon for a small number $\delta>0$ so small, that we can neglect higher order contributions of $\delta$ in the calculation, to the precision $\epsilon>0$ that we are working.

II) Concerning the derivation of Noether's theorem via the trick of an $x$-dependent $\omega(x)$, see this Phys.SE post.

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$^1$ We have dropped the index $a$ in $\omega_a(x)$ for simplicity.

$^2$ Technically, it may be convenient to further assume that the function $f(x)$ is differentiable with bounded derivative, perhaps even of compact support. The function $f(x)$ has a role not unlike a test function.

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