0
$\begingroup$

I'm trying to understand Noether's theorem in QFT. I stumbled across one small doubt, Please explain why is there a negative sign instead of plus sign in the following:

enter image description here

$\endgroup$

2 Answers 2

5
$\begingroup$

If $f(x)$ has its maximum at $x=0$, then $f(x-a)$ has its maximum at $x=a$, so displacing a function through a displacement $\delta x=a$ requires $$ f(x)\mapsto (T_af)(x)= f(x-a)\approx f(x)- a \partial_x f. $$

$\endgroup$
0
$\begingroup$

Mike stone's answer can be explained in a more detailed way. The transformation acts on the coordinates, not the field. So the field at a point in the spacetime remains the same after the transformation.

For any transformation $x\to Tx$, the field after the transformation is marked as $\phi'(x)$, and before transformation it is $\phi(T^{-1}x)$, where $T^{-1}$ is the inverse of the transformation.

In OP's case, $Tx=x+a$ and $T^{-1}x=x-a$. Thus $$ \phi(x)\to\phi'(x)=\phi(x-a)=\phi(x)-\frac{\partial\phi}{\partial x_\nu}a_\nu. $$

$\endgroup$
1
  • $\begingroup$ Thanks a lot. Your explanation solves my query. $\endgroup$
    – Abhinav
    Aug 29, 2022 at 2:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.