Why the topological term F\til{F} is scale independent?

1. sinc

4
why the topological term in gauge theory, $ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ}$ ,is scale-independent?

2. samalkhaiat

1,196
Do you know how the field tensor transforms under scale transformation?

3. sinc

4
if the spacetime coordinates scales as $x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}$,then the potential vector scales as follow: $A_{0}→S^{a+d}\bar{A_0}, A_{i}→S^{b+d}\bar{A_i}$

4. samalkhaiat

1,196
Why do space and time scale differently? Can you tell me your background in physics?

5. sinc

4
We are considering the most general rescaling, so space and time scale differently. This is especially true for nonrelativistic case.

6. samalkhaiat

1,196
No, not “especially”. Time and space scale differently ONLY in non-relativistic theory. But, your original question is meaningless in the non-relativistic domain. This is why I asked you about your background in physics.
Any way, in relativistic field theories, the coordinates scale according to
$$x^{ \mu } \rightarrow \bar{ x }^{ \mu } = e^{ - \lambda } x^{ \mu } ,$$
and the field transforms as
$$F ( x ) \rightarrow \bar{ F } ( \bar{ x } ) = e^{ \lambda \Delta } F ( x )$$
where $\Delta$ is the scaling dimension of the field. In D-dimensional space-time:
$$\Delta = \frac{ D - 2 }{ 2 }, \ \ \mbox{ for } \ \ A_{ \mu } (x) ,$$
and
$$\Delta = \frac{ D }{ 2 } , \ \ \mbox{ for } \ \ F_{ \mu \nu } ( x ) .$$

See (for more detailed description) the link below

7. sinc

4
Thank you for your reply. The link given by you is so long that I need some time to follow. However, I donn't agree with your point that
"Time and space scale differently ONLY in non-relativistic theory". It is the textbook's convention that space and time scale the same in relativistic region, but it is not the principle law. Anyway, I don't want to get invoked into this aspect. There is quick question I want to ask you: How do you determine the scaling dimension of a field? By dimension analysis and keep kinetic term dimension-D?
This textbook's answer doesn't make sense...

8. samalkhaiat

1,196
I’m afraid you have to agree with me. This is not an opinion, it is a “principle law”. Scaling has to preserve the light-cone structure. You violate relativity principles if you scale space and time differently.

Yes, you get it either from the action integral or by demanding scale-invariance of the equal-time commutation relations. This is explained in the thread I linked above. See equations (10.13) to (10.17).