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

[sinc]

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

 

[samalkhaiat]

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

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

 

[sinc]

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

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: $A_{0}→S^{a+d}\bar{A_0}, A_{i}→S^{b+d}\bar{A_i}$

 

[samalkhaiat]

if the spacetime coordinates scales as $x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}$

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

 

[sinc]

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

 

[samalkhaiat]

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

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

www.physicsforums.com/showthread.php?t=172461

 

[sinc]

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...

 

[samalkhaiat]

However, I donn't agree with your point that
"Time and space scale differently ONLY in non-relativistic theory"... but it is not the principle law.

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.

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...

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).