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

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sinc
#1
Jun20-14, 07:07 AM
P: 4
why the topological term in gauge theory, [itex]ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ} [/itex] ,is scale-independent?
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samalkhaiat
#2
Jun20-14, 09:55 AM
Sci Advisor
P: 892
Quote Quote by sinc View Post
why the topological term in gauge theory, [itex]ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ} [/itex] ,is scale-independent?
Do you know how the field tensor transforms under scale transformation?
sinc
#3
Jun22-14, 03:29 AM
P: 4
Quote Quote by samalkhaiat View Post
Do you know how the field tensor transforms under scale transformation?
if the spacetime coordinates scales as [itex]x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}[/itex],then the potential vector scales as follow: [itex]A_{0}→S^{a+d}\bar{A_0}, A_{i}→S^{b+d}\bar{A_i}[/itex]

samalkhaiat
#4
Jun22-14, 01:29 PM
Sci Advisor
P: 892
Why the topological term F\til{F} is scale independent?

Quote Quote by sinc View Post
if the spacetime coordinates scales as [itex]x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}[/itex]
Why do space and time scale differently? Can you tell me your background in physics?
sinc
#5
Jun22-14, 09:29 PM
P: 4
We are considering the most general rescaling, so space and time scale differently. This is especially true for nonrelativistic case.
samalkhaiat
#6
Jun23-14, 06:34 AM
Sci Advisor
P: 892
Quote Quote by sinc View Post
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
[tex]x^{ \mu } \rightarrow \bar{ x }^{ \mu } = e^{ - \lambda } x^{ \mu } ,[/tex]
and the field transforms as
[tex]F ( x ) \rightarrow \bar{ F } ( \bar{ x } ) = e^{ \lambda \Delta } F ( x )[/tex]
where [itex]\Delta[/itex] is the scaling dimension of the field. In D-dimensional space-time:
[tex]\Delta = \frac{ D - 2 }{ 2 }, \ \ \mbox{ for } \ \ A_{ \mu } (x) ,[/tex]
and
[tex]\Delta = \frac{ D }{ 2 } , \ \ \mbox{ for } \ \ F_{ \mu \nu } ( x ) .[/tex]

See (for more detailed description) the link below

www.physicsforums.com/showthread.php?t=172461
sinc
#7
Jun23-14, 09:51 PM
P: 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...
samalkhaiat
#8
Jun24-14, 06:20 AM
Sci Advisor
P: 892
Quote Quote by sinc View Post
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).


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