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

In summary, the topological term in gauge theory, ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ}, is scale-independent because the field tensor and potential vector both transform under scale transformation in a specific way, determined by the scaling dimension of the field. This is a fundamental principle in relativistic field theories, where space and time must scale the same way to preserve the light-cone structure. The scaling dimension can be determined through dimension analysis or by demanding scale-invariance of the equal-time commutation relations.
  • #1
sinc
4
0
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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  • #2
sinc said:
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?
 
  • #3
samalkhaiat said:
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]
 
  • #4
sinc said:
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?
 
  • #5
We are considering the most general rescaling, so space and time scale differently. This is especially true for nonrelativistic case.
 
  • #6
sinc said:
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
 
  • #7
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
sinc said:
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).
 

1. What is the topological term F∼F and why is it important in science?

The topological term F∼F, also known as the topological susceptibility, is a quantity that measures the response of a physical system to changes in its topology, or shape. It is important in science because it provides valuable information about the underlying structure and behavior of different systems, such as materials, fluids, and even the universe itself.

2. How is the topological term F∼F related to scale invariance?

The topological term F∼F is scale invariant, meaning that it remains the same regardless of the size or scale of the system being studied. This is because topology, unlike other physical properties, is independent of scale and remains unchanged even if the system is stretched, compressed, or distorted.

3. Why is it significant that the topological term F∼F is scale independent?

The scale independence of the topological term F∼F is significant because it allows scientists to study and compare systems of different sizes and scales without having to account for the effects of scale on the topological properties. This makes it a useful tool in understanding and predicting the behavior of complex systems.

4. Can the topological term F∼F be used to study and understand different physical phenomena?

Yes, the topological term F∼F has been used in various fields of science, such as condensed matter physics, cosmology, and high-energy physics, to study and understand different physical phenomena. It has been particularly useful in understanding the behavior of phase transitions, where the topology of the system plays a crucial role.

5. How is the topological term F∼F measured experimentally?

The topological term F∼F can be measured experimentally using various techniques, such as lattice simulations, quantum field theories, and Monte Carlo methods. These methods involve studying the response of the system to changes in its topology, such as the introduction of defects or boundary conditions, and calculating the topological susceptibility from these measurements.

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