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sinc
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why the topological term in gauge theory, [itex]ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ} [/itex] ,is scale-independent?
sinc said:why the topological term in gauge theory, [itex]ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ} [/itex] ,is scale-independent?
samalkhaiat said:Do you know how the field tensor transforms under scale transformation?
sinc said:if the spacetime coordinates scales as [itex]x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}[/itex]
sinc said:We are considering the most general rescaling, so space and time scale differently. This is especially true for nonrelativistic case.
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.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.
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...
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.
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.
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.
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.
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.