# Fixed point and scale invariance

• Einj
In summary, The concept of fixed points in the context of QFT refers to a specific value of the running coupling, denoted as ##\lambda^*##, where the theory remains meaningful at arbitrarily high energy scales. In the Wilsonian point of view, changing the energy scale introduces a new theory with a new Lagrangian, and the UV fixed point is the Lagrangian towards which all other Lagrangians converge as the energy scale increases. This is due to the fact that the beta function for the coupling is zero at the fixed point, making it scale independent. However, it is worth noting that RGevo may not necessarily be at the fixed point.
Einj
Hello everyone. I'm studying the fixed point of theory in the context of QFT. First of all, let me say what I think I understood about fixed points and then I'll state my question.
Suppose we have a theory with a certain running coupling ##\lambda(\mu)##. If we have, for example, an UV fixed point, say ##\lambda^*##, this means that when the energy scale increases the coupling will converge towards this value and hence the theory is defined at arbitraty high energy since it remains meaningful.
In the Wilsonian point of view, everytime that we change our energy scale we are introducing a new theory with a new Lagrangian. In this languange a UV fixed point is that Lagrangian towards which every other Lagrangian converge when the energy scale increases.

First of all: is this correct?

Secondly, I found in my places that the theory at the fixed point is scale invariant. Can anyone explain to me why?

Thanks a lot
Cheers

In this example, The fixed point is the position where the beta function for the coupling is zero.

Therefore it is scale independent by definition!

RGevo said:
In this example, The fixed point is the position where the beta function for the coupling is zero.

Therefore it is scale independent by definition!

RGevo is presumably not at the fixed point? :D

## 1. What is a fixed point in scientific research?

A fixed point in scientific research refers to a point or value that remains unchanged under a certain operation or transformation. This means that regardless of the input or conditions, the output or result will always be the same. In other words, the value is invariant.

## 2. How is fixed point related to scale invariance?

Fixed point and scale invariance are closely related concepts in scientific research. Scale invariance refers to the property of a system or phenomenon that remains unchanged when its size or scale changes. A fixed point, as mentioned earlier, is a value that remains invariant under certain operations. In essence, scale invariance can be thought of as a type of fixed point, where the system or phenomenon remains invariant regardless of the scale at which it is observed.

## 3. What are some examples of fixed points in scientific research?

Fixed points can be found in various areas of scientific research, including physics, biology, and economics. Some examples include the fixed point in population dynamics, where a stable population size is reached regardless of initial conditions, and the fixed point in thermodynamics, where the state of a system remains unchanged under certain conditions.

## 4. How are fixed points and scale invariance used in data analysis?

Fixed points and scale invariance are important tools in data analysis, particularly in the study of complex systems. They can help identify patterns or relationships that are invariant or remain unchanged under different conditions. This can provide insights into the underlying mechanisms and dynamics of the system being studied.

## 5. What are the practical applications of fixed point and scale invariance?

Fixed point and scale invariance have practical applications in various fields, including physics, biology, and economics. These concepts can help us understand and predict the behavior of complex systems, such as stock markets, population dynamics, and climate change. They can also be used in data analysis to identify patterns and relationships that are invariant, which can aid in decision-making and problem-solving.

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