Symmetric limit in Peskin's and Schroeder's (page 655)

In summary, the definition of a symmetric limit is when all components of a four-vector approach 0 in a consistent way, as opposed to a simple limit where only one variable approaches 0. This is relevant because the path taken to make all components 0 can affect the outcome. In calculus, we also have limits of several variables, but the definition is different and involves a metric. In the case of anomalies, the choice of regularization/renormalization is dictated by gauge invariance.
  • #1
MathematicalPhysicist
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What is exactly the definition of symmetric limit?
It's the first place in the book that I see this notation, and they don't even define what it means.

How does it a differ from a simple limit or asymptotic limit?
I found a few hits in google, but it doesn't seem to help:
https://physics.stackexchange.com/questions/294126/symmetric-limit-in-peskin-and-schroeder
https://en.wikipedia.org/wiki/Symmetric_derivative

But what is a symmetric limit?!
 
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  • #2
I would take (19.23) as their definition of the symmetric limit i.e., it means to take all components of the four-vector ##\epsilon## to ##0##, in a way that is consistent with those equations (in the links you posted there is some discussion whether the actually printed equations are correct, I don't have any input there). In a simple limit you would usually only have one variable approaching 0, but here there are all four components of ##\epsilon##, and it is relevant what path is taken to make all of them 0, similar to how there could be separate left- and right sided limite where it matters if you approach a limit from smaller or larger values.
 
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  • #3
Dr.AbeNikIanEdL said:
I would take (19.23) as their definition of the symmetric limit i.e., it means to take all components of the four-vector ##\epsilon## to ##0##, in a way that is consistent with those equations (in the links you posted there is some discussion whether the actually printed equations are correct, I don't have any input there). In a simple limit you would usually only have one variable approaching 0, but here there are all four components of ##\epsilon##, and it is relevant what path is taken to make all of them 0, similar to how there could be separate left- and right sided limite where it matters if you approach a limit from smaller or larger values.
In calculus 2-3 we have also limits of several variables; But I understand that simple limit definition, usually we have a metric (topological not the metric from differential geometry), which says the following:
##\lim_{\vec{x}\to \vec{x_0}} f(\vec{x})=\vec{v} \Leftrightarrow \forall \epsilon>0 \exists \delta >0 \forall d(\vec{x},\vec{x_0})< \delta \rightarrow \bar{d}(f(\vec{x}),\vec{v})<\epsilon##; but here it's not clear to me what we are exactly doing here?

There's also the distinction between a repeated limit and the limit itself, which is discussed in every course in calculus 2-3.
 
  • #4
Anomalies are a tricky business. In the case of the axial anomaly you can shuffle the anomaly at will between the vector and the axialvector current when looking at it simply as a question of how to regularize/renormalize a loop diagram (e.g., the triangle loop diagram involving one axial-vector and two vector currents as needed when looking at the decay ##\pi^0 \rightarrow \gamma+\gamma##).

But here how to choose the correct regularization/renormalization prescription is dictated by gauge invariance, i.e., the vector current must stay conserved as a consequence of local gauge invariance. This tells you, how to take the limit (or use any other regularization scheme, e.g., in dim. reg. there's the 't Hooft-Veltman rule about the commutation of ##\gamma^5## to ensure the conservation of the axial current rather than that of any current which is a linear combination of axial and vector currents).
 
  • #5
MathematicalPhysicist said:
In calculus 2-3 we have also limits of several variables; But I understand that simple limit definition, usually we have a metric (topological not the metric from differential geometry), which says the following:
lim→x→→x0f(→x)=→v⇔∀ϵ>0∃δ>0∀d(→x,→x0)<δ→¯d(f(→x),→v)<ϵlimx→→x0→f(x→)=v→⇔∀ϵ>0∃δ>0∀d(x→,x0→)<δ→d¯(f(x→),v→)<ϵ\lim_{\vec{x}\to \vec{x_0}} f(\vec{x})=\vec{v} \Leftrightarrow \forall \epsilon>0 \exists \delta >0 \forall d(\vec{x},\vec{x_0})< \delta \rightarrow \bar{d}(f(\vec{x}),\vec{v})

Not really sure what to say without repeating my previous post. One is basically interested in cases where the limit as you defined it does not necessarily exist (otherwise it should not be necessary to specify the path at all). As you brought up iterated limits: Think about the case where iterated limits in different orders do not give the same results, so the double limit does not exits. We now still want to talk about one particular order of taking the limits. Basically the same is happening here, just that we want to talk about one particular path for ##\epsilon## to approach 0, that is not necessarily given by first taking one component to 0 and then the others.
 
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1. What is the symmetric limit in Peskin's and Schroeder's (page 655)?

The symmetric limit in Peskin's and Schroeder's (page 655) refers to a specific mathematical approach used in quantum field theory to study the behavior of particles in a system. It involves taking the limit where the mass of a particle goes to zero while keeping all other parameters fixed.

2. Why is the symmetric limit important in quantum field theory?

The symmetric limit is important in quantum field theory because it allows for the study of massless particles, which play a crucial role in many physical phenomena, such as the behavior of photons in electromagnetic interactions.

3. How is the symmetric limit calculated?

The symmetric limit is calculated using a mathematical technique called dimensional regularization, which involves analytically continuing the number of spacetime dimensions from 4 to a complex value. This allows for the cancellation of divergences that arise in the calculations.

4. What are some applications of the symmetric limit in physics?

The symmetric limit has many applications in physics, particularly in the study of quantum chromodynamics (QCD), the theory of strong interactions. It is also used in the study of electroweak interactions and the Higgs mechanism.

5. Are there any limitations to the use of the symmetric limit?

While the symmetric limit is a powerful tool in quantum field theory, it does have some limitations. For example, it cannot be applied to systems with non-zero temperatures or to systems with non-trivial boundary conditions. Additionally, the results obtained from the symmetric limit may not always be physically meaningful and may require further interpretation.

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