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

[MathematicalPhysicist]

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?!

 

[Dr.AbeNikIanEdL]

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.

 

[MathematicalPhysicist]

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.

 

[vanhees71]

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).

 

[Dr.AbeNikIanEdL]

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.