Does anybody know any physical argument for using zeta function regularization?
Or it is an open question?
Thanks.
Zeta function regularization (like any regularization procedure) is a choice about mathematics, since the limit can't depend on the procedure; the physics comes after the fact.
My modest contribution can be found here
Preprint Chaotic dynamics of an electron
Hi Stam Nicolis
Thank you for your reply,
but how does an analytic continuation of the zeta function lead to a result that is compatible with experiments like the Casimir effect, do scientists know how this happened or not?
Thanks.
Hi Igor Bayak
Thank you for sharing your article, but please can you help me to understand where is your answer to my question in your article?
Thanks.
If chaotic dynamics is not a physical argument for you, then just ignore my previous message.
Igor Bayak
Can you please apply your idea to the Casimir effect?
Thanks.
Note: I found one article that may help us to answer this question...
Preprint Justification for zeta function regularization
For the Casimir effect and related problems, yes, of course. The calculation is the difference between two infinite quantities, but the important point is that, while both quantities are infinite, their difference is finite. So it can be calculated by imposing any sort of regularization; zeta-function regularization is just one of many.
Hi Stam Nicolis
You said:
"but the important point is that, while both quantities are infinite, their difference is finite"
So as I said before we don't know yet how this is work in fact, right?
Thanks.
We do know, because we can show why the difference must be finite: It's because to define the infinite quantities it's necessary to introduce cutoffs, perform in this way the calculations in terms of finite quantities, and show that the results don't depend in how the cutoffs are defined.
Hi Stam Nicolis
Yes, for the Casimir effect we can perform the calculation without the need to use zeta regularization by introducing the cutoffs, etc...
but my question is how we can transform (when we use the zeta regularization) the infinite sum:
Sum wn to Sum wn * |wn |-s
then we use the analytic continuation of the Riemann zeta function for s=0.
How is this technic compatible with our calculation without regularization?
In other words, how does the pure mathematical concept "analytic continuation" give us the right result?
Thanks.
Dear Lena J-T Strömberg
About plates:
The Casimir effect happened between two close parallel uncharged conducting plates.
In literature:
The appearing force is caused by the vacuum fluctuation of the electromagnetic field.
I think the vacuum fluctuation is an illusion, anyway, this is another story...
Thanks.
The short answer is that the(ill-defined, since divergent) Σ w_n is defined as the value of the (well-defined, since convergent) Σ w_n f(w_n), where f is chosen so that the series converges. The non-trivial part of the process is the proof that the finite value is independent of the choice of the function f.
https://www.google.com/url?sa=t&source=web&rct=j&url=https://dc.etsu.edu/cgi/viewcontent.cgi%3Farticle%3D5409%26context%3Detd&ved=2ahUKEwisoorz6K_5AhXM6aQKHYQxAWM4FBAWegQIDxAB&usg=AOvVaw3n_6NdJ4REgn3cO9cENxH1
This is a useful reference.
Dear Mazen Khoder ,
Changing s to z in the infinite sum, which gives the Riemann z-function, does not give the analytic continuity, automatically. For s=1, it looks smooth, at the plots. You can use s=1, perhaps, directly? I did some analysis, e.g. rewriting the sum into a halfinteger sum denoted sum(3/2), and then it factorise like
sum= (1-exp(-zln2))^-1(sum(3/2)+1)
so 'the thing' at s=0 is seen to appear in the denominator. Close to s=0, it fluctuates 'much', in numerical calculations, if you do not impose something else.
Regards, good luck
Dear Lena J-T Strömberg
For good visualization of the analytic continuation of the zeta function I suggest this beautiful video:
https://www.youtube.com/watch?v=sD0NjbwqlYw
Thanks.
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