Re:Something is wrong in the state of QED [pdf]
by gnramires on Saturday February 25, 2023 @11:38AM
(#34937949)
Something not very intuitive is there's no ab initio (i.e. "blank slate", a priori, purely theoretical, etc.) notion of "legitimacy" in mathematics, or even physics. There was a lot of debate on whether complex numbers were "legitimate". The truth is, they work well in several physical models; that's where their legitimacy comes from. We assume that any crazy axiomatic system can model reality, well, almost any actually. There are some properties excepted of any axiomatic system, like self-consistency, and of course the ability to generate some kind of complete (or locally complete at some domain) model of reality. What we then do is try to come up with models and infer the correct or most useful ones from experiments.
This means if a physical model yields a sum like 1+2+3+..., we of course expect the result to be unphysical and the model invalid. However, if experimental data is consistent with a result like -1/12, that would be a clue to explore a different arithmetical system for our model, one which generalizes summations (to Ramanujam summation or other kinds of sums).
In mathematics the situation is even more interesting: the legitimacy is driven by taste of researchers, which will blend physical and philosophical relevance, and even aesthetics, curiosity and interest in part of the mathematician. (This is why we understand mathematics to be almost art, although a very particular art of course)
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Q. Is Borel summation legitimate?
A. Borel summation is a mathematical technique for summing certain types of divergent series. While it is a useful tool for obtaining asymptotic information about a function defined by a divergent series, the legitimacy of Borel summation is a subject of ongoing debate and depends on the context in which it is used.
In some cases, Borel summation can be used to obtain accurate results for certain physical or mathematical problems, such as in the context of perturbative quantum field theory. However, there are also cases where Borel summation can give incorrect or inconsistent results, and caution must be exercised when using it.
The main issue with Borel summation is that it involves an analytic continuation of the original divergent series to a complex plane, and the properties of this continuation can be difficult to analyze and control. As a result, it is important to carefully consider the assumptions and limitations of the method before applying it to a particular problem.
In summary, Borel summation can be a useful technique for summing certain types of divergent series, but its legitimacy depends on the context in which it is used and the care taken to ensure its validity.
Q. But doesn't a divergent series have no finite sum, by definition?
A. Yes, that's correct. A divergent series is defined as a series whose partial sums do not converge to a finite limit. In other words, a divergent series has no finite sum.
Borel summation is a technique for assigning a value to a divergent series by considering a certain integral transform of the series, known as the Borel transform. This technique can be used to obtain a "sum" for certain types of divergent series, but this sum is not a usual sum in the sense of convergent series.
Instead, the sum obtained through Borel summation is often an asymptotic expansion that captures the leading behavior of the series as the terms become large. This expansion can be used to obtain approximate results or estimates for certain physical or mathematical problems, even in cases where the original series is divergent.
However, it is important to keep in mind that the "sum" obtained through Borel summation is not a true sum in the sense of convergent series, and its validity and usefulness depend on the specific context in which it is used.
Q. What do you mean by "leading behavior of the series"?
A. The "leading behavior" of a series refers to the behavior of the terms of the series that dominate as the index of the terms becomes large. In other words, it refers to the behavior of the series as the terms approach infinity.
For example, consider the divergent series 1 + 2 + 4 + 8 + ... . The partial sums of this series diverge to infinity, so the series has no finite sum. However, we can still talk about the leading behavior of the series by considering the growth rate of the terms as the index of the terms becomes large. In this case, we see that the terms of the series grow exponentially, with each term being twice as large as the previous one. So the leading behavior of the series is exponential growth.
In the context of Borel summation, the idea is to use the Borel transform to extract the leading behavior of the divergent series, and express it as an asymptotic series in powers of some small parameter. This asymptotic series can be used to obtain approximate results or estimates for certain physical or mathematical problems, even in cases where the original series is divergent.
However, it is important to keep in mind that the leading behavior of a series may not capture all of its important features, and in some cases, the asymptotic series obtained through Borel summation may only be valid in a limited range of parameters or under certain assumptions.