I don't defend physicists mathematical laxity, we are very lazy, but I feel like we can't get dinged for something that came out of Ramanujan, he's solidly in the math canon. Arguing that we're using it out of context, maybe, but I've never heard his work should be considered unsound.

It's unsound because it assumes that a divergent series has a finite value. It's a kind of fanciful fiction, no matter who came up with it.

In other words, if 1 + 2 + 3 + ... had a finite value, then that value would be -1/12. But it doesn't have a finite value, because it diverges.

It's like saying that if pigs had wings, then they could fly. That's not something I'm happy to base physics on.

There is solid reasoning behind it though. Divergent perturbation series can be summed in a meaningful way depending on context. Eg https://en.wikipedia.org/wiki/Borel_summation

The mathematically sounds footing for this is unfortunately routinely not taught in undergrad physics.

I'm not familiar with Borel summation, so I asked ChatGPT about it, and found the conversation helpful. Thank you for the link.

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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.

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)

its only unsound if you are using standard summation. That is not the only kind, nor is it the only useful kind. The same can be said about zero or negative numbers, or the real number line. Those are also "fanciful fictions"

Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results. I think that's different from zero, negative numbers, real numbers, etc., which aren't self-contradictory.

I mean, do you really think the sum of the positive integers is -1/12? It's very easy to prove that it isn't. (The sum of any two positive integers is guaranteed to be greater than either one of them. Therefore, no sum of positive integers can be negative.)

> Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results.

It's self-contradictory in the definition that you're using for sum. It'd be like saying:

1 apple + 1 orange != 2 fruits because I only defined addition across the same units.

Also, if finite sums of integers have a property, that does not mean that infinite sums of integers have the same property. Ex:

1. The sum of any two integers is guaranteed to be bounded.

2. Therefore, all sums of integers are bounded.

one thing to keep in mind, is that infinity as a physical entity may not even exist. we have constructed infinity from ZFC, and deduced consequences of those axioms.

Many many examples in Math go against natural intuition but they are still useful

for example, you say that the real number isn't "self-contradictory". Look at the banach-tarski paradox, that certainly contradicts intuition. A large part of mathematics is foregoing natural intuitive, and just follow the logical deductions of axioms. That is how we got hyperbolic geometry, which is essential to special relativity.

It is not unsound, and it does not assume that a divergent series has a finite value.

It is true, that on real numbers addition is an RxR->R function, and you can only add 2 numbers, and you can't add 3 or even infinitely many numbers. But then you can define an another function that can take a whole series as an argument, and call it addition too.

You may forbid mathematicians naming another function 'addition', and talk about infinite sums, but then who cares. You can even die on this hill. Again, who cares.

you're essentially passing the ball to the mathematicians:

seems like you're saying "we physicist got this from the math people, take it up with them"

but what irks me, is the attitude to disregard understanding things which you're making use of (in this sentence: 'you' is a informally defined: 'physicists from community')

Sure, but I've always viewed Ramanujan's summation of divergent series to be a curio, I'd never have expected to see it turn up in physics. Pondering this a little, I think what disturbs me about this is the irreducible discontinuity it introduces. If you were to tell me that because of reasons, F = ma is wrong, it should have been F = m^{1.00000000001}a all along, then the resulting model of physics doesn't change that much; but if the Casimir equation should use ζ(-1+ε) rather than ζ(-1), then the sum is finite but as large as you like and the smaller that ε is, the larger this term is.

> Cashmir

Surely just a typo, but, just to make sure, it's Casimir (https://en.wikipedia.org/wiki/Hendrik_Casimir)—nothing to do with Kashmir (https://en.wikipedia.org/wiki/Kashmir).

Yes, of course (too late to edit that out now though), thanks