Does -1/12 Protect Us From Infinity? - Numberphile

physics of quantum field theory? Quantum field theory is the theory that describes how, basically, it is the best theory of fundamental particles that we have. fundamental fabric of the universe, okay, it tells you, you know how you know particles move and frown, how they interact with each other, this is the theory we use when you do quantum field theory, you get infinities spitting out, they just go out there Are there things like there are infinite sums but they are also infinite integrals and you have to control them somehow? So you have these control mechanisms to try to do it and normally you have to throw away an Infinity, what we did was we decided to develop a new way to control these integrals based on this inside of TOS and we applied it to Quantum F, then we asked the following question: Is there one that is preferred in any way now in the Quantum filter?

In particle physics, the most important thing, what underpins all of particle physics is something called symmetry, so you think about symmetry like you know Reflections, you know something looks the same, you look the same under a reflection, so there is a kind of reflection symmetry or maybe you have rotational symmetry, something like a sphere, you rotate it and it looks the same. You can extend these ideas to particle physics and symmetry dictates the structure of fundamental particles and how particle physics works. Which is fascinating. The thing about this is that generally when you make a quantum field there, when you add quantum corrections to it, you break the symmetries and that's bad, that means you don't have control of the theory, which you don't want to do, what we What we found was that if you use our new way of trying to control the integrals and we ask when you don't break the symmetries and it turns out that it is precisely when you use those Regulators those waiting functions like this, then it is the ones that give you minus a 12 without divergence without

infinity

they have somehow something to do with how nature

protect

s us from

infinity

on the level of particle physics and we don't understand why this happens, well we wonder if there is something about this kind of fibrous nature of how these interactions develop, how String Theory

protect

s us on an even more fundamental level from these Infinities, when we write this article that will be published this week, we are really excited right now.

I just pointed out the coincidence that somehow you have these ways of controlling these infinite sums that give you minus 12 and somehow they have something to do with how nature protects us from breaking symmetry. it protects us from that happening and there's something more to say about it maybe there is maybe you don't know it but if you know it's tempting when you see these coincidences it's hard to ignore them you want to explore them more and yeah we're very excited about that , most people on the street if you ask them about the sum of all natural numbers up to Infinity mhm they say Well, it must be equal to Infinity, that is the intuitive answer, even you would say that it is the intuitive one, how when you have this discussion with someone you say well, actually it could also be 12.

How do you explain that to them without talking about regularization and string theory and what is the only thing you tell them that justifies saying it? it could be minus 12. I mean, I don't think so. I don't think it's possible to explain it to anyone in an intuitive sense because I don't think anyone has an intuitive understanding of it. It's not intuitive. I mean, that's it. as simple as that I think the best thing you can say is look this is an infinite summer an infinite series you know this is what it is and you don't understand the Infinite because you live in a finite world and I live in a finite world we don't have intuition for infinity you don't really know you know this goes back this is infinity went canel crazy right you know this is not a place that that that that you think you understand and many, many, uh, mathematicians and physicists I just said: stay away from that, almost as if you had no right to have an intuition.

Yeah, and the firefighter, of course, the firefighter won the Nobel Prize for controlling some of these infinities in the pH of the particles, but he openly said let's sweep them under the rug. Okay, and what we're trying to do is try to find ways to understand it. Well, let's not stay blank. Let's try to understand how the journey to Infinity is known in some sense. Is there a favorite journey to Infinity? And I think that's what it is. What I would say is happening here is that you have these ways of taking this sum towards infinity and there are potentially preferred Journeys that take you to a finite place and we can see that that is indeed the case in the system of partial sums, well, that's how it is.

I'm not going to take you to a finite place. You know it, but it's just one of the. You know an infinite number of ways you could have taken that Journey correctly, and there are an infinite number of Journeys you can take. which will take you directly to minus 12 and those trips seem to be special and I think they are worth exploring in more detail. Definitely people will always consider it as the first group of terms and then they will try to make that group bigger and bigger and bigger, they will do partial sums, but as Towers faces, if you do partial sums, you are doing a very drastic event, you're doing something very precise, that's actually, you know. it's a zero measures option uh in the kind of things that you could do and why just because your intuition tells you to do this

does

n't mean mathematically that you should do that or even physically that you should do that and are there other possibilities? and they give more interesting answers that maybe tell us something profound about how the universe works and how numbers work upwards and the third one, of course, will be the one that really interests us, which is 1 + 2. + 3 plus four and so on. successively, so let's evaluate all these three different sums.

Now the first one is very easy to evaluate and then you forget, so you put it in a box somewhere and you put it You put it in a closet and you put it, you classify it into things that you have already understood, but I think you don't really understand it. we completely understand.