• Question: What is Renormalization? I have read that its a technique used to remove infinities but I didn't understand a bit about it.

    Asked by rajathjackson to Jack, David, Dave, Chris on 19 Jun 2013.
    • Photo: Jack Miller

      Jack Miller answered on 19 Jun 2013:

      Hi Rajathjackson,

      You sound like a budding particle physicist! Renormalisation is indeed a way of getting rid of pesky infinities, and frankly I don’t understand it too well. There are actually many different specific mathematical tools that renormalise something that goes on, but the basic idea is as follows.

      Let’s say we’ve got a particle that’s doing absolutely nothing. It’s just whizzing along in free space — I measure it at a time t=0 at position x=0, and again at time t=t1, and position x=x1, and I know it’s just moved some distance in space over some time. I can draw a diagram of that particle’s path — called Feynman diagram — that shows this. That particle, if it just whizzed along in space, almost certainly did nothing along the way. Yet, we don’t _know_ that. It’s possible that something happened to that particle along the way — if it’s an electron, let’s say that it met up with a photon that gave it some energy, which it then emitted later before I measured it — and there’s no way I’d be able to tell the difference.

      If you think about it for a bit, there are infinitely many things that could happen to that particle while I’m not looking at it. You can draw more and more unlikely Feynman diagrams that could all be true. My one electron whizzing around in space (or, as far as the electron’s concerned, sitting around in space) could have infinitely many different events happen to it before I measure it, and find it’s the same electron again. And there’s no way I’d know.

      To calculate quantities in particle physics, we basically have to add up all the different Feynman diagrams, which kind of give me a probabilistically-weighted answer of what will happen overall given the lots of different ways it could happen. It also turns out there’s a rigorous mathematical way of translating between these diagrams and the relevant maths you need to do (though it’s far from easy…). You get a giant integral (which you’ll learn about in sixth form — ask your maths teacher when he looks bored) that gives quantities that can be related to things you can measure.

      If the Feynman diagrams are simple — just a straight line — we can do the integral by hand. But, on the other hand, if they contain loops or structures that are harder for us to interpret, the integrals break down, and the probability that something occurs becomes _infinite_. Now, I don’t know what a probability greater than one is (let alone infinity), if not a mistake. Renormalisation is a wonderful fudge, where you basically take this great big integral, and multiply it by one. There are many different forms of one — 1, x divided by x, cos^2 x + sin^2 x — and it’s often a handy mathematical trick to multiply or divide by one if you’re a bit stuck (if you can find the right form of one to use!). We multiply the stuff within the integral with a very, VERY complicated form of “one”, and, if we make things happen just right, we get an answer out that’s not infinity any more, but rather something finite we can measure.

      When we do the experiment, and compare the results with these theories, they agree. To one part in 10^13.

      This is basically renormalisation — a sort of mathematical fudge really to get rid of all these pesky infinities that cloud your theory.

      Hope this helps!

      — Jack

    • Photo: David Freeborn

      David Freeborn answered on 20 Jun 2013:

      Hi rajathjackson,

      That’s another great question. Jack’s answered this pretty well already. As he says, it’s a mathematical fudge to remove the infinities from our theories. These infinities arise from Feynman’s approach to quantum mechanics- the method of adding up every possible path that the particle can travel along.

      For example, sometimes it’s possible for a particle to radiate another particle, and then reabsorb that particle later. In fact, this is always possible, and there are an infinite number of ways this can happen, and that leads to infinities in our equations.

      Renormalisation is one way to get rid of these infinities. Sometimes the method used is very simple: we refuse to take into account any particles above a certain energy, which we call the “scale” of the theory. What this really means is that we think our theory is only an approximation, and it stops working above this energy scale.

      All modern fundamental physics theories are called “gauge field theories” and these all require renormalisation.The only way to get rid of renormalisation would be with a completely different type of theory. So what we need is an entirely new type of physical theory. Nobody knows what that is going to look like yet. We need new ideas!

Comments

  • Photo: rajathjackson

    rajathjackson commented on :

    But why this method cannot be used to remove infinities that arise while trying to combine gravity and other forces of nature?

  • Photo: Jack

    Jack commented on :

    Another excellent question! It depends on the specific problem at hand, and the full, detailed answer frankly requires a level of understanding beyond mine — David uses these sorts of tools on an everyday basis, so he’s perhaps better posed to answer than I am. That being said, I’ll give it a shot…

    The basic answer is that sometimes these tricks don’t work! What we actually do is a process equivalent to taking limits. Let me just explain what that is — ask if you don’t understand anything!

    Consider the curve sin(x)/x (which is often called ‘sinc(x)’) — you probably know what a sine wave looks like, and if it’s being divided by its argument, its amplitude drops off as x increase — in fact, it looks a bit like this: http://en.wikipedia.org/wiki/File:Si_sinc.svg. We can calculate values of this curve at every point other than x=0 — sin(x) is a number, x is a number, and so sin(x)/x is another number. However, at x=0, sin(x) is 0, and x is 0 — so we’re left with the expression ‘0/0’, which isn’t defined (there are other quantities that aren’t defined — the most relevant here is infinity-infinity). Yet, we can actually find out what value sin(x)/x takes _as_ x approaches zero — if you try on your calculator to work out sin(0.0001)/0.0001, you’ll get something close to one; sin(0.0000001)/(0.0000001) is _even closer_ to one. This strongly implies that sin(0)/0 is one, and there’s a formal mathematical process, called ‘finding limits’, to show that as x approaches zero from above (written x->0+) or below (x->0-), sin(x)/x tends to one. We usually then take the leap of faith and say that sin(0)/0 IS one — which is what you can see on the graph I linked to earlier.

    In this example, if I told you that sin(x) is approximately equal to x if x is small (and measured in another unit of angles, called radians, rather than degrees), then you could see that this is true another way — if x is small, then sin(x) is approximately equal to x, and so sin(x)/x is the same as x/x — which IS one. However, if I didn’t tell you that rule, you’d have a harder job to rigorously show that sin(x)/x is one at x=0.

    This is analogous to the problem of renormalising theories of quantum gravity — the usual methods we have for renormalisation don’t work, and as the problem is essentially equivalent to infinity minus infinity, we have to do it properly to avoid getting any arbitrary number out the other side. If we can’t find a mathematical technique that works when we try to find the limit, we’re essentially stuffed. Moreover, a lot of these theories have inherently “more infinity” that crops out in your answer, and it’s difficult to get rid of it using techniques we know about at the moment.

    A lot of people see all of these odd infinities as a fundamental problem with quantum electrodynamics — Feynman included (and he developed it). You end up with very odd consequences, like, in quantum electrodynamics the ‘base’ or ‘bare’ charge of an electron isn’t -1 — it’s infinite. This is because the processes involving the measurement of the physical properties themselves are subject to these sorts of problems — and it’s only once you renormalise the ‘bare’ quantity do you end up with something you can measure. It’s pretty damn odd, but it seems to work.

    If you’re interested in learning more, and hopefully in a more coherent fashion than what I can manage, then I strongly suggest you read a book called ‘QED: A strange theory of light and matter’, by that great Nobel laureate himself, Feynman.

    Keep asking these questions!

    — Jack