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Turning Ab Into My Personal Blog, #2: A Somewhat Ambitious Attempt To Explain One Of The Key Difficulties With Particle Physics Calculations
Two months into my stay, and a little over a month-and-a-half into my work, it seems time for another update. Has to be said that the weather has meant for difficult working conditions, but the good news is that there's no major time pressure and I've got at least one supportive colleague that it's been enjoyable to work with.
But anyway. What I wanted to talk about today is very central to understanding why the computation I'll be doing is difficult, but I'm also going to avoid physics altogether for this post (and later try and connect it).
Consider the following sum:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
If we carry on forever, what does this equal? Does this question even make sense? I think it's a reasonable first intuition that adding together infinitely many things would give an answer that is infinite. But you'll see that this isn't true for the sum above. You can think of this at taking one step towards 2, then half what's left, half of what's left again, and so on... so that, firstly, you'll see that you can never get further than two steps away from where you started; and, secondly, that if you keep doing this forever you'll also get closer, and closer, and closer still... to 2 steps away. So the answer is that, in effect,
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2
and, if there are infinitely many terms on the left, then this is exact.
So, we refine our intuition a bit, and say that, yes, adding infinitely many things together in general gets bigger, but if they are small enough then you get something finite after all.
But how small is "small enough"? Here's another sequence:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...
What does this tend towards? It's clear that everything is getting smaller, and it's also fun to evaluate this for the first, say, hundred, thousand, million, etc, terms, to see what happens. Very roughly, you get the following:
-- sum of first 10 terms = 3
-- sum of first 100 terms = 5
-- sum of first 1000 terms = 7.5
-- sum of first million terms = 14
-- sum of first billion terms = 21
-- sum of first trillion terms = 28
-- sum of first googol terms = 231
It keep growing, excruciatingly slowly, but basically every time you add a thousand times more terms than you had, the total gets around 7 bigger. This never stops, and amazing, despite crawling along at a snail's pace, this sum grows to infinity. This is the famous result that "the Harmonic series is divergent", by the way, for those who want to do more reading. There are several cute and amazing results associated with this, but for my purposes the point is that whether infinite sums blow up or not is not always instantly clear.
Particle Physics is filled with such sums, is the other point. The key difference, and where the real difficulty comes in, is that in the above examples there was a very simple rule for working out the next term: in the first, divide the previous term by two; in the second, add one to the number on the bottom of the fraction. But in Particle Physics it's never so neat, and, whilst we very often have to try and compute these sums, we have to work out each term one at a time!
To be continued...
But anyway. What I wanted to talk about today is very central to understanding why the computation I'll be doing is difficult, but I'm also going to avoid physics altogether for this post (and later try and connect it).
Consider the following sum:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
If we carry on forever, what does this equal? Does this question even make sense? I think it's a reasonable first intuition that adding together infinitely many things would give an answer that is infinite. But you'll see that this isn't true for the sum above. You can think of this at taking one step towards 2, then half what's left, half of what's left again, and so on... so that, firstly, you'll see that you can never get further than two steps away from where you started; and, secondly, that if you keep doing this forever you'll also get closer, and closer, and closer still... to 2 steps away. So the answer is that, in effect,
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2
and, if there are infinitely many terms on the left, then this is exact.
So, we refine our intuition a bit, and say that, yes, adding infinitely many things together in general gets bigger, but if they are small enough then you get something finite after all.
But how small is "small enough"? Here's another sequence:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...
What does this tend towards? It's clear that everything is getting smaller, and it's also fun to evaluate this for the first, say, hundred, thousand, million, etc, terms, to see what happens. Very roughly, you get the following:
-- sum of first 10 terms = 3
-- sum of first 100 terms = 5
-- sum of first 1000 terms = 7.5
-- sum of first million terms = 14
-- sum of first billion terms = 21
-- sum of first trillion terms = 28
-- sum of first googol terms = 231
It keep growing, excruciatingly slowly, but basically every time you add a thousand times more terms than you had, the total gets around 7 bigger. This never stops, and amazing, despite crawling along at a snail's pace, this sum grows to infinity. This is the famous result that "the Harmonic series is divergent", by the way, for those who want to do more reading. There are several cute and amazing results associated with this, but for my purposes the point is that whether infinite sums blow up or not is not always instantly clear.
Particle Physics is filled with such sums, is the other point. The key difference, and where the real difficulty comes in, is that in the above examples there was a very simple rule for working out the next term: in the first, divide the previous term by two; in the second, add one to the number on the bottom of the fraction. But in Particle Physics it's never so neat, and, whilst we very often have to try and compute these sums, we have to work out each term one at a time!
To be continued...
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Multiplication only makes things bigger if the number you are multiplying by is bigger than 1. To see why this is, note that multiplying something by 1 can't change a thing at all. So, if you multiply by something bigger than one it gets bigger, and if you multiply by something smaller than one it gets smaller.
Put another way, multiplying by a number smaller than one is the same as dividing by a larger number. What happens if you divide by a larger number?
Put another way, multiplying by a number smaller than one is the same as dividing by a larger number. What happens if you divide by a larger number?
Yes, perhaps I should have done a post 1.5 on what the meaning of infinite is. Here goes.
* * * * * *
The word "infinite" is in a sense overloaded in maths, it has a lot of meanings.
Meaning 1: "divergent" -- in this sense, if something never stops growing, then, if left long enough, it would get infinitely large.
Meaning 2: "Countably infinite" -- this applies when there are infinitely many objects, but there is also a clear sense in which you could number them off. For example, there are infinitely many numbers 1,2,3..., but you can also count them.
Meaning 3: "uncountably infinite" -- this time, there are infinitely many things, but there is no hope at all of being able to count them. This corresponds to, for example, all the infinitely many points on a straight line.
There are probably more meanings, but these are the three main ones. In the above, the sums are "countably infinite", as you have infinitely many terms but you can count them off one-by-one. The first sum is, however, not divergent: if you do the addition, you eventually get 2. The second sum is both countably infinite and divergent.
* * * * * *
The word "infinite" is in a sense overloaded in maths, it has a lot of meanings.
Meaning 1: "divergent" -- in this sense, if something never stops growing, then, if left long enough, it would get infinitely large.
Meaning 2: "Countably infinite" -- this applies when there are infinitely many objects, but there is also a clear sense in which you could number them off. For example, there are infinitely many numbers 1,2,3..., but you can also count them.
Meaning 3: "uncountably infinite" -- this time, there are infinitely many things, but there is no hope at all of being able to count them. This corresponds to, for example, all the infinitely many points on a straight line.
There are probably more meanings, but these are the three main ones. In the above, the sums are "countably infinite", as you have infinitely many terms but you can count them off one-by-one. The first sum is, however, not divergent: if you do the addition, you eventually get 2. The second sum is both countably infinite and divergent.
Why are they silly? This is standard in mathematics. For example, it's useful and necessary to distinguish between things that can be listed, and things that cannot, so you need two different types of infinity to label them. It's also useful to thing of quantities that blow up in your face, rather than just get very large but still finite.
It's not imbecilic to ask such questions, that's how we learn after all.
This is only meant to be a small introduction, so you'll have to do more reading around if you want to understand this, but rest assured that it's on a firm foundation and all of this makes clear and logical mathematical sense. It just takes a while to get used to. For example, some of the most famous early paradoxes in maths concern precisely this sort of problem, eg Achilles and the Tortoise.
Infinity is hard.
This is only meant to be a small introduction, so you'll have to do more reading around if you want to understand this, but rest assured that it's on a firm foundation and all of this makes clear and logical mathematical sense. It just takes a while to get used to. For example, some of the most famous early paradoxes in maths concern precisely this sort of problem, eg Achilles and the Tortoise.
Infinity is hard.
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