Turning Ab Into My Personal Blog, #2: A Somewhat Ambitious Attempt To Explain One Of The Key Difficulties With Particle Physics Calculations

[jim360]

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...

[Lie-in King]

Is this...

An infinite number of mathematicians walk into a pub.

"I'll have a pint", says the first.
"I'll have a half", says the second.
"I'll have a quarter", says the third.
"I'll have an eighth", says the fourth.

After a few more requests, the barman puts 2 pints on the counter & says, "You mathematicians need to know your limits."

Jim - I do enjoy your posts, even though most of the maths tends to make a "whooshing" sound as it passes over my head :-)

[THECORBYLOON]

JIM, you already said, "Multiplication only makes things bigger if the number you are multiplying by is bigger than 1" and in my example, the multiplying factor is greater than one.

[webbo3]

Two chemists walk into a pub.
The first one says I'll have glass of water please.
The second one says I'll have a H20 too please, he drinks it and dies.

[jim360]

It's perhaps awkward English, but I think it's intuitive to most people that "x times smaller" is the same as "multiplied by 1/x".

[Canary42]

I thought the OP series was the basis of Calculus - am I not right. Or did I learn nothing at school?

P.S. I quote a similar case with the hands of a clock: As the minute hand approaches the hour hand, the latter has moved forward, therefore it will never catch up and pass the hour hand ?

[jim360]

Before going on to my next point, a few interesting videos on the nature of infinity:

[jim360]

[jim360]

[pixie374]

Well, it does, as it's moving faster.

[Jackdaw33]

When I was 10 I was taught to divide by fractions by simply inverting the numerator and denominator and then multiplying. I had no problem with this but I was 14 before I realised why.

[beso]

A friend who was much smarter than me once told me that there were many infinities and some were larger than others.

I never understood this. Was my friend correct?

[jim360]

Yes. This is explained in, among other places, the videos I linked to above. The most important is to recognise the difference between "countable" and "uncountable" infinity. The first is the size of the "counting" numbers 1,2,3... , while the second is the size of the number line -- ie, the answer to the question "How many numbers are there between zero and one?" The key concept is that "countable" infinities are those where you could conceivably count them off, one-by-one. Even though you never stopped, there would be some sense of making progress through the list.

It's possible to prove that one is literally bigger than the other by Cantor's diagonal argument, which should appear in two of the videos linked to above. It sort of runs like this, though:

1. Imagine you have a list of every number between zero and one.
2. This list is countable, by definition (because it's in a list form, so you can count off the entries).
3. On the other hand, you can easily construct a number that doesn't appear in the list, in the following way: change the first digit of the first number in the list by one, the second digit of the second number in the list by one, etc.
4. This new number is therefore different from every one of the numbers on the list.
5. You can always do this, so, no matter how many numbers there are on the list, you can always find at least one more that wasn't on it.

Therefore, there are more numbers between zero and one than can ever be listed, and the infinity is hence literally bigger than countable. The video probably explains it better, but in any case this is the sort of thing that requires a few run-throughs to appreciate, so please don't feel bad if you don't "get" it instantly.