# Mathematical Proof

1. Sep 23, 2008

### Unredeemed

Hi there,
I am giving a talk to my class entitled "Beyond the standard model" - guess where I got that name from :p

And I was wondering if there are any interesting mathematical proofs I can give to a group of Year 11's (GCSE) who don't have a huge amount of mathematical knowledge. I'm going to be explaining things like the Higg's Boson, Graviton, Gravitino etc. in layman's terms so adding in some maths which people could understand would help a lot!

Thanks,
Jamie

2. Sep 23, 2008

### marcus

So this is a class of 14-16 year olds, maybe in Northern Ireland for example, judging from what Wikipedia says
http://en.wikipedia.org/wiki/GCSE

When you say "my class", are you their teacher or are you one of the 14-16 year-olds yourself? It sounds somehow like it is a student talk, not just a regular lecture by the teacher.

You say you would like to spice the talk up with some understandable maths.

Other people will have different ideas but I would say that at the very most basic level, the standard particle model is a predictive structure that doesn't give you predictions unless you plug some experimentally determined numbers into it.

It takes about two dozen numbers----dimensionless parameters.

A lot of them are masses of various particles, like the mass of the electron expressed as a ratio to the natural mass unit (the Planck mass).

One dream of how the model might be improved is if it could be more predictive with less inputs-----if you only had to plug in one dozen numbers to make it go, instead of two dozen.
Or (a fantastic idea) if you only had to plug in one number, and all the other masses and forces and stuff would come out!

As some elementary math, I might show them how to calculate the natural units that are used with the standard model. The Planck units.

Make sure they know what c, G, and hbar are.

natural units are defined by setting their values to unity: c=G=1

So tell me what the natural unit of force is. type c^4/G into google and press search

3. Sep 23, 2008

### Unredeemed

I'm one of the students and we're all 16. Google says 1.21049134 × 10^44 newtons

I think that's a great idea actually explaining the constants and how they are found. Thanks a lot!

But I'd quite like to "wow" my teacher, is there perhaps a way of linking the planck units to the theoretical particles like the Higgs Boson, Graviton, Gravitino, Selectron etc???

4. Sep 23, 2008

### humanino

That would "wow" not only your teacher, as Marcus was saying earlier.

5. Sep 23, 2008

### marcus

OK, tell me what you get when you put in
c*hbar
and press search

and also when you put this in:
c*hbar in newton meter^2

The latter is where you force google to give you the answer not in
joules meter
but in the equivalent units of newton meter^2

this is the natural measure of an inverse-square law attraction (like with a central body and a test particle some distance from it)
if you have a newton-meter^2 quantity and you divide by the square of the distance to a test particle
then you get the force, in newtons, exerted on the test particle

So force-area is a good kind of quantity to have a natural unit for.
==================

Now you might consider dividing the natural unit of force-area by the natural unit of force, which you handled earlier: c^4/G

Dividing (c*hbar)/(c^4/G) gives c*hbar*G/c^4
You can simplify that or you can put it into google as it is. You will get the natural unit of area---the Planck length squared.

Last edited: Sep 23, 2008
6. Sep 23, 2008

### marcus

Unredeem, get a practical operational grasp of what 1/137 is, in terms of the natural force unit c^4/G which you met already.

If you have two electrons (idealized as point particles) and they are 10^25 planck lengths apart, what is the force between them?

The force is 1/137 times the natural force unit c^4/G, except you have to divide by the square of the distance, which is 10^50.

1/137 tells you, in terms of the natural force unit, what is the force between two idealized point charges which are separated by unit distance. But Planck length is very small so the force would come out unimaginably strong. To make it more realistic we take a wider separation and get correspondingly less force.

The number which is approximately 1/137 is called alpha the fine structure constant. You likely know of it. It is a pure (unitless) number. It is in a sense exemplary of the numbers which have to be plugged in to the standard model, to get it to work. Many of the other two-dozen numbers represent particle masses---expressed as ratios to the Planck mass and thus as pure numbers (not GeV or kilogram quantities)

Last edited: Sep 23, 2008
7. Sep 23, 2008

### atyy

One interesting approach that I have heard, but don't remember the details of, so I'm probably garbling something important here, goes something like this:

1) Why do we need Planck's constant?
Experiments show that energy levels in the hydrogen atom are qunatized. So we need a model for this experimental result. We start naively with our classical planetary model of an electron orbiting a proton, and we find that we need a new law that says that the radius of the orbit is quantized. Now, how would we get a formula for the quantized length? We can do dimensional analysis as a first guess. We take all the classical constants like G, c (fundamental speed), e (fundamental charge), and see if we can get a "fundamental length" by combining them. It turns out that we cannot, and we need to introduce a new constant with units different from the classical constants. This turns out to be Planck's constant, which is necessary if we are to do quantum mechanics.

2) Now that we have Planck's constant, what would be a "natural" "fundamental mass"?
With Planck's constant, we can now imagine that there is such a thing as "fundamental length" or "fundamental mass" or "fundamental energy". ("Fundamental length" is the Planck length marcus mentioned. "Fundamental mass" and "fundamental energy" would be related by E=mc2). If we calculate them, we get ridiculously large numbers - all our "fundamental particles" are much lighter than the "natural" "fundamental mass". So the real problem is:

3) Why are all the "fundamental particles" of the standard model so much lighter than the "natural" "fundamental mass"? Is there some mechanism (symmetry?) that prevents them from having their "natural" "fundamental mass"?

BTW, a common introduction to dimensional analysis is to use it to calculate the speed of sound with "no physics", and show that we miraculously get a result that's pretty good, even though we did it so stupidly. There should be many other examples if you hunt around.

8. Sep 24, 2008

### atyy

http://math.ucr.edu/home/baez/lengths.html

In the above, John Baez gives an example of what I was thinking about. His first example is the Bohr radius, which is exactly the quantized "radius of the orbit" I was talking about. In his example, he assumes that Planck's constant exists, but if you play around you will find that you can't get a length from electron mass and charge, which are the only other parameters one might reasonably assume affect the electron's orbit around a proton.

After getting Planck's constant, we realise that the Bohr radius is special for the hydrogen and all of chemistry, which particle physicists claim is not that special since the proton is made of quarks, and there are many other fundamental particles not even used in chemistry. So instead of the Bohr radius, our fundamental length should be the Planck length, which is the fourth length that Baez discusses. He notes that the Planck length is "ridiculously small", and that the Planck mass is huge by particle physics standards.

So maybe none of our fundamental particles are fundamental? Maybe we would only see the true fundamental particles if we did experiments at the Planck energy? Well, that would make sense why we are having so much trouble getting gravity into the standard model - it isn't really a fundamental model at all!

OK, but if we are using the wrong "fundamental particles", then how can the standard model work so well? The idea is that sometimes in physics we are lucky, that we can find laws that work well enough, even though the "true" physics is hidden from us. For example, in modelling air flow over an aircraft's wings, we have laws which work well enough even though they don't take into account the atomic nature of air molecules! Or Newton's laws of mechanics which work well enough even though they don't take into account that everything is really a wave, and there is no such thing as a particle with definite position and momentum! Maybe the standard model is just another lucky theory - this is (roughly speaking) the view of effective field theory: http://en.wikipedia.org/wiki/Effective_field_theory

Now, what then is the "true theory" underlying the standard model? This is exactly what marcus and humanino have been talking about, which if we had, we could in principle derive all of the standard model from. Just as we can already in principle (it is claimed) derive all of chemistry from the standard model! :rofl:

Last edited: Sep 24, 2008
9. Sep 25, 2008

### Unredeemed

Thanks very much Marcus and atyy for your help. I think I'll start off with explaining how to find some of the Planck units and then how they are flawed etc.

Correct me if I'm wrong but I think a few of the predicted particles (Higgs boson, graviton etc.) have some roughly predicted charicteristics and I was wondering if there was any way of showing that?

Also, all the Planck units are effectively different equations relating h, G and c. Do we know why the equation must be arranged as c^4/G for the natural unit of force? Or root (hbar*c/G) for the Planck mass?

What do you mean here?

Thanks

10. Sep 25, 2008

### marcus

Don't get me wrong, I don't know what to suggest you could do for your talk (supposed to be about Beyond SM?)

Here's why I mentioned Planck units: they are basic to understanding how the SM is formulated. If you want a feel for beyond SM then you first need a feel for SM itself.
So natural units are a place to start.

I don't know what to recommend you actually talk about. or what would go down well with the 16-yearold audience. I only think, how can you get an intuitive handle on the SM yourself. What would help?

11. Sep 25, 2008

### Unredeemed

Well yes, for my talk I was going to talk about the Higgs, Graviton and then Super Symetric Particles. So starting with explaining the standard model and the Planck units would work well. But could I lead into the Higgs, Graviton or Super Symetric particles mathematically? Or would it be too difficult for my peers and I?

12. Sep 25, 2008

### marcus

Unredeemed,
Here is a good resource for you to know about
http://pdg.lbl.gov/2008/tables/contents_tables.html
These are the summary tables of the PARTICLE DATA GROUP. They have all the latest figures on the masses of particles and tables that lay the information out.
Here is the main homepage too, they have other stuff besides the tables
http://pdg.lbl.gov/

I just simply don't know! I don't know how you would do the transition, with that audience.
Off-hand I would guess you need to do a search on the web for a nice graphical presentation of the Standard Model, with pictures and diagrams----and basically forget about Planck units, as far as your talk goes.

I just thought I'd mention them to you personally, because it is some simple mathematics that has some connection with fundamental physics. Just a footnote to your main topic really.
You asked about uniqueness. Why does the unit of force have to be c^4/G ?

Because the idea of the units is JUST USE hbar, c, and G, and there is only one way to combine those three quantities so that the result is a force!

All the planck units are unique in that way. If all you have to work with is hbar, c, G and you want to make a quantity of area, then whatever algebraic combination of those three elements you try---whatever turns out to be an area is automatically going to be
hbar*G/c^3
There is no other possibility.

Unless you count multiplying in constant factors, like 2, so you get 2*hbar*G/c^3
(but that is trivial) or unless you count things that are algebraically the same,
like c*hbar*G/c^4
but that is really the same, you can cancel the c out and get back to hbar*G/c^3.

So the units are essentially unique. (up to constant factors like 2 or pi...)

How familiar are you with the fine structure constant---the famous 1/137 number?

It is an example of a coupling constant. The Standard Model contains coupling constants as well as particle masses. It is not only just an orderly listing of particles.
(though in your talk you may want to stick to particles, just to keep things simple)

Last edited: Sep 25, 2008
13. Sep 25, 2008

### Haelfix

You won't be able to talk about them in any depth to your peers unfortunately, besides simply laying down a periodic like table of the particles. Eg the higgs mediates mass, matter comes in generations, these are the fundamental particles we know off, the graviton carries the gravitational force like the photon carries the em force and supersymmetry is a doubling of particle species.

The mathematics and physics that go along with actually showing what they are is advanced and well out of reach of high school students.

You can explain certain elements of symmetry (for instance the eight fold way) and maybe you can copy the standard model lagrangian down and the path integral (to give people an idea of the sophistication of what the full quantum thing really looks like), but yea natural units are nice b/c it gives people an idea about the scales involved.

You could also briefly talk about cross sections and what people actually measure in a lab, which is a nice way to bring quantum mechanics into the picture.

14. Sep 25, 2008

### atyy

Take a look at the Problem/Discussion Set 4 of the Assigments for this course:
http://ocw.mit.edu/OcwWeb/Health-Sciences-and-Technology/HST-750Spring-2006/CourseHome/index.htm

One famous example they mention is the radius of a black hole, which of course requires General Relativity to calculate correctly. But if we just think, well a black hole is just something whose mass *M* is so densely packed in such a small radius *r* that it produces gravity *G* so strong that not even light *c* can escape from it, then there should be some equation relating these quantities.

Instead of looking at the quantities, let's look at their units ([x] means "SI units of x"):
[M]=kg
[r]=m
[G]=m3 kg-1 s-2
[c]=m s-1

We see that this is a correct equation for their units:
[r]=[G][M]/[c]2

Note that it is not at all guaranteed to be a correct equation for the quantities, since that involves physics! But we stupidly go ahead and say that since it's true for the units, it must be true for the quantities, so we stupidly predict that the radius of a black hole is:
r=GM/c2

Now we check what the correct equation is by carefully working it out using General Relativity, and find that the correct equation is:
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/blkhol.html

So we were wrong by 50% - or we can think that by sheer stupidity we already solved 50% of the problem, and that this is a very efficient form of stupidity! Everything marcus was talking about and the article above by Baez is about dimensional analysis. (I should mention that dimensional analysis can fail, and Kenneth Wilson got a Nobel Prize for figuring out a situation where dimensional analysis doesn't work.)

15. Sep 26, 2008

### Unredeemed

I'm not familiar with it at all. I know what it is but don't really understand it. I know that a coupling constant is a constant which governs the interaction of things - for example the Gravitational constant.

Is there a reason that we get this? Or is it just one of those things?

Thanks a lot for everyone's help, my talk preparation is going extremely well so far.

16. Sep 26, 2008

### marcus

The number 1/137 is the size of the electromagnetic interaction expressed in natural units, something you can see if you think of the force between two point particles following an inverse square law. (that is only one way to look at it, alpha plays many roles, and inverse square is an oversimplification that works at ordinary scale but not very small scale).

You can find the force between two electrons separated by a distance R if you just take 1/137 and divide by the square of the distance expressed in planck lengths. That will give you the force expressed in natural force units (the c^4/G you calculated earlier).

If the separation is 10^25 planck lengths, then you just put 1/137 into your calculator and divide by the square, which is 10^50. And that will give you the force, expressed in natural force units.

You already calculated the force unit to be 1.2 x 10^44 newton, so if you divide that by 10^50 you get 1.2 micronewton. So if you do the arithmetic you will get that the force between the electrons is 1/137 x 1.2 micronewton------which is 1.2/137 micronewton, on the order of 10 nanonewton.

What more can you ask from a number that is supposed to tell you the size of the electromagnetic interaction? It tells you the force between two charged particles (in natural units terms) in the simplest way imaginable.

It does a lot lot more, turning up in all kinds of physics contexts, including the lines in the hydrogen atom spectrum---the wavelengths of light that hydrogen glows when heated---and much other stuff. It's the basic electrodynamics number. So it gets into nearly everything. But this is one basic situation.

I am glad to hear that your preparation for the talk is going well. I still advise focusing mainly on the particles, getting graphics that represent the standard model pictorially, and going light on the actual mathematics. for example talking about alpha, the fine structure constant, to that audience might get you in trouble. just a wee bit too technical, I would guess. Another thing, the conventional way physicists TALK about particle masses is in terms of MeV and GeV. That is the way you should have memorized the masses of the main things. They always talk in terms of electron volt energy equivalents.

The planck units are most of the time sitting hunched over in the back room doing the actual work and the smiling faces in the front of the store are all electron volts.
Most working physicists in my experience act as if they are unable to think in planck units and their day-to-day lingo is all GeV. So for your talk to sound right you have to be conversant in GeV terms. that is my personal opinion. if I am wrong I hope someone corrects me on this.

Last edited: Sep 26, 2008
17. Sep 27, 2008

### atyy

Good question. I recommend you try at least 3 or 4 of the other problems in that assignment and see how accurate dimensional analysis is - the turkey problem looks interesting. Generally, dimensional analysis brings us within a power or two of 10, and much more importantly, gives us the right form of the equation. In a celebrated case where it failed spectacularly by not even getting form of the equation right, it pointed to the interesting new viewpoint of the "renormalization group" developed by Leo Kadanoff and Kenneth Wilson, among others. But let's try to answer the question in the context of Beyond the Standard Model.
I'm going to go for glory and devise the theory that all these guys have been talking about. To explain all the masses of the standard model, I postulate that there's only one fundamental particle, and everything else is made of it. Since I'm a biologist and know no physics, I will use stupid dimensional analysis as usual.

Since my theory will be the most fundamental theory, its only inputs will be the fundamental constants G,c,h. The number that marcus is dreaming about will be the mass of the fundamental particle mf, from which we will be able to get the masses of all other particles. Again, just looking at the dimensions:
[mf]=kg
[G]=m3 kg-1 s-2
[c]=m s-1
[h]=m2 kg s-1

This is a correct equation for the units:
[mf]=[h]1/2[c]1/2[G]-1/2

As usual, I boldly and stupidly predict it is the correct equation for the quantities themselves:
mf=h1/2c1/2G-1/2

Plugging in the numbers I get mf=2 × 10-8 kg.

To convert to particle physicist units, I use the conversion machine at http://physics.nist.gov/cuu/Constants/energy.html, and get mf=1 × 1028 eV = 100 × 1017 GeV.

Well, that was easy! I name my great discovery of the fundamental mass An Exceptionally Stupid Theory of Everything.

Well, to be a little serious, this mass is actually the Planck mass that marcus has been talking about. Looking at his link http://pdg.lbl.gov/2008/tables/contents_tables.html, it seems that the heaviest particle, the top quark, ~ 200 GeV, is 1017 times lighter than the Planck mass. If the Planck mass represents a truly fundamental mass out of which the standard model particles are "made", then it appears that lighter particles are "made" of heavier particles - can that even begin to make sense? It seems that dimensional analysis is off not by one or two powers of 10, but an astounding 17 powers. But if we believe that dimensional analysis is only exceptionally stupid, and not truly stupid, then an interesting question is, as Frank Wilczek says, "Why is the proton's mass so small?" (Wilczek comment is in http://arxiv.org/PS_cache/hep-ph/pdf/0201/0201222v2.pdf, recommended by granpa on marcus's thread https://www.physicsforums.com/showthread.php?t=254612)

Last edited: Sep 27, 2008