Witten's dog - as drawn in Futurama the best cartoon in history

What is it about The Simpsons, and Futurama? Why do these shows seem to be loved and hated with equal conviction? I set myself the task to find out just why these cartoons appeal so strongly to such a wide range of people.

My kids laugh with insanity at some parts. The younger of the two likes the slapstick, but I've noticed that my older boy is laughing at some of the more sophisticated jokes. Myself? I get a chuckle from some of the more esoteric gems and I'm embarrassed to admit to most people that these jewels are uber-geeky. So let's start from the top, if only to make the journey easier as it's all downhill from here.

In episode 11 of Futurama, entitled 'Mars University', Farnsworth invented a talking monkey. Fry, Leela, Bender and Farnsworth have to go to Mars University and poor Fry has to share a room with the monkey. But that's not the funny part... well, it is for my younger boy, and the older one, and also for me but that's not the point I'm trying to make. The really funny part flashes by so fast that you have to capture the moment using a time-stopping machine. A frame-grabber on a computer does a good job of that, or you could gather over 200,000 computers and analyse their data. I chose the latter because it's much easier (thanks to Google). Hence the nice picture now seen here, downloaded from the 'net.

Witten was born August 26, 1951 and is an American. He is a mathematical physicist and a professor at the institute for advanced study. Have you heard of him? He is widely regarded as one of the most talented physicists living today. Perhaps you know of Ed Witten if your forte is physics – or mathematics. Then you just might recognise that he has the highest h-index among physicists today. According to reference [6] in the bibliography, his index is 110. Stephen Hawking's h-index is 62 which is about 13th in the list.

The h-index is a measure of the distribution of citations received. A citation is a pointer from one scientific research publication to another. But this is not just an indication of the volume of publications, it is an indication of how influential are those papers. You could publish thousands of papers that are never read and receive no recognition among your peers. Do you think this is a little like the way the Internet is connected? Please see the inset.

Everyone knows Google. I mean 'everyone' in the sense of “everyone who should know about Google, knows about Google, and even if they don't need to, then they probably still do.” I'm sure some people living underground or in caves couldn't care less, but let's take the Engineers' definition of 'everyone' and neglect the small terms.

Google is actually a serendipitous misspelling. It was named after the number known as a googol which is 10 followed by 100 zeros, written as 10¹⁰⁰. This number is so large, that it is larger than any known estimate of the number of atoms in the universe.

The founders of Google were involved in the scientific community, and I think they knew that the Internet was going to be really big. That's why they picked the name 'Googol'. But they also knew something else. They understood the way that scientific publications were rated. To get the best papers, it was efficient to pick those most cited. They based their Internet search tool on the exact analog of citations for web links. This is why you could write a thousand web pages that none or few will read. Google won't find them until you either bribe them with money or many other popular web sites decide to link to yours.

Ok – back on track. We have scratched the surface of 'Witten' (more later) so now move on to 'dog'.

You will, no doubt notice that the squiggly drawing vaguely resembles the shape of a dog. That is actually relevant, as is the tubular-shape – but we have yet to get to that part. 'Dog' is conventionally opposite to 'Cat' even though most cats and dogs don't, in real life, hate each other as much as they do in folklore. What has 'Witten's dog' to do with 'Cat'? Well, Witten is a physicist, and so was another famous character. Schrödinger.

The Austrian Nobel Prize physicist, Erwin Rudolf Josefchen Alexanderchen Schrödinger lived from 1887 to 1961. He contributed significantly to a very important theory known as quantum mechanics and actually has an equation named after him.

Here it is:

Probably not many of you are salivating at this point, but hopefully, at least one person is drooling since it plays a role in quantum mechanics analogous to Newton's second law in classical mechanics, and that makes it pretty exciting for the uber-mega-geek. For the rest of us, note the vertical bar, and the right hand angle bracket. This is part of a notation known as Bra-ket notation.

It is the standard notation for describing quantum states in the theory of quantum mechanics.

Schrödinger posed a very famous thought experiment to illustrate a paradox related to an early interpretation of quantum mechanics. This 'experiment' is called Schrödinger's Cat. He posed this argument because in the sub-atomic world, things are very weird, and the mathematics insists that a particle can be simultaneously in multiple states all at once and also time can go forwards and backwards. This multiple-stateness is known as superposition.

There are very reliable experiments that support this crazy idea and they all involve photons or electrons and other sub-atomic doodads.

The particular mathematical interpretation in question insists that this idea extended to the macroscopic world, i.e. Cat-sized things. I don't know about Schrödinger's relationship with cats, but he either loved them or hated them. If he loved them, then he proposed a thought experiment to put one in a box and kill it. If he hated them, then he proposed the same experiment. Presumably, if he loved cats, then he hoped to revolt like-minded cat-loving people and stir up interest in his paradox thus making them not want to perform the experiment. And if he hated cats... well, it's self explanatory.

Since an atomic particle can be in multiple states at the same time, it's tempting to take a peek at it to see what that looks like. The astounding result is that we never see it in superposition when we actually take a peek. According to the Copenhagen Interpretation It's 'wave function' collapses at that moment. However, if we don't take a peek, but just infer the state of the system as a whole, then the corresponding mathematics shows that it is in superposition.

Schrödinger's thought experiment goes like this:

Seal a cat in a sectioned box. In there, place a canister of deadly gas and link its trigger to a wire that comes out of the section where the cat is and into a sealed radioactive device. The radioactive device is set up to make one single atomic decay happen with an exact 50:50 probability of decaying or remaining stable in ½ an hour. If it decays, a gas is released, and the cat dies. No-one will know whether the atomic nucleus decayed until the box is opened.

So when the box is opened, we take a peek at the atomic state and at that point the wave function is supposed to collapse. By direct implication, the cat is now definitely either dead or alive, and by the same reasoning, it was both dead and alive before that point – which is absurd.

So that's Schrödinger's Cat and now you understand the Futurama joke of “Witten's Dog”.

Phew. We've done 'Witten' and 'Dog' Lets move on to the funny looking diagram.

This dog has a few interesting features. Firstly, it's a tube. And a tube is made of a circle of something that is stretched out. If you join two tubes together, then you might get something like the illustration. The join of these tubes are where two particles interact. 

If you cut off a small circle, and slice that circle, then you get a long thin thing. A long thin thing is either called 'Twiggy' or 'String'. Let's settle for 'String' as most people under 40 won't get the 'Twiggy' joke.

This illustration is a common picture used in popular visualisations of a theory known as string theory. Ed Witten is expert in string theory. In addition to string theory as an idea, physicists realise that there are many flavours. One such flavour might deal only with the sub atomic things that transmit forces (bosons). These are the mysterious gluons, phonons, W , Z, Higgs, and even the photons. Another flavour – (Raspberry perhaps?) includes 'stuff' that makes up what you see around you, and this includes the electron and its antiparticle the positron, also the muon and tau lepton. These are all called fermions. Other fermions include the neutrino and anti neutrino. String theories that include fermions (matter) are called superstring theories.

The big thing about those little fermions is that they just won't bump into each other. They hold their distance – something known as 'Pauli exclusion principle'. It is this refusal to occupy the same state as another that makes matter physical. In supersymmetric theories, every fundamental fermion has a bosonic superpartner and vice versa.

Farnsworth has gone one better than Witten and illustrates superDUPERsymmetric string theory.

Hold that thought. We need to digress yet again and talk about Richard Feynman before being able to explain the neutron encrusted steaming hot dark matter which, it should be obvious, is really a pile of doggy-do.

Nobel prize winner, Richard Feynman (1918-1988 ) famously fixed a long standing problem in Quantum Electrodynamics and made it much more accurate and useful.

In the process of doing this, he invented some useful diagrams that allow you to discuss the interactions between photons ( a massless Boson ) and electrons ( a charged fermionic mass ). With Q.E.D. you can explain all of physics involving electrically charged particles. It is extremely accurate and agrees to several decimal places with practical experiments. In QED, particles interact with each other and in doing so, exchange particles. Feynman diagrams make this clear.

Here is one of many possible scenarios showing interaction between an electron and a photon. Time runs UP the page. The wiggly lines are the paths of photons though time. The solid lines are paths of electrons through time, and the points where they meet are 'junctions'. There is much more to this diagram, but for the purpose here, this simple explanation will do.

Oh – I'll slip in a clue about one of the 'gems' that you might find in either Simpsons or Futurama. The junction is a signal to the physicist to include a mathematical constant known as the 'fine structure constant'. No one really knows WHY it has the the value that it has, but it is measured and calculated to be 1/137.03599911(46) where the numbers in parentheses are uncertain. It's my bet that the number '137' or 1/137 or 137.03599911 appears somewhere in Futurama and/or Simpsons. This constant is thought to vary by less than 0.6 parts per million over the last 10 billion years.

Getting back on track now, you can see the similarity between Witten's dog, and Feynman's diagrams, especially when you know the symbol for an electron is e^- and for a proton is p.

Both of these are 'fed' to Witten's dog (WD).

The thing coming out the dog's bum is a Greek letter nu with a little negative e subscript, so this is an electron-neutrino. (There are three kinds of neutrinos).

The equation just under the dog is

e^- + p (something) + nu_e

The (something) appears to be an n with a squiggle underneath. The n is a neutron, but I am not sure what the squiggle means in this case. In supersymmetry a squiggle above the symbol means it is the supersymmetric partner. So I am guessing that the squiggle underneath is supposed to indicate the superdupersymmetric partner. The arrow is used to indicate decay.

So this equation, and the diagram says, “An electron combines with a proton and produces a superdupersymmetric neutron and a neutrino”.

Electron capture is a form of radioactivity where a parent nucleus happens to capture one of its own orbital electrons. This is one process where unstable atoms can become more stable. During electron capture, an electron in an atom's inner shell is drawn into the nucleus where it combines with a proton, forming a neutron and a neutrino. The neutrino is ejected from the atom's nucleus.

In general terms, this change could be written:

e^- + p n + nu

Which is very similar to the Witten's dog formula – apart from the neutron being a superdupersymmetric neutron, and the neutrino is specifically an electron neutrino.

Way up until 1998, it was unconfirmed whether the neutrino has any mass or was completely massless like the photon. Eventually, the Super-Kamiokande collaboration announced evidence of non-zero neutrino mass at the Neutrino '98 conference. I'll bet that was a geek-fest! Thus, neutrinos make up one possible contribution to the so called 'missing mass' of the universe. This is commonly called dark-matter because you can't see it. Most of the universe is known to consist of dark matter from simple observations of the way that stars move in galaxies, compared to how much visible mass we can count by looking through telescopes.

So that explains the 'Neutron Encrusted Steaming Hot Dark Matter' note on the blackboard. Oh

– except for the 'hot' part.

There are two contributions to dark matter: baryonic and non-baryonic matter. Of the non-baryonic stuff, the Cold stuff is low energy, and then you have hot stuff. Hot stuff includes fast moving things with lots of energy. Stuff that vibrates really really rapidly has a lot of energy and this includes neutrinos, but astrophysicists refer to the hot stuff as neutralinos which includes the neutrino and other dark hot stuff too – like the Hale Beri-ino.

All that's left now is that beautiful looking equation.

This is going to be tough. (Especially as it's contrived). Research on the Internet suggests that it is of similar form to an equation that constrains the mass density of neutrinos in the universe, in other words it puts an upper limit on the proportion of dark matter that is made of neutrinos.

Ω is 'omega' and it is used to say 'Density'. Density is a measure of how much stuff is packed into a given volume. The letter that you put next to it says what it is that you are measuring. So the equation is the density of neutrinos in the universe.

This is a tiny, but significant number. The eV together means 'electronvolts' and is actually a unit of energy. A single electronvolt is equal to the energy required to move one electron through a potential difference of 1 volt. Electrons are really tiny, so you can imagine that 1 eV is also tiny.

In fact 1 eV is only 1.602 176 53 (14)×10−19. or in decimal notation, .0000000000000000000160217653 and the last two digits (14) are uncertain. 93 eV is therefore also tiny, but it's not zero, and that's very significant because there are many truckloads of neutrinos in the universe. Even a very small number, if multiplied by a very large number can give a significant result.

Just above the 93eV, is m_nu which means “mass of a neutrino” (remember it's really tiny but non-zero?) At this point we need to know whether m_nu divided by 93eV is big or small. We can find this out because a proposed value for m_nu divided by 93eV has been published.

This figure is at least 0.1

The z in the superdupersymmetry equation means 'redshift'. So now I need to tell you what this is.

All electromagnetic radiation is really just photons vibrating at various frequencies. Slow vibrations are low energy, and fast vibrations are high energy. Visible light is also part of this spectrum, and in fact there is no difference between microwaves, gamma rays, and X-rays, apart from their energy.

Within visible light, it ranges from low energy red to higher energy blue. Got it? Good. 'Cause that's just the start.

First imagine a punch to the face when your opponent is simply standing in front of you. Ouch. Then imagine the same punch if is is also traveling towards you on a moving walkway. Double ouch. That one has more energy. If he was going backward on the moving walkway, the punch would be less severe as it has less energy. That's redshift. If you look into the night sky and observe several stars, then some will be redder than others. Most will be redder than they should be if space itself was not expanding. Those that are redder must be moving away from us. From these observations, we know that space is expanding.

<ΩbWD>²

It certainly looks like a 'b' and 'D', and the D is likely because we

should not be surprised that Witten's dog WD is in the equation as

it's the topic on the blackboard.

In <ΩbWD>² we see the omega symbol again, which will mean

density, and the 'b' must be linked somehow to 'brightness'. How do I know this? Because one of the tests for the Big Bang theory, versus static universe is called the Tolman surface brightness test . According to the Big Bang theory, we need to multiply a galaxy's brightness by a factor of 1/(1 + z)⁴ to get its intrinsic value. This has to be done in an expanding universe to compensate for the effect of the expansion on our observations. So that means the the top part of the equation must be a measure of brightness. In fact Ω_b refers to the density of 'baryonic' matter. The term baryonic includes atoms of any sort. Luminous matter including stars and gas clouds is part of Ω_b.

So now we can decode some of Ω_bW_D as something like:

“ The density of baryonic matter of the universe multiplied by Witten's dog ”. Now it gets interesting. (As if it was boring to this point; and I know it's not, because you are STILL reading!)

The angle brackets gave me some grief. At first I thought it was to do with a shorthand notation for something to do with ground-state. But after much word munching, I concluded that they are angle brackets meaning 'expectation value' – and there is a good reason for this. An expectation value involves a probability. It's used in gambling to work out your average payout. If you gamble, say, 4 times and bet $5, $6, $2 and 7$ to throw a six on a standard die, and if your banker pays 2 times your bet if you win, and nothing otherwise, then these are all the possibilities:

lose $5 or win $(5 x 2) meaning 5/6 chance to lose $5 and 1/6 chance to win $10

lose $6 or win $(6 x 2) meaning 5/6 chance to lose $6 and 1/6 chance to win $12

lose $2 or win $(2 x 2) meaning 5/6 chance to lose $2 and 1/6 chance to win $4

lose $7 or win $(7 x 2) meaning 5/6 chance to lose $7 and 1/6 chance to win $14

This can be written:

56

−5−6−2−716

1012414

This is −16.666.66 = an expectation of a $ 10 loss.

Σ means 'sum'. That means to add up each term. But it is just shorthand, and is required because sometimes there are thousands, millions, billions or an infinite number of terms and it would be very boring to write them all one by one.

Each term is indexed by an integer. And in this case, the index is 'i' and it starts at 1. This is why, on the bottom it says, i=1. The upper limit is on top of the summation symbol, but in the superdupersymmetric string theory equation, I can't make it out. Usually, the index also appears in the equation, but it is not present here. As it turns out, this is of no real consequence – after all it's not a 'real' equation.

Or is it?

Take a look at illustration 8 from a physics paper: [ Ref 35 ]

Notice the term Ω_v h² . It explains that h is the Hubble expansion constant. This is a measure of how space is expanding. Notice also that the author stated which units he was using. This is because it's common in physics to set constants to convenient values by manipulating the measuring system. For example, If I said I was going to buy 144 eggs, I could also say I was going to buy eggs in lots of a dozen-dozen, calling that measure a buncha (or something) – say , B for short. One trip to the shops would result in 1 buncha eggs. If I was going to get 1 buncha eggs, for every can of meat C, then you would know that when I bought two cans of meat, then I also bought 2 buncha eggs. So in any formula involving C cans of meat, I can tell you up front that there is also C buncha eggs. 

Therefore, the value of B has been chosen to be 1 buncha, rather than 144 eggs. The result of this is that the B is 1, so I don't need to display it in my shopping list formula. The same applies here. If units were chosen so that h took the value of 1, then it would not need to appear in the formula. 

This means that we can also take

Ωv =   ∑mv

     (93ev)

and providing the units are chosen so that h=1, it is an equivalent statement to that in the paper above.

Finally, I can tell you the joke.

Since z is always greater than zero for a redshift, (z+1) can never be zero for any value of z, and all values in the second term are properly defined. By this, I mean that there is no tendency for that term to try to whiz of to an infinite value.

<ΩbWD>2

(z+1)4

is never undefined. This is a good thing. Now, remember that the angle brackets < > means expectation value' ? It means this: for every indexed discrete item inside the brackets, you apply some kind of function to each value, and multiply it by a probability and add up all the results. This gives you an expected outcome. It can range, of course, from zero to whatever.

What is the expectation value of

<ΩbWD>2 ?

It's REALLY easy – and funny. W D is Witten's dog isn't it? And it's described by a decay of a proton and an electron interacting to produce a superdupersymmetric neutron superduperparter, and a neutrino. How likely is that? What's the probability of that occurring? EXACTLY ZERO ! It's just make believe.

So

<ΩbWD>2

(z+1)4

is zero, – this whole term equals zero.

What are we left with if we use units where h=1 ?

Hey presto! behold a really important equation. It's the one from the physics paper above.

[1] http://www.tv.com/futurama/mars-university/episode/1544/summary.html

[2] http://www.geocities.com/theneutralplanet/transcripts/season2/1ACV11.html

[3] http://www.tv.com/futurama/show/249/episode_listings.html&season=0

[4] http://www.theage.com.au/news/tv--radio/sum-thing-to-do-with-maths-genius/2006/01/05/1136387564868.html

[5] http://en.wikipedia.org/wiki/Edward_Witten

[6] http://en.wikipedia.org/wiki/H-index

[7] http://en.wikipedia.org/wiki/Googolhttp://en.wikipedia.org/wiki/Quantum_electrodynamics

[8] http://answers.google.com/answers/threadview?id=352789

[9] http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

[10] http://en.wikipedia.org/wiki/Bracket

[11] http://en.wikipedia.org/wiki/Schrodingers_cat

[12] http://www.columbia.edu/cu/record/archives/vol23/vol23_iss18/14.html

[13] http://www.superstringtheory.com/

[14] http://en.wikipedia.org/wiki/Bosons

[15] http://en.wikipedia.org/wiki/Quantum_electrodynamics

[16] http://www.ps.uci.edu/~superk/

[17] http://www.eso.org/outreach/press-rel/pr-2001/pr-28-01.html

[18] http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-11687.pdf

[19] http://xxx.lanl.gov/pdf/astro-ph/0310911

[20] http://www.calstatela.edu/faculty/rcarr/a151/darkmtr.html

[21] http://scienceworld.wolfram.com/physics/ElectronVolt.html

[22] http://en.wikipedia.org/wiki/Neutron

[23] http://www.ndt-ed.org/EducationResources/HighSchool/Radiography/nuclearreactions.htm

[24] http://hyperphysics.phy-astr.gsu.edu/Hbase/particles/proton.html#c1

[25] http://en.wikipedia.org/wiki/Surface_brightness

[26] http://www.wonderquest.com/star-distances-iii.htm

[27] http://mathworld.wolfram.com/ExpectationValue.html

[28] http://mathworld.wolfram.com/SimonsProblems.html

[29] http://www.bautforum.com/archive/index.php/t-34779.html

[30] http://www.bautforum.com/archive/index.php/t-33841.html

[31] http://sirad.pd.infn.it/people/rando/dati/talks/dark_matter.pdf

[32] http://www.uni-muenster.de/Physik.KP/AGWeinheimer/documents/DesignReport2004-12Jan2005.pdf

[33] http://astro.berkeley.edu/~mwhite/darkmatter/hdm.html

[34] http://en.wikipedia.org/wiki/Electron-volt

[35] http://www.citebase.org/cgi-bin/fulltext?format=application/pdf&identifier=oai:arXiv.org:hep-ph/9704315

[36] http://www.fynu.ucl.ac.be/librairie/theses/gustaaf.brooijmans/node4.html