June 1, 2003

This Week's Finds in Mathematical Physics (Week 196)

John Baez

Today I'd like to talk about the Big Bang and Pythagorean spinors. But first, a book! If you want to start learning general relativity without first mastering the intricacies of tensors, here's a way to get going:

1) James B. Hartle, Gravity: an Introduction to Einstein's General Relativity, Addison-Wesley, San Francisco, 2003.

Hartle is an expert on general relativity, but here he avoids showing off. He gets to the physics of general relativity as quickly and simply as possible, avoiding the usual route of doing huge amounts of math first. In particular, he works out the physics of specific solutions of Einstein's equations - like those describing black holes and the Big Bang - before he introduces the equations. This puts off the hard abstract stuff until later, when the student has more feeling for it. The purists may grumble, but I have a feeling it's pedagogically sound.

Now... what happened in the first second after the Big Bang?

This may sound like an insanely ambitious question, but in fact we seem to have a fairly good idea of what happened all the way back to the first microsecond - unless, of course, there's some important physics we're missing. This paper tells the story quite nicely:

2) Dominik J. Schwarz, The first second of the universe, available as astro-ph/0303574.

But, since the physics gets weirder as we approach the Big Bang, I'll tell this story in reverse order - and I'll start from now, just to set the stage.

So, here's a quick reverse history of the universe:

According to data from the Wilkinskon Microwave Anisotropy Probe (WMAP), our best estimate of the age of the universe is 13.7 billion years, plus or minus 150 million years or so. Previous estimates were similar, but with an uncertainty of about half a billion years.

The temperature listed here is that of the cosmic microwave background radiation. Slight deviations from this average figure were first detected in 1992 by the Cosmic Background Explorer (COBE). This satellite-based experiment found hot and cold patches in the microwave background that differ from the mean temperature by an amount on the order of 30 microkelvin. WMAP is a more refined experiment along the same lines, which came out with a lot of exciting new results in February, 2003.

"Reionization" is the name for when the hydrogen in the universe, which had cooled after the Big Bang, became hot and ionized again. The most likely cause is radiation from the very first stars. So, we now think the first stars ignited about 200 million years after the Big Bang.

Again this is a result from WMAP. The error bars are quite large, so the numbers above could be off by a factor of two or so. Nonetheless, a lot of people were surprised by this result, since they thought that the clumping of matter due to gravity would have taken much longer to form stars!

There's a lot we don't know about star and galaxy formation, because the current conventional wisdom requires that the gravitational clumping was seeded by "cold dark matter" - but nobody knows what this stuff is.

"Recombination" is the usual stupid name for when the universe cooled down enough for electrons and protons to stick together and form hydrogen atoms. It should really be called "combination", because the electrons and protons were never combined before this time. Before this time the universe was always full of plasma - that is, electrically charged particles running around loose. Afterwards it was full of electrically neutral hydrogen... at least until the stars lit up and reionized a lot of this hydrogen.

Plasma absorbs light of all frequencies, while electrically neutral gases tend to be transparent except for certain frequency bands. Thus, recombination was the first time when light could start travelling for long distances without getting absorbed! For this reason, the cosmic background radiation we see now consists of the photons emitted right at the time of recombination. When emitted it had a temperature of about 3000 kelvin, but it has cooled with the expansion of the universe.

The era between recombination and the ignition of the first stars goes by a romantic name: the Dark Ages. Adding to the romance, Pfenniger and Puy hypothesized that hydrogen could have frozen into crystalline flakes before the stars lit up and began warming the universe again. A cold, dark, eerie universe full of hydrogen snowflakes... the thought sends shivers right up my spine! Unfortunately, if stars formed as early as WMAP says they did, the universe would probably not get cold enough for these crystals to form.

By the way, the figure of 380 thousand years for the time of recombination is another result from WMAP, consistent with previous estimates, but presumably more accurate.

Before this time, the energy density due to light exceeded that due to matter, so we say the universe was "radiation-dominated". Afterwards the universe became "matter-dominated" - at least until considerably later, when matter spread out so thin that the dominant form of energy became "dark energy", as it seems to be now.

(The best estimate due to WMAP says that currently the energy in the universe is 4% ordinary matter, 23% cold dark matter and 73% dark energy. For more on the latter two concepts, see "week167".)

The end of the radiation-dominated era is important because this is when gravity began to amplify small fluctuations in the density of matter. In other words, this is when stuff began to form clumps of various sizes, eventually leading to stars, galaxies, galaxy clusters, and so on. During the radiation-dominated era, density fluctuations were mainly made of light, and these could not grow because the light was moving too fast to form clumps. People believe that as soon as this era ended, cold dark matter began clumping up under its own gravity. Ordinary matter started clumping up later, after recombination - since before that it was in the form of plasma, which stayed smoothed out by its interaction with light.

A lone neutron is not a stable particle: with a mean lifetime of 886 seconds, it will decay into a proton, electron and antineutrino. So, any neutrons created early in the history of the universe must fuse with protons to form nuclei by roughly this time, or they are doomed to decay.

At about this time, the temperature dropped to the point where a proton and neutron could stick together forming a deuterium nucleus, and the process of "nucleosynthesis" began, in which deuterium nuclei stick together to form nuclei of helium. This is responsible for the fact that even before the stars started processing hydrogen into heavier elements, the universe was about 25% helium, the rest being almost all hydrogen.

Apart from the neutrinos and the photon, the lightest particle in nature is the electron. The rest mass of an electron corresponds to an energy of 511 keV, so it only takes twice that much energy to create an electron-positron pair. If we multiply 511 MeV by Boltzmann's constant, we get a temperature of roughly 5 billion kelvin. That means that at this temperature, two particles colliding head-on will often have enough kinetic energy to create a electron-positron pair. So, when it's this hot or hotter, collisions between particles generate a thick stew of electrons and positrons!

But as temperatures cool below this point, the density of this stew drops off exponentially: electron-positron pairs annihilate each other, leaving radiation. This happened roughly 10 seconds after the Big Bang.

Neutrinos can easily zip through light-years of lead, but the very early universe was so compressed that they interacted vigorously with other forms of matter. But around a second after the Big Bang, the density of the universe decreased to about 400,000 times that of water, and neutrinos "decoupled" from other matter.

Since these neutrinos were not reheated by nucleosynthesis, they should now be cooler than the cosmic microwave background radiation - about 2 kelvin instead of 2.726 kelvin. We are currently unable to detect such unenergetic neutrinos, but detecting them would be a major confirmation that our theories of the early universe are correct.

Particles made of quarks and antiquarks are called "hadrons", and they interact via the strong nuclear force. The only hadrons we encounter in daily life are protons and neutrons, made of 3 quarks each. But the lightest hadrons are the pions, which come in positive, negative and neutral forms. The positive and negative ones are antiparticles of each other, while the neutral one is its own antiparticle. They all have mass on the order of 100 MeV.

Just as I described for electron-positron pairs, at a high enough temperature everything is always awash in a sea of pions, while below this temperatures the pions quickly disappear by annihilation. To estimate the relevant temperature, we can just convert its mass to a temperature following the rough rule 1 MeV ~ 10 billion kelvin. So, when the temperature of the early universe dropped below 1 trillion kelvin, pions went away. This happened around 100 microseconds after the Big Bang. Before this, hadrons ruled!

At normal temperatures, quarks and antiquarks are confined within hadrons by the strong force. The strong force is carried by gluons, so you can vaguely visualize a hadron as a bag-like thing in which quarks and antiquarks wiggle about, constantly exchanging virtual gluons, which also exchange virtual gluons, quarks and antiquarks of their own. The details are described by "quantum chromodynamics", or QCD. Since QCD says the strong force gets stronger with increasing distance, if you try to pull a quark out of this bag, it takes enough energy to create a whole new bag!

But if you have a bunch of hadrons at temperatures above 2 trillion kelvin or so, they'll be smashing into each other so furiously that the distinction between the "bags" and the "space between the bags", never completely sharp, dissolves entirely. At this point, all you've got is a bunch of quarks, antiquarks and gluons zipping around. This is a new state of matter: a "quark-gluon plasma". In "week76" and "week117" I described how how people at the Relativistic Heavy Ion Collider in Brookhaven are making quark-gluon plasmas by smashing nuclei at each other at high speeds.

A lot of Dominik Schwarz's paper is about the "QCD phase transition" which happened about 50 microseconds after the Big Bang, when the universe cooled down enough for the quark-gluon plasma to condense into the confined phase. Though the subject is controversial, most people think this phase transition is a "first-order" transition, meaning that heat is emitted as the transition happens, just as when water vapor condenses to form liquid droplets. If so, the quark-gluon plasma would probably supercool until small bubbles of hadron phase formed. As these bubbles grew, latent heat would be emitted. This would tend to reheat the quark-gluon plasma, limiting the speed at which the bubbles expand. Heat would mainly be dispersed by means of neutrinos and acoustic waves - i.e., sound.

At high enough temperatures, there should be no difference between the electromagnetic force and weak force. This difference should only arise when things cool down enough for the Higgs field to settle into a fixed position, breaking the symmetry between these forces - a bit like how ice crystallizes, breaking the rotational symmetry of liquid water. At higher temperatures the Higgs field wiggles around too much to settle down. Or in the language of particles rather than fields: collisions between particles create a stew of Higgs bosons!

The mass of Higgs seems to be somewhere around 130 GeV. If so, the electroweak phase transition would have occured roughly 10-11 seconds after the Big Bang. But, since we haven't actually gotten direct evidence for the Higgs boson yet, this is still a bit speculative. Right now people say the Large Hadron Collider at CERN will come online and start looking for the Higgs in 2007. But the LHC project has gotten some nasty budget cuts recently, so I wouldn't be surprised if there were delays. If and when the Higgs is found, maybe I'll return to this topic and say what people think happened before the electroweak phase transition. People have thought about this a lot. But for now, I'll quit here!

If you want to learn more about the early universe, start with this classic:

3) Steven Weinberg, The First Three Minutes, Basic Books, New York, 1977.

Then catch up with recent developments by reading these websites:

3) Ned Wright's Cosmology Tutorial, http://www.astro.ucla.edu/~wright/cosmolog.htm

4) Martin White, The Cosmic Rosetta Stone, http://astron.berkeley.edu/~mwhite/rosetta/rosetta.html

To dig deeper, try these books:

5) P. Coles and F. Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure, Wiley, New York, 1995.

6) Edward W. Kolb and Michael Turner, The Early Universe, Addison-Wesley, Reading, Massachusetts, 1990.

For a detailed description of some of WMAP's results, try these:

7) C. L. Bennett et al, First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results, available as astro-ph/0302207.

8) D. N. Spergel et al, First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters, available as astro-ph/0302209.

These are two of thirteen related papers produced by the WMAP team! Both of them have lots of coauthors, one of which is Ned Wright, author of the nice website mentioned above.

Here is Pfenniger and Puy's paper on hydrogen "snowflakes":

9) D. Pfenniger and D. Puy, Possible flakes of molecular hydrogen in the early Universe, available as astro-ph/0211393.

I should also thank Ted Bunn for telling me some stuff about converting between times, temperatures and redshifts. Cosmologists of the early universe use "redshift z" to stand for the time when the universe was 1/(z+1) times as big as it is now - by which I mean, distances between faraway objects were multiplied by this factor. Equivalently, this is the time when the temperature of the background radiation was z+1 times as big as it is now. So, converting between temperatures and redshifts is easier. Converting these to times is less trivial, and indeed the times listed above are more likely to suffer from inaccuracies than the temperatures.

By the way, some people say it's confusing to use numbers like "billion", "trillion" and "quadrillion" to mean 109, 1012 and 1015, respectively - because these are American usages, and Europeans (they claim) use "milliard", "billion" and "billiard" for these numbers. These people say that, for example "gigakelvin", "terakelvin" and "exakelvin" are less ambiguous than "billion kelvin", "trillion kelvin" and "quadrillion kelvin".

This is probably true, but it's also true that fewer people know what an "exakelvin" is than a "quadrillion kelvin". Since I was trying to explain cosmology to the unwashed masses, I opted to use number words above, and explain them here. I was using the American system... and I'm sort of betting this system will take over, because I've never heard anyone use the word "billiard" to mean 1015. More importantly, I really hope that some system takes over, because it's a bit sad not to be able to use words for numbers.

Speaking of numbers... now for some math!

The volume in honor of Penrose's 65th birthday is full of fun stuff about spin networks, twistors, and so on - but I particularly liked this paper by Trautman on "Pythagorean spinors":

10) Andrzej Trautman, Pythagorean spinors and Penrose twistors, in The Geometric Universe: Science Geometry and the Work of Roger Penrose, eds. Huggett, Mason, Tod, Tsou and Woodhouse, Oxford U. Press, Oxford, 1998. Also available at http://www.fuw.edu.pl/~amt/amt.html

If you're a physicist you'll have heard about Dirac spinors, Weyl spinors, Majorana spinors, and maybe even Majorana-Weyl spinors. I've you haven't, you can read my explanations in "week93". But what in the world are "Pythagorean" spinors? The basic idea is that from two spinors you can make a vector, and Trautman points out that a special case of this idea gives a famous old formula for getting Pythagorean triples - that is, integers a,b,c with

a2 + b2 = c2.

I think I'll explain this in detail....

Spinors are used to describe spin-1/2 particles, so-called because they don't come back to where they were when you turn them around 360 degrees - you have to rotate them twice to get back where you started! Thus, mathematically, spinors are representations of the double cover of the rotation group, or the double cover of the Lorentz group if you take special relativity into account.

In 4d spacetime, the double cover of the Lorentz group is SL(2,C), the group of 2×2 complex matrices with determinant 1. We can take a spinor to be just a pair of complex numbers, but there are actually two ways such a thing can transform under SL(2,C). One way is obvious, but for the other we take the complex conjugate of the matrix before letting it act on the spinor. We get two sorts of spinors, called left- and right-handed "Weyl spinors". In physics, we use these to describe massless particles that spin either clockwise or counterclockwise along their line of motion as they zip along at the speed of light.

In 3d spacetime, the double cover of the Lorentz group is SL(2,R), the group of 2×2 real matrices with determinant 1. In this dimension, we can take a spinor to be a pair of real numbers. But since we don't have complex conjugation at our disposal, we don't get left- and right-handed versions of these spinors, and we don't call them Weyl spinors. Since they are real, we call them "Majorana spinors".

Since Pythagoras had a strong fondness for number theory, if he were alive today he might want to simplify things even further and consider SL(2,Z), the group of 2×2 integer matrices with determinant 1. This acts on "Pythagorean spinors", namely pairs of integers.

We could also go up to higher dimensions using the quaternions and octonions: SL(2,H) is the double cover of the Lorentz group in 6d spacetime, and SL(2,O) is the double cover of the Lorentz group in 10d spacetime. But I explained this in my octonion webpage:

11) John Baez, OP1 and Lorentzian geometry, http://math.ucr.edu/home/baez/octonions/node11.html

so I won't talk about it now.

In each case, there's a trick for turning a spinor into a lightlike vector. In 4 dimensions we do it like this: we take a left-handed spinor ψ, take its conjugate transpose to get a right-handed spinor ψ*, and form

ψ ⊗ ψ*
which we can think of as a 2×2 hermitian matrix. If you're a fancy mathematical physicist you know that the space of 2×2 hermitian matrices is the same as 4d Minkowski spacetime, with the matrices of determinant zero corresponding to the lightlike vectors, so you're done! Otherwise, you can work out the above matrix explicitly:
ψ = (a)                                         (a column vector)

ψ = (a*, b*)                                    (a row vector)

ψ ⊗ ψ* = (aa* ab*)                              (a 2×2 matrix)
         (ba* bb*)                  
This matrix is hermitian, so you can write it as a real linear combination of Pauli matrices:
ψ ⊗ ψ* = t σt + x σx + y σy + z σz
σt = (1  0)
     (0  1)

σx = (0  1)
     (1  0)

σy = (0 -i)
     (i  0)

σz = (1  0)
     (0 -1)
You get a vector in Minkowski spacetime, (t,x,y,z). If you check that this vector is lightlike:
t2 = x2 + y2 + z2 
you'll be done.

The trick in 3 dimensions is just the same except that now the components of ψ are real numbers, so things simplify: we don't need complex conjugation, and ψ ⊗ ψ* will be a real hermitian matrix. Real hermitian matrices are the same as vectors in 3d Minkowski spacetime, since we can write them as linear combinations of the three Pauli matrices without i's in them - namely, all of them except σy. So, we get a lightlike vector in 3d Minkowski spacetime: say, (t,x,z) with

t2 = x2 + z2

Now for the really fun part: the trick works the same with Pythagorean spinors except now everything in sight is an integer...

... so (t,x,z) is a Pythagorean triple!!! And in fact, we get every Pythagorean triple this way, at least up to an integer multiple. And in fact, this trick was already known by Euclid.

Explicitly, if

ψ = (a)
2 ψ ⊗ ψ* = (2a2  2ab) 
           (2ab  2b2)                  

         = (a2 + b2) σt + 2ab σx  + (a2 - b2) σz
so we get the Pythagorean triple
(t,x,z) = (a2 + b2, 2ab, a2 - b2)
For example, if we take our spinor to be
ψ = (2)
we get the famous triple
(t,x,z) = (5,4,3)  
By the way, you'll notice I had to insert a fudge factor of "2" in that formula up there to get things to work. I'm not sure why.

Addendum: Thanks to Andy Everett for catching a typo. Noam Elkies sent me the following:

Dear J. Baez,

In article  you write:

>[......] so (t,x,z) is a Pythagorean triple!!!   And in fact,
>we get every Pythagorean triple this way, at least up to
>an integer multiple.  And in fact, this trick was already
>known by Euclid.

Are you sure of this?  The formula was surely known by Euclid's time --
I've even seen claims that it must have been known by the Babylonians
(perhaps not a coincidence, since Pythagoras spent some time in Babylonia)
-- but did Euclid have anything like this interpretation, or the proof
that every "Pythagorean triple" is proportional to one of this form?

[For that matter there's apparently some controversy about just who
Pythagoras might have been and what he might have known, believed,
and/or proved, since secrecy was one of the Pythagoreans' tenets.]

>By the way, you'll notice I had to insert a fudge factor of "2"
>in that formula up there to get things to work.  I'm not sure why.

This is presumably an artifact of the ambiguity in the
"symmetric square of Z".  In general, the symmetric square
of a module M can be formed as either a submodule or a quotient
module of the tensor square of M.  These two symmetric squares
are (canonically) isomorphic if 2 is invertible, but not in general.

The parametrization of Pythagorean triples is also closely related
with the "half-angle substitution" of elementary calculus.
For yet another interpretation, see


[also .pdf instead of .dvi].

The dissections that illustrate the Pythagorean theorem
can be generalized to the law of cosines; see


for the acute and obtuse cases respectively.

--Noam D. Elkies

I replied saying that by Euclid knowing "this trick", I only meant he knew this formula for Pythagorean triples:

(a2 + b2, 2ab, a2 - b2)
I don't know who proved it gives all of them (up to multiples).

Rob Johnson then noted:

There is not any high powered math involved in showing that these are
all the pythagorean triples up to scalar multiples.  See


Nothing more than the fundamental theorem of arithmetic is used (to
justify the statement "Since b^2 = 4MN and gcd(M,N) = 1, both M and N
must be perfect squares."), and Euclid knew that.

So my guess is that Euclid probably knew that this formula gives all
pythagorean triples up to scalar multiples, but it is just a guess.

Rob Johnson

© 2003 John Baez