December 8, 2003

This Week's Finds in Mathematical Physics (Week 199)

John Baez

I've had a really busy quarter, teaching 3 courses that all require serious thought on my part, so it's been a long while since I've been able to write an issue of This Week's Finds. But, back in September I went to a conference on homotopy theory and its applications at the University of Western Ontario, run by Dan Christensen and Rick Jardine. There were some really cool talks at this conference - my favorite was one by Jack Morava about elliptic cohomology, and I'm really sorry I missed his lectures on Galois theory, since I've been studying that lately. But, instead of trying to describe the talks, I think it would be better if I said a bit about "spectra", which are an important tool in homotopy theory.

The word "spectrum" has a lot of different meanings in mathematics and physics. In experimental physics it refers to the frequencies of light, sound or any other sort of wave emitted by an object. For example, if you send the light emitted by hydrogen through a spectrometer, you'll see a bunch of sharp lines at specific frequencies - the "discrete" spectrum" - along with a diffuse glow at all frequencies - the "continuous spectrum". The German high school teacher Balmer noticed that the sharp lines correspond to light with frequencies proportional to

1/n2 - 1/m2
where n,m = 1,2,3,...

These days, in theoretical physics the "spectrum" of something is the set of frequencies at which it can vibrate - or in quantum theory, the set of energies it can have, since an energy is just a frequency times Planck's constant. For example, Bohr took Balmer's formula and realized that a hydrogen atom must have a discrete set of allowed energy levels

When the atom hops from one energy level to another, it emits or absorbs light with energy equal to the difference of two such numbers! This accounts for the discrete spectrum of light emitted by hydrogen. The atom can also have any positive energy, and this accounts for the continuous spectrum.

In quantum mechanics, observables like energy are described as self-adjoint operators on a Hilbert space. The "spectrum" of an observable A is the set of values it's allowed to have, and mathematically this is the set of numbers x such that the operator A - x has no inverse. For example, if A is a "Hamiltonian", the operator that describes the energy of a quantum system, its spectrum is just the set of allowed energies! The simplest case is when x is an eigenvalue of A: the eigenvalues of an operator form its "discrete spectrum". But, there can also be numbers in the spectrum that aren't eigenvalues, and these form the "continuous spectrum".

In mathematical physics, people talk about the spectrum not just of one observable but of a whole bunch of commuting observables, since commuting observables can be measured simultaneously without the Heisenberg uncertainty principle kicking in to limit the precision. The nice way to think of the spectrum of a bunch of operators uses the concept of "C*-algebra". If we've got a bunch of bounded operators on a Hilbert space that's closed under addition, multiplication and scalar multiplication, closed under taking adjoints and also closed in the norm topology, it's called a "C*-algebra". The "spectrum" of a C*-algebra A is the set of all homomorphisms

x: A → C, 
where C is the complex numbers. Though it's not immediately obvious, this sort of spectrum reduces to the previous one when A is the C*-algebra of operators generated by a single self-adjoint operator. So, it's a nice way to define the spectrum of a whole bunch of observables. This generalization is not very useful when the C*-algebra is noncommutative, since then it may not have many homomorphisms to the complex numbers. But if it's commutative, we know everything about it once we know its spectrum!

This amazing fact is called the Gelfand-Naimark theorem. Here's the idea. There's an easy way to make the spectrum of a commutative C*-algebra A into a topological space: we say xi → x precisely when

xi(a) → x(a)
for all elements a of A. With this topology any element a of A gives a continuous complex function on the spectrum, defined by this clever formula:

a(x) = x(a).

The physicist Chris Isham says he couldn't sleep all night when he first saw this formula, it's so darn clever! And, it turns out that any continuous function on the spectrum comes from an element of A via this formula! So, if you hand me the spectrum Spec(A) of a commutative C*-algebra A, I can recover A (up to isomorphism) by forming the C*-algebra of all continuous functions on Spec(A).

As you can see, the concept of spectrum is getting more abstract - but it still has close ties to the original idea. What once was a bunch of lines on a spectrometer has now become a topological space associated to a commuting collection of observables. The idea is that each point in this space is a way of assigning values to all these observables... just like each line in the spectrometer represents a particular frequency of light!

But the abstraction process doesn't stop here. In algebraic geometry, people want to think of any commutative ring as consisting of functions on some sort of space. For example, the commutative ring of real polynomials in two variables mod the relation

x2 + y2 = 1

is just another way of thinking about polynomial functions on the circle. How do we get the circle back from this commutative ring? Simple: just form the space of all homomorphisms from it to the real numbers!

It would be nice to have a recipe to take any commutative ring A and extract a space from it: its "spectrum". As we've seen, one option is to take the spectrum to consist of all homomorphisms to the complex numbers:

x: A → C
Another would be to use the real numbers:
x: A → R.
Both the real and complex numbers are "fields": commutative rings where we can divide by anything nonzero. But there are a lot of other fields, like Z/p where p is any prime number. So, instead of picking one field, a more evenhanded approach is to use all possible fields, and say a homomorphism to any one of these should give a point of the spectrum.

Actually, since there are zillions of fields out there, a more manageable option is to look not at the homomorphism itself but its kernel: the set of elements a in A with

x(a) = 0.
The kernel of a homomorphism from A to any other ring is an "ideal": a set closed under addition and also multiplication by all elements of A. Even better, the kernel of a homomorphism from A to a field is a "prime" ideal, meaning it's not not all of A, and whenever the product of two elements of A lies in the ideal, at least one of them must be in the ideal. Conversely, given a prime ideal in A, there's always a field k and a homomorphism
x: A → k 
whose kernel is that prime ideal. So, it's reasonable to define the spectrum of A, Spec(A) to be the set of all prime ideals in A.

This turns out to exactly match the previous definition of spectrum when A is a C*-algebra. But why the word "prime"? Well, in the commutative ring of integers, Z, most prime ideals come from prime numbers. If we take all the multiples of any prime number, we get a prime ideal, which is the kernel of the obvious homomorphism

x: Z → Z/p
There's just one other prime ideal in Z, namely all the multiples of 0. In other words, the set consisting of just 0 alone! This is the kernel of the homomorphism from Z into the rationals. For some fascinating reason I'd rather not explain now, this prime ideal is often called "the prime at infinity". It's different from all the rest, but the wise know it's usually good to keep it in.

So, the spectrum of the integers is just the set of ordinary primes together with the "prime at infinity":

Spec(Z) = {2, 3, 5, 7, 11, ... ∞}
We seem to have gotten pretty far from physics by now, but in fact many people believe that taking this spectrum seriously from a physical viewpoint will be crucial to proving the Riemann hypothesis - a famous open conjecture related to the distribution of prime numbers. I don't have time to do justice to this, but the basic idea goes as follows.

Suppose we have a quantum system whose Hamiltonian has this spectrum:

{ln 2, ln 3, ln 5, ln 7, ln 11, ....}
We can think of these as energy states of some sort of particle: the "primon".

Now let's second quantize this system. The idea of second quantization is that we form a new system consisting of an arbitrary finite collection of noninteracting indistinguishable copies of the original system. For example, if the original system was some sort of particle, a state of the new system would consist of an arbitrary number of particles of this sort, treated as identical bosons. If second quantize our "primon", we'll get a system with energy levels that are arbitrary sums of entries from the above list. If we write them in increasing order, they look like this:

{0, ln 2, ln 3, ln 2 + ln 2, ln 5, ln 2 + ln 3, ln 7, ln 2 + ln 2 + ln 2, ....}
or in other words, just
{ln 1, ln 2, ln 3, ln 4, ln 5, ln 6, ln 7, ln 8, ....}
since every whole number can be built from primons in a unique way! Bernard Julia calls this new system the "free Riemann gas", since it's made of noninteracting primons - and in a minute we'll see it's related to the Riemann hypothesis.

To see this, let's do some statistical mechanics with the free Riemann gas! As usual, at any temperature T the probability that this system will be in a state of energy E is proportional to

where β = 1/kT and k is Boltzmann's constant. But to get these numbers to add up to one as probabilities should, we have to normalize them, dividing by their sum, which goes by the name of the "partition function". The partition function for the free Riemann gas is:
 ∑ exp(-β ln n)   =   ∑  n
 n                    n
the so-called "Riemann zeta function". It's well-defined for β > 1 - that is, low temperatures - but it blows up when β = 1. This means that the free Riemann gas has a "Hagedorn temperature": a temperature that it can't go above, because doing so would take an infinite amount of energy.

Nonetheless we can analytically continue the Riemann zeta function around β = 1, and the Riemann hypothesis says that it can only vanish if β is a negative even integer or a number with real part equal to 1/2. And, precisely because the free Riemann gas is made of primons, this hypothesis has a lot to do with prime numbers! For example, it's equivalent to the assertion that the number of primes less than x differs from

Li(x) =  ∫  dt/ln t  
by less than some constant times ln(x) √x.

All this is lots of fun. I urge the physicist reader to compute the free energy and specific heat of the free Riemann gas, and also to investigate the system where we treat the primons as fermions. But, the big question is whether we can use physics-inspired reasoning to prove the Riemann hypothesis!

In 1995, a step in this direction was taken by Bost and Connes. I'm not ready to really explain it, so I'll just tantalize you by dangling their abstract in front of you:

In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one-parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*-dynamical system, which admits the zeta function as partition function and the Galois group Gal(Qcycl/ Q) of the cyclotomic extension Qcycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β = 1.
Here's the reference:

1) J.-B. Bost and Alain Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

The idea of the free Riemann gas was introduced most clearly by Julia, though there were many precursors:

2) Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293. Summarized by Matthew Watkins in

Matthew Watkins has a lot of other fascinating material about prime numbers and physics on his website:

3) Matthew Watkins,

so this is the best place to start if you're a beginner wanting to learn more about this stuff. There are also a bunch of new popular books on the Riemann hypothesis, so if you're looking for good Christmas gifts, you might try one of these:

4) Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins, 2003.

5) Karl Sabbagh, The Riemann Hypothesis: the Greatest Unsolved Problem in Mathematics, Farrar Strauss & Giroux, 2003.

6) John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem Mathematics, Joseph Henry Press, 2003.

I haven't read any of them, but from reviews it sounds like the third one focuses on Riemann while the first two talk more about modern developments.

If you want something quite a bit more substantial but still not requiring a PhD, try this:

7) Jeffrey Stopple, A Primer of Analytic Number Theory: from Pythagoras to Riemann, Cambridge U. Press, Cambridge, 2003.

This is the only introduction to analytic number theory that's so simple that I feel I have a good chance of reading it all the way through.

There's also a lot of interesting work relating the Riemann zeta function to quantum chaos. Alas, I don't know how this is related to the "free Riemann gas" idea! But here's a nice easy introduction:

8) Barry Cipra, A prime case of chaos, in What's Happening in the Mathematical Sciences, vol. 4, American Mathematical Society. Also available at

Finally, if you get stuck on the fermionic version of the free Riemann gas, read Julia's paper or this one:

9) Donald Spector, Supersymmetry and the Moebius inversion function, Communications in Mathematical Physics 127 (1990) 239-252.

Anyway, all this post up to now has been just a big joke - although everything I said is true. The joke is that all this stuff about different meanings of "spectrum" has nothing to do with the sort of "spectra" they were talking about at that conference on homotopy theory! Topologists like to study a completely different sort of spectrum... so now let me talk about those.

In topology, a "spectrum" is defined to be a sequence of pointed topological spaces, each of which is homeomorphic to the space of all based loops in the next. So, each space in a spectrum is an "infinite loop space": a space of loops in a space of loops in a space of loops in....

In "week149" I described how this sort of spectrum gives a generalized cohomology theory, and I mentioned a bunch of examples. I gave some more examples in "week150" and "week197". But I never described the cool way to construct spectra that Graeme Segal came up with - so let me do that now.

There's a cute way to get a space from a category that goes like this. First create a simplicial set from your category, with one 0-simplex for each object:

one 1-simplex for each morphism:
                x             y
one 2-simplex for each composable pair of morphisms:
                      / \
                     /   \
                    /     \
                  f/       \g
                  /         \
                 /           \
                /             \
               /      fg       \
              x                 z

and so on ad infinitum. This is called the "nerve" of the category. Then, think of this simplicial set as a topological space - i.e., take its "geometric realization". The result is called the "classifying space" of the category. By the way, I described this construction in a lot more detail in "week117". I also explained how you can get every space, up to homotopy equivalence, as the classifying space of some category! But what I didn't say is this:

If you start with a monoidal category, the group completion of its classifying space will be a loop space.
You can get any loop space this way.

If you start with a braided monoidal category, the group completion of its classifying space will be a double loop space.
You can get any double loop space this way.

If you start with a symmetric monoidal category, the group completion of its classifying space will be an infinite loop space.
You can get any infinite loop space this way.

Huh? There are lots of terms here that I haven't defined yet....

For starters, a "loop space" is the space of based loops in some pointed topological space. A "double loop space" is the space of based loops in the space of based loops in some pointed topological space, and so on. Secondly, all the above statements are only true up to homotopy equivalence. Third, I'm talking about various sorts of category here. A monoidal category is roughly a category with a tensor product. This gives its classifying space a product, making it into a topological monoid; turning this into a group by throwing in inverses is called "group completion". A braided monoidal category is roughly a monoidal category with an isomorphism

Bx,y: x ⊗ y → y ⊗ x

for any pair of objects; we require this isomorphism satisfy some rules motivated by thinking it as a "braiding", like this:

              x            y
               \          /
                \        /
                 \      /
                  \    /
                   \  /
                   /  \
                  /    \
                 /      \
                /        \
               /          \
              y            x
A symmetric monoidal category is a braided monoidal category for which Bx,y is the inverse of By,x. Some more details on these category-theoretic notions can be found in "week121".

Symmetric monoidal categories abound in mathematics, so we can easily use them to get lots of nice infinite loop spaces - and thence spectra and generalized cohomology theories!

For example, if we take the category of finite sets, with disjoint union as the "tensor product", and the obvious braiding, the group completion of its classifying space will be

Ω S  =  lim  Ωk Sk
         k → ∞
the limit of taking the kth loop space of the k-sphere! The corresponding spectrum is called the "sphere spectrum" and the corresponding generalized cohomology theory is called "stable homotopy theory".

If we take the category of finite-dimensional complex vector spaces, with direct sum as the "tensor product", and the obvious braiding, the group completion of its classifying space will be

BU(∞) =   lim        BU(k)
              k → ∞
where BU(k) is the classifying space of the group of k x k unitary matrices! The corresponding spectrum is called the "spectrum for connective complex K-theory" and the corresponding generalized cohomology theory is called "connective complex K-theory". (Here "connective" refers to the fact that unlike some other K-theory you may be familiar with, the cohomology groups Ki with i negative have been set to zero.)

More generally, we can take the category of finitely generated projective modules of a ring R, again with direct sum as the tensor product and the obvious braiding. This gives something called "algebraic K-theory". More precisely, the homotopy groups of the resulting infinite loop space are called the algebraic K-theory groups Ki(R).

Yet another example comes from taking the category of finite CW complexes, with disjoint union as the "tensor product" and the obvious braiding. This gives a generalized cohomology theory called "A-theory", due to Waldhausen.

I would like to say more about this stuff sometime. There's a lot more to say! For example, there are some cool relations between the algebraic K-theory groups of the integers, Ki(Z), and the Riemann zeta function at odd integers, ζ(2n+1). (Hmm, so maybe the different sort of spectra are related!) There's also a lot of nice stuff about how algebraic K-theory is related to topology. You can learn about that here:

10) Jonathan Rosenberg, K-theory and geometric topology, available at

But, I'll stop here for now. For more on how different sorts of category can be used to get ahold of n-fold loop spaces, see:

11) C. Balteanu, Z. Fiedorowicz, R. Schwaenzl, and R. Vogt, Iterated monoidal categories, available at math.AT/9808082.

Addenda: Here's my reply to some questions, and also some comments by my friend Squark about my use of the term "the prime at infinity".

Rene Meyer wrote:

 John Baez wrote:

 > The "spectrum" of a C*-algebra A is the set of all homomorphisms
 > x: A → C,
 > where C is the complex numbers.  
 > There's an easy way to make the spectrum of a commutative
 > C*-algebra A into a topological space: we say xi → x precisely when
 > xi(a) → x(a)
 > for all elements a of A.  With this topology any element a of A gives
 > a continuous complex function on the spectrum, defined by this clever
 > formula:

 I don't understand what you mean by
 xi → x
 xi(a) → x(a)
That's a way of saying that the sequence xi converges to x, or the sequence xi(a) converges to x(a).
 What has the index i to do with this?
It's the index for some sequence of homomorphisms, xi.
 xi and x are the above mentioned homomorphisms, right?
x is a homomorphism, xi is a sequence of homomorphisms, and I'm telling you when the sequence xi converges to x.
 Could you explain in a little more detail?
I was describing how to make the spectrum of a C*-algebra into a topological space. One way to do this is to say when a sequence xi of points in the spectrum converges to some point x. So, I took a sequence of homomorphisms
 xi: A → C
and told you when it converges to a homomorphism
 x: A → C
And here's what I said: xi converges to x precisely when the sequence of numbers xi(a) converges to the number x(a) for all a in A.

[Experts will know that now I'm lying slightly. In general, to specify the topology of a space, it's not really good enough to just say when sequences converge; you need to say when nets converge. A net is like a sequence, but the index i can range over an arbitrary "directed set". I don't feel like defining a directed set right now; one can find this in any good introduction to point-set topology. The point is that there are some spaces that are not "first countable", meaning that some points don't have a countable base of neighborhoods. A countable sequence just isn't long enough to converge to such a point, unless it equals that point for all sufficiently large i. So in general we need nets, though for metric spaces sequences are sufficient. Luckily, the notation and basic theorems concerning nets look almost like those for sequences! So, I was actually talking about nets in my post above - but I was hoping that people who only knew about sequences would think I was talking about sequences, in which case they'd be slightly wrong, but not too far off.]

Squark wrote:

John Baez wrote:

> the commutative
> ring of integers, Z, most prime ideals come from prime numbers.  If we
> take all the multiples of any prime number, we get a prime ideal, which 
> is the kernel of the obvious homomorphism
> x: Z → Z/p
> There's just one other prime ideal in Z, namely all the multiples of
> 0.  In other words, the set consisting of just 0 alone!  This is the
> kernel of the homomorphism from Z into the rationals.  For some
> fascinating reason I'd rather not explain now, this prime ideal is
> often called "the prime at infinity".

I don't quite agree. The 0 ideal corresponds merely to the generic
point of Spec Z, a usual thing for schemes. The "prime at infinity",
as far as I understand, comes from viewing Spec Z as an "affine
line" over some mysterious impossible field and then completing
it into a "projective line".

In more detail, for any actual affine line Spec K[x] where x is a
field one can use each point x0 in K to define a norm on K(x),
the field of rational functions over K. This is the
non-Archimedean norm ||f|| = q(degx0 f) where degx0 f
is the degree of the pole f has at x0 (or minus the degree of the
zero). I think it's possible to prove K(x) has exactly one norm
except this one: qdeg f where deg f is just the rational function
degree. This norm corresponds to the "point at infinity", adding it
gives us the projective line KP1 (deg f is precisely the degree
of the pole at the point at infinity). Note that the product of all of
these norms is 1.

The rational functions over Spec Z is Q. Each prime gives us a
norm on Q which turns out to be the p-adic norm (modulo the
choice of q, which is a subtler issue, but also solvable, I think).
However Q has another norm on it: the usual, Archimedean
norm! Since it is Archimedean, it cannot come out of the qdeg
construction (in more fancy terms it doesn't correspond to any
local ring with Q its field of fractions). However, one can play the
"as if" game and imagine it does correspond to a point at infinity
lying in some weird completion of Spec Z. The generic point, on
the other hand, is present already for Spec K[x], and it is a
distinct point from the point at infinity for KP1.

There are other interesting things related to this. In particular, the
Cauchy completion procedure is possible to formulate in purely
algebraic terms. For algebraic curves such as Spec K[x] it gives
a ring of formal series around the given point - a sort of
improvement of the usual local ring. This is something useful on its
own in algebraic geometry, for instance this "improved local ring"
(I don't remember the real name :-) ) is the same for the
self-intersection point of the "figure 8 curve" and the curve
consisting of two intersecting lines. The usual local ring
distinguishes between the two cases, so it's in some sense "not
local enough".

For Spec Z we get the p-adic numbers Qp at the prime points
and we should get R at the point at infinity. This would be very
cool, since otherwise R seems to be an entirely analytic object,
impenetrable by algebra.

Best regards,
By the way, the reason Squark pointed out that the product of all norms of an element of K(x) equals 1, is that the same is true for the product of all p-adic norms of a rational together with its usual norm. So, the analogy is good.

But anyway, I guess I should have spoken of "the generic point" instead of "the prime at infinity" when talking about the prime ideal {0} in Z. The "prime at infinity" is a more mysterious thing. To learn more about it, read this book:

12) M. J. Shai Haran, The Mysteries of the Real Prime, Oxford U. Press, Oxford, 2001.

It touches upon lots of interesting relations between number theory and mathematical physics.

Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice's world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there. - Marcus du Sautoy

God may not play dice with the universe, but something strange is going on with the primes. - Paul Erdös

© 2003 John Baez