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December 9, 2007

This Week's Finds in Mathematical Physics (Week 259)

John Baez

This week I'll talk about the "field with one element" - even though

it doesn't exist. It's a mathematical phantom.

But first: the Egg Nebula.

In "week257" and "week258" I talked about interstellar dust.

As I mentioned, lots of it comes from "asymptotic giant branch"

stars - stars like our Sun, but later in their life, when they're

big, red, pulsing, and puffing out elements like hydrogen, helium,

carbon, nitrogen, and oxygen.

The pulsations grow wilder and wilder until the star blows off its

entire outer atmosphere, forming a big cloud of gas and dust

misleadingly called a "planetary nebula". It leaves behind

its dense inner core as a hot white dwarf. Intense radiation from

this core eventually heats the gas and dust until they glow.

Back in "week223" I showed my favorite example of a planetary

nebula: the Cat's Eye. And I quoted the astronomer Bruce Balick

on what will happen here when our Sun becomes a planetary nebula

6.9 billion years from now:

Here on Earth, we'll feel the wind of the ejected gases

sweeping past, slowly at first (a mere 5 miles per second!),

and then picking up speed as the spasms continue - eventually

to reach 1000 miles per second!! The remnant Sun will rise as

a dot of intense light, no larger than Venus, more brilliant

than 100 present Suns, and an intensely hot blue-white color

hotter than any welder's torch. Light from the fiendish blue

"pinprick" will braise the Earth and tear apart its surface

molecules and atoms. A new but very thin "atmosphere" of free

electrons will form as the Earth's surface turns to dust.

Eerie!

Here's a "protoplanetary nebula" - that is, a planetary nebula that's

just getting started:

1) Rainbow image of a dusty star, NASA,

http://hubblesite.org/newscenter/archive/releases/nebula/planetary/2003/09/

It's called the "Egg Nebula". You can see layers of dust coming out

puff after puff, shooting outwards at about 20 kilometers per second,

stretching out for about a third of a light year. The colors - red,

green and blue - aren't anything you'd actually see. They're just

an easy way to depict three different polarizations of light. I

don't know why the light is polarized that way.

You can see a dark disk of thicker dust running around the star. It

could be an "accretion disk" spiralling into the star. The "beams"

shining out left and right are still poorly understood. Maybe they're

jets of matter ejected from the north and south poles of the disk?

This idea seem more plausible when you look at this photo taken by

NICMOS, which is Hubble's "Near Infrared Camera and Multi-Object

Spectrometer":

2) Raghvendra Sahai, Egg Nebula in polarized light, Hubble Heritage

Project, http://heritage.stsci.edu/2003/09/supplemental.html

This near-infrared image also shows a bright spot called "Peak A" about

500 AU from the central star. An "AU", or astronomical unit, is the

distance from the Sun to the Earth.

Nobody knows what this bright spot is. Some argue that it's just a

clump of dust reflecting light from the main star. Others advocate a

more exciting theory: it's a white dwarf orbiting the main star, which

exploded in a "thermonuclear burst" after accreting a bunch of dust.

3) Joel H. Kastner and Noam Soker, The Egg Nebula (AFGL 2688): deepening

enigma, to appear in Asymmetrical Planetary Nebulae III, eds. M. Meixner,

J. Kastner, N. Soker, and B. Balick, ASP Conference Series. Available

as arXiv:astro-ph/0309677.

I hope to say more about planetary nebulae in future Weeks, mainly

because they're so beautiful.

But now: the field with one element!

A field is a mathematical structure where you can add, multiply,

subtract and divide in ways that satisfy these familiar rules:

x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

xy = yx (xy)z = x(yz) 1x = x

x(y + z) = xy + xz

every element x has an element -x with x + (-x) = 0

every element x that's not 0 has an element 1/x with x (1/x) = 1

You'll note that the last clause is the odd man out. Addition,

subtraction and multiplication can all be described as everywhere

defined operations. Division cannot, since we can't divide by 0.

This is the funny thing about fields, which is what causes the

problem we'll run into.

Everyone who has studied math knows three examples of fields:

the rational numbers Q, the real numbers R and the complex numbers C.

There are a lot more, too - for example, function fields, number

fields, and finite fields.

Let me say a tiny bit about these three kinds of fields.

The simplest sort of "function field" consists of rational functions

of one variable - that is, ratios of polynomials, like this:

(2z^3 + z + 1)/(z^2 - 7)

Here the coefficients of your polynomials should lie in some field

F you already know about. The resulting field is called F(z).

If F is the complex numbers, we can think of F(z) = C(z) as consisting

of functions on the Riemann sphere. In "week201", I explained how

the symmetries of this field form a group important in special

relativity: the Lorentz group!

It's also very interesting to study the field of functions on a

surface fancier than the sphere, but still defined by algebraic

equations, like the surface of a doughnut or n-holed doughnut.

Number theorists and algebraic geometers spend a lot of time

thinking about these fields, which are called "function fields

of complex curves".

For example, different ways of describing the surface of a doughnut

by algebraic equations give different "elliptic curves". This is a

great example of terminology that's bound to mislead beginners!

They're called "curves" even though it's 2-dimensional, because it takes one

*complex* number to say where you are on a little patch of a surface,

just as it takes one *real* number to say where you are on an

ordinary curve like a circle. That's the origin of the term

"complex curve". And, they're called "elliptic" because they first

showed up when people were studying elliptic integrals, which are

generalizations of trig functions from circles to ellipses.

I explained more about elliptic curves in "week13" and "week125".

Lurking behind this, there's a lot of fascinating stuff about

function fields of elliptic curves.

The simplest sort of "number field" comes from taking the rational

numbers and throwing in the solutions of a polynomial equations.

For example, in "week20" I talked about the "golden field", which

consists of all numbers of the form

a + b sqrt(5)

where a and b are rational.

One of the most beautiful ideas in math is the analogy between

number fields and function fields - the idea that numbers are like

functions on some sort of "space". I began explaining this in

"week205", "week216" and "week218", but there's much more to say

about what's known... and also many things that remain mysterious.

In particular, it's pretty well understood how number fields

resemble function fields of complex curves, and how this relates

number theory to *2-dimensional* topology. But, there are also many

analogies between number theory and *3-dimensional* topology, which

I began listing in "week257". It seems these analogies are doomed to

remain mysterious until we get a handle on the field with one element.

But more on that later.

The simplest sort of "finite field" comes from choosing a prime number

p and taking the integers modulo p. The result is sometimes called

Z/p, especially when you're just concerned with addition. But when you

think of it as a field, it's better to call it F_p.

The reason is that there's a finite field of size q whenever q is a

*power* of a prime, and this field is unique - so it's called F_q. You

build F_q sort of like how you build the complex numbers starting from

the real numbers, or number fields starting from the rational numbers.

Namely, to construct F_{p^n}, you take F_p and throw in the roots of a

well-chosen polynomial of degree n: one that doesn't have any roots in

F_p, but "wants" to have n different roots.

Okay: that was a tiny bit about function fields, number fields and finite

fields. But now I need to point out some slight lies I told!

I said there was a finite field with q elements whenever q was a prime

power. You might think this should include q = 1, since 1 is the

*zeroth* power of *any* prime.

So, is there a field with one element?

If so, it must have 1 = 0. That doesn't violate the definition of

a field that I gave you... does it? The definition said any element

that's not 0 has a reciprocal. In this particular example, 0 also has

a reciprocal, since we can set 1/0 = 1 and not get into any contradictions.

But that's not a problem: in usual math practice, saying "we can divide

by anything that's not zero" doesn't deny the possibility that we can

divide by 0.

Unfortunately, allowing a field with 1 = 0 causes nothing but grief.

For example, we can define vector spaces using any field (people say

"over" any field), and there's a nice theorem saying two vector spaces

are isomorphic if and only if they have the same dimension. And normally,

there's one vector space of each dimension. But the last part isn't true

for a field with 1 = 0. In a vector space over such a field, every

vector v has

v = 1 v = 0 v = 0

So, every vector space is 0-dimensional!

To prevent such problems, people add one extra clause to the definition

of a field:

1 is not equal to 0

This clause looks even more tacked-on and silly than the clause

saying everything *nonzero* has a reciprocal... but it works fairly

well.

However, the field with one element still wants to exist! Not the

silly field with 1 = 0, but something else, something more mysterious...

something that Gavin Wraith calls a "mathematical phantom":

4) Gavin Wraith, Mathematical phantoms,

http://www.wra1th.plus.com/gcw/rants/math/MathPhant.html

What's a mathematical phantom? According to Wraith, it's an object

that doesn't exist within a given mathematical framework, but

nonetheless "obtrudes its effects so convincingly that one is forced

to concede a broader notion of existence".

Like a genie that talks its way out of a bottle, a sufficiently powerful

mathematical phantom can talk us into letting it exist by promising to

work wonders for us. Great examples include the number zero, irrational

numbers, negative numbers, imaginary numbers, and quaternions. At one

point all these were considered highly dubious entities. Now they're

widely accepted. They "exist". Someday the field with one element

will exist too!

Why?

I gave a lot of reasons in "week183", "week184", "week185", "week186"

and "week187", but let me rapidly summarize.

It's all about "q-deformation". In physics, people talk about q-deformation

when they're taking groups and turning them into "quantum groups". But it

has a closely related aspect that's in some ways more fundamental. When

we count things involving n-dimensional vector spaces over the finite field

F_q, we often get answers that are polynomials in q. If we then set q = 1,

the resulting formulas count analogous things involving n-element sets!

So, finite sets want to be finite-dimensional vector spaces over the

(nonexistent) field with one element... or something like that. We can

be more precise after looking at some examples.

Here's the simplest example. Say we count lines through the origin

in an n-dimensional vector space over F_q. We get the "q-integer"

q^n - 1

------- = 1 + q + q^2 + ... + q^{n-1}

q - 1

which I'll write as [n] for short.

Setting q = 1, we n. This is the number of points in an n-element set.

Sure, that sounds silly. But, I'm trying to make a point here! At

q = 1, stuff about n-dimensional vector spaces over F_q reduces to stuff

about n-element sets, and the q-integer [n] reduces to the ordinary

integer n.

This may not be the best way to understand the pattern, though.

Lines through the origin in an n-dimensional vector space are the

same as points in an (n-1)-dimensional projective space. So, the

real analogy may be between "points in a projective space" and

"points in a set".

Here's a more impressive example. Pick any uncombed Young diagram D

with n boxes. Here's one with 8 boxes:

X 1 box in the first row

X X 3 boxes in the first two rows

X X X 6 boxes in the first three rows

X X 8 boxes in the first four rows

Then, count the "D-flags on an n-dimensional vector space over F_q".

In our example, such a D-flag is:

a 1-dimensional subspace

of a 3-dimensional subspace

of a 6-dimensional subspace

of a 8-dimensional vector space over F_q

If you actually count these D-flags you'll get some formula, which is

a polynomial in q. And when you set q = 1, you'll get the number of

"D-flags on an n-element set". In our example, such a D-flag is:

a 1-element subset

of a 3-element subset

of a 6-element subset

of a 8-element set

For details, and a proof that this really works, try:

5) John Baez, Lecture 4 in the Geometric Representation Theory

Seminar, October 9, 2007. Available at

http://math.ucr.edu/home/baez/qg-fall2007/qg-fall2007.html#f07_4

These examples can be generalized. In "week187" I showed how to get

one example for each subset of the dots in any Dynkin diagram! This

idea goes back to Jacques Tits, who was the first to suggest that there

should be a field with one element. Dynkin diagrams give algebraic

groups over F_q... but he noticed that these groups reduce to "Coxeter

groups" as q -> 1. And, if you mark some dots on a Dynkin diagram you

get a "flag variety" on which your algebraic group acts... but as q -> 1,

this reduces to a finite set on which your Coxeter group acts.

If you don't understand the previous paragraph, don't worry - it's

over now. It's great stuff, but my main point is that there seems to

be an analogy like this:

q = 1 q = a power of a prime number

n-element set (n-1)-dimensional projective space over F_q

integer n q-integer [n]

permutation groups S_n projective special linear group PSL(n,F_q)

factorial n! q-factorial [n]!

This opens up lots of questions. For example, if projective spaces over

F_1 are just finite sets, what should *vector spaces* over F_1 be?

People have thought about this, and the answer seems to be "pointed

sets" - sets with a distinguished point, which you can think of as

the "origin". A pointed set with n+1 elements seems to act like

an n-dimensional vector space over F_q.

For more clues, and an attempt to do algebraic geometry using the

field with one element, try this:

6) Christophe Soule, On the field with one element, Talk given at the

Arbeitstagung, Bonn, June 1999, IHES preprint available at

http://www.ihes.fr/PREPRINTS/M99/M99-55.ps.gz [Broken]

Soule tries to define "algebraic varieties" over F_1, namely curves

and their higher-dimensional generalization. And, he talks a lot about

zeta functions for such varieties. He goes into more detail here:

7) Christophe Soule, Les varietes sur le corps a un element, Moscow

Math. Jour. 4 (2004), 217-244, 312.

The theme of zeta functions - see "week216" - is deeply involved in

this business. For more, try these papers:

8) N. Kurokawa, Zeta functions over F_1, Proc. Japan Acad. Ser. A

Math. Sci. 81 (2006), 180-184.

9) Anton Deitmar, Remarks on zeta functions and K-theory over F1,

available as arXiv:math/0605429.

But instead of talking about zeta functions, I'd like to talk about

two approaches to giving a formal definition of the field with one

element. Both of them involve taking the concept of field and

modifying it so it doesn't necessary involve the operation of addition.

The first one, due to Deitmar, simply throws out addition! The second,

due to Anton Durov, allows for a wide choice of operations - and thus

a wide supply of "exotic fields".

For Deitmar's approach, try these:

10) Anton Deitmar, Schemes over F1, available as arXiv:math/0404185.

F1-schemes and toric varieties, available as arXiv:math/0608179.

The usual approach to fields treats fields as specially nice commutative

rings. A "commutative ring" is a gadget where you can add and multiply,

and these rules hold:

x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

xy = yx (xy)z = x(yz) 1x = x

x(y + z) = xy + xz

every element x has an element -x with x + (-x) = 0

Deitmar throws out addition and treats fields as specially nice

commutative monoids. A "commutative monoid" is a gadget where you can

multiply, and these rules hold:

xy = yx (xy)z = x(yz) 1x = x

For Deitmar, the field with one element, F_1, is just the commutative

monoid with one element, namely 1. A "vector space over F_1" is just

a set on which this monoid acts via multiplication... but that amounts

to just a plain old set. The "dimension" of such a "vector space"

is just its cardinality.

All this so far is quite trivial, but Deitmar makes a nice attempt at

redoing algebraic geometry to include this field with one element.

One reason to do this is to understand the mysterious 3-dimensional

aspect of number theory.

To explain this, I need to say a bit about "schemes". In ordinary

algebraic geometry, we turn commutative rings into spaces to think

about them geometrically. I explained this back in "week199" and

"week205", but let me review quickly, and go further:

We can think of elements of a commutative ring R as functions on

certain space called the "spectrum" of R, Spec(R). This space has

a topology, so we can also talk about functions that are defined,

not on all of Spec(R), but just *part* of Spec(R) - namely some open set.

Indeed, for each open set U in Spec(R), there's a commutative ring O(U)

consisting of those functions defined on U. These commutative

rings are related in nice ways:

A) If the open set V is smaller than U, we can restrict functions from

U to V, getting a ring homomorphism O(U) -> U(V)

B) If U is covered by a bunch of open sets U_i, and we have a function

f_i in each O(U_i), such that f_i and f_j agree when restricted to

the intersection of U_i and U_j, then there's a unique function f in O(U)

that restricts to each of these functions f_i.

Something satisfying condition A) is called a "presheaf" of commutative

rings; something also satisfying condition B) is called a "sheaf" of

commutative rings.

So, Spec(R) is not just a topological space, it's equipped with a

sheaf of commutative rings. People call this a "ringed space".

Whenever we have a ringed space, we can ask if it comes from a

commutative ring R in the way I just sketched. If so, we call

it an "affine scheme". Affine schemes are just a fancy geometrical way

of talking about commutative rings!

More interestingly, whenever we have a ringed space, we can ask if

it's *locally* isomorphic to one coming from a commutative ring.

In other words: does every point have a neighborhood that, as a

ringed space, looks like Spec(R) for some commutative ring R?

Or in other words: is our ringed space *locally* isomorphic to an

affine scheme? If so, we call it a "scheme".

A classic example of a scheme that's not an affine scheme is the

Riemann sphere. There aren't any rational functions defined on the

whole Riemann sphere, except for constants - the rest all blow up

somewhere. So, it's hopeless trying to think of the Riemann sphere

as an affine scheme.

But, for any open set U there's a commutative ring O(U) consisting of

rational functions that are defined on U. So, the Riemann sphere

becomes a ringed space. And, it's *locally* isomorphic to the complex

plane, which is the affine scheme corresponding to the commutative ring

of complex polynomials in one variable. So, the Riemann sphere is a

scheme!

For more on schemes, try this nice introduction, which actually

has lots of pictures:

11) David Eisenbud, The Geometry of Schemes, Springer, Berlin, 2000.

Now, we can talk about schemes "over a field F", meaning that each

commutative ring O(U) is also a vector space over F, in a well-behaved

way, giving us a "sheaf of commutative rings over F". For example,

the Riemann sphere is a scheme over C.

There's a secret 3-dimensional aspect to the affine scheme Spec(Z),

where Z is the commutative ring of integers. As explained in the

Addenda to "week257", we might understand this if we could see Spec(Z)

as a scheme over the field with one element! For more, see this:

12) 7) M. Kapranov and A. Smirnov, Cohomology determinants and

reciprocity laws: number field case, available at

http://wwwhomes.uni-bielefeld.de/triepe/F1.html [Broken]

So, we really need a theory of schemes over the field with one element.

The problem is, F_1 isn't really a field. In Deitmar's approach, it's

just a commutative monoid.

So, let me sketch how Deitmar gets around this. In a nutshell, he takes

advantage of the fact that a lot of basic algebraic geometry only requires

multiplication, not addition!

He starts by defining a "commutative ring over F_1" to be simply a

commutative monoid. The simplest example is F_1 itself.

Now, watch how he gets away with never using addition:

He defines an "ideal" in a commutative monoid R to be a subset I for

which the product of something in I with anything in R again lies in I.

He says an ideal P is "prime" if whenever a product of two elements in

R is in P, at least one of them is in P.

He defines the "spectrum" Spec(R) of a commutative monoid R to be the

set of its prime ideals. He gives this the "Zariski topology". That's

the topology where the closed sets are the whole space, or any set of

prime ideals that contain a given ideal.

He then shows how to get a sheaf of commutative monoids on Spec(R).

He defines a "scheme" to be a space equipped with a sheaf of

commutative monoids that's *locally* isomorphic to one of this sort.

If you know algebraic geometry, these definitions should seem very

familiar. And if you don't, you can just replace the word "monoid"

by "ring" everywhere in the previous three paragraphs, and you'll get

the standard definitions in algebraic geometry!

Deitmar shows how to build a scheme over F_1 called the "projective

line". The projective line over C is just the Riemann sphere. The

projective line over F_1 has just two points (or more precisely, two

closed points). This is good, because the projective line over the

field with q elements has

[2] = 1 + q

points, and we're doing the q = 1 case.

Deitmar's construction seems like a lot of work to get ahold of the

2-element set, if that's all it secretly is. But, I need to think

about this more. After all, he doesn't just get a space; he gets a

sheaf of commutative monoids on this space! And what's that like?

I should work it out.

Deitmar also shows how to relate schemes over F_1 to the usual sort

of schemes.

From a commutative ring, we can always get a commutative monoid just

by forgetting the addition. This process has a kind of reverse, too.

Namely, from a commutative monoid, we can get a commutative ring

simply by taking formal integral linear combinations of elements.

Using this, Deitmar shows how we can turn ordinary schemes into

schemes over F_1... and conversely. He says an ordinary scheme is

"defined over F_1" if it arises in this way from a scheme over F1.

Okay, that's a taste of Deitmar's approach. For Durov's approach,

try this mammoth 568-page paper:

13) Anton Durov, New approach to Arakelov geometry, available as

arXiv:0704.2030.

or read our discussions of it at the n-Category Cafe, starting here:

14) David Corfield, The field with one element,

http://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html

Durov defines a "generalized ring" to be what Lawvere much earlier

called an "algebraic theory". What is it? Nothing scary! It's just a

gadget with a bunch of abstract n-ary operations closed under composition,

permutation, duplication and deletion of arguments, and equipped with

an identity operation.

So, for example, if our gadget has a binary operation

(x,y) |-> f(x,y)

we can compose this with itself to get the ternary operation

(x,y,z) |-> f(f(x,y),z)

and the 4-ary operation

(w,x,y,z) |-> f(w,f(x,f(y,z)))

and so on. We can then permute arguments in our 4-ary operation

to get one like this:

(w,x,y,z) |-> f(z,f(x,f(w,y)))

or duplicate some arguments to get a binary operation like this

(x,y) |-> f(x,f(x,f(y,y)))

From this we can then form a 3-ary operation by deleting an argument,

for example like this:

(x,y,z) |-> f(x,f(x,f(y,y)))

If you know about "operads", this kind of gadget is just a specially

nice operad where we can duplicate and delete operations.

Now, a generalized ring is said to be "commutative" if all the

operations commute in a certain sense. (I'll let you guess what

it means for an n-ary operation to commute with an m-ary operation.)

We get an example of a commutative generalized ring from a commutative

ring R if we let the n-ary operations be "n-ary R-linear combinations",

like this:

(x_1, ..., x_n) |-> r_1 x_1 + ... + r_n x_n

We also get a very similar example from any "commutative rig", which is

a gizmo satisfying rules like those of a commutative ring, but without

negatives:

x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

xy = yx (xy)z = x(yz) 1x = x

x(y + z) = xy + xz

And, we get an example from any commutative monoid, where we only

have 1-ary operations, coming from multiplication by elements of

our monoid:

(x_1) |-> r x_1

So, Durov's framework generalizes Deitmar's! But, it includes a lot

more examples: exotic hothouse flowers like the "tropical rig", the

"real integers", and more. He develops a theory of schemes for all

these generalized rings, and builds it "up to construction of algebraic

K-theory, intersection theory and Chern classes" - fancy things that

algebraic geometers like.

What I don't yet see is how either Deitmar's or Durov's approach

helps us understand the secret 3-dimensional nature of Spec(Z).

I may just need to read their papers more carefully and think about

them more.

Finally, here's yet another approach to the field with one element:

15) Bertrand Toen and M. Vaquie, Under Spec(Z), available as

arxiv:math/0509684.

In short, a mathematical phantom is gradually materializing before our

very eyes! In the process, a grand generalization of algebraic

geometry is emerging, which enriches it to include some previously

scorned entities: rigs, monoids and the like. And, this enrichment

holds the promise of shedding light on some otherwise impenetrable

mysteries: for example, the deep inner meaning of q-deformation, and

the 3-dimensional nature of Spec(Z).

-----------------------------------------------------------------------

Quote of the Week:

"The analogy between number fields and function fields finds a basic

limitation with the lack of a ground field. One says that Spec(Z)

(with a point at infinity added, as is familiar in Arakelov geometry)

is like a (complete) curve, but over which field?" - Christophe Soule

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# This Week's Finds in Mathematical Physics (Week 259)

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