# September 19, 1998 {#week123}

> It all started out as a joke. Argument for argument's sake. Alison
> and her infuriating heresies.
>
> "A mathematical theorem," she'd proclaimed, "only becomes true
> when a physical system tests it out: when the system's behaviour
> depends in some way on the theorem being *true* or *false*.
>
> It was June 1994. We were sitting in a small paved courtyard, having
> just emerged from the final lecture in a one-semester course on the
> philosophy of mathematics --- a bit of light relief from the hard grind
> of the real stuff. We had fifteen minutes to to kill before meeting
> some friends for lunch. It was a social conversation --- verging on mild
> flirtation --- nothing more. Maybe there were demented academics,
> lurking in dark crypts somewhere, who held views on the nature of
> mathematical truth which they were willing to die for. But were were
> twenty years old, and we *knew* it was all angels on the head of a
> pin.
>
> I said, "Physical systems don't create mathematics. Nothing
> *creates* mathematics --- it's timeless. All of number theory would
> still be exactly the same, even if the universe contained nothing but
> a single electron."
>
> Alison snorted. "Yes, because even *one electron*, plus a space-time
> to put it in, needs all of quantum mechanics and all of general
> relativity --- and all the mathematical infrastructure they entail. One
> particle floating in a quantum vacuum needs half the major results of
> group theory, functional analysis, differential geometry --- "
>
> "OK, OK! I get the point. But if that's the case... the events in
> the first picosecond after the Big Bang would have 'constructed'
> every last mathematical truth required by *any* physical system, all
> the way to the Big Cruch. Once you've got the mathematics which
> underpins the Theory of Everything... that's it, that's all you
> ever need. End of story."
>
> "But it's not. To *apply* the Theory of Everything to a particular
> system, you still need all the mathematics for dealing with *that
> system* --- which could include results far beyond the mathematics the
> TOE itself requires. I mean, fifteen billion years after the Big Bang,
> someone can still come along and prove, say... Fermat's Last
> Theorem." Andrew Wiles at Princeton had recently announced a proof of
> the famous conjecture, although his work was still being scrutinised
> by his colleagues, and the final verdict wasn't yet in. "Physics
> never needed *that* before."
>
> I protested, "What do you mean, 'before'? Fermat's Last Theorem
> never has --- and never will --- have anything to do with any branch of
> physics."
>
> Alison smiled sneakily. "No *branch*, no. But only because the class
> of physical systems whose behaviour depend on it is so ludicrously
> specific: the brains of mathematicians who are trying to validate the
> Wiles proof."
>
> "Think about it. Once you start trying to prove a theorem, then even
> if the mathematics is so 'pure' that it has no relevance to any
> other object in the universe... you've just made it relevant to
> *yourself*. You have to choose *some* physical process to test the
> theorem --- whether you use a computer, or a pen and paper... or just
> close your eyes and shuffle *neurotransmitters*. There's no such
> thing as a proof which doesn't rely on physical events, and whether
> they're inside or outside your skull doesn't make them any less
> real."

And this is just the beginning... the beginning of Greg Egan's tale of
an inconsistency in the axioms of arithmetic --- a "topological defect"
left over in the fabric of mathematics, much like the cosmic strings or
monopoles hypothesized by certain physicists thinking about the early
universe --- and the mathematicians who discover it and struggle to
prevent a large corporation from exploiting it for their own nefarious
purposes. This is the title story of his new collection, "Luminous".

I should also mention his earlier collection of stories, named after a
sophisticated class of mind-altering nanotechnologies, the
"axiomatics", that affect specific beliefs of anyone who uses them:

1) Greg Egan, _Axiomatic_, Orion Books, 1995.

    Greg Egan, _Luminous_, Orion Books, 1998.

Some of the stories in these volumes concern math and physics, such as
"The Planck Dive", about some far-future explorers who send copies of
themselves into a black hole to study quantum gravity firsthand. One
nice thing about this story, from a pedant's perspective, is that Egan
actually works out a plausible scenario for meeting the technical
challenges involved --- with the help of a little 23rd-century technology.
Another nice thing is the further exploration of a world in which
everyone has long been uploaded to virtual "scapes" and can easily
modify and copy themselves --- a world familiar to readers of his novel
"Diaspora" (see ["Week 115"](#week115)). But what I really like
is that it's not just a hard-science extravaganza; it's a meditation
on mortality. You can never really know what it's like to cross an
event horizon unless you do it....

Other stories focus on biotechnology and philosophical problems of
identity. The latter sort will especially appeal to everyone who liked
this book:

2) Daniel C. Dennett and Douglas R. Hofstadter, _The Mind's I: Fantasies and Reflections on Self and Soul_, Bantam Books, 1982.

Among these, one of my favorite is called "Closer". How close can you
be to someone without actually *being them*? Would temporarily merging
identities with someone you loved help you understand them better?
Luckily for you penny-pinchers out there, this particular story is
available free at the following website:

3) Greg Egan, _Closer_, `http://www.eidolon.net/old_site/issue_09/09_closr.htm`

Whoops! I'm drifting pretty far from mathematical physics, aren't I?
Self-reference has a lot to do with mathematical logic, but.... To
gently drift back, let me point out that Egan has a website in which he
explains special and general relativity in a nice, nontechnical way:

4) Greg Egan, _Foundations_, `http://www.netspace.net.au/~gregegan/FOUNDATIONS/index.html`

Also, here are some interesting papers:

5) Gordon L. Kane, "Experimental evidence for more dimensions reported", _Physics Today_, May 1998, 13--16.

    Paul M. Grant, "Researchers find extraordinarily high temperature superconductivity in bio-inspired nanopolymer", _Physics Today_, May 1998, 17--19.

    Jack Watrous, "Ribosomal robotics approaches critical experiments; government agencies watch with mixed interest", _Physics Today_, May 1998, 21--23.

What these papers have in common is that they are all works of science
fiction, not science. They read superficially like straight science
reporting, but they are actually the winners of Physics Today's
"Physics Tomorrow" essay contest!

For example, Grant writes:

> "Little's concept involved replacing the phonons --- characterized by
the Debye temperature --- with excitons, whose much higher characteristic
energies are on the order of 2 eV, or 23,000 K. If excitons were to
become the electron-pairing 'glue', superconductors with $T_c$'s as
high as 500 K might be possible, even under weak coupling conditions.
Little even proposed a possible realization of the idea: a structure
composed of a conjugated polymer chain (polyene) dressed with highly
polarizable molecule (aromatics) as side groups. Simply stated, the
polyene chain would be a normal metal with a single mobile electron per
C-H molecular unit; electrons on separate units would be paired by
interacting with the exciton field on the polarizable side groups."

Actually, I think this part is perfectly true --- William A. Little
suggested this way to achieve high-temperature superconductivity back in
the 1960s. The science fiction part is just the description, later on in
Grant's article, of how Little's dream is actually achieved.

Okay, enough science fiction! Time for some real science! Quantum
gravity, that is. (Stop snickering, you skeptics....)

6) Laurent Freidel and Kirill Krasnov, "Spin foam models and the classical action principle", preprint available as [`hep-th/9807092`](https://arxiv.org/abs/hep-th/9807092).

I described the spin foam approach to quantum gravity in
["Week 113"](#week113). But let me remind you how the basic idea
goes. A good way to get a handle on this idea is by analogy with Feynman
diagrams. In ordinary quantum field theory there is a Hilbert space of
states called "Fock space". This space has a basis of states in which
there are a specific number of particles at specific positions. We can
visualize such a state simply by imagining a bunch of points in space,
with labels to say which particles are which kinds: electrons, quarks,
and so on. One of the main jobs of quantum field theory is to let us
compute the amplitude for one such state to evolve into another as time
passes. Feynman showed that we can do it by computing a sum over graphs
in spacetime. These graphs are called Feynman diagrams, and they
represent "histories". For example,
$$
  \begin{tikzpicture}[thick]
    \draw (0,0) to node[rotate=225,label={[label distance=-2mm]left:{u}}]{$\blacktriangle$} (1,-1);
    \draw (1,-1) to node[rotate=-225,label={[label distance=-3.2mm]left:{d}}]{$\blacktriangle$} (0,-2);
    \draw [decorate, decoration=snake] (1,-1) to node[label=above:{W}]{} (3,-1);
    \draw (4,0) to node[rotate=-225,label={[label distance=-2mm]right:{e}}]{$\blacktriangle$} (3,-1);
    \draw (3,-1) to node[rotate=225,label={[label distance=-2mm]right:{$\nu$}}]{$\blacktriangle$} (4,-2);
  \end{tikzpicture}
$$
would represent a history in which an up quark emits a W boson and turns
into a down quark, with the W being absorbed by an electron, turning it
into a neutrino. Time passes as you march down the page. Quantum field
theory gives you rules for computing amplitudes for any Feyman diagram.
You sum these amplitudes over all Feynman diagrams starting at one state
and ending at another to get the total amplitude for the given
transition to occur.

Now, where do these rules for computing Feynman diagram amplitudes come
from? They are not simply postulated. They come from perturbation
theory. There is a general abstract formula for computing amplitudes in
quantum field theory, but it's not so easy to use this formula in
concrete calculations, except for certain very simple field theories
called "free theories". These theories describe particles that don't
interact at all. They are mathematically tractable but physically
uninteresting. Uninteresting, that is, *except* as a starting-point for
studying the theories we *are* interested in --- the so-called
"interacting theories".

The trick is to think of an interacting theory as containing parameters,
called "coupling constants", which when set to zero make it reduce to
a free theory. Then we can try to expand the transition amplitudes we
want to know as a Taylor series in these parameters. As usual, computing
the coefficients of the Taylor series only requires us to to compute a
bunch of derivatives. And we can compute these derivatives using the
free theory! Typically, computing the $n$th derivative of some transition
amplitude gives us a bunch of integrals which correspond to Feynman
diagrams with n vertices.

By the way, this means you have to take the particles you see in Feynman
diagrams with a grain of salt. They don't arise purely from the
mathematics of the interacting theory. They arise when we *approximate*
that theory by a free theory. This is not an idle point, because we can
take the same interacting theory and approximate it by *different* free
theories. Depending on what free theory we use, we may say different
things about which particles our interacting theory describes! In
condensed matter physics, people sometimes use the term
"quasiparticle" to describe a particle that appears in a free theory
that happens to be handy for some problem or other. For example, it can
be helpful to describe vibrations in a crystal using "phonons", or
waves of tilted electron spins using "spinons". Condensed matter
theorists rarely worry about whether these particles "really exist".
The question of whether they "really exist" is less interesting than
the question of whether the particular free theory they inhabit provides
a good approximation for dealing with a certain problem. Particle
physicists, too, have increasingly come to recognize that we shouldn't
worry too much about which elementary particles "really exist".

But I digress! My point was simply to say that Feynman diagrams arise
from approximating interacting theories by free theories. The details
are complicated and in most cases nobody has ever succeeded in making
them mathematically rigorous, but I don't want to go into that here.
Instead, I want to turn to spin foams.

Everything I said about Feynman diagrams has an analogy in this approach
to quantum gravity. The big difference is that ordinary "free
theories" are formulated on a spacetime with a fixed metric - usually
Minkowski spacetime, with its usual flat metric. Attempts to approximate
quantum gravity by this sort of free theory failed dismally. Perhaps the
fundamental reason is that general relativity doesn't presume that
spacetime has a fixed metric --- au contraire, it's a theory in which the
metric is the main variable!

So the idea of Freidel and Krasnov is to approximate quantum graivty
with a very different sort of "free theory", one in which the metric
is a variable. The theory they use is called "$BF$ theory". I said a lot
about $BF$ theory in ["Week 36"](#week36), but here the main point is
simply that it's a topological quantum field theory, or TQFT. A TQFT is
a quantum field theory that does not presume a fixed metric, but of a
very simple sort, because it has no local degrees of freedom. I very
much like the idea that a TQFT might serve as a novel sort of "free
theory" for the purposes of studying quantum gravity.

Everything that Freidel and Krasnov do is reminscent of familiar quantum
field theory, but also very different, because their starting-point is
$BF$ theory rather than a free theory of a traditional sort. For example,
just as ordinary quantum field theory starts out with Fock space, in the
spin network approach to quantum gravity we start with a nice simple
Hilbert space of states. But this space has a basis consisting, not of
collections of 0-dimensional particles sitting in space at specified
positions, but of $1$-dimensional "spin networks" sitting in space. (For
more on spin networks, see ["Week 55"](#week55) and
["Week 110"](#week110).) And instead of using $1$-dimensional Feynman
diagrams to compute transition amplitudes, the idea is now to use
2-dimensional gadgets called "spin foams". The amplitudes for spin
foams are easy to compute in $BF$ theory, because there are a lot of
explicit formulas using the so-called "Kauffman bracket", which is an
easily computable invariant of spin networks. So then the trick is to
use this technology to compute spin foam amplitudes for quantum gravity.

Now, I shouldn't give you the wrong impression here. There are lots of
serious problems and really basic open questions in this work, and the
whole thing could turn out to be fatally flawed somehow. Nonetheless,
something seems right about it, so I find it very interesting.

Anyway, on to some other papers. I'm afraid I don't have enough energy
for detailed descriptions, because I'm busy moving into a new house, so
I'll basically just point you at them....

7) Abhay Ashtekar, Alejandro Corichi and Jose A. Zapata, "Quantum theory of geometry III: Non-commutativity of Riemannian structures", preprint available as [`gr-qc/9806041`](https://arxiv.org/abs/gr-qc/9806041).

This is the long-awaited third part of a series giving a mathematically
rigorous formalism for interpreting spin network states as "quantum
3-geometries", that is, quantum states describing the metric on
3-dimensional space together with its extrinsic curvature (as it sits
inside $4$-dimensional spacetime). Here's the abstract:

> "The basic framework for a systematic construction of a quantum theory
of Riemannian geometry was introduced recently. The quantum versions of
Riemannian structures --- such as triad and area operators --- exhibit a
non-commutativity. At first sight, this feature is surprising because it
implies that the framework does not admit a triad representation. To
better understand this property and to reconcile it with intuition, we
analyze its origin in detail. In particular, a careful study of the
underlying phase space is made and the feature is traced back to the
classical theory; there is no anomaly associated with quantization. We
also indicate why the uncertainties associated with this
non-commutativity become negligible in the semi-classical regime."

In case you're wondering, the "triad" field is more or less what
mathematicians would call a "frame field" or "soldering form" --- and
it's the same as the "B" field in $BF$ theory. It encodes the
information about the metric in Ashtekar's formulation to general
relativity.

Moving on to matters $n$-categorical, we have:

8) Andre Hirschowitz, Carlos Simpson, "Descente pour les $n$-champs" (Descent for $n$-stacks), approximately 240 pages, in French, preprint available as [`math.AG/9807049`](https://arxiv.org/abs/math.AG/9807049).

Apparently this provides a theory of "$n$-stacks", which are the
$n$-categorical generalization of sheaves. Ever since Grothendieck's
600-page letter to Quillen (see ["Week 35"](#week35)), this has
been the holy grail of $n$-category theory. Unfortunately I haven't
mustered sufficient courage to force my way through 240 pages of French,
so I don't really know the details!

For the following two $n$-category papers, exploring some themes close to
my heart, I'll just quote the abstracts:

9) Michael Batanin, "Computads for finitary monads on globular sets", preprint available at `http://www.ics.mq.edu.au/~mbatanin/papers.html`

> "This work arose as a reflection on the foundation of higher
dimensional category theory. One of the main ingredients of any proposed
definition of weak $n$-category is the shape of diagrams (pasting scheme)
we accept to be composable. In a globular approach \[due to Batanin\]
each $k$-cell has a source and target $(k-1)$-cell. In the opetopic approach
of Baez and Dolan and the multitopic approach of Hermida, Makkai and
Power each $k$-cell has a unique $(k-1)$-cell as target and a whole
$(k-1)$-dimensional pasting diagram as source. In the theory of strict
$n$-categories both source and target may be a general pasting diagram.

The globular approach being the simplest one seems too restrictive to
describe the combinatorics of higher dimensional compositions. Yet, we
argue that this is a false impression. Moreover, we prove that this
approach is a basic one from which the other type of composable diagrams
may be derived. One theorem proved here asserts that the category of
algebras of a finitary monad on the category of n-globular sets is
*equivalent* to the category of algebras of an appropriate monad on the
special category (of computads) constructed from the data of the
original monad. In the case of the monad derived from the universal
contractible operad this result may be interpreted as the equivalence of
the definitions of weak $n$-categories (in the sense of Batanin) based on
the 'globular' and general pasting diagrams. It may be also considered
as the first step toward the proof of equivalence of the different
definitions of weak $n$-category.

We also develop a general theory of computads and investigate some
properties of the category of generalized computads. It turned out, that
in a good situation this category is a topos (and even a presheaf topos
under some not very restrictive conditions, the property firstly
observed by S. Schanuel and reproved by A. Carboni and P. Johnstone for
2-computads in the sense of Street)."

10) Tom Leinster, "Structures in higher-dimensional category theory", preprint available at `http://www.dpmms.cam.ac.uk/~leinster`

> "This is an exposition of some of the constructions which have arisen
in higher-dimensional category theory. We start with a review of the
general theory of operads and multicategories. Using this we give an
account of Batanin's definition of $n$-category; we also give an informal
definition in pictures. Next we discuss Gray-categories and their place
in coherence problems. Finally, we present various constructions
relevant to the opetopic definitions of $n$-category.
> New material includes a suggestion for a definition of lax cubical
$n$-category; a characterization of small Gray-categories as the small
substructures of $2\text{-}\mathsf{Cat}$; a conjecture on coherence theorems in higher
dimensions; a construction of the category of trees and, more generally,
of $n$-pasting diagrams; and an analogue of the Baez-Dolan slicing process
in the general theory of operads."

Okay --- now for something completely different. In
["Week 122"](#week122) I said how Kreimer and Connes have teamed up
to write a paper relating Hopf algebras, renormalization, and
noncommutative geometry. Now it's out:

11) Alain Connes and Dirk Kreimer, "Hopf algebras, renormalization and noncommutative geometry", preprint available as [`hep-th/9808042`](https://arxiv.org/abs/hep-th/9808042).

Also, here's an introduction to Kreimer's work:

12) Dirk Kreimer, "How useful can knot and number theory be for loop calculations?", Talk given at the workshop _Loops and Legs in Gauge Theories_, preprint available as [`hep-th/9807125`](https://arxiv.org/abs/hep-th/9807125).

Switching over to homotopy theory and its offshoots... when I visited
Dan Christensen at Johns Hopkins this spring, he introduced me to all
the homotopy theorists there, and Jack Morava gave me a paper which
really indicates the extent to which new-fangled "quantum topology"
has interbred with good old-fashioned homotopy theory:

12) Jack Morava, "Quantum generalized cohomology", preprint available as [`math.QA/9807058`](https://arxiv.org/abs/math.QA/9807058) and `http://hopf.math.purdue.edu/`

Again, I'll just quote the abstract rather than venturing my own
summary:

> "We construct a ring structure on complex cobordism tensored with the
rationals, which is related to the usual ring structure as quantum
cohomology is related to ordinary cohomology. The resulting object
defines a generalized two- dimensional topological field theory taking
values in a category of spectra."

Finally, Morava has a student who gave me an interesting paper on
operads and moduli spaces:

13) Satyan L. Devadoss, "Tessellations of moduli spaces and the mosaic operad", preprint available as [`math.QA/9807010`](https://arxiv.org/abs/math.QA/9807010).

> "We construct a new (cyclic) operad of 'mosaics' defined by polygons
with marked diagonals. Its underlying (aspherical) spaces are the sets
$M_{0,n}(\mathbb{R})$ of real points of the moduli space of punctured Riemann
spheres, which are naturally tiled by Stasheff associahedra. We
(combinatorially) describe them as iterated blow-ups and show that their
fundamental groups form an operad with similarities to the operad of
braid groups."

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

> *Some things are so serious that one can only jest about them.*
> 
> --- Niels Bohr.