The Physics of Wall Street: a brief History of Predicting the Unpredictable

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you strike north into russia. (now, at least, you get to walk forward.) 
You head all the way to the Arctic circle, without turning your body. 
When you reach the north Pole, you see that new York is directly in 
front of you, far to the south. You keep moving forward into north-
ern canada, and then work your way down the Hudson until you 
return to new York. But this time, when you return to the place where 
you began, you’re facing a different direction: due south! What’s gone 
wrong? You didn’t turn your body at any point of the journey, and 
yet at the end you’re facing the opposite way from the direction you 
started facing — and from the direction you were facing at the end of 
your first journey.
the reason you end up facing in a different direction after your sec-
ond round-the-world trip is that the globe is a curved surface (see fig-
ure 5). A city block, meanwhile, is flat. (At least to a first approxima-
tion — real city blocks lie on the surface of the earth, which of course 
is curved. But you don’t see the effects of this curvature over short 
distances.) If you imagine an ant trying to perform the same experi-
ment on a kitchen table, you would find that, no matter what route the 
ant took, it would always end up facing in the same direction. this is 
what a mathematician means when he says that a surface, or a shape, 
is flat: it exhibits “path independence of parallel transport” (parallel 
transport because the goal is to try to keep your body parallel to its 
last direction at all times). for curved surfaces, meanwhile, the direc-
tion an arrow points at the end of a journey is “path dependent.” on a 
curved surface, different routes can lead to different results.
the connection between path dependence and curvature may be 
unfamiliar to non-mathematicians. But the basic idea of path depen-
dence isn’t. It is easy to find examples in day-to-day life of things that 
are path dependent, and things that are path independent. If you go 
to the store and buy groceries, the amount of milk you have when you 
get back home is path independent. the amount of milk isn’t going to 
change if you take a different route home from the store. the amount 
of gasoline in your tank, however, is path dependent. If you take the 
direct route home, you will usually have more gasoline left when you 
arrive than if you had taken the scenic route. Path dependence of par-

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allel transport is just a special case of the more general fact that some-
times, things depend not just on where you start and where you end 
up, but also on the road you take to get there.
einstein’s theory of general relativity makes essential use of the 
fact that space and time are curved in the sense that parallel trans-
port is path dependent. But Weyl thought that einstein hadn’t gone 
far enough. In general relativity, if you begin with an arrow at one 
place and then move it around a path that brings it back to the start-
ing point, it might face a different direction. But it will always have 
the same length. Weyl thought this was an arbitrary distinction that 
couldn’t have physical meaning, and so he came up with an alterna-
tive theory in which length, too, was path dependent, so that if you 
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figure 5: If you move an arrow along a path on a curved surface, being careful to keep 
the arrow pointing in the same direction at all times, the direction that the arrow points 
at the end of the path will depend on the path taken. Mathematicians call this property 
of curved surfaces “path dependence of parallel transport.” In this figure, there are two 
paths around a sphere. the first path takes the arrow from point A around the equa-
tor and back to point A. At the end of the trip, the arrow faces the same direction as 
when it began. the second trip again starts at point A and travels around the equator, 
but only halfway. on the other side of the sphere, the path moves up over the north 
Pole and returns to point A that way. At the end of this trip, the arrow is pointing in 
the opposite direction from when the trip began. Weyl observed that it was possible 
to construct physical theories in which not only was direction path dependent, but so 
was the length of an arrow. the physical world doesn’t actually work that way, but in 
the years since Weyl first came up with his theory — which he called a gauge theory
— many physicists and mathematicians have adapted the mathematics he invented to 
other problems, with much more success.

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A New Manhattan Project 
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moved a ruler around two different closed paths, it would have differ-
ent lengths when it returned to the starting point, depending on the 
path it took.
Weyl called his new theory a gauge theory. It was the first time the 
term had been used, and it was based on the idea that there was no 
universal, once-and-for-all way to “gauge,” or measure, the length of 
a ruler. Suppose you and your neighbor are both about to leave your 
driveway in the morning on the way to work. Imagine you drive iden-
tical cars, and you both work at the same location. What would you 
say if someone stopped you and asked which car would have more 
gasoline in the tank when you both got to work, yours or your neigh-
bor’s? You might glance at your gas gauge and see that you have a full 
tank, and then ask your neighbor how much gas he has. But this isn’t 
enough information to answer the question. the answer will depend 
on the paths you and your neighbor take to work: you might take a 
direct route, while the neighbor takes the scenic route. Your neighbor 
might take a highway, while you stick to city streets. Whatever the case 
may be, how much gasoline each of you has left at the end of your 
journeys will depend on the paths you take to work. comparing some 
path-dependent quantities does not yield a straightforward answer.
this was the sense in which, in Weyl’s theory, there was no uni-
versal way to measure a ruler, since there was no path-independent 
way to compare two rulers in different locations. But Weyl realized 
that this wasn’t necessarily a problem: if you wanted to compare the 
length of a ruler in chicago to the length of a ruler in copenhagen, or 
on Mars, all you needed to do was figure out a way to bring the rulers 
to the same place so you could hold them up next to each other. this 
wouldn’t be path independent, but that was oK, as long as you could 
figure out how the change in length would depend on the path you 
took. In other words, Weyl realized that what really mattered to his 
theory was identifying a mathematical standard by which compari-
sons of length could be made — a way of “connecting” different points 
in a principled way, so that you could compare rulers, even though 
length was path dependent. Mathematically, what Weyl accomplished 
was to show how to compare two otherwise incomparable quantities, 

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by moving them to a common location where their properties (in this 
case, their lengths) could be compared directly.
Weyl’s theory wasn’t a success. einstein quickly pointed out that it 
was inconsistent with some well-known experimental results, and it 
was soon relegated to the dustbins of scientific history. But Weyl’s basic 
idea about gauge — that to determine if two quantities are equal in a 
physical theory, you need a standard of comparison that accounts for 
possible path dependence — was destined to be far more important 
than the theory that led to it. Gauge theory was resurrected in the 
1950s by a pair of young researchers at Brookhaven national Labora-
tory named c. n. Yang and robert Mills. Yang and Mills took Weyl’s 
theory one step further: If it was possible to construct a theory in 
which length was path dependent, was it possible to construct theo-
ries in which still other quantities were path dependent? the answer, 
they realized, was yes. they went on to develop a general framework 
for much more complicated gauge theories than the one Weyl had 
imagined.
these theories, now known as Yang-Mills theories, spawned what is 
sometimes called the gauge revolution. Beginning in the 1961, funda-
mental physics was rewritten in terms of gauge theory — a process that 
only accelerated when Yang, in collaboration with Jim Simons of re-
naissance, realized a deep connection between Yang-Mills gauge theo-
ries and modern geometry later that decade. Gauge theories proved