edge | The modern theory of the quantum has only recently come to be understood
to be even more exquisitely geometric than Einstein's General
Relativity. How this realization unfolded over the last 40 years is a
fascinating story that has, to the best of my knowledge, never been
fully told as it is not particularly popular with some of the very
people responsible for this stunning achievement.

To set the stage, recall that fundamental physics can be divided into
two sectors with separate but maddeningly incompatible advantages. The
gravitational force has, since Einstein's theory of general relativity,
been admired for its four dimensional geometric elegance. The quantum,
on the other hand encompasses the remaining phenomena, and is lauded
instead for its unparalleled precision, and infinite dimensional
analytic depth.

The story of the geometric quantum begins at some point around
1973-1974, when our consensus picture of fundamental particle theory
stopped advancing. This stasis, known as the 'Standard Model', seemed
initially like little more than a temporary resting spot on the
relentless path towards progress in fundamental physics, and theorists
of the era wasted little time proposing new theories in the expectation
that they would be quickly confirmed by experimentalists looking for
novel phenomena. But that expected entry into the promised land of new
physics turned into a 40-year period of half-mad tribal wandering in an
arid desert, all but devoid of new phenomena.

Yet just as particle theory was failing to advance in the mid 1970s,
something amazing was quietly happening over lunch at the State
University of New York at Stony Brook. There, Nobel physics laureate CN
Yang and geometer (and soon to billionaire) Jim Simons had started an
informal seminar to understand what, if anything, modern geometry had to
do with quantum field theory. The shocking discovery that emerged from
these talks was that both geometers and quantum theorists had
independently gotten hold of different collections of insights into a
common structure that each group had independently discovered for
themselves. A Rosetta stone of sorts called the Wu-Yang dictionary was
quickly assembled by the physicists, and Isadore Singer of MIT took
these results from Stony Brook to his collaborator Michael Atiyah in
Oxford where their research with Nigel Hitchin began a geometric
renaissance in physics inspired geometry that continues to this day.

While the Stony Brook history may be less discussed by some of today's
younger mathematicians and physicists, it is not a point of contention
between the various members of the community. The more controversial
part of this story, however, is that a hoped for golden era of
theoretical physics did not emerge in the aftermath to produce a new
consensus theory of elementary particles. Instead the interaction
highlighted the strange idea that, just possibly, Quantum theory was
actually a natural and elegant self-assembling body of pure geometry
that had fallen into an abysmal state of pedagogy putting it beyond
mathematical recognition. By this reasoning, the mathematical basket
case of quantum field theory was able to cling to life and survive
numerous near death experiences in its confrontations with mathematical
rigor only because it was being underpinned by a natural infinite
dimensional geometry, which is to this day still only partially
understood.

In short, most physicists were trying and failing to quantize Einstein's
geometric theory of gravity because they were first meant to go in the
opposite and less glamorous direction of geometrizing the quantum
instead. Unfortunately for Physics, mathematicians had somewhat dropped
the ball by not sufficiently developing the geometry of infinite
dimensional systems (such as the Standard Model), which would have been
analogous to the 4-dimensional Riemannian geometry appropriated from
mathematics by Einstein.

This reversal could well be thought of as Einstein's revenge upon the
excesses of quantum triumphalism, served ice cold decades after his
death: the more researchers dreamed of becoming the Nobel winning
physicists to quantize gravity, the more they were rewarded only as
mathematicians for what some saw as the relatively remedial task of
geometrizing the quantum. The more they claimed that the 'power and
glory' of string theory (a failed piece of 1970s sub-atomic physics
which has mysteriously lingered into the 21

^{st}century) was the 'only game in town', the more it suggested that it was the string theory-based unification claims that, in the absence of testable predictions, were themselves sinking with a glug to the bottom of the sea.
What we learned from this episode was profound. Increasingly, the
structure of Quantum Field Theory appears to be a purely mathematical
input-output machine where our physical world is but one of many natural
inputs that the machine is able to unpack from initial data. In much
the way that a simple one-celled human embryo self-assembles into a
trillion celled infant of inconceivable elegance, the humble act of
putting a function (called an 'action' by physicists) on a space of
geometric waves appears to trigger a self-assembling mathematical
Rube-Goldberg process which recovers the seemingly intricate features of
the formidable quantum as it inexorably unfolds. It also appears that
the more geometric the input given to the machine, the more the
unpacking process conspires to steer clear of the pathologies which
famously afflict less grounded quantum theories. It is even conceivable
that sufficiently natural geometric input could ultimately reveal the
recent emphasis on 'quantizing gravity' as an extravagant mathematical
misadventure distracting from Einstein's dream of a unified physical
field. Like genius itself, with the right natural physical input, the
new geometric quantum now appears to many mathematicians and physicists
to be the proverbial fire that lights itself.

Yet, if the physicists of this era failed to advance the standard model,
it was only in their own terms that they went down to defeat. Just as
in an earlier era in which physicists retooled to become the first
generation of molecular biologists, their viewpoints came to dominate
much of modern geometry in the last four decades, scoring numerous
mathematical successes that will stand the tests of time.

Likewise their
quest to quantize gravity may well have backfired, but only in the most
romantic and elegant way possible by instead geometrizing the venerable
quantum as a positive externality.

But the most important lesson is that, at a minimum, Einstein's minor
dream of a world of pure geometry has largely been realized as the
result of a large group effort. All known physical phenomena can now be
recognized as fashioned from the pure, if still heterogeneous, marble of
geometry through the efforts of a new pantheon of giants. Their
achievements, while still incomplete, explain in advance of unification
that the source code of the universe is overwhelmingly likely to
determine a purely geometric operating system written in a uniform
programming language. While that leaves Einstein's greater quest for the
unifying physics unfinished, and the marble something of a
disappointing patchwork of motley colors, it suggests that the leaders
during the years of the Standard Model stasis have put this period to
good use for the benefit of those who hope to follow.

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