### Still a physicist! Thanks, Emmy Noether

Recently while browsing my FaceBook feed, I was tempted to
take one of the BuzzFeed quizzes that regularly pop up. Usually, I'm immune to
this kind of clickbait, not really being interested in "Which American
Idol judge are you?" or "Which Game of Thrones character are you
like?" (Though as a frequent traveler, I do often do the ones that ask,
"How many states have you visited?" or "How many of the top 150
world travel sites have you seen?") This one asked, "Are you more of
a physicist, biologist, or chemist?" This was clearly a quiz for
scientists and, though I'm a lifelong physicist (practicing for 50 years), I've
always been a "biology appreciator", collecting Wildlife Stamps as a
boy, and reading Stephen J. Gould, E. O. Wilson, Konrad Lorenz, and lots of
other as an adult. And for the past half dozen years or so, I've been holding
many conversations with multiple biologists and learning some serious bio in
the service of carrying out a deep reform on algebra-based physics to create an
IPLS (Introductory Physics for Life Scientists) class – NEXUS/Physics. I wondered whether I had
been sufficiently infected with biology memes to have gone over to the dark
side.

I needn't have worried. As expected, I came out
"Physicist". Their description of a physicist was one I liked and
that describes my favorite physicists (and I hope me too): "You’re a
thinker who loves nothing more than getting stuck into a good intellectual
challenge. You love to read, and you’ve got so much information (useless and
otherwise) stored in your brain that everyone wants to have you on their pub
quiz team. Physics suits you because it lets you spend your time contemplating
some of the smallest and biggest things in the universe, and tackle some really
huge questions while you’re at it."

But I particularly found one item in the quiz interesting:
"Select a real scientist." They offered three female scientists: Emmy
Noether, Jane Goodall, and Rosalind Franklin. Although I assume that they
matched Emmy to Physics, Jane to Biology, and Rosalind to Chemistry, I think of
both Goodall and Franklin as biologists. I have read some of both of their work
– one of Jane Goodall's books on chimpanzees (and I regularly contribute to her save the chimps foundation), and Rosalind Franklin's paper on X-ray diffraction from DNA
crystals. I've never read any of Emmy Noether's original writings, but her work
was introduced into my physics classes in junior year and had a powerful impact
on my thinking about the world and about physics. That's what I want to talk
about here.

[But first, I'm inspired to make one of my typical academic
digressions about a topic I've been thinking about: the structure of biological
research. Reading E. O. Wilson's memoir,

*Naturalist*, clarified for me a lot of what I have been seeing in my recent conversations with multiple biologists. I refer to this as "the Wilson/Watson abyss". About 1960, E. O. Wilson and J. D. Watson were both new Assistant Professors in the Harvard Biology Department. Over the next few years they engaged in a fierce battle for the soul of biology. What were the key issues for biology research for the next few decades? E. O., a field biologist rapidly becoming the world's greatest expert on ants, argued vigorously for a holistic approach: looking at whole animals, their behavior, how they interacted with others and their environments. J. D., fresh off his success in deciphering the structure of DNA and offering a molecular model for evolution, argued vigorously for a reductionist approach: studying the molecular mechanism of biology and the genome. The result was a split into two departments, and, essentially, a victory for Watson. Although there is excellent research in both areas, for the past half century, the strongest focus has been on microbiology and molecular models. Premier biology research institutes are often entirely focused on molecular and cellular biology and far more funding goes into that area. I personally think this is a problem and that the critical biological problems for the next half century are going to be that we HAVE to understand the systemic aspects of ecology – both for our interaction with the planet and even for medicine (through consideration of the human as an ecosystem by including our microbiome and the implications of social and environmental interactions on it).
Of course this digression is inspired by the choices of Jane
Goodall – a premier field biologist in the Wilson model (though she came
through anthropology as a student of Louis Leakey), and of Rosalind Franklin –
a premier biochemist in the Watson model (and her work was critical in allowing
the Watson-Crick breakthrough).

An interesting point for another post, is to note that
evolution is the bridge that spans the Wilson/Watson abyss. Evolution is not a
hypothesis or even really a theory, but rather a conclusion that grows out of a
number of fundamental principles based strongly in observation and experiment:
heredity (through DNA and its copying mechanism), variation, morphogenesis (the
building of a phenotype – the individual organism – from the genomic info), and
natural selection. (One might choose a different set, but this is one I like so
far.) The first lies firmly on the Watson side, the last on the Wilson side.
You can't make sense of evolution unless you are willing to consider both
ends.]

We now return to our main program. Why did I pick Emmy over
Jane and Rosalind, both of whose work I have actually read and I think are
immensely important?

The reason is because for me as a physicist, Emmy Noether's
result was a total game changer for me in the way I think about physics, the
epistemology of physics, and how the world works. To state her result crudely
in a way that the non-mathematician might understand, Noether's theorem says:

*If you have a system of interacting objects whose behavior in time is governed by a set of equations that have a symmetry, then you can find a conserved quantity.*

By a "symmetry", she means that you can change
something about your description that doesn't change the math. By a "conserved
quantity" she means something you can calculate that doesn't change as the
system changes through time. (Of course Noether's theorem is a mathematical
statement and there are conditions and a process to find the conserved
quantity, but that requires a lot of math to elaborate. I refer you to the Wikipedia article
on Noether's theorem for those who want the details. Warning: It requires
knowledge of Lagrangians and Hamiltonian – junior level physics.)

This is a little dense. Let's take an example or three to
see just what it means.

Suppose I have a set of interacting objects – something like
the planets in the solar system interacting via gravity, or a set of atoms and
molecules interacting via electric forces. We can describe these interactions
either using forces or energy. (These approaches can be shown to be
mathematically equivalent, though each tends to foreground different ways of
thinking about the system.) The key is that the interactions of the objects
only depend on the distances between them. This means that I can choose any
coordinate system to describe the system: I can put my reference point – the 0 of my coordinates or

*origin*– anywhere I want. Whatever origin I choose, the distance between two objects is the difference of the positions of those two objects and when you subtract their positions to get their relative distance, the position of the origin cancels.
This is a

*symmetry*. The equations that describe the motion of the system do not change depending on the position of the origin of the coordinate system. You can choose it as you like – and we typically pick an origin that will make the calculation simpler. This symmetry is called*translation invariance*. It means you can shift (translate) the origin freely without anything changing.
But what Noether's theorem shows is the symmetry doesn't
just mean we are allowed to choose the coordinate system that makes the
calculation simpler, it says there is a

*conserved quantity*and it allows you to find and calculate it.
In the case of translation invariance, Noether's conserved
quantity is

*momentum*– in most cases, the product of the mass and velocity for each object. You calculate the momentum of each object in the system, add them up at one time, and for any later time you will always get the same answer, no matter how the objects have moved, even though the motions may be amazingly complicated – and may involve billions of particles!
This is immensely important and has powerful practical
implications. One technical example is, "How can you figure out how protons move
inside a nucleus or electrons move inside an atom?" In the case of
protons, you don't actually know exactly what the force law between two protons
is (though there are lots of models), but we are pretty sure that they only
depend on the distance between them.* But we can shoot very fast protons
at a nucleus. Sometimes they will strike a proton moving in the nucleus and
knock it out. If we measure the momenta of the two outgoing protons, and since
we know the momentum of the incoming proton, we can infer the initial momentum
of the struck proton inside the nucleus using momentum conservation. We then do
a lot of these scatterings and get a probability distribution for the velocities of protons
inside the nucleus.

Since we do know the force between electrons and the
nucleus (the electric force), this technique is extremely powerful for studying
the structure of atoms and molecules. While this seems rather technical, we'll
see that there are even more important implications that providing a
measurement tool for difficult to observe quantum systems.

Two other fairly obvious symmetries in our description of
systems are:

- · Time translation invariance
- · Rotational invariance

The first, time translation, means that it doesn't matter
when you start your clock (what time you take as 0 of time). This is true for most
dynamic models in physics. Gravitational forces don't depend on time and neither
do electrical ones. Since these are the two forces that dominate everything
bigger than a nucleus, this symmetry holds for everything from atoms up to galaxies
(where there are some as yet unsolved anomalies). Emmy's theorem says that due
to the time translation symmetry there is a conserved quantity – in this case

*energy*.
The second, rotational invariance, means that it doesn't
matter in which direction you point your axes. You can take the positive x direction
as being towards the north star or towards the middle star of Orion's belt.
(You want your coordinates to be fixed in space, not rotating with the earth or
you introduce fake forces like centrifugal force and Coriolis forces.) The
conserved quantity that goes with this is

*angular momentum*, another useful principle (though more complicated to use because of more vectors).
OK. That tells us what Noether's theorem tells us – about
important conservation laws like (linear) momentum, energy, and angular
momentum. But we learn about these in introductory physics classes without
needing a sophisticated theorem. What does it add?

For me, it adds something deeply epistemological – something
fundamental about what we know in physics and how we know it. It shows that two
very different things are tightly related: how we are allowed to describe the
system at a given instant of time without changing anything (where we can choose our space and time
coordinates) – a purely static statement about what kinds of forces or energies
we have – and how the system moves in time – a dynamic statement
about how things change.

This is immensely powerful. This means that if I have
created a mathematical model of a system and I find that energy is NOT
conserved, I know that either I have made a mistake, or I have assumed
interactions that change with time. If I find that momentum is NOT conserved, I
know that I must have tied something to a fixed origin rather than to a
relative coordinate between two objects.

Now this isn't always wrong or bad. If I have a particle
moving through a vibrating fluid I might want to treat the fluid like a fixed
time dependent potential energy field. What this will mean is that the energy
of my particle will not be conserved and where the energy goes (into the fluid)
will not be correctly represented in this model.

A more common example is projectiles
or falling bodies. Since the earth is so much larger than our projectiles we
take the origin of our coordinates as a fixed point on the earth instead of
taking the force as depending (as it actually does) on the distance between the
center of the earth and the projectile. This means we won't see momentum
conserved since we have fixed the earth. Momentum transfer to it will not be
correctly represented. This might not matter depending on what we want to focus
on.

But what Noether's theorem shows us is that there are
powerful – and absolute – links between two distinct ways of thinking about
complex systems: the structure of the mathematical models we set up to describe
the evolution of systems and characteristics of how those systems evolve in
time. And that the result can be something as powerful and useful as a
conservation law blew me away. More, that we now know exactly what
characteristics of a mathematical model leads to a conservation law! There is
nothing analogous to this in biology or chemistry – except as it is inherited
from Noether's theorem in mathematical models biologists or chemists build or
as they use energy or charge conservation. But as far as I can tell they rarely
pay attention to conservation laws – even when they might do them some good.

It also showed me that when you build mathematical models
you occasionally hit the jackpot: you get out more than you thought you put in.
Extensions of Noether's theorem to other symmetries have become a powerful tool
in constructing new models of dynamics. Instead of trying to invent new force
laws, we look experimentally for conservation laws, find symmetries that can give those
conservation laws, and construct new dynamical models by putting together
variables that fit the symmetry. This is the way much of particle physics has
functioned for the past 50 years.

So that question on the quiz is probably the best selector
of the "physicist" category. Goodall and Franklin both did essential
and pivotal work in their fields; but Noether's was a core pillar for all of 20

^{th}century physics and for me, won hands down. Thanks Emmy!