Authored by John Baez

I will begin with a thoroughly fictionalized account of the quest in
physics to find bigger and bigger symmetry groups. Then I will say a
bit about how that quest has led to some interesting applications of
category theory.

Once upon a time up was up, down was down, so the symmetry group of
the world was E(2), the Euclidean group in two dimensions. This is a
3-dimensional group since it is generated by:

• 
  translations in the x direction,
  
• 
  translations in the y direction, and
  
• 
  rotations in xy plane.
  

Then someone shook up the world by pointing out that it has as
symmetries the group E(3), since up and down are in fact merely
conventional concepts and one man’s up is another woman’s down. This
bigger symmetry group include rotations that mix up and down! It is
6-dimensional since in addition to the above it includes

• 
  translations in the z direction,
  
• 
  rotations in the xz plane, and
  
• 
  rotations in the yz plane.
  

The laws of physics were thus thought to be symmetric under E(3).

In classical physics – or to be precise, "Hamiltonian mechanics"
or "Lagrangian
mechanics" – each symmetry of the laws of physics
gives rise to a conserved quantity. This is a wonderful result called
Noether’s theorem.
From the translational symmetries we get conservation of momentum
in the x, y, and z directions, while from the rotational symmetries we get
angular momentum in the xy, xz, and yz planes.

The laws of physics are also invariant under time evolution itself.
Time evolution is a symmetry of spacetime given by

t → t + c
x → x
y → y
z → z

and the corresponding conserved quantity is energy.
Ultimately, this is because
energy tells you how fast things are wiggling around as time passes!

The group consisting of E(3) plus time evolution (i.e. the direct sum
E(3)+R) doesn’t have any common name, but it’s the simplest group of
spacetime symmetries. Simplest in the sense of most naive, that is.

Galileo pointed out that the laws of physics are the same in a boat
moving on a constant speed (at least on a calm sea!). This gives rise to the
notion of Galilei transformations:

t → t
x → x + vt
y → y
z → z

(and similarly for y and z) which express how to transform coordinates
into a frame of reference moving at velocity v in the x direction. The
Galilei group is a group containing E(3)+R but also these:

• 
  Galilei transformations in the x direction,
  
• 
  Galilei transformations in the y direction, and
  
• 
  Galilei transformations in the z direction.
  

Thus this group, sometimes called G (a fancy script G, please!) is
10-dimensional. This is the group of symmetries of classical mechanics.

Maxwell’s equations, alas, were not symmetric under the group G! How
could they be: they say that light moves at the same speed – let’s call
it "1" – no matter what unaccelerated frame of reference one
is in! This is inconsistent with how velocities transform under the
Galilei transformations.

Poincaré realized that the symmetry group of Maxwell’s equations was (at
least) the Poincaré group. "My group!" he cried,
"Amazing – the laws of nature just happen to be symmetric under a
group named after me!"

His group is just like the Galilei group
except that the formula for the Galilei transformations gets changed to
the following "Lorentz transformations":

t → (cosh s)t + (sinh s)x
x → (sinh s)t + (cosh s)x
y → y
z → z

(and similar ones for y and z). These express how one transforms
coordinates into a different unaccelerated coordinate system in special
relativity. Here s, the "rapidity", is related to the
ordinary velocity v by v = tanh s. Note that this means |v| is less
than 1 for real rapidities – the speed limit is 1!

Here, of course, I’m working
in units where the speed of light is 1. People only started using
such units after they realized that Lorentz transformations mixed
up space and time. Before that, people foolishly used different units
for space and time – like “meters” and “seconds” – which gave the
false impression that the speed of light was actually an interesting
physical constant. It’s a bit like how the Egyptians, when building pyramids,
used to measure horizontal distances in “palms” and vertical distances
in “cubits”. Seven palms per cubit – did they think this was an
interesting physical constant? (Thanks go to Michael Weiss for digging
this fact out of Eli Maor’s book Trigonometric Delights.)

The formula for the Lorentz tranformations looks awfully
like the formula for rotations:

t → t
x → (cos θ)x + (sin θ)y
y → -(sin θ)x + (cos θ)y
z → z

except that the trig functions have been replaced by hyperbolic trig
functions and that ugly minus sign is gone! I won’t explain why, but any
decent physicist can.

In any event, the Poincaré group is 10 dimensional like the Galilei
group. What, one may ask, are the conserved quantities corresponding
to the Lorentz transformations? This is a good riddle to test
undergraduates. If you get stuck, go here.

But Maxwell’s equations have even more symmetries than those in the
Poincaré group. The reason is that these equations have have no
characteristic length scale. In other words, anything you can make out
of light, you can make a replica of that is just a scaled-up or
scaled-down version. Thus in addition to the Poincare group Maxwell’s
equations are invariant under scale transformations, or
"dilations":

t → ct
x → cx
y → cy
z → cz

The group consisting of the Poincaré group and dilations is sometimes
called the "Weyl group".
(Beware: mathematicians also call a different
sort of group a Weyl group.) The Weyl group is 11-dimensional.

Only massless particles are invariant under the Weyl group. A mass
sets a length scale. Huh? To see this one needs to use relativity
and quantum mechanics. Mass has dimensions of M. Length has
dimension L. In relativity we have a constant, the speed of light,
with dimensions L/T, and in quantum mechanics we have a constant,
Planck’s constant, with dimensions ML²/T = energy × time =
momentum × position. These two constants enable us to trade units
of mass for dimensions of inverse length. In other words:

M = (ML2/T)(T/L)1/L = hbar/c 1/L.

In nature it appears that the only massless particles are the photon
and the (hypothesized) graviton. The neutrinos aren’t very heavy, but
these days it’s looking more and more like they do have a nonzero mass.

Now massless particles are in fact symmetric under an even bigger
group than the Lorentz group – namely, the conformal group. This
group doesn’t act as symmetries of Minkowski spacetime, but under a
(mathematically useful) completion, the "conformal compactification of
Minkowski space". This group is 15-dimensional and it’s just the
group SO(2,4), or if you prefer, the covering group SU(2,2)!
I have used this group a lot in my study of nonlinear partial
differential equations. Maxwell’s equations are invariant under the
conformal group but there aren’t too many other equations that are.
The massless Dirac equation is one – that’s for the neutrinos. There
are other linear equations for free particles of any spin. There
are not too many nonlinear equations invariant under the conformal
group but there is a very famous one – the Yang-Mills equations.
These of course are used to describe the weak and strong interactions.
(In practice various sneaky things – infrared slavery and Higgses -
conspire to cloak the conformal invariance of the fields in these
cases, so the practical relevance of conformal symmetry is limited.)
There also is a nice baby version of the Yang-Mills equations, namely
the massless phi⁴ theory. I spent a bunch of time studying these 2 equations
using conformal symmetry. Here one uses the amazing fact that in
conformally invariant theories one can define an alternate notion of
energy coming from an alternate "time evolution" symmetry contained in
the conformal group – the "Einstein energy".

Now I should point out that for each of the symmetry groups I’ve
listed above, people have worked out the "representations"
of these groups. A representation of a group is a way to think of its
elements as operators, and this is what we need to understand
symmetries in quantum physics. The representation theory
of the Poincar´ group dominates relativistic physics, while the
representation theory of the Galilei group dominates nonrelativistic
physics. I encourage everyone to learn the derivation of Schrödinger’s
equation straight from the representation theory of the Galilei
group! It’s cool. (I think it appears in the books by Mackey and Jauch
listed here.)

In any event, we can ask for still more symmetry than conformal
symmetry. We can ask for symmetry under all smooth coordinate
transformations! The first to demand this effectively was Einstein, who
got his wishes when he devised general relativity as a theory of
gravity. So gravity is the most symmetrical of field theories so far -
it’s "generally covariant", or invariant under the group
Diff(M) all diffeomorphisms of a spacetime M! One can also come up with
a (classical) theory of gravity coupled to Yang-Mills fields, or
whatever fields you like.

In these theories one may ask what sort of quantity we get associated to
the symmetries under all smooth coordinate transformations. We get
exactly the stress-energy tensor, which is a symmetric 2-tensor field
whose divergence is zero. This is what becomes of conservation of
energy and momentum in general relativity. There is generally no
conserved "total energy of the universe" or "total
momentum", since there is no canonical choice of time or space
translations in a wiggly spacetime. There is only a divergence-free
stress-energy tensor – which is actually nicer since it expresses the
locality of the notion of energy and momentum.

Okay, is that enough symmetry yet? Well, there can be a lot more! But
we’re in trouble already, because curiously it is the extremely high
degree of symmetry of general relativity that has made it hard to
quantize gravity. That seems odd – that too much symmetry could make a
problem hard! The point is that we don’t know how to use so much
symmetry. We are not used to the fact that in quantizing gravity there
is no "background metric" with which to establish the relative
positions of points. All the techniques of physics (n-point Greens
functions, Hamiltonians) are adapted to a situation in which you can
measure times and distances with respect to a fixed metric, not a metric
that can wiggle around in many ways and is one of the fields one is
trying to quantize! Moreover, the representation theory of
diffeomorphism groups is hard and still poorly understood in 4
dimensions. Plus, it’s not clear that the representations of the
diffeomorphism group are all that relevant… unlike the previous
groups, the diffeomorphism group is often regarded as a
"gauge" group, that is, a group expressing the fact that the
mathematics you have used to express the physics contains redundancies -
i.e., two spacetimes with metrics that are equivalent under some
diffeomorphism should really be regarded as the same physical system.
So to get to the real physics one should simply mod out by the
diffeomorphsim group. That’s what people often say, anyway – this is a
somewhat controversial and confused subject.

In any event, a quantum field theory that is invariant under all
diffeomorphisms of spacetime is called by physicists a "topological
quantum field theory," or TQFT. It has only been a few years since
people have been seriously studying worked-out examples of TQFTs. The
understandable examples so far have been in 2- and 3-dimensional
spacetimes, not our own lovely 4-dimensional spacetime. But these
examples are still amusing and perhaps enlightening. They also have a
lot to do with knots – but that’s another story.

What’s a topological quantum field theory, mathematically? It’s a
functor. Namely, it is a functor from the category Cob to the category
Hilbert. The category Cob is the category whose objects are
(n-1)-dimensional manifolds ("space") and whose morphisms are
n-dimensional manifolds ("spacetime") having one
(n-1)-dimensional manifold as "incoming" and another as
"outgoing" boundary. We say that the n-manifold is a
cobordism between the two (n-1)-manifolds. For example, in n = 2, the
"trinion" or "pair of pants" is a cobordism
M from a
space S consisting of two circles to a space S’
consisting of one:

We can think of this as a spacetime where two 1-dimensional circular
universes collide and form one! Or we can think of it as a tubular
Feynman diagram. Weird, huh?

In any event, a TQFT is a "functor" from the cobordism category
nCob (for
some dimension n) to the category Hilb of Hilbert spaces.
That is, a TQFT assigns
to each (n-1)-dimensional manifold S a Hilbert space of states
Z(S). This Hilbert space desscribe all the
the states our theory has when space is the manifold S.
Also, a TQFT assigns to any cobordism

M: S → S’

between such manifolds an operator

Z(S): Z(S) → Z(S’).

This operator describes time evolution: how states on S
turn into states on S’.
Since there is no such thing as waiting "a certain amount of
time" in a general TQFT, this time evolution operator only depends
on the topology of the cobordism S. That’s why it’s
called a topological quantum field theory!

People have worked out interesting examples of TQFT’s for n = 2 and n
= 3, the latter being more interesting since 3-dimensional manifolds
are quite sneaky. The most famous 3-dimensional TQFT is Chern-Simons
theory. I won’t explain this now, but simply attempt to lure the
reader into studying it by noting:

1. 
   The first example of a 3-dimensional TQFT was discovered by Gauss.
   Gauss showed that if you had two loops that were linked, and ran a
   unit current around one and did the line integral of the magnetic
   field around the other, one gets an invariant of the link called the
   linking number, which simply counts how many times one loop wraps
   around the other. This aspect of magnetism is what would now be
   called "U(1) Chern-Simons theory."

   
   Here is a beautiful related result. Say one is in the vacuum
   and has two linked loops. Let E denote the integral of the electric
   field through the surface spanned by one loop, and let B denote the
   integral of the magnetic field through the surface spanned by the
   other. One can use quantum electrodynamics to show that

   
   ΔE ΔB   ≥   (hbar/2) L
   

   
   where L is the linking number of the two loops! Thus the canonical
   commutation relations for E and B in this form are
   diffeomorphism-invariant – which is curious because quantum
   electrodynamics is not! I think this
   result is due to Ashtekar and Corichi.

   
   

2. 
   Wilson loops in the SU(2) Chern-Simons theory allow us to compute
   the Jones polynomial of knots and links, which is a generalization
   of the linking number!

   
   

3. 
   Wilson loops in Chern-Simons theory with other gauge groups gives
   rise to many other link invariants.

   
   

4. 
   Chern-Simons theory is the source of the best-understood states of
   quantum gravity in the loop representation.
   

The last part is the most exciting to me, since it says that not only
are TQFT’s mathematically interesting, they may also shed light on
real-world physics, namely, quantum gravity. It’s a long story
that’s not completely understood yet by any means!

Note that a TQFT is a "representation of a category," that is, a
functor from some category to a category of vector spaces (or
Hilbert spaces). Thus, the concept of symmetry
in topological quantum field theories
generalize that in earlier theories. Earlier
theories only dealt with group representations, while TQFTs are
category representations!

So, in modern physics categories are gradually beginning to take
over where groups left off, expanding our concept of symmetry.
To understand how this actually works, you should probably bone up a bit on
categories, and then check out
my website on Higher-dimensional algebra and
Planck-scale physics.