Newsgroups: sci.physics,sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Symmetries, groups, and categories
Date: Thu, 13 Aug 92 18:24:11 GMT
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
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
rotations in the yz plane.
The laws of physics were thought to be symmetric under E(3). In
classical physics (to be precise, "Hamiltonian mechanics" or "Lagrangian
mechanics"), to each symmetry which commutes with time evolution
one associates a conserved quantity. This is the brilliant Noether's theorem.
To 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 conserved quantity correspoding to time translation is ENERGY.
In short, 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 (on a calm sea!). This gives rise to the
notion of Galileo 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 speed z in the x direction. The
Galilei group is a group containing E(3)+R but also
Galilei transformation in the x direction
Galilei transformation in the y direction
Galilei transformation 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.
Poincare realized that the symmetry group of Maxwell's equations was
(at least) the Poincare group. "My group!" he cried, "amazing, the
laws of nature just happen to be symmetric under a group named after
me!" This 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)x + (cosh s)t
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!)
Note that the formula for the Lorentz tranformations looks awfully
like the formula for rotations
t -> t
x -> (cos theta)x + (sin theta)y
y -> -(sin theta)y + (cos theta)x
z -> z
except that the trig functions have been replaced by hyperbolic trig
functions and that ugly - sign is gone! I won't explain why, but any
decent physicist can.
In any event, the Poincare 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 with.) Well, the answer is, they exist, but people are
rarely interested in them because they are NOT, in fact, conserved!
Huh? Well, the point is that Lorentz transformations do not commute
with time evolution. Only symmetries that commute with time evolution
give quantities that are conserved under time evolution! Any symmetry
gives a quantity of some interest, but for some reason the
quantities associated to Lorentz transformations hasn't been given a name.
In any event, as someone recently pointed out, Maxwell's equations
have no characteristic length scale. Anything you can make out of
LIGHT, in other words, 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 Poincare 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 ML^2/T = energy times time =
momentum times position. These two constants enable us to trade units
of mass for dimensions of inverse length. I.e.:
M = (ML^2/T)(T/L)1/L = hbar/c 1/L.
In nature it appears that the only massless particles are the photon,
the neutrinos (maybe) and the (hypothesized) graviton. The neutrinos,
in any event, aren't very heavy.
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 practicla relevance of conformal symmetry is limited.)
There also is a nice baby version of the Yang-Mills equations, namely
the massless phi^4 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 (the
interesting ones anyway) of these groups. The representation theory
of the Poincare group dominates relativistic physics, while the
representation theory of the Galilei group dominates nonrelativistic
physics. I encourage everyone to learn the derivation of Schrodinger's
equation straight from the representation theory of the Galilei
group! It's cool.
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 from a space consisting
of 2 circles to a space consisting of 1:
O\
\ \
\ \___
\ ___O
/ /
/ /
O/
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 Cob (for
some dimension n) to the category Hilb of Hilbert spaces (with not necessarily
unitary operators as morphisms!). That is, a TQFT would assign to each
(n-1)-dimensional manifold a Hilbert space of states representing the
states the system can take on that manifold, which represents space at a
given time. And given a cobordism between (n-1)-dimensional manifolds
we obtain a linear "time evolution" operator between the corresponding
Hilbert spaces. 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 between two manifolds.
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 another 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 QED to calculate
Delta E Delta B >= hbar 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 QED is not! This
result is due to Ashtekar, apparently.
2) Wilson loops in the SU(2) Chern-Simons theory allow us to compute
the Jones polynomial of knots and links!
3) Wilson loops in Chern-Simons theory with other gauge groups gives
rise to a wide class of link invariants which can also be obtained via
quantum groups.
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 a category to a category of vector spaces (actually
Hilbert spaces). The symmetries in topological quantum field theories
generalize the symmetries of earlier theories, thus, since earlier
theories only dealt with *group* representations, while TQFTs are
category representations.
Is quantum gravity a 4-dimensional TQFT? This is a natural question.
The answer SEEMS to be "no," although I certainly can't yet prove it's
"no".
In that case, the question is, what is quantum gravity? Louis Crane's
proposal is that it's a representation of a 2-category! What's a
2-category? Stay tuned....