- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week251.html
May 5, 2007
This Week's Finds in Mathematical Physics (Week 251)
John Baez
I learned some fun stuff about the foundations of quantum mechanics
last week at Les Treilles, so I want to mention that before I forget!
I'll take a little break from the Tale of Groupoidification... though
if you've been following carefully, you may see it lurking beneath the
surface.
Lately people have been developing "foils for quantum mechanics":
theories of physics that aren't classical, but aren't ordinary
quantum theory, either. These theories can lack some of the
weird features of quantum theory... or, they may have "supra-quantum"
features, like the Popescu-Rohrlich box I mentioned last week.
The idea is not to take these theories seriously as models of our
universe - though one can always dream. Instead, it's to explore
the logical possibilities, so we can see quantum mechanics and
classical mechanics as just two examples from a larger field of
options, and better understand what's special about them.
Rob Spekkens is a young guy who used to be at the Perimeter Institute;
now he's at DAMTP in Cambridge. At Les Treilles he gave a cool talk
about a simple theory that mimics some of features of quantum mechanics:
1) Evidence for the epistemic view of quantum states: a toy theory,
Phys. Rev. A 75, 032110 (2007). Also available as quant-ph/0401052.
The idea is to see how far you get using a very simple principle,
namely: even when you know as much as you can, there's an equal
amount you don't know.
In this setup, the complete description of a physical system involves
N bits of information, but you can only know N/2 of them. When you do
an experiment to learn more information than that, the system's state
changes in a random way, so something you knew become obsolete.
The fraction "1/2" here is chosen for simplicity: it's just a toy
theory. But, it leads to some charming mathematical structures
that I'd like to understand better.
In this theory, the simplest nontrivial system is one whose state
takes two bits to describe - but you can know at most one. Two bits
of information is enough to describe four states, say states 1, 2, 3,
and 4. But, since you can only know one bit of information, you can't
pin down the system's state completely. At most you can halve the
possibilities, and know something like "the system is in state 1 or 3".
You can also be completely ignorant - meaning you only know "the
system is in state 1, 2, 3 or 4".
Since there are 3 ways to chop a 4-element set in half, there are
3 "axes of knowledge", namely
Is the system's state in {1,2} or {3,4}?
Is the system's state in {1,3} or {2,4}?
Is the system's state in {1,4} or {2,3}?
You can only answer one of these questions.
This has a cute resemblance to how you can measure the angular
momentum of a spin-1/2 particle along the x, y, or z axis, in
each case getting two choices. Spekkens has a nice picture
in his paper:
{1,2}
|
| {2,4}
| /
|/
{1,4}-----{1,2,3,4}-----{2,3}
/|
/ |
{1,3} |
|
{3,4}
This octahedron is a discrete version of the "Bloch ball" describing
mixed states of a spin-1/2 particle in honest quantum mechanics. If
you don't know about that, I should remind you:
A "pure state" of a spin-1/2 particle is a unit vector in C^2, modulo
phase. The set of these is just the Riemann sphere!
In a pure state, we know as much as we can know. In a "mixed state",
we know less. Mathematically, a mixed state of a spin-1/2 particle
is a 2x2 "density matrix" - a self-adjoint matrix with nonnegative
eigenvalues and trace 1. These form a 3-dimensional ball, the "Bloch
ball", whose boundary is the Riemann sphere.
The x, y, and z coordinates of a point in the Bloch ball are the
expected values of the three components of angular momentum for a
spin-1/2 particle in the given mixed state. The center of the Bloch
ball is the state of complete ignorance.
In honest quantum mechanics, the rotation group SO(3) acts as symmetries
of the Bloch ball. In Spekken's toy version, this symmetry group is
reduced to the 24 permutations of the set {1,2,3,4}. You can think
of these permutations as acting on a tetrahedron whose corners are the
4 states of our system. The 6 corners of the octahedron above are the
midpoints of the edges of this tetrahedron!
Since Spekkens' toy system resembles a qubit, he calls it a "toy bit".
He goes on to study systems of several toy bits - and the charming
combinatorial geometry I just described gets even more interesting.
Alas, I don't really understand it well: I feel there must be some
mathematically elegant way to describe it all, but I don't know what
it is.
Just as you can't duplicate a qubit in honest quantum mechanics - the
famous "no-cloning" theorem - it turns out you can't duplicate a toy bit.
However, Bell's theorem on nonlocality and the Kochen-Specker theorem on
contextuality don't apply to toy bits. Spekkens also lists other
similarities and differences.
All this is fascinating. It would be nice to find the mathematical
structure that underlies this toy theory, much as the category of
Hilbert spaces underlies honest quantum mechanics.
In my talk at Les Treilles, I explained how the seeming weirdness of
quantum mechanics arises from how the category of Hilbert spaces
resembles not the category of sets and functions, but a category
with "spaces" as objects and "spacetimes" as morphism. This is
good, because we're trying to unify quantum mechanics with our best
theory of spacetime, namely general relativity. In fact, I think
quantum mechanics will make more sense when it's part of a theory
of quantum gravity! To see why, try this:
2) John Baez, Quantum quandaries: a category-theoretic perspective,
talk at Les Treilles, April 24, 2007, http://math.ucr.edu/home/baez/treilles/
For more details, see my paper with the same title (see "week247").
This fun paper by Bob Coecke gives another view of categories and
quantum mechanics, coming from work on quantum information theory:
3) Bob Coecke, Kindergarten quantum mechanics, available as
quant-ph/0510032.
To dig deeper, try these:
4) Samson Abramsky and Bob Coecke, A categorical semantics of
quantum protocols, quant-ph/0402130.
5) Peter Selinger, Dagger compact closed categories and completely
positive maps, available at
http://www.mscs.dal.ca/~selinger/papers.html#dagger
Since the category-theoretic viewpoint sheds new light on the no-cloning
theorem, Bell's theorem, quantum teleportation, and the like, maybe
we can use it to classify "foils for quantum mechanics". Where would
Spekkens' theory fit into this classification? I want to know!
Another mathematically interesting talk was by Howard Barnum,
who works at Los Alamos National Laboratory. Barnum works on a general
approach to physical theories using convex sets. The idea is that
in any reasonable theory, we can form a mixture or "convex linear
combination"
px + (1-p)y
of mixed states x and y, by putting the system in state x with
probability p and state y with probability 1-p. So, mixed states
should form a "convex set".
The Bloch sphere is a great example of such a convex set. Another
example is the octahedron in Spekken's theory. Another example is
the tetrahedron that describes the mixed states of a classical
system with 4 pure states. Spekken's octahedron is a subset of
this tetrahedron, reflecting the limitations on knowledge in his
setup.
To learn about the convex set approach, try these papers:
6) Howard Barnum, Quantum information processing, operational
quantum logic, convexity, and the foundations of physics, available
as quant-ph/0304159.
7) Jonathan Barrett, Information processing in generalized
probabilistic theories, available as quant-ph/0508211.
8) Howard Barnum, Jonathan Barrett, Matthew Leifer and Alexander Wilce,
Cloning and broadcasting in generic probabilistic theories,
available as quant-ph/0611295.
Actually I've been lying slightly: these papers also allow mixtures
of states
px + qy
where p+q is less than or equal to 1. For example, if you prepare
an electron in the "up" spin state with probability p and the
"down" state with probability q, but there's also a chance that you
drop it on the floor and lose it, you might want p+q < 1.
I'm making it sound silly, but it's technically nice and maybe even
conceptually justified. Mathematically it means that instead
of a convex set of states, you have a vector space equipped with
a convex cone and a linear functional P such that the cone is
spanned by the "normalized" states: those with P(x) = 1. This
is very natural in both classical and quantum probability theory.
Quite generally, the normalized states form a convex set.
Conversely, starting from a convex set, you can create a vector
space equipped with a convex cone and a linear functional with
the above properties.
So, I was only lying slightly. In fact, a complicated
web of related formalisms have been explored; you can learn
about them from Barnum's paper.
For example, the convex cone formalism seems related to the Jordan
algebra approach described in "week162". Barnum cites a paper by
Araki that shows how to get Jordan algebras from sufficiently nice
convex cones:
9) H. Araki, On a characterization of the state space of quantum
mechanics, Commun. Math. Phys. 75 (1980), 1-24.
It's a very interesting paper but a wee bit too technical for me to
feel like summarizing here.
Some nice examples of Jordan algebras are the 2x2 self-adjoint matrices
with real, complex, quaternionic or octonionic entries. Each of these
algebras has a cone consisting of the nonnegative matrices, and the trace
gives a linear functional P. The nonnegative matrices with trace = 1
are the mixed states of a spin-1/2 particle in 3, 4, 6, and 10-dimensional
spacetime, respectively! In each case these mixed states form a convex
set: a round ball generalizing the Bloch ball. Similarly, the pure states
form a sphere generalizing the Riemann sphere.
Back in "week162" I explained how these examples are related to
special relativity and spinors in different dimensions. It's so cool
I can't resist reminding you.
Our universe seems to like complex quantum mechanics. And, the space
of 2x2 self-adjoint complex matrices - let's call it h_2(C) - is
isomorphic to 4-dimensional Minkowski spacetime! The cone of positive
matrices is isomorphic to the future lightcone. The set of pure
states of a spin-1/2 particle is the Riemann sphere CP^1, and this is
isomorphic to the "heavenly sphere": the set of light rays through a
point in Minkowski spacetime.
This whole wonderful scenario works just as well in other dimensions
if we replace the complex numbers (C) by the real numbers (R), the
quaternions (H) or the octonions (O):
h_2(R) is 3d Minkowski spacetime, and RP^1 is the heavenly sphere S^1.
h_2(C) is 4d Minkowski spacetime, and CP^1 is the heavenly sphere S^2.
h_2(H) is 6d Minkowski spacetime, and HP^1 is the heavenly sphere S^4.
h_2(O) is 10d Minkowski spacetime, and OP^1 is the heavenly sphere S^8.
So, it's all very nice - but a bit mysterious. Why did our universe
choose the complex numbers? We're told that scientists shouldn't ask
"why" questions, but that's not really true - the main thing is to do
it only to the extent that it leads to progress. But, sometimes
you just can't help it.
String theorists occasionally think about 10d physics using the octonions,
but not much. The strange thing about the octonions is that the
self-adjoint nxn octonionic matrices h_n(O) only form a Jordan algebra
when n = 1, 2, or 3. So, it seems we can only describe very small
systems in octonionic quantum mechanics! Nobody knows what this means.
People working on the foundations of quantum mechanics have also
thought about real and quaternionic quantum mechanics. h_n(R), h_n(C)
and h_n(H) are Jordan algebras for all n, so the strange limitation
afflicting the octonions doesn't affect these cases. But, I wound up
sharing a little cottage with Lucien Hardy at Les Treilles, and he
turns out to have thought about this issue. He pointed out that
something interesting happens when we try to combine two quantum
systems by tensoring them. The dimensions of h_n(C) behave quite
nicely:
dim(h_{nm}(C)) = dim(h_n(C)) dim(h_m(C))
But, for the real numbers we usually have
dim(h_{nm}(R)) > dim(h_n(R)) dim(h_m(R))
and for the quaternions we usually have
dim(h_{nm}(H)) < dim(h_n(H)) dim(h_m(H))
So, it seems that when we combine two systems in real quantum mechanics,
they sprout mysterious new degrees of freedom! More precisely, can't
get all density matrices for the combined system as convex combinations
of tensor products of density matrices for the two systems we combined.
For the quaternions the opposite effect happens: the combined system
has fewer mixed states than we'd expect.
I think this observation plays a role in this paper of his:
10) Lucien Hardy, Quantum theory from five reasonable axioms,
available as quant-ph/0101012.
Here are some more references, kindly provided by Rob Spekkens. The
pioneering quantum field theorist Stueckelberg wrote a bunch of papers
on real quantum mechanics. Spekkens recommends this one:
10) E. C. G. Stueckelberg, Quantum theory in real Hilbert space,
Helv. Phys. Acta 33, 727 (1960).
This is a modern review:
11) Jan Myrheim, Quantum mechanics on a real Hilbert space, available
quant-ph/9905037.
What I find most fascinating is the connection between real quantum
mechanics and time reversal symmetry. In ordinary complex quantum
mechanics, time reversal symmetry is sometimes described by a
conjugate-linear (indeed "antiunitary") operator T with T^2 = 1.
Such an operator is precisely a "real structure" on our complex
Hilbert space: it picks out a real Hilbert subspace of which our
complex Hilbert space is the complexification.
It's worth adding that in the physics of fermions, another possibility
occurs: an antiunitary time reversal operator with T^2 = -1. This is
precisely a "quaternionic structure" on our complex Hilbert space: it
makes it into a quaternionic Hilbert space!
For more on these ideas try:
12) Freeman J. Dyson, The threefold way: algebraic structure of
symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3
(1962), 1199-1215.
13) John Baez, Symplectic, quaternionic, fermionic,
http://math.ucr.edu/home/baez/symplectic.html
>From all this one can't help but think that complex, real, and quaternionic
quantum mechanics fit together in a unified structure, with the complex
numbers being the most important, but other two showing up naturally
in systems with time reversal symmetry.
Stephen Adler - famous for the Adler-Bell-Jackiw anomaly - spent
a long time at the Institute for Advanced Studies working on
quaternionic quantum mechanics:
14) S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields,
Oxford U. Press, Oxford, 1995.
A problem with this book is that it defines a quaternionic vector
space to be a *left* module of the quaternions, instead of a *bimodule*.
This means you can't naturally tensor two quaternionic vector spaces
and get a quaternionic vector space! Adler "solves" this problem
by noting that any left module of the quaternions becomes a right module,
and in fact a bimodule, via
xq := q*x
But, when you're working with a noncommutative ring, you really need
to think about left modules, right modules, and bimodules to
understand the theory of tensor products. And, the quaternions have
more bimodules than you might expect: for example, for any invertible
quaternion a there's a way to make H into an H-bimodule with the
obvious left action and a "twisted" right action:
qx := qx
but
xq := x f(q)
where f is any automorphism of the quaternions. Since the automorphism
group of the quaternions is SO(3), there turn out to be SO(3)'s worth of
nonisomorphic ways to make H into an H-bimodule!
For an attempt to tackle this issue, see:
14) John Baez and Toby Bartels, Functional analysis with quaternions,
available at http://math.ucr.edu/home/baez/toby/
However, it's possible we'll only see what real and quaternionic
quantum mechanics are good for when we work in the 3-category Alg(R)
mentioned in "week209", taking R to be the real numbers. Here:
there's one object, the real numbers R.
the 1-morphisms are algebras A over R.
the 2-morphisms M: A -> B are (A,B)-bimodules.
the 3-morphisms f: M -> N are (A,B)-bimodule morphisms.
This could let us treat real, complex and quaternionic quantum
mechanics as part of a single structure.
Dreams, dreams...
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
May 5, 2007
This Week's Finds in Mathematical Physics (Week 251)
John Baez
I learned some fun stuff about the foundations of quantum mechanics
last week at Les Treilles, so I want to mention that before I forget!
I'll take a little break from the Tale of Groupoidification... though
if you've been following carefully, you may see it lurking beneath the
surface.
Lately people have been developing "foils for quantum mechanics":
theories of physics that aren't classical, but aren't ordinary
quantum theory, either. These theories can lack some of the
weird features of quantum theory... or, they may have "supra-quantum"
features, like the Popescu-Rohrlich box I mentioned last week.
The idea is not to take these theories seriously as models of our
universe - though one can always dream. Instead, it's to explore
the logical possibilities, so we can see quantum mechanics and
classical mechanics as just two examples from a larger field of
options, and better understand what's special about them.
Rob Spekkens is a young guy who used to be at the Perimeter Institute;
now he's at DAMTP in Cambridge. At Les Treilles he gave a cool talk
about a simple theory that mimics some of features of quantum mechanics:
1) Evidence for the epistemic view of quantum states: a toy theory,
Phys. Rev. A 75, 032110 (2007). Also available as quant-ph/0401052.
The idea is to see how far you get using a very simple principle,
namely: even when you know as much as you can, there's an equal
amount you don't know.
In this setup, the complete description of a physical system involves
N bits of information, but you can only know N/2 of them. When you do
an experiment to learn more information than that, the system's state
changes in a random way, so something you knew become obsolete.
The fraction "1/2" here is chosen for simplicity: it's just a toy
theory. But, it leads to some charming mathematical structures
that I'd like to understand better.
In this theory, the simplest nontrivial system is one whose state
takes two bits to describe - but you can know at most one. Two bits
of information is enough to describe four states, say states 1, 2, 3,
and 4. But, since you can only know one bit of information, you can't
pin down the system's state completely. At most you can halve the
possibilities, and know something like "the system is in state 1 or 3".
You can also be completely ignorant - meaning you only know "the
system is in state 1, 2, 3 or 4".
Since there are 3 ways to chop a 4-element set in half, there are
3 "axes of knowledge", namely
Is the system's state in {1,2} or {3,4}?
Is the system's state in {1,3} or {2,4}?
Is the system's state in {1,4} or {2,3}?
You can only answer one of these questions.
This has a cute resemblance to how you can measure the angular
momentum of a spin-1/2 particle along the x, y, or z axis, in
each case getting two choices. Spekkens has a nice picture
in his paper:
{1,2}
|
| {2,4}
| /
|/
{1,4}-----{1,2,3,4}-----{2,3}
/|
/ |
{1,3} |
|
{3,4}
This octahedron is a discrete version of the "Bloch ball" describing
mixed states of a spin-1/2 particle in honest quantum mechanics. If
you don't know about that, I should remind you:
A "pure state" of a spin-1/2 particle is a unit vector in C^2, modulo
phase. The set of these is just the Riemann sphere!
In a pure state, we know as much as we can know. In a "mixed state",
we know less. Mathematically, a mixed state of a spin-1/2 particle
is a 2x2 "density matrix" - a self-adjoint matrix with nonnegative
eigenvalues and trace 1. These form a 3-dimensional ball, the "Bloch
ball", whose boundary is the Riemann sphere.
The x, y, and z coordinates of a point in the Bloch ball are the
expected values of the three components of angular momentum for a
spin-1/2 particle in the given mixed state. The center of the Bloch
ball is the state of complete ignorance.
In honest quantum mechanics, the rotation group SO(3) acts as symmetries
of the Bloch ball. In Spekken's toy version, this symmetry group is
reduced to the 24 permutations of the set {1,2,3,4}. You can think
of these permutations as acting on a tetrahedron whose corners are the
4 states of our system. The 6 corners of the octahedron above are the
midpoints of the edges of this tetrahedron!
Since Spekkens' toy system resembles a qubit, he calls it a "toy bit".
He goes on to study systems of several toy bits - and the charming
combinatorial geometry I just described gets even more interesting.
Alas, I don't really understand it well: I feel there must be some
mathematically elegant way to describe it all, but I don't know what
it is.
Just as you can't duplicate a qubit in honest quantum mechanics - the
famous "no-cloning" theorem - it turns out you can't duplicate a toy bit.
However, Bell's theorem on nonlocality and the Kochen-Specker theorem on
contextuality don't apply to toy bits. Spekkens also lists other
similarities and differences.
All this is fascinating. It would be nice to find the mathematical
structure that underlies this toy theory, much as the category of
Hilbert spaces underlies honest quantum mechanics.
In my talk at Les Treilles, I explained how the seeming weirdness of
quantum mechanics arises from how the category of Hilbert spaces
resembles not the category of sets and functions, but a category
with "spaces" as objects and "spacetimes" as morphism. This is
good, because we're trying to unify quantum mechanics with our best
theory of spacetime, namely general relativity. In fact, I think
quantum mechanics will make more sense when it's part of a theory
of quantum gravity! To see why, try this:
2) John Baez, Quantum quandaries: a category-theoretic perspective,
talk at Les Treilles, April 24, 2007, http://math.ucr.edu/home/baez/treilles/
For more details, see my paper with the same title (see "week247").
This fun paper by Bob Coecke gives another view of categories and
quantum mechanics, coming from work on quantum information theory:
3) Bob Coecke, Kindergarten quantum mechanics, available as
quant-ph/0510032.
To dig deeper, try these:
4) Samson Abramsky and Bob Coecke, A categorical semantics of
quantum protocols, quant-ph/0402130.
5) Peter Selinger, Dagger compact closed categories and completely
positive maps, available at
http://www.mscs.dal.ca/~selinger/papers.html#dagger
Since the category-theoretic viewpoint sheds new light on the no-cloning
theorem, Bell's theorem, quantum teleportation, and the like, maybe
we can use it to classify "foils for quantum mechanics". Where would
Spekkens' theory fit into this classification? I want to know!
Another mathematically interesting talk was by Howard Barnum,
who works at Los Alamos National Laboratory. Barnum works on a general
approach to physical theories using convex sets. The idea is that
in any reasonable theory, we can form a mixture or "convex linear
combination"
px + (1-p)y
of mixed states x and y, by putting the system in state x with
probability p and state y with probability 1-p. So, mixed states
should form a "convex set".
The Bloch sphere is a great example of such a convex set. Another
example is the octahedron in Spekken's theory. Another example is
the tetrahedron that describes the mixed states of a classical
system with 4 pure states. Spekken's octahedron is a subset of
this tetrahedron, reflecting the limitations on knowledge in his
setup.
To learn about the convex set approach, try these papers:
6) Howard Barnum, Quantum information processing, operational
quantum logic, convexity, and the foundations of physics, available
as quant-ph/0304159.
7) Jonathan Barrett, Information processing in generalized
probabilistic theories, available as quant-ph/0508211.
8) Howard Barnum, Jonathan Barrett, Matthew Leifer and Alexander Wilce,
Cloning and broadcasting in generic probabilistic theories,
available as quant-ph/0611295.
Actually I've been lying slightly: these papers also allow mixtures
of states
px + qy
where p+q is less than or equal to 1. For example, if you prepare
an electron in the "up" spin state with probability p and the
"down" state with probability q, but there's also a chance that you
drop it on the floor and lose it, you might want p+q < 1.
I'm making it sound silly, but it's technically nice and maybe even
conceptually justified. Mathematically it means that instead
of a convex set of states, you have a vector space equipped with
a convex cone and a linear functional P such that the cone is
spanned by the "normalized" states: those with P(x) = 1. This
is very natural in both classical and quantum probability theory.
Quite generally, the normalized states form a convex set.
Conversely, starting from a convex set, you can create a vector
space equipped with a convex cone and a linear functional with
the above properties.
So, I was only lying slightly. In fact, a complicated
web of related formalisms have been explored; you can learn
about them from Barnum's paper.
For example, the convex cone formalism seems related to the Jordan
algebra approach described in "week162". Barnum cites a paper by
Araki that shows how to get Jordan algebras from sufficiently nice
convex cones:
9) H. Araki, On a characterization of the state space of quantum
mechanics, Commun. Math. Phys. 75 (1980), 1-24.
It's a very interesting paper but a wee bit too technical for me to
feel like summarizing here.
Some nice examples of Jordan algebras are the 2x2 self-adjoint matrices
with real, complex, quaternionic or octonionic entries. Each of these
algebras has a cone consisting of the nonnegative matrices, and the trace
gives a linear functional P. The nonnegative matrices with trace = 1
are the mixed states of a spin-1/2 particle in 3, 4, 6, and 10-dimensional
spacetime, respectively! In each case these mixed states form a convex
set: a round ball generalizing the Bloch ball. Similarly, the pure states
form a sphere generalizing the Riemann sphere.
Back in "week162" I explained how these examples are related to
special relativity and spinors in different dimensions. It's so cool
I can't resist reminding you.
Our universe seems to like complex quantum mechanics. And, the space
of 2x2 self-adjoint complex matrices - let's call it h_2(C) - is
isomorphic to 4-dimensional Minkowski spacetime! The cone of positive
matrices is isomorphic to the future lightcone. The set of pure
states of a spin-1/2 particle is the Riemann sphere CP^1, and this is
isomorphic to the "heavenly sphere": the set of light rays through a
point in Minkowski spacetime.
This whole wonderful scenario works just as well in other dimensions
if we replace the complex numbers (C) by the real numbers (R), the
quaternions (H) or the octonions (O):
h_2(R) is 3d Minkowski spacetime, and RP^1 is the heavenly sphere S^1.
h_2(C) is 4d Minkowski spacetime, and CP^1 is the heavenly sphere S^2.
h_2(H) is 6d Minkowski spacetime, and HP^1 is the heavenly sphere S^4.
h_2(O) is 10d Minkowski spacetime, and OP^1 is the heavenly sphere S^8.
So, it's all very nice - but a bit mysterious. Why did our universe
choose the complex numbers? We're told that scientists shouldn't ask
"why" questions, but that's not really true - the main thing is to do
it only to the extent that it leads to progress. But, sometimes
you just can't help it.
String theorists occasionally think about 10d physics using the octonions,
but not much. The strange thing about the octonions is that the
self-adjoint nxn octonionic matrices h_n(O) only form a Jordan algebra
when n = 1, 2, or 3. So, it seems we can only describe very small
systems in octonionic quantum mechanics! Nobody knows what this means.
People working on the foundations of quantum mechanics have also
thought about real and quaternionic quantum mechanics. h_n(R), h_n(C)
and h_n(H) are Jordan algebras for all n, so the strange limitation
afflicting the octonions doesn't affect these cases. But, I wound up
sharing a little cottage with Lucien Hardy at Les Treilles, and he
turns out to have thought about this issue. He pointed out that
something interesting happens when we try to combine two quantum
systems by tensoring them. The dimensions of h_n(C) behave quite
nicely:
dim(h_{nm}(C)) = dim(h_n(C)) dim(h_m(C))
But, for the real numbers we usually have
dim(h_{nm}(R)) > dim(h_n(R)) dim(h_m(R))
and for the quaternions we usually have
dim(h_{nm}(H)) < dim(h_n(H)) dim(h_m(H))
So, it seems that when we combine two systems in real quantum mechanics,
they sprout mysterious new degrees of freedom! More precisely, can't
get all density matrices for the combined system as convex combinations
of tensor products of density matrices for the two systems we combined.
For the quaternions the opposite effect happens: the combined system
has fewer mixed states than we'd expect.
I think this observation plays a role in this paper of his:
10) Lucien Hardy, Quantum theory from five reasonable axioms,
available as quant-ph/0101012.
Here are some more references, kindly provided by Rob Spekkens. The
pioneering quantum field theorist Stueckelberg wrote a bunch of papers
on real quantum mechanics. Spekkens recommends this one:
10) E. C. G. Stueckelberg, Quantum theory in real Hilbert space,
Helv. Phys. Acta 33, 727 (1960).
This is a modern review:
11) Jan Myrheim, Quantum mechanics on a real Hilbert space, available
quant-ph/9905037.
What I find most fascinating is the connection between real quantum
mechanics and time reversal symmetry. In ordinary complex quantum
mechanics, time reversal symmetry is sometimes described by a
conjugate-linear (indeed "antiunitary") operator T with T^2 = 1.
Such an operator is precisely a "real structure" on our complex
Hilbert space: it picks out a real Hilbert subspace of which our
complex Hilbert space is the complexification.
It's worth adding that in the physics of fermions, another possibility
occurs: an antiunitary time reversal operator with T^2 = -1. This is
precisely a "quaternionic structure" on our complex Hilbert space: it
makes it into a quaternionic Hilbert space!
For more on these ideas try:
12) Freeman J. Dyson, The threefold way: algebraic structure of
symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3
(1962), 1199-1215.
13) John Baez, Symplectic, quaternionic, fermionic,
http://math.ucr.edu/home/baez/symplectic.html
>From all this one can't help but think that complex, real, and quaternionic
quantum mechanics fit together in a unified structure, with the complex
numbers being the most important, but other two showing up naturally
in systems with time reversal symmetry.
Stephen Adler - famous for the Adler-Bell-Jackiw anomaly - spent
a long time at the Institute for Advanced Studies working on
quaternionic quantum mechanics:
14) S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields,
Oxford U. Press, Oxford, 1995.
A problem with this book is that it defines a quaternionic vector
space to be a *left* module of the quaternions, instead of a *bimodule*.
This means you can't naturally tensor two quaternionic vector spaces
and get a quaternionic vector space! Adler "solves" this problem
by noting that any left module of the quaternions becomes a right module,
and in fact a bimodule, via
xq := q*x
But, when you're working with a noncommutative ring, you really need
to think about left modules, right modules, and bimodules to
understand the theory of tensor products. And, the quaternions have
more bimodules than you might expect: for example, for any invertible
quaternion a there's a way to make H into an H-bimodule with the
obvious left action and a "twisted" right action:
qx := qx
but
xq := x f(q)
where f is any automorphism of the quaternions. Since the automorphism
group of the quaternions is SO(3), there turn out to be SO(3)'s worth of
nonisomorphic ways to make H into an H-bimodule!
For an attempt to tackle this issue, see:
14) John Baez and Toby Bartels, Functional analysis with quaternions,
available at http://math.ucr.edu/home/baez/toby/
However, it's possible we'll only see what real and quaternionic
quantum mechanics are good for when we work in the 3-category Alg(R)
mentioned in "week209", taking R to be the real numbers. Here:
there's one object, the real numbers R.
the 1-morphisms are algebras A over R.
the 2-morphisms M: A -> B are (A,B)-bimodules.
the 3-morphisms f: M -> N are (A,B)-bimodule morphisms.
This could let us treat real, complex and quaternionic quantum
mechanics as part of a single structure.
Dreams, dreams...
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html