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This Week's Finds in Mathematical Physics (Week 251)

  1. May 8, 2007 #1
    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
     
  2. jcsd
  3. May 24, 2007 #2
    John Baez wrote:
    [..]

    > 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.


    Reminds me of some thoughts I had:

    Suppose you build a Universe simulated of your computer, and your
    simulation requires N bits of information to describe its state. Now
    suppose your dynamical laws posses certain symmetries. One of these
    symmetries might be a group of order p. An element of this group maps a
    state S onto another state gS, in such a way that the dynamics of gS is
    very similar to S. To be a bit more precise, if U is the time evolution
    operator, then U(S) = [g^-1 U g](S). This is saying that you can
    transform a state using the symmetry operator, time evolve it, then
    transform it back using the inverse of the symmetry operator, and get
    just the same result as time evolving the original state.

    The key is that someone inside your universe (who I like to call an
    'internal observer'), cannot distinguish between the states that are
    equivalent by the symmetry group: All his laws that he could use to do
    experiments are invariant under the group! So he can know at most N/p
    bits (ignoring the fact that he has to store them in a subsystem).

    Quantum weirdness arises when he considers a subsystem that is more or
    less isolated from its environment. (eg a stable elementary particle) In
    that case this subsystem will be like a small piece of isolated
    universe, and appear to have symmetry just like the total universe, with
    p possible states. Only when he does an 'experiment', he will couple the
    isolated subsystem to the total system, which has already made a choice
    out of p possible instances of the symmetry. This is the collapse of the
    wave function.

    Gerard
     
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