an apology, and some category theory

I apologize for being so absent from the blogosphere. My computer died — it moved from epileptic through comatose to completely blind — in January, and since then I've been spending maybe ten minutes a day on the internet. I'm not taking classes, so my math exposure consists mostly of reading (I'm currently enjoying Hartry Fields' Science Without Numbers); instead, my time is consumed by an almost complete dance project (our main performance is Friday), running the hiring committee for next year's staff in my dorm, and getting into (and soon visiting) graduate schools around the country.

I did find the time recently to catch up on This Week's Finds, and was particularly interested in the discussion in Week 244. In it, Baez discusses an article by Tom Leinster, called The Euler characteristic of a category, which I skimmed and need to go read more thoroughly.

I certainly haven't grokked most of the material, but I did want to make two comments. First, a small one: Baez wonders at the coincidence that the character 1 has Euler characteristic 1, whereas the character 0 has Euler characteristic 0. I suggest that this has, in fact, real meaning: 1 presumably comes from a has mark, a line, a dot; any natural character for 1 ought to be readily contractible to a point. But 0, on the other hand, should represent exactly an absence: it must be filled with a point (equivalently a disk) to become a disk (equivalently a point). And this exactly defined its Euler characteristic to be 0. Somehow, the ancients, or perhaps the forces of calligraphic natural selection, understand enough naive topology to build Euler characteristics into the system.

My second comment is a little more substantive. In defining the Euler characteristic of a (for now, finite) category, we're trying to generalize

1. Euler characteristic of topological space. If our category is a groupoid (all morphisms are iso), then equivalently our category is the homotopy groupoid of the space formed by including a point for each object in our category, an edge for each morphism, a disk for each commuting triangle, a 3-ball for each commuting tetrahedron, etc.
   
2. Euler characteristic, as defined by Rota, of a poset. This motivates the idea of defining on our category such objects as a ζ function, a μ function, convolution of functions, etc., with which the Euler characteristic is defined.
   

Which is all well and good. But then Leinster et al decide on what seems to me to be a bizarre idea: a category with two objects and one morphism (between them), i.e.
o---->----o
should also (along with o--<--->--o) become an interval, and so should have Euler characteristic 1.

See, in a poset, this was exactly the right thing to do. But I claim that we're then wrong to think of a poset as a category, at least in the normal way. In a category, unlike a poset, you can have morphisms both directions. An isomorphism is really two morphisms, glued together by a commuting disk. There's a difference in a category between that and a unidirectional morphism. On the other hand, in a poset, there's no chance at morphisms going both ways. Instead, a poset is really just a space, but one with a (transitive) sense of "up" and "down".

It seems really bizarre to throw away so much information from the category by ignoring the direction of morphisms when taking Euler characteristics. Instead, why not follow Baez's suggestion: "If we were willing to make up new kinds of numbers, we could make up a new number for the size of this category."

I'm willing. What properties should our new number have? I thought at first that an isomorphism was the sum of two inverse morphisms. If this were so, then we'd want to assign some value X to a "forward" morphism, and some value X^{-1} to a "backwards" one. (Of course, forward and backward are local properties, so these values might want to live locally, say as a field rather than a function, but we are also going to want to sum them.) Then, I thought, we should want X + X^{-1} to be an isomorphism, which we know has Euler characteristic -1 (either object has characteristic 1). We can, of course, solve this: X+X^{-1}=-1 means that X = -(1/2)\pm(i\sqrt{3}/2). And, I said to myself, this is cool: the "Op" operation that reverses categories is just like complex conjugation!

Alas, however, this method fails a horrible death. Because an isomorphism is not the sum of two morphisms, but the sum of two morphisms and a commuting disk. And if we decide that disks should have characteristic 1, then each morphism really should have characteristic -1: we're solving X+X^{-1}+1=-1. And a commuting disk, at least the kind in an isomorphism, really really ought to have characteristic 1, because it can go in any direction. (I am interested by those pairs of morphisms for which, e.g., fg=1 but gf is not 1. I've never totally understood how to treat them and their commuting diagrams.)

So, yes, what should our method be? Certainly I want to assign weights to morphisms that are sensitive to their direction. Perhaps I should even attend to composition and the like: in a one-object category, a nilpotent morphism should be different from an isomorphism? So I want to build some sort of field on my category, and then integrate it against some sort of measure? I definitely want Op to act like "complex conjugation". Perhaps we'll end up with choices — certainly we'll write down natural (local) conditions that the values of our field should satisfy, but their values will depend on what field algebraic space (I'm using "field" in the physicists' sense) we solve the equations in — but then what should happen is that we get some sort of "free" object that can map to, for instance, the reals. And so the Euler characteristic, as described by Leinster, is one (and possible the most natural) representation. But in the same way that knot theorists find more powerful tools when they look for representations that attend to crossings, we should look for representations that attend to the direction of our morphisms.

Poisson brackets and scalar fields

Today's* ODEs class was rather amazing. In standard Yasha style, the topics darted around (he once explained that "It is impossible to be lost in my class. Because topic changes every two minutes, so if you are lost, you won't be lost."): a taste of KM theory, a dash of geodesics on ellipsoids, and a little infinite-dimensional Hamiltonian systems.

It is this last discussion that I found rather mind-blowing. We did not say anything new; rather, we started defining what will eventually give us the quantum behavior of fields, from an entirely classical viewpoint.

Consider one PDE \d u(x,t)/\d t = F(u), where u is a function and F is some map from functions to functions. Let's say, for example, that we're interested in complex functions on the circle: u:S\to\C is what the physicists would call a (complex) scalar field. Let V be the set of all scalar fields; I will not be precise what conditions I want (presumably some smoothness conditions, say complex-analytic, and perhaps some convergence conditions, for instance that the square integral converges. I will call maps from V \to \R "functionals" and those from V \to V "operators"; V is a (complex) vector space, so it makes sense to talk about (real-, complex-, anti-) linear functionals and operators. For instance, \d/\dx is a linear operator; (1/2\pi) \int_0^{2\pi} -- g(x) dx is a linear functional. (I will from now on write \int for \int_0^{2\pi}; consistent with half of mathematics, the volume of the circle is 2\pi.) Rather than thinking of my problem as a PDE, I should think of it as an ODE in this infinite-dimensional space.

Imposing analyticity, etc., conditions on V curtails the freedom of functions: the value of a function at a point largely determines the value at nearby points. We ought to perform a change-of-basis so that we can better tell functions apart: let's assume, for instance, that each function has a Fourier expansion

u(x) = u_0 + \sum p_k e^{-ikx} + q_k e^{ikx}

(Given, of course, by

u_0 = (1/2\pi) \int u(x) dx; p_k = (1/2\pi) \int u(x) e^{ikx} dx; q_k = (1/2\pi) \int u(x) e^{-ikx} dx.)

((Sometime soon I will figure out what I believe to be the proper class of functions with which to do physics. One might hope for (complex) holomorphic, or for complex analytic, or for real C^\infty, or real analytic, or maybe just real integrable. But the Fourier transformation is vital to modern physics, and none of these is particularly the class of things with natural Fourier transforms, because I often end up with \delta functions. I ought to study the Fourier transform a bit more, because I don't understand something basic: it seems that the \delta functions supply a continuum of basis states, whereas the fourier modes are countable. But every delta function can fourier-transform, and modulo convergence this should be a change-of-basis. Perhaps it has something to do with the fact that the Fourier transform doesn't care about individual values of functions, just their integrals. We really ought to pick a class of "functions" so that Fourier really is a legitimate "change-of-basis" in the appropriate infinite-dimensional sense.))

Then our manifold V is, more or less, an odd-dimensional manifold. We cannot hope to put a symplectic structure on it. On the other hand, the ps and qs so naturally line up that we really want to write down \omega = \sum a_k dp_k \wedge dq_k, and we can do so on the even-dimensional subspace \{u_0 = 0\}. (The coefficients a_k have yet to be determined; \omega is a symplectic form for any choice of a_k, and we should choose judiciously to match other physics. Throughout this entry, I eschew the Einstein summation conventions.)

Well, almost. This \omega is almost certainly not going to converge if you feed it most pairs of functions, and I think that to restrict our functions to those that have Fourier expansions that converge rapidly enough is premature, especially since we have yet to determine the a_k. Rather, Yasha suggests, we should discuss the Poisson bracket, which is what really controls physics.

How so? you ask. And what, fundamentally, is a Poisson bracket? Consider our original Poisson bracket \{F,G\} = \omega(X_G,X_F), where X_F is defined by \omega(X_F,-) = dF(-). Then unwrapping definitions gives that \{F,G\} = dF(X_G) = X_G[F], where on the RHS I'm treating X_G as a differential operator. Our Poisson bracket knows exactly the information needed: given a Hamiltonian H, the flow is generated by the vector field X_H = \{-,H\}.

In general, a Poisson bracket is any bracket \{,\} satisfying
(a) Bilinearity (for now \R-linear; perhaps we will ask for \C-linear soon?)
(b) Anti-symmetry and Jacobi (i.e. \{,\} is a Lie bracket)
(c) It behaves as a (first-order) differential operator in each variable: \{FG,H\} = F\{G,H\} + \{F,H\}G.

Condition (c) guarantees that a Poisson bracket never cares about overall constants in F, etc.: \{F,G\} depends only on dF and dG. Symplectic forms, being non-degenerate, are dual to antisymmetric _0^2-tensors (two raised indices), and Poisson brackets are exactly of this form. But Poisson brackets need not be nondegenerate to give us physics. Indeed, the Poisson bracket as described is plenty to define a vector field to be the "derivative" of each function. (Not quite the gradient, but an antisymmetric version of one.) And this is what's needed.

So, to describe the physics of our system, it suffices to pick an appropriate Poisson bracket. Returning now to the system we were originally interested in, of scalar fields on the circle, our Hamiltonians should be functionals of u(x). Let's assume that every functional (at least all the physical ones) can be written in Taylor expansion as polynomials in p_k and q_k. Then to define the Poisson bracket, and continuing to ignore issues of convergence, it suffices to define the brackets between our various coefficients p_k, q_l, and u_0. Given \omega = \sum a_k p_k \wedge q_k (on our subspace of u_0=0), we get \{p_k,q_k\} = a_k^{-1}, and all other brackets are 0.

What would be nice now is to find some physical reason to pick particular a_k, or to even motivate a Poisson bracket of this form at all. Perhaps you vaguely recall a physicists telling you that the Hamiltonian density functional for free scalar fields should look something like h(u(x)) = 1/2 (u^2 + m^2 u'^2), where u'(x) = du/dx. Then the Hamiltonian would be the integral of this: H(u) = \int_0^{2\pi} h(u(x)) dx = u_0^2/2 + \sum_{k=1}^\infty (1-m^2k^2) p_k q_k. With the Poisson bracket given, we can solve this explicitly: p_k(t) = p_k(t=0) e^{(m^2k^2-1) a_k t} and q_k(t) = q_k(t=0) e^{(1-m^2k^2) a_k t}. But this doesn't particularly match physical expectations --- why, for m^2 small and positive and a_k positive, should we expect fields to tend to develop lots of p_k oscillation for large k and lots of q_k oscillation for small k, and for the rest of the oscillation to die?

I'll come back to this kind of physical justification, and perhaps argue my way to a Hamiltonian and a Poisson bracket from the other direction, later. First, I want to explain why Yasha likes this set-up, via mathematical, rather than physical, elegance.

Many interesting Hamiltonians, Yasha observes, are of forms similar to the one I considered in the previous example: the functional H(u) is defined as the integral of some density functional h(x,u(x),u'(x),...). Let's consider this case. Yasha, in fact, uses a particularly restricted case: isotropy is easy (no x dependence), but Yasha also says no derivatives: h = h(u). Then H(u) = (1/2\pi) \int h(u(x)) dx

Then what happens? Well, \dot{q_k} = \{H,q_k\} = a_k \dH/\dp_k = (1/2\pi) \int a_k \dh/\dp_k, and \dot{p_k} = (-1/2\pi) \int a_k \dh/\dq_k/. Plugging these into \dot{u}(x) = \sum \dot{p_k} e^{ikx} + \dot{q_k} e^{-ikx} gives

\dot{u}(x) = (1/2\pi) \int \sum [-a_k \dh/\dq_k e^{ikx} + a_k \dh/\dp_k e^{-ikx}].

Can we recognize this as anything simpler? Recall the chain rule: \dh/\du = \sum (\dh/\dq_k \dq_k/\du + \dh/\dp_k \dp_k/\du). Then, since q_k = (1/2\pi) \int e^{ikx} u(x) dx, we see that \dq_k/\du(x) = (1/2\pi) e^{ikx}. So we see that

h'(u)(x) = \dh/\du(x) = (1/2\pi) \sum [ \dh/\dq_k e^{ikx} + \dh/\dp_k e^{-ikx} ]

This isn't quite what we want, because we have opposite signs on the two a_k. But if we differentiate h'(u) with respect to x, we get

\d/\dx [h'(u)(x)] = (1/2\pi) \sum [ ik \dh/\dq_k e^{ikx} + -ik \dh/\dp_k e^{-ikx} ]

And so, if we pick a_k = k, we see that

\dot{u}(x) = i \d/\dx [h'(u)]

I've been a bit sloppy with this calculation, and you may have trouble following the factors of 2\pi. In particular, I write \int, when I probably should have written \int_{y=0}^{2\pi} dy. But then I would have had to keep track of what's a function of what. Anyway, somewhere in here there's a delta-function, and the formula is correct.

Yasha doesn't do this calculation, preferring one a bit more general and rigorous, which I will sketch:

He observes that since we're a vector space, we can identify points and tangent vectors. Then what are the cotangent vectors? Our vector space of functions has a natural dot product: <u,v> = (1/2\pi) \int_0^{2\pi} u(x) v(x) dx; where now I'm thinking of u and v not as points but as tangent vectors at 0. So to each vector u(x) I can identify the linear functional (1/2\pi) \int u(x) -- dx, which Yasha calls \delta u. Then, knowing that he's chosen {p_k,q_k} = k, Yasha guesses a Poisson bracket:

P(\delta u,\delta v) = (1/2\pi i) \int u v' dx

Recalling the expressions for p_k and q_k as functions of u, we can recognize dp_k = \delta[e^{-ikx}] and dq_k = \delta[e^{ikx}]. Then we can check whether P is the proper Poisson bracket (really the tensor form, eating the derivatives of the functions we would feed into \{,\}) by evaluating it on our ps and qs:

\{p_k,q_k\} = P(e^{-ikx},e^{ikx}) = (1/2\pi i) int e^{-ikx} ik e^{ikx} dx = k; all other brackets are 0, and P is antisymmetric by integration by parts, so must be correct.

Then if F and G are two (real- or complex-valued) functionals on our space of scalar fields, what is their bracket \{F,G\}(u)? Let's say that F and G have the nice form F(u) = (1/2\pi) int f(u(x)) dx (and G is similar). Then \{F,G\} = P(dF,dG) ... what is dF? Well, it's the linear part of F. If \epsilon v is a small change in u, then

(1/\epsilon) (F(u + \elpsilon v) - F(u)) = (1/2\pi) \int f'(u(x)) v(x) dx

where f'(u) is the functional derivative of f with respect to u. So we can recognize dF(u) as \delta[f'(u)], and conclude that

\{F,G\}(u) = (1/2\pi i) \int (f'(u)) (g'(u))' dx

where on the second multiplicand, the inside prime is w.r.t. u, and the outside is w.r.t. x.

Then \dot(u)? Well, for any function(al) F(u), we have \dot{F} = \{H,F\}. u is not a functional, of course, but it is a vector of functionals: u(y) = (1/2\pi) \int u(x) 2\pi \delta(x-y) dx. So du(y) is the covector field (in x) given by \delta[2\pi \delta(x-y)]. (And I'm unfortunately using \delta for too many things, because I want it is an operator and as a Dirac delta function.) So, all in all,

\dot{u}(y) = -\{u(y),H\} = (-1/2\pi i) \int 2\pi \delta(x-y) (h'(u))' dx = i d/dx [h'(u)].


I would like to complete my discussion of scalar fields from entirely a different direction. Given a unit circle, I'd like to describe the propagation of "free scalar fields" around the circle, where now I'm thinking of these as some sort of wave. Remembering very little physics, I can imagine two different interesting dynamics. Either all waves move the same speed, regardless of their shape, or waves propagate at different speeds, with the "high-energy" ones moving faster.

Let's write down some differential equations and see what happens. I'm interested in \dot{u} = some functional of u. Of course, we should demand some isotropy, so x should not appear explicitly. What are the effects of different terms? Keeping everything linear — I want free field propagation, so everything should superimpose — we could ask about \dot{u} = cu, for constant c, but this is boring: the value of the field at a point never cares what the value is ant neighboring points. (Indeed, the whole field evolves by multiplication by e^{ct}. If, for instance, c=i, then sure, different "modes" move at different "speeds", but this is the wrong analysis, since really the whole field is just rotating by some time-varying phase.)

More interesting is if \dot{u} = -u', say. Then expanding u(x) = \sum_{-\infty}^\infty u_k e^{ikx}, we can solve and conclude that \dot{u_k} = -ik u_k, and \dot{u(x)} = u(x-t). So this is what happens if waves travel all at constant velocity.

But let's say that the kinds of waves we care about are surface waves. For instance, we might have a taught string, and waves are small oscillations. Then really physics should act to even out curvatures: we should expect an upwards pull on any point where the field has positive curvature. If we don't remember freshman mechanics, we might write down \dot{u} = u'', which gives us u_k(t) = e^{-k^2 t} u_k(0). This isn't bad: different modes move with velocity proportional to t. It's not quite perfect, though, because really it's the force in that direction, not the derivative, so really we should have acceleration \ddot{u} = u''. Then we get back our original waves, except we have left-movers and right-movers. (More generally, we can add a mass term, and get H(u) = (1/2\pi) \int (1/2) [\dot{u}^2 - (u')^2 + m^2 u^2] = 1/2 \sum_{-\infty}^\infty [\dot{u_k}\dot{u_{-k}} + (k^2 + m^2) u_k u_{-k}], and the modes really do move at different velocities.)

Anyway, the point is that I really do expect, in this world, to have 2\infty total dimensions: \Z worth of "position" coordinates u_k and \Z momentum coordinates, not the \Z/2 of each that Yasha was considering. By just inventing conjugate momentum coordinates v_k to position coordinates u_k, we can get, for the free field, such simple equations of motion as \dot{u_k} = v_{-k} and \dot{v_k} = (k^2+m^2) u_{-k}.

So why does every quantum field theory course start quantizing the (real) scalar field by expanding in Fourier modes and imposing a nontrivial bracket between the coefficients? Because the (free) equations of motion, not the original setup, demand relationships between the fourier modes of u and \dot{u}, and the nontrivial bracket is between u and \dot{u}.


Perhaps next time I will venture into the realm of quantum mechanics. I'd really like to understand how the classical Poisson bracket becomes the quantum Lie bracket, and where the hell that i\hbar comes from. First, of course, I will have to talk more about sets, Hilbert spaces, and the like, and I'll probably stay finite-dimensional for a while. Eventually, of course, I want to describe Feynman diagrams, and tie them back to the Penrose birdtracks and the tensors that started this series of entries.

That is, of course, if I ever get that far. I tend to be distracted by other time-consuming tasks: I am only a few hours of work away from being done applying to graduate schools.


*Of course, I started this entry a few weeks ago.

Liouville's Theorem

A long time ago, in a 2n-dimensional symplectic manifold M, with form \omega_{\mu\nu} (and dual form \omega^{\mu\nu}), far, far away...

Prologue: Our hero, a young Hamiltonian H: M\to\R, defines a "hamiltonian flow" via a vector field (X_H)^\nu = dH_\mu \omega^{\mu\nu}. We can understand H as, for instance, the total energy, and M as the phase space. H has a friend, G, which is preserved by the hamiltonian flow (e.g. momentum in some direction). This happens exactly when X_H[G] = (dG).(X_H) = 0 (thinking of X_H in the first line as a differential operator). But dG.X_H = (dG)_\nu (X_H)^\nu = (X_G)^\mu \omega_{\mu\nu} (X_H)^\nu = (dG)_\nu \omega^{\mu\nu} (dH)_\mu. So H's flow preserves G if and only if G's flow preserves H: being friends is a reflective relationship.

Following the classical mechanists, we say that H and G are "in involution" if indeed \omega(X_H,X_G) = 0. More generally, we can define the "Poisson Bracket" \{H,G\} = \omega(X_H,X_G) = (dH)_\mu (dG)_\nu \omega^{\mu\nu}. Then clearly \{H,G\} = -\{G,H\}, and in particular H preserves itself (energy is conserved). Indeed, \{,\} behaves as a Lie bracket out: it satisfies the Jacobi identity \{\{G,H\},K\} + \{\{H,K\},G\} + \{\{K,G\},H\} = 0, and \{,\} is \R-linear. (Thus C^\infty(M) is naturally a Lie algebra; the corresponding Lie group is the space of "symplectomorphisms", or diffeomorphisms on M that preserve \omega.) Moreover, X_{\{G,H\}} = [X_G,X_H] where [,] is the (canonical) Lie bracket on vector fields. (The Hamiltonian fields X_H are exactly the differentials of symplectomorphisms, hence the identification in the previous parenthetical.)

((Actually, it's not C^\infty(M) that's tangent to the symplecto group of M, but C^\infty(M) / \R, when M is connected. Our \{,\} depends only on the differential of Hamiltonian functions, and so ignores constant terms: there are \R possible constant terms for each connected component of M. We can, of course, equip \R with the trivial Lie algebra, and then C^\infty(M) is, as a Lie algebra, T(symplectomorphisms) \times \R. The physicists would say this by observing that energy is defined only up to a total constant; this constant cannot affect our physics because it appears only in commutators. The physicists try to use this observation to justify introducing infinite constants into their expressions.))

One day our hero H met another function F, but this one unpreserved by H's flow. How does F change? In our setup, where H and F have no explicit time dependence, and we're just flowing via \dot{x} = X_H, we have that dF/dt = X_H[F] = {F,H}.

When H and G are buddies (in involution), then each of H and G is preserved by X_H: the flow stays in the common level set H = H(0) and G = G(0). Assuming that H and G are independent, in the sense that dH and dG are linearly independent (so we're not in the G = H^2 case, for instance), this common level set is (2n-2)-dimensional.

The story: As a young and attractive Hamiltonian, our hero H was particularly popular: there were n-1 other Hamiltonians H_2,...,H_n so that, along with H_1=H, all were pairwise in involution (\{H_i,H_j\} = 0), and all independent (the set of dH_i is linearly independent at each point in M, or at least at each point in some common level set, and so in a neighborhood of that level set). This is the most friends any Hamiltonian can have: the common level set is n-dimensional, and the tangent space contains n independent vectors X_i = X_{H_i}. Because, the X_i spanned each tangent, and because \omega of any two was zero, the common level set was for all to see a Lagrangian submanifold.

Being very good friends, the H_i never got in each other's way. X_i could flow, and X_j could flow, and because of the relationship between Poisson and Lie brackets, their flows always commuted. The gang used this to great affect: by giving each friend an amount to flow, the crowd defined an \R^n action on the common level set. The friends set out to explore this countryside: Hamiltonian flow is volume-preserving, since it preserves the symplectic form (whose nth power is a volume form), and a volume preserving \R-action is onto connected components.

The friends, returning home by some element of the stabilizer subgroup, understood the landscape: the only discrete subgroups of \R^n are lattices, and so the common level set was necessarily a torus (in the compact case). Picking standard coordinates q^j for a torus, the friends observed an isotropy: at every point, X_i = a_i^j \d/\dq^j with a constant "frequency" matrix a.

Our hero's life was solved. If all the frequencies a_1^j were rational multiples of each other, what the Greeks called "commensurate", H's paths were closed. Otherwise, H's flow would be dense in some subtorus, and either way, physics was simple. Indeed, because every Lagrangian manifold has a neighborhood symplectomorphic to its tangent bundle, there were "momentum" coordinates p_i conjugate to the angular position coordinates q^i, and these p_i depended only on the H_j. Indeed, in p,q coordinates, X_i was by definition -\dH_i/\dp_j \d/\dq^j + \dH_i/\dq^j \d/\dp_j, so the H_i knew themselves: H_i = -a_i^j p_j + const.

It was in this way that our hero the Hamiltonian understood how to flow not only in the level set, but in some neighborhood. The friends lived happily ever after.

Epilogue: Sadly, not all Hamiltonians can have as nice a life as our hero, because many do not have so many friends. It has been shown that the three-body problem is not Liouville-integrable, as this property of having enough mutual friends (and hence admitting Lagrangian tori) came to be called. Much analysis has gone into studying perturbations of Liouville systems — weakly-interacting gravitating bodies, for instance — but I do not know this material, and so will not exposit on it here. In my next entry, I hope to speak more about the Poisson bracket, and how it turns classical into quantum systems.

Edit: The matrix a_i^j may depend, of course, on H, or equivalently on p. What is actually true is that, up to a constant, H_i(p) = \int_0^{p} a_i^j(p') dp'_j. It is by solving this equation that one may find the conjugate p coordinates. That H_i = -a_i^j p_j + const. is true only to first order, and to first-order we cannot know whether, for instance, entries in a_i^j remain commensurate, and so whether paths stay closed as the momentum changes. Generically, Hamiltonian flows are not closed, and instead a single path is dense in the entire torus. In the general three-body problem, the flow is dense in a space greater than the dimension of any Lagrangian submanifold.

Tensors and Hamiltonians

I seem to have fallen way behind in writing about my classes. In particular, it may be a while yet before I do any quantum mechanics; I'm more excited by my classical geometry. But perhaps I will move into the quantum world soon. I almost understand it.

In the last few weeks, my classes have defined forms and fields, integration, chains, Lie groups, and Riemannian manifolds; quantum fields, fermions, SUSY, and Feynman diagrams; structural stability, Anosov flow, and a world worth of material in Yasha Eliashberg's class. Yasha pointed out in one lecture, "It is impossible to be lost in my class, because I keep changing topics every two minutes."

But I'm trying to provide a unified account of such material in these pages, and I last left you only with fields of tangent vectors. So, today, tensors, differential forms, and Hamiltonian mechanics.

Remember where we were: we have a smooth manifold M with local coordinates x^i, over which we can build two extremely important bundles, called T(M) and T^*(M). T(M) is the space of "derivations at a point": on each fiber we have a basis \d/\dx^i and coordinates \dot{x}^i. T^*(M) is dual to T(M) fiberwise: its basis is dx^i and its coordinates are p_i. But from these we can build all sorts of tensor bundles.

I touched on tensors in my last post, but hardly defined them. They are, however, a straightforward construction. A tensor is two vectors set next to each other. Or almost. If I have two vectors v\in V and w\in W, I can take their tensor product v\tensor w: I define \tensor to be multilinear, and that's all. V\tensor W is then generated by all the possible v\tensor w. More precisely, V\tensor W is the universal object so that bilinear maps from V\times W factor through it: there's a canonical bilinear map V\times W \to V\tensor W so that any bilinear from V\times W to \R factors through this map and some linear V\tensor W to \R.

If you haven't seen tensors before, this definition probably only made things worse, so let me say some other words about tensors. (i) The product \tensor is multilinear: (av)\tensor w = a(v\tensor w) = v\tensor(aw), and (v_1 + v_2)\tensor w = v_1\tensor w + v_2\tensor w, and the same on the other side. Thus \R\tensor V is canonically isomorphic to V. (ii) if {v_i} and {w_j} are bases for V and W, then {v_i\tensor w_j} is a basis for V\tensor W. It is in this way that \tensor correctly generalizes \times from Set to Vect.

We will be primarily interested in tensors comprised only of V and V^*, i.e. of vectors and dual vectors. Even more, when we move to bundles, we will be interested only in tensors over T(M) and T*(M). Of course, we can canonically commute V* past V, so all our tensors might as well live in V \tensor ... \tensor V \tensor V* \tensor ... \tensor V*, for various (possibly 0) numbers of Vs and V*s. Some notation: if there are n Vs and m V*s, I will write this (tensor) product as \T^n_m(V). \T^0_0 = \R; \T^1_0 = V.

How should you write these? As birdtracks, a name which Penrose even seems to be adopting. For these, and in my (and many physicists') notation, draw vectors with upward-pointing "arms" (since we write them with raised x^i) and dual vectors with downward-pointing "legs" (indices are lowered). The order of the arms matters, as does the order of the legs, but the canonical commutation referred to in the previous paragraph is explicit. To multiply two vectors, just draw them next to each other; in general, any tensor is the sum of products of vectors, but not necessarily just a product, so in general tensors are just shapes with arms and legs.

Birdtracks are an exquisite way to keep track of what's called "tensor contraction". See, what's important about dual vectors is that they can "eat" vectors and "spit out" numbers: there is a canonical pairing from V\tensor V* = \T^1_1 \to \R. We can generalize this contraction to any tensors, if we just say which arm eats which leg. In these notes, drawing birdtracks is hard; I will use the almost-as-good notation of raised and lowered indices. We can define \T^{-1} as being basically the same as \T_1, except that it automatically contracts in tensor products; this breaks associativity.

So, our basic objects are sections of \T^n_m(T(M)), by which I mean fields of tensors. A few flavors of tensors deserve special mention.

To \T^n(V) we can impose various relationships. In particular, there are two important projection operators, Sym and Ant. There is a canonical action of S_n on \T^n(V): \pi\in S_n sends a basis element e_{i_1}\tensor...\tensor e_{i_n} to e_{\pi(i_1}\tensor...\tensor e_{\pi(i_j)}. Extending this action linearly, we can construct Sym and Ant by
Sym(\omega) = (1/n!) \sum_{\pi\in S_n} \pi(\omega)
Ant(\omega) = (1/n!) \sum_{\pi\in S_n} \sgn(\pi) \pi(\omega)
where \sgn(\pi) is 1 if \pi is an even permutation, -1 if it is odd. These definitions, of course, also work for T_n. These are projection operators — Ant^2 = Ant and Sym^2 = Sym — so we can either quotient by their kernels or just work in their images, it doesn't matter. Define \S^n = Sym(\T^n) and \A^n = Ant(\T^n), and similarly for lowered indices. We have symmetric and "wedge" (antisymmetric) multiplication by, e.g., \alpha\wedge\beta = Ant(\alpha\tensor\beta); each is associative. One can immediately see that, if dim(V)=k, then dim(\S^n(V)) = \choose{k+n}{n} and dim(\A^n(V)) = \choose{k}{n}; of course, dim(\T^n) = k^n. Of particular importance: \T^2 = \S^2 + \A^2, where I will always use "+" between vector spaces to simply mean "direct sum" (which correctly generalizes disjoint union of bases).

From now on, V will always be T(M), and I'm now interested in fields. I will start using \T^_, \A, and \S to refer to the spaces of fields. We will from time to time call on tensors fields in \S_n, but those in \A_n end up being more important: we can understand them as differential forms. I may in a future entry try to understand differential forms better; for now, the following discussion suffices.

To each function f\in C^\infty(M) = \T_0 = \A_0 we can associate a canonical "differential": remembering that v\in \T^1 acts as a differential operator, we can let df\in\T_1 be the dual vector (field) so that df_i v^i = v(f). Even more simply, f:M\to\R, so it has a ("matrix") derivative Df: TM\to T\R. But T\R is trivial, and indeed we can canonically identify each fiber just with \R. So df = Df composed with this projection T\R \to \R along the base (preserving only the fiber). In coordinates, df = \df/\dx^i dx^i. It's tempting to follow the physicists and write d_i = \d/\dx^i, since the set of these "coordinate vector fields" "transforms as a dual vector".

From this, we can build the "exterior derivative" d:\A_i\to\A_{i+1} by declaring that d(df) = 0, and that if \alpha\in\A_r and \beta\in\A_s, then d(\alpha\wedge\beta) = d\alpha \wedge \beta + (-1)^i \alpha \wedge d\beta. This will not be important for the rest of my discussion today; I may revisit it. But I may decide that I prefer the physicists' \d_i = \d/\dx^i which acts on tensors of any stripe and satisfies the normal Leibniz rule. We'll see.

So, what can we do with tensors? We could pick a nondegenerate positive definite element of S_2; such a critter is called a "Riemannian metric". I won't talk about those right now (I may come back to them in a later entry). Instead, I'm interested in their cousins, symplectic forms: nondegenerate fields \omega in \A_2. By nondegenerate, I mean that for any non-zero vector y there's a vector x so that \omega(x,y)\neq 0. Thus, symplectic forms, live metrics, give isomorphisms between V and V*, and so \omega_{ij} has an inverse form \omega^{ij}\in\A^2 so that \omega_{ij}\omega^{jk} = \delta_i^k \in \A_1^1 = \T_1^1.

Some observations about symplectic forms are immediate, and follow just from linear algebra. (i) Symplectic forms only live in even-dimensions. (ii) We can find local coordinates p_i and q^i so that \omega = dp_i\wedge dq^i. (iii) In 2n-dimensional space, the form (\omega)^n \in \A_{2n} is a nowhere-zero volume form on space, so symplectic forms only live in orientable manifolds.

It's observation (ii) that gives us our first way of tackling symplectic forms, because we have a canonical example: the cotangent bundle of a manifold. If dim(M) = n, then T*M is 2n-dimensional and has a canonical form \omega = dp_i \wedge dq^i, where the q^i are "position" coordinates in M and the p_i are the "conjugate momentum" coordinates in the fibers. You can show that this coordinate formula for \omega transforms covariantly with changes of coordinates, so the form is well-defined; better is an invariant geometric meaning. And we can construct one. Let v be a vector in T(T*M), based at some point x,y\in T*M, i.e. x\in M and y is a covector at x. Then \pi: T*M \to M projects down along fibers, so we can pushforward v to \pi_*(v) \in TM. Now let y act on this tangent vector. This defines a form \alpha\in T*(T*M) = \A_1 (T*M): \alpha(v_{x,y}) = y.\pi_*(v). In local coordinates, \alpha = p_i dq^i. Then we can differentiate \alpha to get \omega = d\alpha \in \A_2; one can show that d\alpha is everywhere nondegenerate.

Why do we care? Well, let's say we're solving a mechanics problem of the following form: we have a system, in which the position of a state is given by some generalized coordinates q^i, and momentum by p_i. Then the total phase space is T(position space). And to each state, let's assign a total energy H = p^2/2m + U(q). We're thinking of p^2/2m as the "kinetic energy" and U(q) as the "potential energy". More generally, we could imagine replacing p^2/2m by any (positive definite symmetric) a^{ij} p_i p_j / 2; of course, we ought to pick coordinates to diagonalize a^{ij}, but c'est la vie. (And, of course, a^{ij} may depend on q.) Then our dynamics should be given by
\dot{q} = p/m = \dH/\dp
\dot{p} = -\dU/\dq = -\dH/\dq

How can we express this in more universal language? A coordinate-bound physicists might be content writing \dot{q}^i = a^{ij}p_j and \dot{p}_i = -\dU/\dq^i. But what's the geometry? Well, we want this vector field X_H = (\dot{p},\dot{q})\in \T^1 = \A^1, and we have the derivative dH \in \A_1 = \T_1. It turns out that the relationship is exactly that for any other v-field Y, dH.Y = \omega(X_H,Y). I.e. dH is X_H contracted with \omega. For the physicists, letting \mu index the coordinates in T*M, we have (X_H)^\mu = \omega^{\mu\nu} \dH/\dx^\mu.

This is an important principle: the Hamiltonian, which knows only the total energy of a state, actually knows everything about the dynamics of the system (provided that the system know which momenta are conjugate to which positions).

From here, we could go in a number of directions. One physics-ish question that deeply interests me is why our universe has this physics. The conventional story is that God created a position space, and that inherent to the notion of "position" is the notion of "conjugate momentum", and thus it would be natural to create a dynamics like we have. But what's entirely unclear to me is why our physics should be second-order at all. Wouldn't it be much more natural for God to start with just a manifold of states and a vector-field of dynamics? Perhaps God felt that a function is simpler than a vector field. But that's no matter: God could have just picked a manifold with a function and any nondegenerate element of \T^2, with which to convert d(the function) into the appropriate field. For instance, we could have had a Riemannian manifold with dynamics given by "flow in the direction of the gradient of some function".

No, instead God started with a symplectic manifold. Well, that's fine; I don't begrudge the choice of anti-symmetry. But then there's a really deep mystery: why, if fundamentally our state is given by a point in some symplectic manifold, do we distinguish between position and momentum? Certainly we can always find local coordinates in which \omega = dp\wedge dq, but for God there's some rotation between p- and q-coordinates that we don't see. The symmetry is broken.

Another direction we could go is to discuss this physics from the opposite direction, and I certainly intend to do so in a future entry. As many of you probably know, alternate to the Hamiltonian formalism is a similarly useful Lagrangian formalism for mechanics. Along with its "least action" principle, Lagrangian mechanics is a natural setting in which to understand Noether's Theorem and Feynman's Sum-Over-Paths QFT. At the very least, sometime soon I will try to explain the relationship between Lagrangians and Hamiltonians; it will draw deeply on such far-afield ideas as projective geometry.

But I think that in my next entry I will pursue yet another direction. I'd like to talk about Liouville's Theorem, which tells you how to solve Hamiltonian diffeqs, by introducing a bracket between Hamiltonian functions. I hope that I will be able to then explain how this relates to QFT and its canonical commutation relations. This last step I don't fully understand yet, but I hope to: I think it's the last bit I need to know before I can understand what it means to "quantize" a classical theory.

Edit: In addition to symplectic forms being nondegenerate antisymmetric, they must also be exact: their (exterior) derivatives should be zero. Most immediately, this fact assures that the symplectic forms be locally equivalent, and, most basically, this allows us to talk about (co)homology of a symplectic. In a more advanced setting, exactness will translate into Jacobi's identity.

Negative Dimensions

Since I'm behind in my series of posts on fields, quantum or otherwise, I will instead talk today about some linear algebra, and not define most of my terms.

The category Vect of vector spaces (over generic field \R = "real numbers") nicely generalizes the category Set of sets. Indeed, there is a "forgetful" functor in which each set forgets that it has a basis. Yes, that's the direction I mean. A vector space generalizes ("quantizes") in a natural way the notion of "set": rather than having definite discrete elements — two elements in a set either are or are not the same — a vector space allows super-positions of elements. A set is essentially "a vector space with a basis": morphisms of sets are morphisms of vector spaces that send basis elements to basis elements. So our forgetful functor takes each set X to the vector space Hom(X,\R) (Hom taken in the category of sets). But, I hear you complain, Hom(-,\R) is contravariant! Yes, but in this case, where I forgot to tell you that all sets are finite and all vector spaces finite-dimensional, we can make F = Hom(-,\R) covariant by F(\phi): f \mapsto g(y) = \sum_{x\in X s.t. \phi(x)=y} f(x). Actually, of course, if I'm allowing infinite sets, then I should specify that I don't quite want X \to Hom(X,\R), but the subspace of functions that send cofinitely many points in X to zero.

Anyhoo, so Set has an initial object 1 = {one element} and a terminal object 0 = {empty set}, and well-defined (up to canonical isomorphism) addition and multiplication (respectively disjoint union and cartesian product). These generalize in Vect to 1 = \R and 0 = {0}, and to direct sum and tensor product; if we identify n = "\R^n" (bad notation, because it's really n\R; I want n-dimensional space with a standard basis, so the space of column vectors), then it's especially clear that sums and products are as they should be. So Vect is, well, not quite a rig (ring without negation), because nothing is defined uniquely, but some categorified version, where all I care is that everything be defined up to canonical isomorphism (so, generically, given by a universal property).

But I can do even better. To each vector space V is associated a dual space V^* = Hom_{Vect}(V,\R), and most of the time V^{**} = V. (I need to learn more linear algebra: I think that there are various kinds of vector spaces, e.g. finite-dim ones, for which this is true, and I think that there's something like V^* = V^{***}. If so, then I should certainly always pass immediately from V to V^{**}, or some such; I really want dualing to be involutive.) By equals, of course, I always mean "up to a canonical isomorphism". Now, V\times V^* = Hom(V,V) is rather large, but there is a natural map Trace:Hom(V,V)\to\R, and this allows us to define a particular product "." which multiplies an element v\in V with w\in V^* by v.w = Tr(v\tensor w). Then . is multi-linear, as a product ought to be, and we can thus consider V.V^* = \R. Indeed, we can imagine some object 1/V that looks like V^* — a physicists wouldn't be able to tell the difference, because their elements are the same — so that V \tensor 1/V = \R. (Up to canonical isomorphism. It's not, of course, clear which copy of V we should contract 1/V with in V\tensor V. But either choice is the same up to canonical isomorphism.) There is even a natural trace from, say, \Hom(2,4) \to 2 — take the trace of the two 2x2 squares that make up the 4x2 matrices — "proving" that 4/2 = 2.

So it seems that, well, Vect is not a division rig, but it naturally extends to one. But what about that n in "ring"? What about negative dimensions? This I don't know.

See, it's an important question. Because, consider the tensor algebra T^{.}(V) = \R + V + V\tensor V + ... — this is an \N-graded algebra of multilinear functions on V^*. This looks an awful lot like the uncategorified 1+x+x^2+..., which we know is equal to 1/(1-x). (Why? Because (1-x)(1+x+...) = 1-x+x-x^2+x^2-... = 1, since every term cancels except for the -x^\infty, which is way off the page.) Anyhoo, so we ought to write the tensor algebra as 1/(1-V).

Which doesn't make any sense at all. 1-V? Well, we might as well define 1-V as dual to the tensor algebra: there should be a natural way to contract any element of 1-V with any multilinear function on V^*. But this has a much shorter algebraic expression, which ought to have Platonic meaning. So, what's a natural object that we can construct out of V that contracts (linearly) with all multilinear functions to give real-valued traces?

If we could answer this, then perhaps we could find out what -V is. How? Not, certainly, by subtracting 1=\R from 1-V. No, I suggest that whatever our proposal be, we then try it on 1-2V = (T^.(V+V))^* = 1/(\R + V+V + (V+V)\tensor(V+V) + ...), and compare. What out to happen is that there should be some natural object W such that 1-2V = W + 1-V, and it should turn out that 1-V = 1 + W. Whatever the case is, there should be a natural operation that "behaves like +" such that 1-V + V = 1. It's certainly not standard direct sum, just like how V \times 1/V is not the standard tensor product. But it should be like it in some appropriate sense. Most necessarily, it should satisfy linearity: if v_1,v_2\in V and w_1,w_2\in W, then v_1+w_1 and v_2+w_2 \in V+W should sum to (v_1+v_2)+(w_1+w_2). And, of course, if you have the right definition, then all the rest of arithmetic should work out: 1/(-V) = -(1/V), -V = -\R \times V, (-V)\times W = -(V\times W), and, most importantly, --V = V (up to canonical isomorphism).

One can go further with such symbolic manipulation. You've certainly met the symmetric tensor algebra S^{.}(V) of multilinear symmetric tensors, and you've probably defined each graded component S^{n}(V) as V^{\tensor n} / S_n, where by "/ S_n" I mean mean "modulo the S_n action that permutes the components in the n-times tensor product." (If you are a physicists, you probably defined the symmetric tensors as a _subspace_ of all tensors, rather than a quotient space, but this is ok, because the S_n identification generates a projection operator Sym: \omega \to (1/n!)\sum_{\pi\in S_n} \pi(\omega), and so the subspace is equal to the quotient. At least when the characteristic of the ground field is 0.) Well, S_n looks an awful lot like n!, so the symmetric algebra really looks like 1 + V + V^2/2! + ... = e^V. Which is reasonable: we can naturally identify S^{.}(V+W) = S^{.}V\tensor S^{.}W.

It's not quite perfect, though. The dimension of S^{.}V, if dim V = n, is not e^n, but 1 + n + n(n+1)/2 + n(n+1)(n+2)/6 + ..., which is only correct in the limit n\to\infty. Well, so why is that the dimension? When we symmetrize v\tensor w to 1/2(vw+wv), we generically identify different tensors. But v^2 symmetrizes to itself. Baez, though, says how to think about this: when a group action does not act freely, we should think of points like v^2 as only going to "half" points. So, for example, the group 2 can act on the vector space \R in a trivial way; we should think of \R/2 as consisting of only "half a dimension".

Anyway, the point is that we can divide by groups, and this is similar to our division by (dual) vector spaces: in either case, we are identifying, in a linear way, equivalence classes (either orbits or preimages).

Now, though, it becomes very obvious that we need to extend what kinds of spaces we're considering. Groups can act linearly in lots of ways, and it's rare that the quotient space is in fact a vector space. Perhaps the physicists are smart to confuse fixed subspaces and quotients: it restricts them just to projection operators. But, for instance, if we mod out \C by 2 = complex conjugation (which is real-linear, although not complex-linear), do we get \R or some more complicated orbifold? Is there a sense in which \R/2 + \R/2 = \R, where 2 acts by negation? \R/2 is the ray, so perhaps the direct sum model works, but you don't naturally get \R, just a one-dim space? To give interesting physics, it would be nice if these operations really did act on the constituent parts of each space. And what about dividing by 3? Every field has a non-trivial square root of 1, but only \C has nontrivial nth roots. So perhaps we really should just work with Vect of \C-linear spaces. Then we can always mod out by cyclic actions, but we don't normally get vector spaces.

Of course, part of the fun of categorifying is that there are multiple categorical interpretations of any arithmetic object: 6 may be the cyclic group C_6 = C_2 \times C_3, but 3! is the symmetric group S_3, and the groups 4 and 2x2 are also unequal. But if we come up with a coherent-enough theory, we ought to be able to make interesting discussion of things like square roots: there's an important sense in which the square root of a Lorentz vector is a spinor, and we should be able to say (1+V)^{1/2} = 1 + (1/2)V + (1/2)(-1/2)V^2/2 + (1/2)(-1/2)(-3/2)V^3/3! + ....

Overall, the move from Set to Vect deserves to be called "quantization" — well, really quantization doesn't yield vector spaces but (complex) Hilbert spaces, so really it should be the forgetful functor Set \to Hilb. If we have a coherent theory of how to categorify from numbers to Set, then it should match our theory of how to categorify from numbers to Hilb. And, ultimately, we should be able to understand all of linear algebra as even more trivial than how we already understand it: linear algebra is simply properly-categorified arithmetic.

Tangent vectors and their fields

Voice-over: "Last time, on Blogging My Classes, Blogging My Fields,"
Screen flashes with images of surfaces and atlases. Main character says something cliche (but stunning because of the background music) about the definition of the manifold. Then screen switches to the final scene: The Scalar Bundle.
Voice-over: "And now, the continuation."

Classically, the tangent bundle T(M) to a manifold M was defined by taking equivalence classes of (parameterized) curves at each point, equivalent if they're tangent there. Slightly more universally, we can take our atlas of patches, and on each patch, consider the (locally trivial) bundle of tangent spaces to \R^n, then modding out by the transition functions between patches. But there is a better, more algebraic way to develop tangent vectors, directly from the sheaf of differentiable functions.

Within the space of linear functionals on C^\infty(M), consider those that are "derivations at the point p": l:C^\infty(M)\to\R should satisfy, for all f,g, l(fg) = f(p)l(g) + g(p)l(f). Of course, derivations at points of constant functions return zero, and one can check that derivations at points don't care about the value of the function outside a nbhd of the point, by considering bump functions. Given a coordinate patch x^i, the m derivations \d/\d x^i |_p (derivative in the i'th direction, evaluated at p) are examples, and it turns out that these form a basis for the (linear) space of derivations at p. (This is not entirely obvious. In coordinates, it comes from the fact that I can write any f(x):\R^n\to\R as f(x) = f(0) + \sum g_i(x) x^i (in a small nbhd of 0), by letting h_x(t) = f(xt) and thus g_i(x) = \int_0^t h_{x^i}(u) du.) So we have, given an n-dimensional manifold, n dimensions worth of derivations at each point.

Now, intuitively, any tangent vector gives a derivation at its basepoint, by differentiating the function "in the direction of the vector". And, intuitively, there are n dimensions worth of tangent vectors. So we can define a tangent vector at p to be a derivation at p.

Thus, it's clear that a vector field is exactly a derivation: a field worth of derivations, one at each point. Indeed, any derivation — an algebraically-defined object, exactly a linear operator L from C^\infty(M) to itself that satisfies the Leibniz rule L(fg) = L(f)g + fL(g) — gives us a derivation at each point: L_p(f) = L(f)(p). (And, by chasing definitions, two derivations agree iff they agree at each point.) More generally, we can talk about the sheaf of (tangent) vector fields, by asking about derivations of functions defined only on various open sets.

It's worth now mentioning cotangent vectors, and specifying some notation. Of course, to any vector space (e.g. T_p(M), the tangent vectors at p), we can define the dual space (of linear functionals). By linear algebra, if dim(V)<\infty, then the dual space has the same dimension; given a basis, we can construct a dual basis. Working now with manifolds, given any function f, I can get a cotangent field df defined by df(v) = v[f], where we think of v as a derivation. In particular, by the claim I made above about being able to write f in some local normal form, given a coordinate system x = {x^i} on a nbhd U, it's clear that the {dx^i} are a basis for the space of sections of T^*(U) as a module over C^\infty(U). (Similarly, the partials \d/\d x^i are a basis of {sections of T(U)} as a module over functions.)

Following the physicists' convention, I will usually just write p_i for the cotangent field p_i dx^i, and similarly I will usually just write q^i for the vector field q^i \d/\d x^i. (Continuing the summation convention.) This works, because dx^i \d/\d x^j = \delta^i_j, so (p_i dx^i)(q^j \d/\d x^j) = p_i q^j \delta^i_j = p_i q^i, so the dot-products work out right. This is only because I happen to be using a basis and its dual basis. Eventually, I may redefine the index conventions truly coordinate-independently, but for now let's maintain the convention that whenever we interpret our formulas in terms of coordinates, we always use dual bases for T and T^*.

Next time, I'd like to talk more about tensors, metrics, and similar structures. In particular, I'd like to define the Lorentz group and classify its representations.

A new class of entries

I think I might like to spend some time thinking about definitions in mathematical physics. What is a quantum field, for instance? Physicists usually give a slightly incoherent answer: a quantum field is a quantum particle at every point, just like a field is a number at every point. You ask them to unpack this a bit, and some might remember that there may be global — what the physicists call "topological" — issues with such a definition, but for now let's only be concerned with the local definition, where a field is a function. So what should a quantum field be?

Conveniently, I'm taking three classes right now on related questions: Differential Geometry, Geometric Methods to ODEs, and Quantum Field Theory. I would like to start a series of entries blogging those classes, and relating it back to such foundational questions. I hope to get to answers involving infinitesimals: Robinson's "Non-standard Analysis", or Kock's "Synthetic Geometry". I don't have the answers yet.

What's most important about fields is their geometric nature. Like the physicists and the classical differential geometers, I may from time to time refer to coordinates, but ultimately I'd like a coordinate-invariant picture — indeed, one without coordinates at all. I also hope to ask and answer issues about how to regularize our fields, by which I mean "how continuous should they be?" This is an extremely non-trivial question: not only is it extremely unclear how to demand that two
"nearby" "quantum particles" be "similar" (we can demand as much of classical fields: for any epsilon, there should be a delta at each point so that within the delta ball at that point the fields don't vary more than epsilon; perhaps we should find the right metric on Schrodinger-quantized particles?), but the physicists don't even want to be stuck with, say, C^\infty fields. They want \delta functions to work within their formalism. And yet they adamantly refuse to consider "pathological" fields that are too "wildly varying".

Eventually, it would be nice also to understand the Lagrangian and Hamiltonian, and this almost-symmetry between position and momentum. For now, I'd like to end this entry with some basic definitions.


Manifolds: There are many equivalent definitions of a manifold. Since the physicists and classical geometers like to work with coordinates (replacing geometry-defined, invariant objects with coordinate-defined, covariant objects), I'll use the definition that mentions coordinates explicitly. A manifold is a (metrizable) topological space M with a maximal atlas — to each "small" open set U in M we assign a module of "coordinate patches" \phi:U\to\R^n, which should be homeomorphisms, subject to some regularity condition: if \phi:U\to\R^n and \psi:V\to\R^n, then \phi\psi^{-1} should be, say, smooth wherever it's defined. In general, modifying the word manifold modifies the condition on \phi\psi^{-1}: a C^\infty manifold has that all the \phi\psi^{-1}'s are C^\infty, for example. I will generally be interested only in C^\infty (aka "smooth") manifolds, although once we understand what kinds of functions the physicists are ok with, we may change that restriction. For a manifold, I demand that the atlas be maximal in the sense that it list all possible coordinatizations consistent with the smoothness condition. It is, of course, sufficient to simply cover our space with (coherent) patches, defining the rest as all other possibilities.

So that we can generalize this definition if we need to, it would be nice to reword this definition in the language of sheaves. The god-given structure on a smooth manifold is exactly enough to tell which functions are differentiable: a sheaf is a topological space along with a ring of "smooth functions" on each open set, so that the function rings align coherently (in full glory, a sheaf is a (contravariant) functor from the category of open sets in the space to the category of commutative \R-algebras whatever your sheaf is of, along with some "local" axioms, which ultimately say that to know a function I need exactly to know it on an open cover). I probably won't use this description, largely because I don't know what other conditions I would want to put on my sheaf in order to make it into something like a smooth manifold. Clearly every manifold generates a sheaf, and I have it on good authority that if two manifolds have the same sheaf, then they are the same manifold.

So what about our most-important of objects: a field? A field is a "section" of a "bundle".

Let's start with the latter of those undefined words. To each point p\in M, we associate a (for now) vector space V_p, called the "fiber at p". And let's (for now) demand some isotropy: V_p should be isomorphic to V_q for any given p and q in M, although not necessarily canonically so. (When we move to the realm of infinite-dimensional fibers, we may demand only that the fibers be somehow "smoothly varying" — I'm not sure yet how to define this. So long as everything is finite-dimensional, the isomorphism class of a fiber is determined by an integer, and integers cannot smoothly vary, so it suffices to consider bundles where the dimension of the fibers is constant.)

There should be some sort of association between nearby fibers: locally (on small neighborhoods U) the bundle should look like U\times V. So I ought to demand that the bundle be equipped with a manifold structure, which aligns coherently with M: a bundle E is a manifold along with a projection map \pi_E : E\to M, such that the inverse image of each point is a vector space. This is the same as saying that among the coordinate patches in E's atlas, there are some of the form \Phi: \pi^{-1}(U) \to \R^(n+k), (where, of course, n is the dimension of M and k is the dimension of each fiber) so that \Phi = (\phi,\alpha), where \phi is a coordinate patch on M and \alpha is linear on each fiber. We can naturally embed M\into E by identifying each point p\in M with (p,0) in E (where 0 is the origin of the fiber at p).

I will soon make like a physicist and forget about global issues, but I do want to provide one example of why global issues are important: the cylinder and the mobius strip are both one-dimensional ("line") bundles over the circle. The latter has a "twist" in it: as you go around the circle, you come back with an extra factor of -1.

So what's a section of a bundle? A (global) section is a map s:M\to E so that \pi s:M\to M is the identity, i.e. a section picks out one vector from each fiber. We will for now think of our sections as being C^\infty.


The most important kinds of fields are "scalar" fields, by which I exactly mean a function, i.e. a number at every point. I want to do this because I want to consider other spaces of fields as modules over the ring of scalar fields, so I need to be able to multiply. Of course, there are many times when I don't want a full-fledged scalar field. The potential energy, for instance, is only defined up to a constant: I will eventually need my formalism to accommodate objects that have fields as derivatives, but aren't fields themselves. Since potentials don't care about constants, we could imagine that after going around a circle we measure a different potential energy than we had to begin with, but that we never picked up any force. The string theorists, in fact, need similar objects: locally, string theory looks like (conformal) field theory on the string's worldsheet. But perhaps the string wraps around a small extra dimension? This is why in the previous paragraph I refer to "global" sections: I really ought to allow myself a whole sheaf of fields, understanding that sometimes I want to work with fields that are only defined in a local area. But the physicists are generally clever about this type of problem, so, at the risk of saying things that we might think generalize but actually don't, I'm going to restrict my attention to scalar fields.

In which case, yes, by "scalar field" I mean "function from M \to \R". "A section of M\times\R". "A number at each point". Those who prefer to start with the sheaf of scalar fields will be happy to know that, when I define tangent vectors and their relatives in the next entry, I will start with these scalar fields.