### Classical, Canonical, Stringy

#### Posted by urs

In the last entry the following question was raised (in slightly different formulation):

Classical mechanics of point particles is governed by symplectic geometry and hence in particular by a symplectic 2-form ${\omega}_{2}$. We know that, morally, stringification lifts form degrees by one.

Hence:

Is there a V.I. Arnold-like generalization of this which replaces the symplectic 2-form by a three form and describes the dynamics of 1-dimensional objects? If so, how?

(Of course we know how to describe field theory using a symplectic 2-form on an infinite-dimensional space. And the dynamics of a 1-dimensional object is just 1+1 dimensional field theory. Hence whatever answer the above question has, it should be reducible to the ordinary setup.)

You might object that this sounds like a typical excercise for a mathematical physicist who is interested in reformulating something that everybody already understands. But there should be more to it.

Anyway, I think that a very good approach to answering this question has been given a couple of years ago in

C. Rovelli
**Covariant hamiltonian formalism for field theory**

gr-qc/0207043 .

I’ll try to review some key ideas.

First quickly recall how symplectic geometry encodes classical dynamics.

Start with the **non-relativistic particle** case. There is configuration space $K$. Consider choosing local coordinates $\{{q}^{a}\}$ on this. It’s cotangent bundle is $\Omega ={T}^{*}K$. Let local coordinates here be $\{{q}^{a},{p}_{a}\}$. There is a canonical 1-form which locally reads

The differential of this is the nondegenerate 2-form

Hence $(\Omega ,\omega )$, known as the *phase space*, is naturally a symplectic space. $\omega $ defines Poisson brackets.

That’s the kinematics. Now choose a dynamics by specifying a function

the *Hamiltonian*. We declare that a ‘solution’ to the dynamics is a curve

whose tangent vector field we call

such that *Hamilton’s equation*

holds.

Next consider describing a **relativistic particle** this way and further unify the formalism. Include the $\tau $ from above into the configuration space and call

the *extended configuration space*.

For later application think of this as the space whose points $(\tau ,{q}^{i})$ define the value of a field (${\gamma}^{i}={q}^{i}$) at a parameter space point ($\tau $).

Again there is a canonical 1-form on $\Omega ={T}^{*}C$, which we suggestively write locally as

Similarly the symplectic form is locally

This is the kinematics. Dynamics is again specified by a hamiltonian

but now one is interested only in the constraint surface

$H$ is known as the *Hamiltonian constraint* and Hamilton’s equation

reduces on $\Sigma $ to

The space of such solutions is the *phase space* $\Gamma $. Since every point in $\Sigma $ belongs to precisely one solution of Hamilton’s equation, there is a projection map

This can be used to pull back the 2-form ${\omega}_{2}{\mid}_{\Sigma}$ to phase space $\Gamma $

The symplectic space

is the usual physical phase space. And, Rovelli points out, it is really this symplectic space (not $(\Omega ,{\omega}_{2})$) which is the right symplectic space to look at. In generalization to the string, the analog of $\Omega $ will cease to be a symplectic space (rather carry a certain 3-form than a 2-form), while there is still a phase space with symplectic structure obtainable from that.

For $C$ finite dimensional, the above describes the relativistic particle. Now on to the string.

Assume parameter space of the relativistic **string** is

with local coordinates $(\tau ,\sigma )$. In a generalization of the above, call

the extended configuration space. A point $(\tau ,\sigma ,{q}^{i})$ in that space encodes the position in target space of the point of the string at parameter value $(\tau ,\sigma )$.

Now pass not to the cotangent bundle, but to the second exterior power of the cotangent bundle of $C$:

Choose local coordinates suggestively as $\{{p}_{\mathrm{ab}},{q}^{a}\}$.

There is a special 2-form on this space which in these coordinates locally reads

Hence we get a closed **3-form** ${\omega}_{3}$, replacing the symplectic 2-form from the ordinary case, which locally reads

Choose for $\pi $ the hamiltonian density of the Polyakov string and convince yourself that Hamilton’s equation in the form

yields the string’s equations of motion. Alternatively, see the example on pp. 13 in Rovelli’s paper.

Fine, now reproduce a symplectic phase space with symplectic 2-form from this setup. Rovelli observes that the former, naïve definition of phase space from above no longer works. Instead, the correct phase space should be the space of initial and final data on a string evolution. Hence let $G$ be the space of “(closed, oriented) 1-dimensional hypersurfaces” in extended configuration space $C$, i.e. essentially

the free loop space over C. (Each point of which describes a string configuration with chosen parameterization.)

One gets a symplectic structure on $G$ by using Rovelli’s equation (62), which is nothing but the prescription that was mentioned in a comment to the previous entry.

Take the 3-form ${\omega}_{3}$ and integrate it over the given bounding loop to get a loop space 2-form

(which is supposed to be shorthand for the precise formula given in the above mentioned comment).

It can be checked without much effort that

is really a (infinite dimensional) symplectic space (‘loop phase space’ if you wish) and that tangent vectors $X$ to physically realized string trajectories (worldsheets) do satisfy

This way, once again, string dynamics in extended config space governed by a closed 3-form is reformulated as point mechanics in loop space governed by a symplectic 2-form.

**Exercise:** Check that turning on a Kalb-Ramond field 3-form field strength ${H}_{3}$ on target space induces a deformation of the symplectic structure on $(G,{\omega}_{G})$ analogous to how a 2-form field strength ${F}_{2}$ deforms the symplectic structure on the phase space of a relativistic particle.

## Re: Classical, Canonical, Stringy

I think my reply got eaten, anyway I just wanted to say thats an extremely interesting and elegant formulation.

Id be interested in seeing how the quantization procedure goes through in the generalized case. Pre quantum 2 bundles, or something nice and esoteric like that?