Classical, Canonical, Stringy

November 2, 2005

Posted by Urs Schreiber

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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.

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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

(1)

\theta_1 = p_a \mathbf{d}q^q \,.

The differential of this is the nondegenerate 2-form

(2)

\omega_2 = \mathbf{d}\theta_1 = \mathbf{d}p_a \wedge \mathbf{d}q^a \,.

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

(3)

H \in C^\infty(\Omega) \,,

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

(4)

\array{ \gamma : P_1 \subset \mathbb{R} &\to& \Omega \\ \tau &\mapsto& \gamma(\tau) }

whose tangent vector field we call

(5)

X = \gamma' = \frac{d\gamma}{d\tau}

such that Hamilton’s equation

(6)

\omega(X,\cdot) = \mathbf{d}H(\cdot)

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

(7)

C = P_1 \times K

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

(8)

\theta_1 = p_a \mathbf{d}q^a = \pi d\tau + p_i \mathbf{d}q^i \,.

Similarly the symplectic form is locally

(9)

\omega_2 = \mathbf{d}\theta_1 = \mathbf{d}\pi\wedge \mathbf{d}\tau + \mathbf{d}p_i\wedge \mathbf{d}q^i \,.

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

(10)

H \in C^\infty(\Omega)

but now one is interested only in the constraint surface

(11)

\Sigma = \{x \in \Omega | H(x) = 0\} \,.

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

(12)

\omega(X,\cdot) = \mathbf{d}H

reduces on $\Sigma$ to

(13)

\omega(X,\cdot) = 0 \,.

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

(14)

\Pi : \Sigma \to \Gamma \,.

This can be used to pull back the 2-form $\omega_2|_\Sigma$ to phase space $\Gamma$

(15)

\omega_\Gamma = \Pi^*\omega_2 \,.

The symplectic space

(16)

(\Gamma, \omega_\Gamma)

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

(17)

P_2 \subset \mathbb{R}^2

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

(18)

C = P_2 \times K

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 $C$ :

(19)

\Omega = \Lambda^2 T^*C \,.

Choose local coordinates suggestively as $\{p_{ab},q^a\}$ .

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

(20)

\theta_2 = p_{ab} \mathbf{d}q^a \wedge \mathbf{d}q^b := \pi \mathbf{d}\tau \wedge \mathbf{d}\sigma + p_{\tau i} \mathbf{d}q^i \wedge d\tau + p_{\sigma i} \mathbf{d}q^i \wedge d\sigma \,.

Hence we get a closed 3-form $\omega_3$ , replacing the symplectic 2-form from the ordinary case, which locally reads

(21)

\omega_3 = \mathbf{d}\theta_2 \,.

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

(22)

\omega_3(X,\cdot) = 0

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

(23)

G = LC

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

(24)

\omega_G(\gamma) = \int_\gamma \omega_3

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

It can be checked without much effort that

(25)

(G,\omega_G)

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

(26)

\omega_G(X,\cdot) = 0 \,.

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.

Posted at November 2, 2005 6:25 PM UTC

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6 Comments & 1 Trackback

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?

Posted by: haelfix on November 3, 2005 4:00 AM | Permalink | Reply to this

Re: Classical, Canonical, Stringy

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I’d 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?

Yes, that’s certainly what one would think of next! There might be some cool things to be found here.

Somebody should take the time and write things out in detail and cleanly. Unfortunately, probably I am supposed to be doing something else… :-)

I know that Robert has a nice discussion of string mechanics in terms of symplectic geometry on loop space (at least somewhat implicitly) in hep-th/0409182, finding in particular the conformal anomaly this way. There should be a nice gerbe on $\Lambda^2 T^* C$ whose class, when pulled back to loops, gives rise to this symplectic structure. The central extension should be even more manifestly visible in the gerbe language, I’d expect.

There are further intriguing patterns here. I note that in the context of generalized geometry we start with bundles/particles described by an algebroid of the form

(1)

T M \oplus \Lambda^0 T^* M = T M \oplus 1

(as on p.37 of Gualtieri’s math/0401221). At the same time we have a 2-form $\omega_2$ on the cotangent bundle of config space

(2)

\Lambda^1 T^* C = \T^* C \,.

Next, in proper generalized geometry, we pass to gerbes and strings and use the algebroid

(3)

T M \oplus \Lambda^1 T^*M = T M \oplus T^*M

(p. 39). At the same time we work now with a 3-form $\omega_3$ on

(4)

\Lambda^2 T^* C

according to what I wrote in the above entry.

We know that this pattern continues. In a $p$ -gerbe generalization of generalized geometry, pertaining to $p$ -branes, we use the algebroid

(5)

T M \oplus \Lambda^p T^* M

(see the remark on the bottom of p. 36) and have a $p+2$ -form $\omega_{p+2}$ on

(6)

\Lambda^{p+1} T^* C

according to Rovelli’s proposal.

This does not really look like a coincidence, does it?

Posted by: Urs Schreiber on November 3, 2005 9:42 AM | Permalink | Reply to this

Re: Classical, Canonical, Stringy

From my point of view, the most interesting thing with Rovelli’s paper is the goal of obtaining a covariant Hamiltonian formulation of GR. Once we have that, the constraint algebra consists of arbitary 4-diffeomorphisms rather than the ugly Dirac algebra, and we know how to build quantum representations. Note that this statement is independent of the precise form of the covariant Hamiltonian formulation.

On multi-symplectic structures: Many years ago, I pondered a kind of 2-algebra generalization of the Moyal algebra, written in a Fourier basis as

exp(im.x) * exp(in.x) = exp(i hbar A(m,n)) exp(i(m+n).x),

where the momenta m and n are vectors in Z^2 and A(m,n) is the area enclosed by the vectors m, n, and m+n. A kind of 2-algebra, introduced at the time by Ruth Lawrence and perhaps others, can be defined by gluing three triangles together to make a new one; denote this 2-product by m(T1,T2,T3). Then the natural 2-algebra generalization of the Moyal product is

*(T1,T2,T3) = exp(iV(T1,T2,T3)) m(T1,T2,T3),

where V is the volume of the enclosed tetrahedron. I never wrote it up, since the construction seemed quite useless to me, but morally it is similar to what you ask for; A is the symplectic 2-form, and V corresponds to a 3-form.

Posted by: Thomas Larsson on November 3, 2005 10:32 AM | Permalink | Reply to this

Re: Classical, Canonical, Stringy

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constraint algebra consists of arbitary 4-diffeomorphisms rather than the ugly Dirac algebra, and we know how to build quantum representations

What confuses me about this statement is that we build quantum reps of constraints by definition only on the phase space. It’s on the phase space that we have Poisson brackets which are promoted to quantum commutators and hence give rise to quantum reps.

One might wonder if there is a way to generalize this away from ordinary symplectic phase space to somehing, maybe, carrying a closed 3-form. But that’s not understood yet, I think.

gluing three triangles together to make a new one

I guess this operation is defined (only) on triples of triangles two of whose whose edges each match pairwise?

What I mean is, given a triangle spanned by two vectors $t^1_1$ and $t^1_2$ and another one spanned by $t^2_1$ and $t^2_2$ and a third one spanned by $t^3_1$ and $t^3_2$ then this product $m(t^1,t^2,t^3)$ is defined only if $t^1_2 = t^2_1$ , $t^2_2 = t^3_1$ and $t^3_2 = t^1_1$ , right?

Posted by: Urs Schreiber on November 3, 2005 12:04 PM | Permalink | Reply to this

Re: Classical, Canonical, Stringy

gluing three triangles together to make a new one
I guess this operation is defined (only) on triples of triangles two of whose whose edges each match pairwise?

This is essentially correct, but one can do slightly better. In the Moyal 1-algebra, one can introduce a trace map which assigns unity to two incident lines with opposite orientation. A triangle with sides m, n and m+n then gets a phase factor exp(i hbar A(m,n)), which is independent of the starting point due to the cyclicity of the trace. Similarly, every closed polygon train, and in the limit every closed curve in momentum space, has the phase exp(i hbar A), where A is the enclosed area. We can now add closed curves with some common parts as the concatenated curve (with common parts deleted), because the phases add.

This generalized nicely to higher dimensions. Each tetrahedron gets a phase factor exp(i hbar V), which is independent of which three triangles we choose to multiply first. We can now define a phase factor exp(i hbar V) for each closed, smooth 2-manifold. This is well defined, because volumes add when we glue 2-manifolds together.

So this aspect of the Moyal algebra generalizes to higher dimensions. I don’t see why it should be useful in physics, though. A crucial property of ordinary Poisson brackets is that you can realize symmetry operations, in particular time evolution, as a canonical transformation. This property is probably lost if you higher-dimensionalize

Posted by: Thomas Larsson on November 3, 2005 5:45 PM | Permalink | Reply to this

Re: Classical, Canonical, Stringy

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Urs wrote, a while back:

“I had a vague recollection from long long ago (it seems) that C. Rovelli once thought about related things….”

Excellent! My student Miguel Carrion got really excited about this idea of Carlo Rovelli’s - it’s also described in section 3.3.2 of his book, perhaps in a more expository way. When I first heard it, it seemed too radical! Gradually I got used to it… and now I really like it!

Of course: our interest in symplectic structures as 2-forms comes from particle physics, where the action (or change in phase!) of a particle moving round a path in phase space is the integral of the 1-form $\alpha = p_i dq^i$ When we integrate this around a little loop, we get the same thing as integrating the symplectic structure $\omega = d\alpha = dp_i \wedge dq^i$ over a surface bounded by that loop.

But when we replace a particle by a higher-dimensional `brane’, or switch from particle mechanics to field theory, we should boost the dimensions of everything here!

In particular, just as the cotangent bundle $T^*M$ has a tautologous 1-form $\alpha$ on it, given in local coordinates above, just because points of $T^*M$ are 1-forms…

… the $p$ th exterior power of the cotangent bundle has a tautologous p-form $\alpha$ on it, because points in this bundle are $p$ -forms! And, the exterior derivative of this $\alpha$ is what’s analogous to the symplectic structure when we go from particle mechanics to $p$ -brane mechanics!

This is just a more mathematical way of describing some of what Rovelli is doing.

Posted by: john baez on November 4, 2005 5:21 AM | Permalink | Reply to this

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November 2, 2005