Poisson Brackets for the Closed String

String Theory without the Baggage, Episode 29

Nothing but trouble coming your way this week.

As we’ve discussed, defining the canonical momenta for any physical system amounts to choosing a symplectic structure on phase space. In practice, all this gets tied up with the actual dynamics, which is why choosing study study a system free of any interaction - that is, without a potential - can be so useful.

For the relativistic string, that’s specified by the now familiar Polyakov action¹

\(S = -\frac{\kappa}{2}\int d^{2}\sigma \, \sqrt{-\gamma}\,\gamma^{ab} \,\partial_{a}X^{\mu}\partial_{b}X_{\mu}.\)

The canonical momentum chosen involves trusting the diffeomorphic embedding X and taking the timelike worldsheet coordinate τ to represent the time-evolution of our dynamics:

\(P_{\mu} = -\kappa \sqrt{-\gamma}\,\gamma^{\tau a}\,\partial_{a}X_{\mu}.\)

In the conformal gauge, this reads:

\(P_{\mu}\rightsquigarrow \dot{X}_{\mu}.\)

The Hamiltonian for the free string in this gauge vanishes, but is related specifically to the constraints:

\(H = \frac{\lambda_{1}}{2}\left(\dot{X}^{2} + X^{\prime 2}\right) + \lambda_{2}\dot{X}^{\mu}X^{\prime}_{\mu}.\)

Lightning Aside on the Relativistic Point Particle

This covariant formulation worked reasonably well for relativistic point particle. There were some complications:

A Different Hamiltonian

but broadly speaking it was easy to assign the worldline parameter to time. From the perspective of the symplectic form, the manifestly covariant approach might have naïvely looked something like

\(\omega = dt \wedge dE + \sum_{i=1}^{3} dx^{i} \wedge dp_{i},\)

but the mass-shell constraint would augment this slightly to something like

\(\omega = \sum_{i=1}^{3} \left(\frac{p_{i}}{|\vec{p}|^{2}}dt + dx^{i}\right) \wedge dp_{i}.\)

So ultimately this still structurally akin to the usual, non-relativistic version

\(\omega = \sum_{i=1}^{3} dx^{I}\wedge p_{i}.\)

With three basic degrees of freedom, there are three independent canonical momenta.

Back to the Worldsheet

Being an extended object, the string is infinitely more complicated than a point particle. While the (noninteracting) string worldsheet is constrained to be smooth, simply connected surface, the individual bits of that surface may of course move in relatively different directions.

There are different ways to interpret this. One way is to consider the string as a collection of N point particles tied to each other with springs, something often employed in the context of field theory.

In this case, we can rethink of the σ coordinate as a vector of individual, ordered points:

\(\sigma \rightsquigarrow ( \sigma_{0} , \sigma_{1} , \sigma_{2} , \cdots , \sigma_{N-1} , \sigma_{N} ).\)

In this way, you might try to come up with N independent coordinates,

\(X^{\mu}_{n}(\tau) = X^{\mu}(\tau,\sigma = \sigma_{n}),\)

and take some kind of limit where N goes to infinity.

Alternatively, we can interpret these coordinates as Fourier modes by resuming this series. In fact, we’ve already done this:

Holomorphicity and Mode Expansions

Using these explicit mode expansion we find, for the closed string:

\(X^{\mu} = x^{\mu} + \ell^{2}p^{\mu}\tau +\frac{i\ell}{2} \sum_{n\neq 0} \frac{1}{n}e^{-2in\tau}\left( e^{2in\sigma}\alpha^{\mu}_{n} + e^{-2in\sigma}\bar{\alpha}^{\mu}_{n} \right).\)

For the open string, we have:

\(X^{\mu} = x^{\mu} + \ell^{2}p^{\mu}\tau +i\ell \sum_{n\neq 0} \frac{1}{n}e^{-in\tau}\alpha_{n}^{\mu}\cos(n\sigma).\)

We already know what the canonical momenta should be, or at least, collectively:

\(\dot{X}^{\mu} = \frac{\partial X^{\mu}}{\partial \tau} = \ell^{2} p_{\mu} +\ell \sum_{n\neq 0} e^{-2in\tau}\left(\alpha_{n}^{\mu}e^{-2in\sigma} + \bar{\alpha}^{\mu}_{n}e^{2in\sigma}\right)\)

and

\(\dot{X}^{\mu} = \ell^{2} p_{\mu} +\ell \sum_{n\neq 0} e^{-2in\tau}\alpha_{n}^{\mu}\cos(n\sigma),\)

respectively.

Clearly the configuration space is infinite-dimensional. Our task is to try to come up with some kind of symplectic form based on these variables. Specifically, we’ll operator with the mode co-efficients.

\(x^{\mu} , \;p^{\mu} ,\; \alpha^{\mu}_{n},\; \bar{\alpha}^{\mu}_{n}.\)

In the linked post above, as well as in

Global Poincaré Invariance

we’ve already seen that x corresponds to a center of mass position of the string and p is its corresponding momentum. Hence we expect to find a symplectic pairing with

\(x^{\mu} \leftrightarrow p_{\mu}.\)

Finding a symplectic pairing for the rest of these coefficients requires some work. They’re all present in the mode expansions of both X and P, so either way the associated Poisson brackets are going to be tricky to handle.

An intuitive choice is to take the reality conditions seriously,

\(\alpha^{\mu \dagger}_{n} = \alpha_{-n}^{\mu} \quad \mathrm{and}\quad \bar{\alpha}^{\mu \dagger}_{n} = \bar{\alpha}_{-n}^{\mu},\)

and identify the symplectic pairing as

\(\alpha^{\mu}_{-|n|} \leftrightarrow \alpha^{\mu}_{|n|},\quad \bar{\alpha}^{\mu}_{-|n|} \leftrightarrow \bar{\alpha}^{\mu}_{|n|}.\)

From our discussion on symplectic forms last time, we can take heart that our choice really won’t matter too much. So long we we’re consistent about choosing the form of our interactions when the time is right², we can probably find some appropriate map to get to a slightly different basis if we so choose. In any case, we’ll see how this choice of structure naturally emerges in the algebraic context of conformal field theory in due course.

Let us pick an ω:

\(\omega = \kappa_{0}\, dx^{\mu} \wedge dp_{\mu} + \sum_{n=1}^{\infty} \kappa_{n}\,d\alpha^{\mu}_{-n} \wedge d\alpha_{n \mu} +\bar{\kappa}_{m} \,d\bar{\alpha}^{\mu}_{-n} \wedge d\bar{\alpha}_{n \mu} \)

Here we have included various normalization coefficients κ_i for later use. You might notice that we haven’t employed any of the constraint equations yet, so our naive version of ω is still subject to those constraints.

Checking the Poisson Brackets

Clever, but now we must prove that this choice is valid! That is, we must use the coordinates to define a Poisson Bracket that satisfies³:

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \ell^{2}\delta^{\mu}_{\nu}\delta(\sigma-\sigma^{\prime}),\)

and of course

\(\left\{ \dot{X}^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \left\{ X^{\mu}(\tau,\sigma) , X_{\nu}(\tau,\sigma^{\prime}) \right\} = 0.\)

Notice the presence of the Dirac delta function. We are comparing the dynamical degrees of freedom at fixed points on the string. The symplectic form based on our mode-expansion coefficients must also conspire to produce this structure. Which while perhaps intuitive from the “beads tied by springs” picture isn’t necessarily obvious using the vibrational modes.

With our choice of ω, a generic Poisson Bracket for the open string is

\(\left\{ A , B \right\} = \kappa_{0}\left(\frac{\partial A}{\partial x^{\lambda}}\frac{\partial B}{\partial p_{\lambda}} - \frac{\partial B}{\partial x^{\lambda}}\frac{\partial A}{\partial p_{\lambda}} \right) + \sum_{n=1}^{\infty}\kappa_{n}\left(\frac{\partial A}{\partial \alpha^{\lambda}_{-n}}\frac{\partial B}{\partial \alpha_{n\lambda}} - \frac{\partial B}{\partial \alpha^{\lambda}_{-n}}\frac{\partial A}{\partial \alpha_{n\lambda}}\right),\)

and the closed string includes the analogous series of “barred” terms as well.

By antisymmetry we immediately see terms like {X,X} and {P,P} vanish, so the meat of this proof is to check the Poisson bracket of X and P.

The Zero Modes

The zeroth order term is easy enough to check.

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{0} = \frac{\partial X^{\mu}}{\partial x_{\lambda}}\frac{\partial \dot{X}_{\nu}}{\partial p_{\lambda}} - 0 = \delta^{\mu}_{\lambda}\delta^{\lambda}_{\nu}\ell^{2} = \ell^{2}\delta^{\mu}_{\nu}.\)

There is no σ dependence to concern ourselves with.

The Closed String σ-dependent terms

The n-th order term is that which we must compute:

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{n} =\kappa_{n}\left( \frac{\partial X^{\mu}(\tau,\sigma)}{\partial \alpha_{-n}^{\lambda}}\frac{\partial \dot{X}_{\nu}(\tau,\sigma^{\prime})}{\partial \alpha_{n\lambda}} - \frac{\partial \dot{X}^{\mu}(\tau,\sigma^{\prime})}{\partial \alpha_{-n}^{\lambda}}\frac{\partial X_{\nu}(\tau,\sigma)}{\partial \alpha_{n\lambda}} \right).\)

For the closed string, working through it we find,

\(\frac{\partial X^{\mu}(\tau,\sigma)}{\partial \alpha_{-n}^{\lambda}} = -\delta^{\mu}_{\lambda}\frac{i\ell}{2n}e^{2in(\tau-\sigma)}\)

and

\(\frac{\partial \dot{X}_{\nu}(\tau,\sigma^{\prime})}{\partial \alpha_{n\lambda}} =\delta_{\nu}^{\lambda} \ell e^{-2in(\tau-\sigma^{\prime})}.\)

With careful attention paid to minus signs and the index on n, we find for the closed string

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{n} =-\delta^{\mu}_{\nu}\frac{i\ell^{2}}{2n}\kappa_{n}(e^{2in(\sigma -\sigma^{\prime})} + e^{-2in(\sigma -\sigma^{\prime})} ) -\delta^{\mu}_{\nu}\frac{i\ell^{2}}{2n}\bar{\kappa}_{n}(e^{-2in(\sigma -\sigma^{\prime})} + e^{+2in(\sigma -\sigma^{\prime})} ).\)

Setting these constants at,

\(\kappa_{0} = 1 ,\quad \kappa_{n} = \bar{\kappa}_{n}= in,\)

these terms combine to gives cosines

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \ell^{2}\delta^{\mu}_{\nu}\left(1 + 2\sum_{n=1}^{\infty} \cos n(\sigma^{\prime}-\sigma) \right).\)

Note that we have used the fact that cosine is an even function of its argument. Resuming that series, so that

\(\delta(x) = 1 + 2\sum_{n=1}^{\infty}\cos(nx) = \sum_{n\in \mathbb{Z}} e^{inx}.\)

we find the answer we were after,

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \ell^{2}\,\delta^{\mu}_{\nu}\,\delta(\sigma-\sigma^{\prime}).\)

Note that this expression for the Dirac delta function is valid specifically for smooth functions only. Functions that satisfy looser conditions can be codified in a similar manner, but that would take us deeper into functional analysis than we should probably dare to venture at this time.

Additionally, this also assumes that the functions are defined on σ from -π to π, which is at least reasonable for the closed string given the periodic boundary conditions.

We’ll have more to say on these details much later, but hopefully this fills in some of the gaps in the computations presented in GSW.

Next time, we’ll repeat this calculation for the open string, which as one additional nuance worth punting for now.

Failed to render LaTeX expression — no expression found

1. We continue to present these familiar objects primarily as a reference for downstream computations in the present text.

2. Actually, that’s the one nice thing about String Theory. The interactions you pick are highly restricted anyway.

3. The normalization here is traditional.