The Expanded Symplectic Structure of Phase Space

String Theory without the Baggage, Episode 16

Hey Friends!

This is the last little bit we’ll need to take our first look at the BRST formalism for the relativistic point particle. Time to remember the basics of Poisson Brackets and the canonical / symplectic structure of phase space.

Sean

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From an autumnal visit to Jackson Hole. This cat has no idea how good he has it.

A Quick Recapitulation

The action for a relativistic point particle is described by it's path through R^3,1:

\(S(\gamma) = -mc\int_{\gamma} ds,\)

where

\(\gamma: \mathbb{R} \rightarrow \mathbb{R}^{3,1},\)

is some specified smooth path or wordlline.

We can make this more explicit, taking x as our coordinates in R^3,1,

\(S = -mc \int_{\mathbb{R}}ds\, \sqrt{\eta_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}}.\)

Here dots represent derivatives with respect to s, which is an explicit parameter on 𝛾, which we now take concretely as a real parameter. We are taking a “mostly minus” signature convention mostly just to annoy the relativists.

We define

\(p_{\mu} = \frac{\delta}{\delta \dot{x}^{\mu}}\left(-mc \sqrt{\dot{x}^{\nu}\dot{x}_{\nu}} \right) = -\frac{mc\dot{x}_{\mu}}{\sqrt{|\dot{x}|^{2}}},\)

which plays the role of a momentum coordinate for x.

We may then write our Hamiltonian as

\(H = p_{\mu}\dot{x}^{\mu} + mc \sqrt{|\dot{x}|^{2}}.\)

Careful expansion of these variables here reveals that H vanishes.

Our experience with using time as the explicit parameter allowed us to derive the mass-shell condition,

\(\eta^{\mu\nu}p_{\mu}p_{\nu} - (mc)^{2}= 0,\)

which we now remember that have to impose by hand. Hence,

\(L \rightarrow L - \frac{\lambda}{2}\left( p_{\mu}p^{\mu} - (mc)^{2}\right),\)

so that

\(H = \frac{\lambda}{2}\left( p_{\mu}p^{\mu} - (mc)^{2}\right).\)

Remember, this is the Hamiltonian used to define dynamics on the covariant phase space, T*R^3,1.

The main issue here is that this system is invariant under reparameterizations of the worldline. That is, diffeomorphisms of R. That is, we are faced with a Gauge Theory, which makes passing to the Quantum Theory complicated. Specifically, we have to fix a gauge to avoid overcounting paths in the path integral.

Imposing the gauge 𝜆 = 1 amounts to including additional, anticommuting ghost fields, 𝜃 ̄ and 𝜃, where now the action in first order form is

\(S = \int ds\; p_{\mu}\dot{x} - \frac{\lambda}{2}\left(p_{\mu}p^{\mu} - (mc)^{2}\right) + i\bar{\theta}\dot{\theta}.\)

We will keep 𝜆 around for show, but remember that this action itself was derived from the Path Integral with a delta function¹.

The Symplectic Structure

Let's rederive the Hamiltonian now inclusive of the ghosts. There is no canonical momentum associated to 𝜃 ̄, but there is for 𝜃,

\(p_{\theta} = \frac{d}{d\dot{\theta}}\left( i\bar{\theta}\dot{\theta}\right).\)

Here it is absolutely crucial to observe that the derivative operator is a Grassmann value, and so anticommutes with 𝜃 ̄, so that

\(p_{\theta} = -i\bar{\theta},\)

Hence

\(H = \frac{\lambda}{2}\left(p_{\mu}p^{\mu} - (mc)^{2}\right) + p_{\theta}\dot{\theta} - i\bar{\theta}\dot{\theta}.\)

That crucial minus sign results in our full Hamiltonian.

\(H = \frac{\lambda}{2}\left(p_{\mu}p^{\mu} - (mc)^{2}\right) -2i\bar{\theta}\dot{\theta}.\)

We can now compute some Poisson brackets to verify the symplectic structure of our combined system.

\(\left\{x^{\mu},H\right\} = \frac{\partial x^{\mu}}{\partial x^{\lambda}} \frac{\partial H}{\partial p_{\lambda}} - \frac{\partial x^{\mu}}{\partial p_{\lambda}} \frac{\partial H}{\partial x^{\lambda}} = \lambda p^{\mu}.\)

This is consistent with our first order picture. Great. Next.

\(\left\{p_{\mu},H\right\} = \frac{\partial p_{\mu}}{\partial x^{\lambda}} \frac{\partial H}{\partial p_{\lambda}} - \frac{\partial p_{\mu}}{\partial p_{\lambda}} \frac{\partial H}{\partial x^{\lambda}} = 0.\)

H just doesn't depend on x, explicitly.

Now for the ghosts, Because the canonical momentum for 𝜃 is −𝑖𝜃 ̄, we'll just use 𝜃 ̄ explicitly, and rescale its associated symplectic form with an 2i.

\(\left\{ \theta , H \right\} = \frac{1}{2i}\frac{\partial\theta}{\partial\theta}\frac{\partial H}{\partial \bar{\theta}} - \frac{1}{2i}\frac{\partial\theta}{\partial\bar{\theta}}\frac{\partial H}{\partial {\theta}} = \dot{\theta},\)

which gives us the conventional normalization. Similarly,

\(\left\{ \bar{\theta} , H \right\} = 0,\)

as H does not depend on 𝜃 explicitly².

We should note that despite the fact that we appeared to separate the Poisson brackets between particle and the ghost sectors, we simply dropped vanishing terms in our computation. In principle one must sum over all symplectic pairs:

\(\left\{A , B\right\} = \frac{\partial A}{\partial x^{\mu}}\frac{\partial B}{\partial p_{\mu}} - \frac{\partial A}{\partial p_{\mu}}\frac{\partial B}{\partial x^{\mu}} + \frac{1}{2i}\left(\frac{\partial A}{\partial\theta} \frac{\partial B}{\partial \bar{\theta}} - \frac{\partial A}{\partial\bar{\theta}} \frac{\partial B}{\partial {\theta}} \right).\)

Notice also the implicit sum over spacetime indices. The other thing you should probably be aware of is that, unless otherwise specified all derivatives act from the left.

And with that, we’re ready to introduce the concepts of BRST quantization. It won’t fully have the character that you might see for a nonabelian Lie algebra - like SU(3) - but it should help us get on board with what we need for the string.

1. Some authors include an additional, auxiliary field in this action to enforce 𝜆 = 1, but this is both confusion and superfluous, since the form of the ghost action itself depends on this choice of gauge.

2. This is also consistent with 𝜃 ̄ not having a time-derivative in the action.