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Extended objects

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Extended objects are generalizations of the concept of point particles, such that, instead of being a $0$-dimensional object moving through space, we have $p$-dimensional objects, such that $p < n$, the spacetime dimension. In modern parlance, the $p$-dimensional object is called a $p$-brane, while the volume it spans moving through time is called its world-volume.

As such, in a relativistic setting (to which we will stick), extended objects describe a $p+1$ submanifold $\Sigma$ of the spacetime $M$. In other words, for some $(p+1)$-manifold, there exists an inclusion function

\begin{equation} X : \Sigma \to M \end{equation}

For objects to be reasonable physical entities, we'll also ask that this submanifold be timelike or null. While it is perfectly well defined to have a spacelike submanifold, this would correspond to a tachyonic object, which are not usually considered. So we require that the normal $(n-p-1)$-form be spacelike (or null) everywhere.

1. Currents

Before we get into anything regarding extended objects, we need to define currents first. As our objects are basically singular objects (defined on sets of measure zero), we need to define first how we'd deal with the integrals of such objects.

Much like for anything regarding distributions, we need to define test functions first.

Therefore, a distribution is simply a current of degree $N$, with the mapping

\begin{equation} \langle T, \phi \rangle = \langle T, \phi dx_1 \wedge \ldots \wedge dx_n \rangle \end{equation}

And just as for distribution, for any $p$-form $T$, there corresponds a current of degree $p$ via the map

\begin{equation} \langle T, \omega \rangle = \int_M T \wedge \omega \end{equation}

Integration over a subset of $M$ also defines a current of degree $0$. Consider some open set $U \subset M$. The integration current $[U]$ is

\begin{equation} \langle [U], \omega \rangle = \int_M \omega \end{equation}

for $\omega$ an $n$-form. More generally, given a map $f : N \to M$ from $N$ a manifold of dimension $n - p$, and $U \subset N$, $U' = f(U) \subset M$ two open sets, we have

\begin{equation} \langle [U'], \omega \rangle = \int_{U'} f^*(\omega) \omega \end{equation}

for $\omega$ an $(n-p)$-form on $N$.

Wedge product : for $\alpha \in \mathscr{D}^{n-p-q}$, $\beta \in \mathscr{D}^{q}$

\begin{equation} \langle T \wedge \alpha, \omega \rangle = \langle T , \alpha \wedge \omega \rangle \end{equation}

Exterior derivative : for $\alpha \in \mathscr{D}^{n-p-1}$

\begin{equation} \langle d T, \omega \rangle = (-1)^{p+1} \langle T, d\omega \rangle \end{equation}

Pushforward : for a map $f : N \to M$, $N$ of dimension $m > p$, for $\alpha \in \mathscr{D}^{n-p}$

\begin{equation} \langle f_* T, \omega \rangle = \langle T, f^*(\omega) \rangle \end{equation}

Boundary operator : for $\partial U$ the boundary of $U$, take the

2. Action of extended objects

As for any physical theory, we require some action for those objects to predict their dynamic. There are quite a number of equivalent ways to describe them, but the two main one are the Nambu-Goto action and the Polyakov action.

1. Harmonic maps

Before getting into specific actions for extended objects, we need to study harmonic maps. Let's consider two manifolds equipped with metrics, $(M, \gamma)$ and $(N, g)$. Coordinates on $M$ will have latin indexes while $N$ will have greek ones. We also define

\begin{equation} f : M \to N \end{equation}

a $C^1$ map between the two.

2. The Nambu-Goto action

The Nambu-Goto action is roughly defined by the volume of the object. That is, for an embedding $X$, we have the induced metric defined by the pushforward

\begin{equation} h = X_* g \end{equation}

And the action is then defined by the volume of the world-volume.

\begin{equation} S[X, g] = T \int_\Sigma d\mu[X_* g] \end{equation}

$T$ here is some factor corresponding to some characteristic of the object, called the brane tension. It will correspond for points and strings to the mass and tension, respectively. In coordinate form, this is

\begin{equation} S[X, g] = T \int_\Sigma \sqrt{\det \left[g_{\mu\nu}(X(\sigma)) (\partial_a X^\mu(\sigma)) (\partial_b X^\nu(\sigma)) \right]} d^p\sigma \end{equation}

If we consider $X$ for now, the variation will give us

\begin{equation} \int_\Sigma \left[\frac{\partial \mathcal{L}}{\partial X^\mu} \delta X^\mu + \frac{\partial \mathcal{L}}{\partial (\partial_a X^\mu)} \delta \partial_a X^\mu\right] d^p\sigma \end{equation} \begin{eqnarray} \frac{\partial \mathcal{L}}{\partial X^\mu} &=& \frac{\partial \sqrt{h}}{\partial h} \frac{\partial h}{h_{cd}} \frac{\partial h_{cd}}{\partial X^\mu}\\ &=& \frac{1}{2} \sqrt{h} h^{cd} g_{\alpha\beta,\mu}(X(\sigma)) (\partial_c X^\alpha(\sigma)) (\partial_d X^\beta(\sigma)) \end{eqnarray} \begin{eqnarray} \frac{\partial \mathcal{L}}{\partial (\partial_a X^\mu)} &=& \frac{\partial \sqrt{h}}{\partial h} \frac{\partial h}{h_{cd}} \frac{\partial h_{cd}}{\partial (\partial_a X^\mu)}\\ &=& \frac{1}{2} \sqrt{h} h^{cd} g_{\alpha\beta}(X(\sigma)) \frac{\partial }{\partial (\partial_a X^\mu)} (\partial_c X^\alpha(\sigma)) (\partial_d X^\beta(\sigma))\\ &=& \frac{1}{2} \sqrt{h} h^{cd} g_{\alpha\beta}(X(\sigma)) [\delta^a_{c} \delta^{\alpha}_\mu (\partial_d X^\beta(\sigma)) + \delta^a_{d} \delta^\beta_\mu (\partial_c X^\alpha(\sigma))]\\ &=& \end{eqnarray}

Now from the formulas of variation, we know that

\begin{equation} \int_\Sigma V^a \delta \partial_a X^\mu d\sigma = - \int_\Sigma \delta X^\mu \partial_a V^a d\sigma + \int_{\partial \Sigma} \delta X V^a \nu_a \end{equation}

with $\nu$ the normal vector to the boundary of $\Sigma$. There are two boundaries which concern us here : first, we will not usually integrate the Lagrangian over the whole worldsheet of our object (it's easy enough to show, using a free point particle for instance, that this integral may diverge), so that we'll integrate between two spacelike hypersurfaces $S_1$ and $S_2$, the boundaries then being $\Sigma \cap S_1$ and $\Sigma \cap S_2$, and the worldsheet itself may have boundaries, such as the case of the open string or some non-compact membrane. The first boundary isn't of much concern to us, as it is the usual case for a Lagrangian, and as we will only consider variations of the field that leave the boundaries invariant, $\delta X$ does vanish here and therefore the usual Euler-Lagrange equation is recovered here. This will be therefore true for any brane without boundaries, such as a point particle or closed string.

If the brane does have boundaries, we will have to require some boundary conditions on them as well. There's two types of boundary conditions we can apply to $X$ here :

• The Neumann boundary condition, where the velocity of the boundary vanishes : $\partial_a X^\mu(\partial \Sigma) = 0$
• The Dirichlet boundary condition, where the position of the brane is constrained at the boundary, much like for the timelike boundary : $\delta X^\mu = 0$

It is possible to mix the boundary conditions for every dimension $\mu$, so that the boundary is Neumann for $p$ dimensions and Dirichlet for $D-p$ dimensions. In which case we say that the

The Polyakov action

The Polyakov action is defined by the harmonic map of the inclusion $X$. Given some map $X : \Sigma \to M$ between two manifolds, with metrics $\gamma$ on $\Sigma$ and $g$ on $M$, we define the norm

\begin{equation} \| dX \|^2 = \langle dX, dX \rangle_{T^*\Sigma \otimes X^{-1} TM} \end{equation} \begin{equation} S[X, \gamma] = \int_\Sigma \| dX \| d\mu[\gamma] \end{equation}

As we will see later on, there is an interpretation of this action as the action of a scalar field theory on a $(1+1)$-dimensional spacetime. To make it more complete, and add terms which will be of use to us for some cases, we will also add the Einstein-Hilbert action equivalent to this :

\begin{equation} S[X, \gamma, g] = \int_\Sigma (\| dX \| + R - \Lambda) d\mu[\gamma] \end{equation}

The curvature part of this action will not usually be of use. For point-particles in $(0+1)$-dimensions, it will be zero, and for strings in $(1+1)$-dimensions, it will only be a constant, although for the quantum theory of string this will act as a weight on the different string topologies. But the cosmological constant here will act as a mass term for our point particles.

The coordinate form of this action will be

\begin{equation} S[X, \gamma, g] = \int_\Sigma (h^{ab}(\sigma)g_{\mu\nu}(X) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma) + R(\sigma) - \Lambda) \sqrt{-\gamma} d\sigma^p \end{equation}

Here we have two fields to consider for the equations of motion. For $X$, only the first part of the Lagrangian matters, and we get

\begin{equation} \frac{\partial \mathcal{L}}{\partial X^\mu} = \end{equation}

Hamiltonian and constraints

The study of extended objects is complicated by its gauge symmetries, which require the use of Hamiltonian constraints. The most important symmetry of both actions is diffeomorphism invariance. For the Nambu-Goto action, this is given by the diffeomorphism $f$

\begin{equation} \sigma \to \sigma' = f(\sigma) \end{equation}

which transforms the measure as

\begin{equation} d\sigma \to d\sigma' = f'(\sigma) d\sigma \end{equation}

...

There are many more possible symmetries, depending on the specific objects we study and the symmetries of the target manifold. Two important cases are the Lorentz invariance and Weyl invariance. If our manifold is simply Minkowski space, the action is obviously invariant under a transformation

\begin{equation} X^\mu \to X'^\mu = \Lambda^\mu_\nu X^\nu \end{equation}

...

Once given all that, let's now consider the Hamiltonian transformation of our actions.

Nambu-Goto Hamiltonian

Let's consider first the momentum $P$ of our object.

\begin{eqnarray} P_\mu = \frac{\partial \mathcal{L}}{\partial \dot{X}^\mu} \end{eqnarray}

Polyakov Hamiltonian

Point particles

The simplest case of extended objects is the $0+1$-dimensional case, which is the movement of a point particle. The Nambu-Goto action corresponds to the classic action of a point particle. In coordinates,

\begin{equation} S[X] = -m \int_{t_1}^{t_2} \sqrt{g_{\mu\nu}\dot{X}^\mu(\tau)\dot{X}^\nu(\tau)} \end{equation}

Where the factor $m$ corresponds to the mass of the particle as we'll see. The Euler-Lagrange equation for this becomes

\begin{equation} x \end{equation}

...

For $0$-particles in Minkowski space, we have the usual free particle equation

\begin{equation} \ddot{X}^\mu(\tau) = 0 \end{equation}

with solution

\begin{equation} X^\mu(\tau) = V_0^\mu \tau + X_0^\mu \end{equation}

Strings

The other common extended object is the $1+1$ dimensional case for the movement of strings.

\begin{equation} S[X] = - T \iint_\Sigma d\tau d\sigma \sqrt{g_{\mu\nu} \partial_b X^\mu(\sigma, \tau) \partial_a X^\nu(\sigma, \tau)} \end{equation}

The equation of motion in Minkowski space is, for the embedding map $X$,

\begin{equation} \partial_\alpha (\sqrt{-\gamma} \gamma^{\alpha\beta} \partial_\beta X^\mu)= 0 \end{equation}

For the submanifold metric $\gamma$, we end up with roughly the same situation as general relativity in two dimensions, varying the action with respect to the metric. As the Einstein tensor is identically zero, this gives us

\begin{eqnarray} \frac{\delta S}{\delta \gamma_{\alpha\beta}} &=& 0\\ &=& -\frac{T}{2} \sqrt{-g} T_{\mu\nu} \end{eqnarray}

While things look bad, we have a few elements in our favor : strings have a lot of symmetries we can exploit, and as a string on a Cauchy surface is simply a map from $[a,b] \to \mathbb{R}^3$, we can define it as a Fourier series.

Conformal gauge :

\begin{equation} \gamma_{\alpha\beta} = e^{2\phi(\sigma, \tau)} \eta_{\alpha\beta} \end{equation}

Weyl transformation :

\begin{equation} \gamma_{\alpha\beta} = \eta_{\alpha\beta} \end{equation}

with those transformations, we are left with the wave equation

\begin{equation} \Box X^\mu = 0 \end{equation}

3. Extended objects in general relativity

Extended objects are used in general relativity for a variety of purpose, from modelling point particles to modelling idealized topological defects (such as cosmic strings and domain walls) to studying thin-shell metrics.

To study this, we'll need first to recast the action under a more suitable form to actually be an action over the whole spacetime manifold, and not just the worldvolume. To do this we're gonna need to remove the dependance of the metric on the particle position, by the use of a Dirac distribution :

\begin{equation} S[X] = -m \int_M d^p\sigma d^nx \delta^{(n)}(x^\alpha - X^\alpha(\sigma)) \sqrt{\det\left[ g_{\mu\nu}(x)\partial_a X^\mu(\sigma)\partial_b X^\nu(\sigma) \right]} \end{equation}

Then we can simply compute the stress-energy tensor the usual way :

4. Supersymmetry

As with most actions, it is possible to extend the action of extended objects to a supersymmetric version.

5. Quantization of extended objects

As with most Lagrangian systems, extended objects can be quantized. This is principally useful in $0+1$ and $1+1$ dimensions, for relativistic quantum mechanics (which can give rise to the worldline formalism) and string theory.

Canonic quantization

The canonical quantization of string theory can be performed in several ways, depending on the action we choose and the field operators considered.

Nambu-Goto quantization

The first choice we have is to simply consider the Nambu-Goto action with the most basic field operators $X$ and $P$, with the Poisson brackets

\begin{equation} \left\{ X^\mu, P^\nu \right\} = \end{equation}

Light cone quantization

Polyakov quantization

Path integral quantization

Quite commonly done for extended objects, we'll just consider the usual path integral

\begin{equation} \langle T \left\{ \mathcal{O}^1(x_1) \mathcal{O}^2(x_2) \ldots \right\} \rangle = \int \mathcal{D}\phi(x) e^{iS} \end{equation}

Nambu-Goto quantization

Polyakov quantization

Bibliography

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Last updated : 2020-03-09 15:51:58