Relativistic Particles, Branes, and the Birth of the String

String theory begins with a geometric idea: the action of a relativistic object should be proportional to the invariant volume swept out by that object. A point particle sweeps out a worldline, a string sweeps out a worldsheet, and a $p$-brane sweeps out a $(p+1)$-dimensional worldvolume. The first surprise is that this innocent principle already contains many of the ingredients that later become central: gauge redundancy, constraints, massless worldvolume fields, and universal long-distance corrections.

We use mostly-plus target-space signature,

$$G_{\text{flat}}{}_{\mu\nu}=\eta_{\mu\nu}=\operatorname{diag}(-,+,\ldots,+),$$

and write

$$\dot X^\mu = \partial_\tau X^\mu, \qquad X^{\prime\mu}=\partial_\sigma X^\mu.$$

For a target-space vector product we use

$$A\cdot B=G_{\mu\nu}(X)A^\mu B^\nu,$$

or $A\cdot B=\eta_{\mu\nu}A^\mu B^\nu$ in flat spacetime. Natural units $\hbar=c=1$ are used throughout.

The relativistic particle as a worldline theory

A free massive particle moving from $x_i$ to $x_f$ traces a curve $X^\mu(\tau)$ in spacetime. The parameter $\tau$ is arbitrary; it is merely a coordinate on the curve. The physical length of a timelike curve is

$$ds = d\tau\,\sqrt{-G_{\mu\nu}(X)\dot X^\mu\dot X^\nu},$$

so the reparametrization-invariant action is

$$S_{\text{pp}}[X] = -m\int ds = -m\int d\tau\,\sqrt{-G_{\mu\nu}(X)\dot X^\mu\dot X^\nu}.$$

The subscript $\text{pp}$ stands for point particle. This is the one-dimensional prototype of the Nambu—Goto action. Its value is the invariant length of the worldline times the particle mass.

The square-root action has a useful immediate consequence. The canonical momentum is

$$p_\mu={\partial L\over \partial \dot X^\mu} ={mG_{\mu\nu}\dot X^\nu\over \sqrt{-\dot X^2}}, \qquad \dot X^2=G_{\mu\nu}\dot X^\mu\dot X^\nu.$$

Therefore

$$G^{\mu\nu}p_\mu p_\nu + m^2=0.$$

This is not an equation of motion in the usual Newtonian sense; it is a constraint. It appears because the parameter $\tau$ has no physical meaning. Reparametrization invariance removes one apparent degree of freedom and forces the particle onto the relativistic mass shell.

The canonical Hamiltonian vanishes identically:

$$H=p_\mu\dot X^\mu-L=0.$$

This is another way to say that evolution in $\tau$ is gauge motion. A re-labeling of points on the same worldline should not change the physics.

The einbein form and the proper-time propagator

The square root is geometrically transparent but awkward in a path integral. A standard trick is to introduce a one-dimensional metric, or einbein, $e(\tau)$, and use the classically equivalent action

$$S[X,e] ={1\over 2}\int d\tau\, \left(e^{-1}G_{\mu\nu}(X)\dot X^\mu\dot X^\nu-e m^2\right).$$

Varying with respect to $e$ gives

$$0={\delta S\over \delta e} =-{1\over 2}e^{-2}\dot X^2-{1\over 2}m^2,$$

so

$$e={\sqrt{-\dot X^2}\over m}$$

for a future-directed timelike trajectory. Substituting this solution back into $S[X,e]$ gives exactly $S_{\text{pp}}[X]$. The einbein is thus an auxiliary field: it has no propagating degree of freedom, but it makes the action quadratic in $\dot X^\mu$.

The worldline path integral for a scalar particle is schematically

$$G_F(x_f,x_i) =\int {\mathcal D X\,\mathcal D e\over \operatorname{Diff}} \exp\left(iS[X,e]\right),$$

with boundary conditions

$$X^\mu(0)=x_i^\mu, \qquad X^\mu(1)=x_f^\mu.$$

Because one-dimensional metrics have no local geometry, gauge fixing leaves only one modulus: the total proper time

$$T=\int d\tau\,e(\tau).$$

One may choose $e(\tau)=T$ on the interval $0\leq \tau\leq 1$. In flat spacetime this produces a Gaussian path integral over $X^\mu$, together with an ordinary integral over $T$. The result is the Schwinger proper-time representation of the scalar propagator,

$$G_F(x_f,x_i) =\int_0^\infty dT\int_{X(0)=x_i}^{X(1)=x_f}\mathcal D X\, \exp\left[{i\over 2}\int_0^1 d\tau\, \left(T^{-1}\dot X^2-Tm^2\right)\right]$$

and, after evaluating the Gaussian integral,

$$G_F(x_f,x_i) =\int {d^Dp\over (2\pi)^D} {i\,e^{ip\cdot(x_f-x_i)}\over p^2+m^2-i\epsilon}.$$

This calculation is worth remembering. Later, the string path integral will be a two-dimensional version of the same story. The point-particle modulus $T$ becomes the moduli of Riemann surfaces, and the mass-shell constraint becomes the Virasoro constraints.

A point particle sweeps out a worldline, a string sweeps out a worldsheet, and a membrane sweeps out a three-dimensional worldvolume. The embedding fields $X^\mu(\sigma)$ tell us where each point of the object sits in spacetime.

From worldlines to worldvolumes

A $p$-brane is a $p$-dimensional spatial object. Its history is a $(p+1)$-dimensional worldvolume $\Sigma_{p+1}$ with coordinates

$$\sigma^\alpha=(\sigma^0,\sigma^1,\ldots,\sigma^p), \qquad \sigma^0=\tau, \qquad \alpha=0,1,\ldots,p.$$

The brane is embedded into a $D$-dimensional spacetime by maps

$$X^\mu=X^\mu(\sigma^0,\ldots,\sigma^p), \qquad \mu=0,1,\ldots,D-1.$$

The spacetime metric induces a metric on the worldvolume:

$$h_{\alpha\beta} =G_{\mu\nu}(X)\,\partial_\alpha X^\mu\partial_\beta X^\nu.$$

This is the pullback of $G_{\mu\nu}$ to the brane. The invariant worldvolume element is

$$d\operatorname{Vol}_{p+1} =d^{p+1}\sigma\,\sqrt{-\det h_{\alpha\beta}},$$

where the minus sign is appropriate for Lorentzian signature, since the worldvolume has one time direction. The natural geometric action is therefore

$$S_p=-T_p\int_{\Sigma_{p+1}} d^{p+1}\sigma\, \sqrt{-\det h_{\alpha\beta}}.$$

The constant $T_p$ is the tension, or energy per unit spatial $p$-volume. In units $\hbar=c=1$,

$$[T_p]=\text{mass}^{p+1}=\text{length}^{-(p+1)}.$$

Special cases are important:

object	spatial dimension	worldvolume	action coefficient
point particle	$p=0$	worldline	$T_0=m$
string	$p=1$	worldsheet	$T_1=T$
membrane	$p=2$	three-dimensional worldvolume	$T_2$

The particle action is precisely the $p=0$ case:

$$S_0=-T_0\int d\sigma^0\sqrt{-h_{00}} =-m\int d\tau\sqrt{-G_{\mu\nu}\dot X^\mu\dot X^\nu}.$$

For a fundamental string one usually writes

$$T={1\over 2\pi\alpha'}, \qquad \ell_s=\sqrt{\alpha'}, \qquad M_s={1\over \sqrt{\alpha'}}.$$

The parameter $\alpha'$ has dimensions of length squared. It sets the intrinsic string length scale and, as we shall see, the slope of Regge trajectories.

Open and closed strings

For a string, the worldvolume is two-dimensional, with coordinates

$$\sigma^a=(\tau,\sigma), \qquad a=0,1.$$

There are two basic possibilities.

For an open string, $\sigma$ ranges over an interval, often chosen as

$$0\leq \sigma\leq \pi.$$

The worldsheet has two boundaries, one at each endpoint of the string. Boundary conditions at these endpoints will later become one of the main engines of the subject: Neumann boundary conditions lead to momentum flow along a brane, while Dirichlet boundary conditions pin the string endpoint to a D-brane.

For a closed string, $\sigma$ is periodic,

$$\sigma\sim \sigma+2\pi,$$

so the embedding obeys

$$X^\mu(\tau,\sigma+2\pi)=X^\mu(\tau,\sigma)$$

in a noncompact target-space direction. In a compact target-space direction, this equation can be modified by winding; that is the beginning of T-duality, but for now we keep the target spacetime noncompact.

An open string sweeps out a strip with boundaries at its endpoints. A closed string sweeps out a cylinder, with $\sigma$ periodically identified.

The choice $0\leq\sigma\leq\pi$ or $0\leq\sigma\leq 2\pi$ is a convention. What matters is the combination of the coordinate range with the normalization of the tension and the oscillator modes. Silent changes of this convention are a common source of stray factors of $2$.

The Nambu—Goto action

For $p=1$, the induced worldsheet metric is

$$h_{ab}=G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu.$$

The string action is the area of the worldsheet times the tension:

$$S_{\text{NG}} =-T\int d\tau d\sigma\,\sqrt{-h}, \qquad h=\det h_{ab}.$$

In flat spacetime,

$$h_{ab} = \begin{pmatrix} \dot X^2 & \dot X\cdot X' \\ \dot X\cdot X' & X'^2 \end{pmatrix},$$

so

$$-h=(\dot X\cdot X')^2-\dot X^2X'^2.$$

Thus

$$S_{\text{NG}} =-T\int d\tau d\sigma\, \sqrt{(\dot X\cdot X')^2-\dot X^2X'^2}.$$

This is the direct two-dimensional analogue of the relativistic particle action. The particle action measures the length of a curve; the Nambu—Goto action measures the area of a surface.

The action has two obvious spacetime symmetries in flat space:

$$X^\mu\to X^\mu+a^\mu, \qquad X^\mu\to \Lambda^\mu{}_{\nu}X^\nu,$$

and one essential worldsheet gauge symmetry:

$$(\tau,\sigma)\to (\widetilde\tau(\tau,\sigma),\widetilde\sigma(\tau,\sigma)).$$

The last symmetry is not optional. It says that the coordinates painted on the worldsheet are not physical. Only the image of the surface in spacetime is physical.

There is a price for this geometric clarity: the square root makes quantization difficult. The next page introduces the Polyakov action, where an auxiliary worldsheet metric removes the square root and makes the two-dimensional gauge symmetries manifest. Before doing that, it is useful to see what the Nambu—Goto action says in a physical gauge.

Static gauge and transverse fluctuations

Consider a long, nearly straight string stretched in the $X^1$ direction. Let its length be $L$, and choose static gauge

$$X^0=\tau, \qquad X^1=\sigma, \qquad 0\leq\sigma\leq L.$$

The remaining coordinates are transverse displacements,

$$X^i=Y^i(\tau,\sigma), \qquad i=2,\ldots,D-1.$$

In this gauge,

$$\dot X^2=-1+\dot Y^i\dot Y^i, \qquad X'^2=1+Y^{\prime i}Y^{\prime i}, \qquad \dot X\cdot X'=\dot Y^iY^{\prime i}.$$

Therefore

$$-h =(1-\dot Y^2)(1+Y'^2)+(\dot Y\cdot Y')^2,$$

where

$$\dot Y^2=\sum_{i=2}^{D-1}\dot Y^i\dot Y^i, \qquad Y'^2=\sum_{i=2}^{D-1}Y^{\prime i}Y^{\prime i}.$$

For small slopes and velocities, expand the square root:

$$\sqrt{-h} =1+{1\over 2}(Y'^2-\dot Y^2)+O((\partial Y)^4).$$

The Nambu—Goto action becomes

$$S_{\text{NG}} =-TL\int d\tau +{T\over 2}\int d\tau\int_0^L d\sigma\, \sum_{i=2}^{D-1} \left[(\partial_\tau Y^i)^2-(\partial_\sigma Y^i)^2\right] +O((\partial Y)^4).$$

The first term is simply the classical energy $TL$ of a stretched string. The second term is the action for $D-2$ massless scalar fields on the worldsheet. These are the transverse oscillations of the string.

In static gauge, a long string stretched along $X^1$ is described by its transverse displacement fields $Y^i(\tau,\sigma)$. The physical low-energy modes are transverse; the longitudinal motion has been removed by reparametrization invariance.

This is a crucial result. The string does not carry independent longitudinal oscillations. A longitudinal ripple can be removed by changing the coordinate $\sigma$ along the string. The physical fluctuations are transverse because they change the actual shape of the embedded surface.

Equivalently, a straight string breaks the target-space translations transverse to it. The fields $Y^i$ are the corresponding Goldstone modes on the worldsheet. For a general $p$-brane in static gauge,

$$X^\alpha=\sigma^\alpha, \qquad X^i=Y^i(\sigma), \qquad \alpha=0,\ldots,p, \qquad i=p+1,\ldots,D-1,$$

there are

$$D-p-1$$

transverse scalar fields. For strings, this gives $D-2$.

It is often useful to introduce canonically normalized transverse fields

$$\Phi^i=\sqrt{T}\,Y^i.$$

Then the quadratic action is

$$S_2={1\over 2}\int d\tau d\sigma\, \sum_i\left[(\partial_\tau\Phi^i)^2-(\partial_\sigma\Phi^i)^2\right],$$

while the interactions are suppressed by powers of $1/T$. A large tension makes the string stiff; a small tension makes the string floppy.

Confining flux tubes as effective strings

The Nambu—Goto action is not only a model for fundamental strings. It is also the universal long-distance action for many string-like objects. A particularly important example is a confining flux tube between a heavy quark and antiquark. At large separation $L$, the color-electric flux cannot spread freely through space; it is squeezed into a tube. The leading energy is linear,

$$V(L)=T_{\text{conf}}L+\cdots,$$

where $T_{\text{conf}}$ is the confining string tension.

The microscopic flux tube is generally a fat string: it has a thickness set by the confinement scale. A fundamental perturbative string is instead thin at distances large compared with $\ell_s$. But at distances much larger than the thickness, both are governed by the same symmetry logic. The only exactly massless modes are the transverse Goldstone fields, so the effective action begins with the static-gauge Nambu—Goto form.

Quantizing the $D-2$ transverse massless fields gives the leading universal correction to the potential of a long open string with fixed endpoints:

$$V(L)=T_{\text{conf}}L+\mu-{\pi(D-2)\over 24L}+O(L^{-3}).$$

The constant $\mu$ depends on short-distance physics near the endpoints. The $1/L$ term is the Lüscher term. Its coefficient is universal as long as the only massless worldsheet fields are the transverse translations.

This is the first example of a theme that runs through the whole subject: a string action can be fundamental, or it can be an effective long-distance description of a more microscopic theory. In either case, the worldsheet viewpoint organizes the physics by symmetry, topology, and fluctuations.

What to remember

The main points of this page are compact but foundational.

idea	formula	meaning
relativistic particle	$S=-m\int ds$	the worldline action is invariant length times mass
einbein action	$S={1\over 2}\int d\tau(e^{-1}\dot X^2-em^2)$	makes the particle path integral Gaussian in $X^\mu$
$p$-brane action	$S_p=-T_p\int d^{p+1}\sigma\sqrt{-\det h}$	the action is invariant worldvolume times tension
string tension	$T=1/(2\pi\alpha')$	$\alpha'$ sets the string length and mass scales
Nambu—Goto action	$S_{\text{NG}}=-T\int d^2\sigma\sqrt{-h}$	the string sweeps out a minimal-area surface
static-gauge spectrum	$D-2$ massless scalars	the physical low-energy oscillations are transverse

The next step is to replace the square-root Nambu—Goto action with the Polyakov action. That replacement is classically equivalent, but it exposes the two-dimensional gauge structure that makes string quantization possible.

Exercises

Exercise 1: mass-shell constraint from the square-root action

Starting from

$$S_{\text{pp}}=-m\int d\tau\sqrt{-\dot X^2},$$

compute the canonical momentum and show that it obeys $p^2+m^2=0$. Then show that the canonical Hamiltonian vanishes.

Solution

The Lagrangian is

$$L=-m\sqrt{-\dot X^2}.$$

The momentum is

$$p_\mu={\partial L\over \partial \dot X^\mu} ={m\dot X_\mu\over \sqrt{-\dot X^2}}.$$

Therefore

$$p^2=p_\mu p^\mu ={m^2\dot X^2\over -\dot X^2} =-m^2,$$

so

$$p^2+m^2=0.$$

The Hamiltonian is

$$H=p_\mu\dot X^\mu-L ={m\dot X^2\over \sqrt{-\dot X^2}}+m\sqrt{-\dot X^2}=0.$$

This vanishing is a consequence of reparametrization invariance: $\tau$-evolution is gauge evolution.

Exercise 2: eliminating the einbein

Show that the einbein action

$$S[X,e]={1\over 2}\int d\tau\left(e^{-1}\dot X^2-em^2\right)$$

is classically equivalent to the square-root particle action for a timelike trajectory.

Solution

Varying the action with respect to $e$ gives

$${\delta S\over \delta e} =-{1\over 2}e^{-2}\dot X^2-{1\over 2}m^2=0.$$

Thus

$$e^2=-{\dot X^2\over m^2}.$$

For a timelike path, $\dot X^2<0$, and choosing the positive einbein gives

$$e={\sqrt{-\dot X^2}\over m}.$$

Substitution into the action gives

$$S[X,e_{\text{cl}}] ={1\over 2}\int d\tau\left( {m\dot X^2\over \sqrt{-\dot X^2}} -m\sqrt{-\dot X^2} \right) =-m\int d\tau\sqrt{-\dot X^2}.$$

Hence the two actions are classically equivalent.

Exercise 3: determinant of the induced string metric

For a string in flat spacetime, show that

$$-h=(\dot X\cdot X')^2-\dot X^2X'^2.$$

Then impose static gauge $X^0=\tau$, $X^1=\sigma$, $X^i=Y^i(\tau,\sigma)$ and derive the quadratic action for the transverse fields.

Solution

The induced metric is

$$h_{ab}=\partial_aX\cdot\partial_bX = \begin{pmatrix} \dot X^2 & \dot X\cdot X' \\ \dot X\cdot X' & X'^2 \end{pmatrix}.$$

Thus

$$h=\dot X^2X'^2-(\dot X\cdot X')^2,$$

so

$$-h=(\dot X\cdot X')^2-\dot X^2X'^2.$$

In static gauge,

$$\dot X^2=-1+\dot Y^2, \qquad X'^2=1+Y'^2, \qquad \dot X\cdot X'=\dot Y\cdot Y'.$$

Therefore

$$-h=(1-\dot Y^2)(1+Y'^2)+(\dot Y\cdot Y')^2.$$

To quadratic order in derivatives of $Y^i$,

$$\sqrt{-h}=1+{1\over 2}(Y'^2-\dot Y^2)+O((\partial Y)^4).$$

The Nambu—Goto action becomes

$$S_{\text{NG}} =-T\int d\tau d\sigma +{T\over 2}\int d\tau d\sigma\, \sum_{i=2}^{D-1} \left[(\partial_\tau Y^i)^2-(\partial_\sigma Y^i)^2\right] +O((\partial Y)^4).$$

Thus the physical low-energy fields are $D-2$ massless transverse scalars.

Exercise 4: dimensions of a $p$-brane tension

Assume the coordinates $X^\mu$ and $\sigma^\alpha$ have dimensions of length in static gauge. Use

$$S_p=-T_p\int d^{p+1}\sigma\sqrt{-\det h}$$

to determine the mass dimension of $T_p$.

Solution

The action is dimensionless in units $\hbar=1$. In static gauge, $d^{p+1}\sigma$ has dimension $L^{p+1}$. The induced metric components are dimensionless if $\sigma^\alpha$ are chosen as spacetime coordinates, since $\partial_\alpha X^\mu$ is dimensionless. Therefore

$$[T_p]L^{p+1}=1,$$

so

$$[T_p]=L^{-(p+1)}=M^{p+1}.$$

For $p=0$, $T_0$ has dimensions of mass, so it is the particle mass. For $p=1$, $T$ has dimensions $M^2$, as expected for energy per unit length.

Exercise 5: the universal $1/L$ correction

Treat the transverse fluctuations of a long open string of length $L$ as $D-2$ free massless scalar fields with Dirichlet endpoints. Their normal-mode frequencies are

$$\omega_n={\pi n\over L}, \qquad n=1,2,\ldots.$$

Using zeta-function regularization, compute the zero-point energy and obtain the Lüscher term.

Solution

Each real oscillator contributes $\omega_n/2$ to the zero-point energy. For $D-2$ transverse fields,

$$E_0={D-2\over 2}\sum_{n=1}^\infty {\pi n\over L} ={\pi(D-2)\over 2L}\sum_{n=1}^\infty n.$$

Zeta-function regularization assigns

$$\sum_{n=1}^\infty n=\zeta(-1)=-{1\over 12}.$$

Therefore

$$E_0=-{\pi(D-2)\over 24L}.$$

The long-string potential takes the form

$$V(L)=T_{\text{conf}}L+\mu-{\pi(D-2)\over 24L}+O(L^{-3}),$$

provided the only massless worldsheet modes are the transverse Goldstone fields. The coefficient of $1/L$ is universal; the constant $\mu$ is not.