The String "Mass-Shell" Condition

String Theory without the Baggage, Episode 22.

Somewhere along I-70 near Denver. This string theory email train just keeps moving!

The relativistic point particle has an elegant action,

\(S_{pp} = -m \int ds\)

which translates minimizing the action to minimizing the length of a particle’s worldline.

Missing perhaps is the information associated with the mass-shell constraint¹

\(-p^{\mu}p_{\mu} = m^{2}c^{4}\)

Where does that constraint equation come from?

We saw through the generalized Polyakov action

\(S = \kappa \int d\sigma \, \frac{1}{2\sqrt{\gamma} }g_{\mu\nu} \partial_{1}X^{\mu}\partial_{1}X^{\nu} + \sqrt{\gamma} \Lambda .\)

that the mass-shell condition emerges naturally from the equation of motion for the auxiliary worldline metric.

The Polyakov Action

That thing - or at least related to that thing - that we’ve called λ.

But where might it come from without the Polyakov action?

Representations of the Poincaré Group

Wigner’s theorem formalizes the rôle of Poincaré invariance in Quantum Mechanics. The details forms a course in its own right. The two Casimir invariants are essentially mass and helicity. More precisely, they are

\(P_{\mu}P^{\mu}\quad\mathrm{and}\quad W_{\mu}W^{\mu},\)

where the Pauli-Lubanski vector is

\(W_{\mu} = \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}J^{\mu\nu}P^{\sigma}.\)

which is a fancy way to encode what, in nonrelativistic terms, we call spin. The eigenvalues of the p^2 Casimir are proportional² to mass:

\(P_{\mu}P^{\mu} = -m^{2}\)

Those of the W^2 are

\(W_{\mu}W^{\mu} = -m^{2}s(s+1).\)

where m is proportional to the mass of the particle and s is proportional to its spin.

Of course, the relativistic point particle we’ve been discussing has implicitly been spin zero, so really the mass-shell constraint amounts to a choice of irreducible representation of the Poincaré group.

Great. What about the string?

Constraints for the String

Last time we derived the Nambu-Goto action using a similar technique. But you might wonder, could we have started with that nonlinear action? Using the principle that minimizing the action should be tantamount to minimizing the area of the worldsheet, could we at least aspire to the same logic as Yoichiro Nambu and Tetsu Goto?

If so, the action would read

\(S =\frac{\kappa}{2} \int d\tau\,d\sigma \sqrt{\dot{X}^{2}X^{\prime 2} - (\dot{X}\cdot X^{\prime})^{2}}\)

but how would we arrive at the constraints equations

\(\begin{eqnarray*} \dot{X}^{2} + X^{\prime 2}&=& 0, \\ \dot{X}^{\mu}X^{\prime}_{\mu}&=& 0\;? \end{eqnarray*}\)

The trouble here is that the string doesn’t have a prescribed mass or spin. In some sense, that’s the whole point of studying string theory.

The mass and spin of a physical state of the string depends on its internal dynamics.

We shall see later how the mass and helicity of the string is related to its vibrational modes and boundary conditions. One way to interpret this fact is to observe that there is a quantum field theory living on the string worldsheet itself. Hence, we are less concerned with a constraint that picks a specific representation of the Poincaré group, and more concerned with constraints that maintain consistency of the worldsheet embedding.

Let us first observe that the p_μ is essentially³ the spacetime tangent vector to the worldline. Hence the mass-shell condition is a constraint that fixes the magnitude of that tangent vector as the particle moves through spacetime.

For the string we have two such tangent vectors:

\(\dot{X}^{\mu}\quad\mathrm{and}\quad X^{\prime \mu}.\)

Previously we saw that the trace of the worldsheet stress-energy tensor fixes the worldsheet ‘cosmological constant’ at zero. Given that it played the rôle of a mass term in point particle action, we can very loosely interpret the first of the string constraints,

\( \dot{X}^{2} + X^{\prime 2} = 0\)

as saying that the worldsheet tangent vectors fall into a massless or null spacetime representation of the Poincaré group.⁴

Indeed, for string with Neumann boundary conditions, the endpoints move at the speed of light.

Great. Now let’s look at the second constraint:

\(\dot{X}^{\mu}X^{\prime}_{\mu}= 0.\)

Evidently the two worldsheet tangent vectors must be orthogonal to each other.

This is required for a consistent embedding of the worldsheet into spacetime, where a local neighborhood about a point on the worldsheet should be two-dimensional, just as its embedding into spacetime. Moreover, one of those directions is spacelike, and one is timelike. A breakdown of this condition could lead to a loss of unitarity⁵. For a maximal case, if at some point

\(\dot{X}^{\mu} \propto X^{\prime \mu},\)

the string would fail to be extended in spacetime. This could happen, for example, if the endpoints both started moving faster than the speed of light. Which is of course is something we aim to avoid.

1. I warned you we’d be chaotic with metric signature choices.

2. That is to say, we are leaving factors of c and ℏ implicit.

3. More precisely it’s a cotangent vector, but whatever. You get the idea.

4. The more I sit with this idea the less confident I am in saying it. This wold be true if the two terms individually vanish, but of course they don’t. One is timelike and the other is spacelike, although it is true that their sum is proportional to the worldsheet cosmological constant - which just happens to vanish.

5. This presents a fun opportunity for an exercise. Suppose we deformed X infinitesimally by a spacetime vector quantity that depends only on σ. Suppose further that, for whatever reason, the tangent vector to this infinitesimal quantity is timelike. Using techniques from previous episodes, verify that such a deformation is not a diffeomorphism of Σ into spacetime. To extend your fun, what other sorts of infinistirmal deformations might you consider? Why aren’t these a problem?