String Theory, Part V: Worldsheets and Currents

EDIT: It has been brought to my attention that I was getting ahead of myself a little bit. At this point let us elaborate on the concept of a worldsheet. We have been looking at worldlines which are the paths of point particles in spacetime. Since we are dealing with strings, we will extend the concept from a zero-dimensional point to a one-dimensional string. The string thus sweeps out an area in spacetime known as the worldsheet of the string. Since stings can either be open or closed, worldsheets may look like ribbons (A) in case of open strings, or tubes (B) in the case of closed strings:

Worldsheets1

Now we must be very careful to pay attention to our coordinates. We have spacetime coordinates X^\mu but we also have worldsheet coordinates, \sigma^\alpha . The worldsheet of the string has only two dimensions, one of space and one of time, so that \alpha = 0\equiv\tau , a timelike dimension, and \alpha = 1\equiv\sigma , a spacelike dimension. (We will later learn that \mu goes from 0 to 25 in Bosonic string theory, and from 0 to 9 when supersymmetry is initially imposed.) As we discussed earlier, we have reparameterization invariance and so we can define our worldsheets to have static gauge which allows X^0 = ct = c\tau . This is illustrated below:

Worldsheets2

We are starting with a flat Minkowski space to start. Curved space will be introduced later.

Currents

From Noether’s Theoem, we know that for a Lagrangian density, \mathcal{L}(\Phi_A) = \mathcal{L}(\Phi_A +\delta\Phi_A) , the variation \frac{\partial\mathcal{L}}{\partial\Phi_A}\delta\Phi_A = 0 implies that

\displaystyle j_{i}^{\alpha}=\frac{\partial\mathcal{L}}{\partial(\partial_\alpha\Phi_A)}\delta\Phi_A\implies\partial_\alpha j_{i}^{\alpha}=0

are conserved currents on the worldsheet giving rise to conserved charges

\displaystyle Q_{i}=\int\limits j_{i}^{\alpha}

EDIT: Here the i index the various currents and \alpha index the components of the currents.

For our purposes, we identify

j_{\mu}^{\alpha}=\mathcal{P}_{\mu}^{\alpha},

where the \alpha = 0, 1 with 0\equiv\tau and 1\equiv\sigma .

We can then define the analogous charges

M_{\mu\nu}=\int_{\gamma}\left(X_{\mu}\mathcal{P}_{\nu}^{\tau}-X_{\nu}\mathcal{P}_{\mu}^{\sigma}\right)\, d\sigma

where \gamma is any curve with ends attached to the boundaries of the worldsheet. Now the components of M_{\mu\nu} give the angular momenta L_i = \frac{1}{2}\epsilon_{ijk}M_{jk}

The only nonvanishing component of angular momentum is M_{12} . So we consider M_{12} for a rotating string:

M_{12}=\int\limits_{0}^{\sigma_1}\left(X_{1}\mathcal{P}_{2}^{\tau}-X_{2}\mathcal{P}_{1}^{\tau}\right)\, d\sigma=\frac{\sigma_1}{\pi}\frac{T_0}{c}\int\limits_{0}^{\sigma_1}cos^{2}\left(\frac{\pi\sigma}{\sigma_1}\right)\, d\sigma = \frac{\sigma_{1}^{2}\, T_{0}}{2\pi c}

Noting that \sigma_1 = E/T_0 , we conclude that

J=M_{12}=\frac{E^{2}}{2\pi T_{0}c}

This is the relation found in the Regge trajectories (mentioned in the first string theory post) that initially led to the idea of strings. Since J has units of \hbar , we divide by \hbar and define a multiplier \alpha^\prime by

\displaystyle\frac{J}{\hbar} = \alpha^\prime E^2

which give us

\displaystyle\alpha^\prime = \frac{1}{2\pi T_0\hbar c}\implies T_0=\frac{1}{2\pi\alpha^\prime\hbar c}.

Plugging this back in, the Nambu-Goto action becomes

\displaystyle S_{NG}= -\frac{1}{2\pi\alpha^{\prime}\hbar c^2}\int_{\tau_i}^{\tau_f}\int_{0}^{\sigma_1}\sqrt{(\dot{X}\cdot X^{\prime})^2 - (\dot{X})^2(X^{\prime})^2}\;d\sigma\, d\tau.

For our purposes, we take \hbar = c = 1 and so we have

\displaystyle S_{NG}= -\frac{1}{2\pi\alpha^\prime}\int_{\tau_i}^{\tau_f}\int_{0}^{\sigma_1}\sqrt{(\dot{X}\cdot X^{\prime})^2 - (\dot{X})^2(X^{\prime})^2}\;d\sigma\, d\tau.

or

\displaystyle S_{NG}= -\frac{1}{2\pi\alpha^\prime}\int\sqrt{-\gamma}\;d\sigma\, d\tau.

Physically, this is the area swept out by the string on the worldsheet, and \sqrt{\alpha^\prime} is a length which determines the string size.

Since for any two-dimensional metric h^{\alpha\beta}h_{\alpha\beta} = 2 , the metric can also be written

\displaystyle S_{string}= -\frac{1}{4\pi\alpha^\prime}\int\sqrt{-\gamma}\gamma^{\alpha\beta}\gamma_{\alpha\beta}\;d\sigma\, d\tau.

which is the classical equivalent of

\displaystyle S_{string}= -\frac{1}{4\pi\alpha^\prime}\int\sqrt{-\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta} X^{\nu}\eta_{\mu\nu}\;d\sigma\, d\tau.

This is called the Polyakov action.

Our momentum densities have become

\displaystyle\mathcal{P}^{\tau\mu} = \frac{1}{2\pi\alpha^\prime}\dot{X}^\mu

and

\displaystyle\mathcal{P}^{\sigma\mu} = -\frac{1}{2\pi\alpha^\prime}{X}^{\mu\prime}

From this we can say that

\displaystyle S_{string}= -\frac{1}{4\pi\alpha^\prime}\int\left(\dot{X}^{\mu}X_{\nu} - \dot{X}_{\nu}X^{\mu}\right)\, d\sigma

Since we are in flat spacetime, we have the following symmetries:

Lorentz: \displaystyle X^{\prime\mu} = \Lambda^{\mu}_{\nu}X^{\nu}

Poincare: \displaystyle X^{\prime\mu} = X^{\mu} + a^{\mu}

and Conformal: \displaystyle \gamma_{\alpha\beta}\to e^{2\phi}\gamma_{\alpha\beta}

This gives us that

\displaystyle \sqrt{det\,\gamma}\to e^{4\phi}\sqrt{det\gamma}

or \displaystyle \sqrt{det\,\gamma}\to e^{2\phi}\sqrt{det\gamma}.

It is important to note that in two dimensions, we have that \gamma_{\alpha\beta}\to e^{2\phi}\eta_{\alpha\beta} , for some \phi ,

so we can just set $latex \gamma_{\alpha\beta} = \eta_{\alpha\beta} $.

We now have

\displaystyle S_{string}= -\frac{1}{4\pi\alpha^\prime}\int\eta^{\alpha\beta}\eta_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta} X^{\nu}\, d^2\sigma (again with \sigma = \sigma^\alpha,\;\alpha =0,1)

This is a free field theory with a constraint from the \gamma_{\alpha\beta} equation of motion:

\displaystyle T_{\alpha\beta} = \frac{1}{\sqrt{-\gamma}}\frac{\delta\mathcal{L}}{\delta\gamma^{\alpha\beta}} = \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} - \frac{1}{2}\gamma_{\alpha\beta}\gamma^{\lambda\kappa}\partial_{\lambda}X^{\mu} \partial_{\kappa}X^{\nu}\eta_{\mu\nu}=0

We may recognize this to be the stress-energy tensor.

EDIT: I have omitted the rest of this post as I think it should be separate. The next post will deal with quantization.

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12 Responses to String Theory, Part V: Worldsheets and Currents

  1. sheabrowne says:

    First off, great post. I’m glad you’re keeping at the string theory. I’m missing a lot here though, I need more explanation. We’ve moved from a classical string in the Lagrangian formulation to its relativistic version. Here we see the introduction of the ‘worldsheet’, which I’d like to see explained a little more. Is the metric which you define just for this two dimensional space? Is the worldsheet embedded in Minkowski (flat) space-time? If yes, I’m missing the motivation.
    Once we have the classical action for the worldsheet, you use Noether’s theorem to construct the conserved currents for translational symmetry, but I don’t see what the index i on j^{\alpha}_{i} is coming from? Usually this “extra” indexing in Noether’s theorem comes from having more than one field; are you treating the time-like and space-like conjugate momenta as separate dynamical degrees of freedom? If so, is your \mathcal{P}_{\mu}^{\alpha} really the stress-energy tensor for the worldsheet.
    The other part I’m confused about is the introduction of the Regge trajectories, which you seemed to have mentioned at some point. Here \hbar is sneaking in without any mention of quantization yet. Is this coming from assuming canonical commutation relation for the angular momentum operators? Once you do quantize, we seem to get two excitations or ‘particles’, are they antiparticles of each other (they look like it, but I haven’t worked it out)? Do all the commutations work out as expected? If we go back and construct M_{\mu \nu} in terms of operators, with the commutation relation be that of the rotation group?
    Sorry for all the questions, but I’m excited to follow along and I wan’t to make sure I understand where you’r going with all of this!

  2. sheabrowne says:

    P.S., Sorry for all the spelling grammar mistakes above, I wrote this at 5am with a 2-year-old looking over my shoulder.

  3. sheabrowne says:

    This is much better, thanks for filling it out! Now I’ve got no excuse not to work through the details!

    • seanparrottwolfe says:

      Not a problem. I haven’t had a lot of experience with this stuff, so teaching it is a little over my head. I’m doing my best, though.

  4. sheabrowne says:

    An interesting personal note, the Polyakov action was first introduced by my Quantum-Mechanics/General Relativity professor at Brandeis, Stanley Deser (see http://inspirehep.net/record/108644 ). It wasn’t called the Polyakov action of course, but it was used in supergravity. When he taught us GR, he used the Einstein-Hilbert action, and even showed us how the linear theory could be quantized…I wish I knew then what I know now, I would have gotten so much more out of it!

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