Proper way to quantize the string in the light-cone gauge

Question

In many books like Polchinski and Green-Schwarz-Witten the light cone quantization is carried out in a fast way. They just use the virasoro constraint in the light-cone gauge to get the ligh-cone action, light-cone hamiltonian and choose the dynamical variables to formulate the theory in terms of the physical degrees of freedom. And then, they quantize as in the simple case of a dynamical system without constraints imposing the usual canonical commutation relations.

I read then in the boook "Basic concepts of string theory" (by Blumenhagen, Lust and Theisen) this footpage on page 42:

"The proper way to go to light-cone gauge would be to use the local symmetries on the world-sheet to fix components of the world-sheet metric and $X^+$. One then has to show that no propagating ghosts are introduced in this process of gauge fixing."

Could you clarify these steps? What would be a correct definition of propagating ghost? If you know give some references, this issues are not obvious for me. Thanks.

Answer 1

This is either a partial answer or partially incorrect. I had to research this, as I did not know the answer immediately.

As I see it, your question has three parts.

1. Why don't the constraints matter for the commutation relations? The derivation of the commutation relations with constraints taken into account is given in the historical reference on the subject$^1$, [1]. As it turns out, the constraints do not need to be taken into account. I suspect it is possible to treat the constraints in terms of Dirac brackets$^2$, but I am not sure. In these lecture notes on page 37 (page 10 of the PDF), Tong gives a very cryptic explanation of why the naive commutation relations are correct.

2. How do we use the local symmetries to fix the metric and $X^+$? It may be shown$^3$ that every two-dimensional manifold without topological obstructions is conformally flat. The string action has Weyl symmetry, i.e. the action is invariant under $h_{\alpha\beta}\rightarrow\mathrm{e}^{2\phi}h_{\alpha\beta}$. We then pick a gauge in which $h_{\alpha\beta}=\eta_{\alpha\beta}$. (We know there exists some $\omega$ such that $h_{\alpha\beta}=\mathrm{e}^{2\omega}\eta_{\alpha\beta}$. Choose the gauge transformation $\mathrm{e}^{-2\omega}$.)

3. What do "ghosts" mean in this context? A general problem with gauge theories is that they can produce states with negative norm, called ghosts. In string theory, when we fix the light cone gauge, we much check two things actually. First we must check that the Lorentz algebra is closed. This leads$^4$ to the critical dimension $D=26$ and normal ordering constant $a=1$. Using this, one may prove$^5$ the so-called no-ghost theorem. This theorem essentially says that the physical string spectrum is free from ghost states iff $D=26$.

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Notes:

$^1$ Link.

$^2$ As in [2], section 7.6.

$^3$ A simple proof is given in [3] using isothermal coordinates (Example 7.9).

$^4$ This is done in [1].

$^5$ See, e.g. [4], sections 2.4 and 2.5.

References:

[1] Goddard, P., Goldstone, J., Rebbi, C., & Thorn, C.B. (1973). Quantum dynamics of a massless relativistic string. Nucl. Phys., B56, 109.

[2] Weinberg, S. (1995). The Quantum Theory of Fields, Vol. 1: Foundations. Cambridge.

[3] Nakahara, M. (2003). Geometry, Topology and Physics. Institute of Physics Publishing.

[4] Becker, K., Becker, M. & Schwarz, J.H. (2007). String Theory and M-Theory. Cambridge.