# Light-cone coordinates to quantize strings?

Zweibach, in A First Course in String Theory said: "We now discuss a coordinate system that will be extremely useful in our study of string theory, the light-cone coordinate system", then he mentioned that this coordinate system will be used to quantize strings.

My question: What's so special about light-cone coordinates/gauge, so they have used them to quantize strings? Why are they "extremely useful"?

• damtp.cam.ac.uk/user/tong/string.html chapter 2 and I strongly recommend a cup of very strong coffee. – user108787 Aug 28 '16 at 20:31
• I am not a physicist, but you can look at Polchinski's first volume on String theory, where he covers that early on. I guess they just make many quantities vanish, or many formulas simpler, or something like that. This is in some sense similar to why Frenet frames are useful. Being adapted to the shape of the curve in $\mathbb{R}^3$, they make formulas nicer looking. – Malkoun Aug 28 '16 at 20:59
• Thank you, this is obviously true. But I want to know exactly what "quantities" have been removed or how they have made the "formulas nicer". I know that the usage of those coordinates have some physical results like getting rid of "ghosts". That's what's my question about, the physical results. – Milou Aug 28 '16 at 21:35
• You can already see simplifications in the analogous, though simpler case, of the quantization of a single particle satisfying an action similar to the Polyakov action (refer to Polchinski's 1st volume). By picking $\tau = X^+(\tau)$, the Lagrangian will contain the time derivative of $X^-$, but not of $X^+$. Here $x^{\pm} = 2^{-1/2}(x^0 \pm x^1)$. – Malkoun Aug 29 '16 at 4:37
• Quoting Polchinski: "We discuss the spectrum of the open string in this section and that of the closed in the next, using light-cone gauge to eliminate the diff × Weyl redundancy. This gauge hides the covariance of the theory, but it is the quickest route to the spectrum and reveals important features like the critical dimension and the existence of massless gauge particles." – Malkoun Aug 29 '16 at 4:39

1. In the World-sheet (WS). To quantize a physical system, one should always start by analyzing the corresponding classical system. The classical equation of motion of the free bosonic string is the wave equation $$\Box X ~\approx~ 0$$ in 1+1D. It is a linear 2nd-order PDE, whose complete solution $$X ~\approx~ X_L(\sigma^+)+X _R(\sigma^-)$$ is a sum of arbitrary left- and right-moving solutions, each of which only depends on one of the two WS LC coordinates $\sigma^{\pm}$. The WS LC coordinates are the characteristics of the PDE. This is essentially the main reason for the prominent role played by the WS LC coordinates $\sigma^{\pm}$.
2. In the target space (TS). The bosonic string action has WS reparametrization gauge symmetry. One can use modern Lorentz-covariant BRST quantization, but it is easier to work in a particular gauge. Any admissible TS gauge choice will in principle do (in the sense that gauge-invariant physical observables do not depend on gauge-choice), but the TS LC gauge $$X^+(\tau,\sigma)~=~f(p^+(\tau))\tau\qquad\text{and}\qquad P^+(\tau,\sigma)~=~p^+(\tau),$$ is a convenient choice to effectively separate physical and unphysical degrees of freedom (DOF) and minimize the needed algebra, cf. e.g. this Phys.SE post.