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Light-cone (LC) coordinates appear in bosonic string theory in two places:

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.

Light-cone (LC) coordinates appear in bosonic string theory in two places:

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 full/general 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.

Light-cone (LC) coordinates appear in bosonic string theory in two places:

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.

Light-cone (LC) coordinates appear in bosonic string theory in two places:

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.

1. 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 world-sheet (WS) light-cone (LC) coordinates $\sigma^{\pm}$. This is essentially the main reason for the prominent role played by the WS LC coordinates $\sigma^{\pm}$.

2. 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 target space (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 and cleanly separate physical and unphysical degrees of freedom (DOF), cf. e.g. this Phys.SE post.

Light-cone (LC) coordinates appear in bosonic string theory in two places:

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.