Given any constrained system such as the string we can either work with the constraints or we can solve them. The former course of action was pursued for the string under the name of the ‘old covariant quantization’ in chapter 3. The latter approach has the advantage that having solved the classical constraints in terms of independent variables, it is then straightforward to quantise the theory. The disadvantage is that when solving the constraints by expressing some variables in terms of others one is left with independent variables which no longer transform in a simple way under the Lorentz group. The Lorentz transformations of the remaining variables become very non-linear and the Lorentz symmetry is no longer manifest. As a result, one must verify explicitly that Lorentz invariance is not broken in the quantisation procedure. Indeed, one finds that this leads to non-trivial conditions even for the free quantum string. For the interacting string, the verification of Lorentz invariance in the quantum light-cone theory is a very non-trivial calculation.

At first sight, solving the constraints of the string looks like a very non-trivial task. However, we are free to use the reparameterisation invariance to choose a gauge and we will find that there exists a particularly useful gauge which reduces the problem of solving the constraints to an almost trivial task. The light-cone approach to the free string was worked out in [2.4].