It follows that we now want to turn our focus to quantising the string. Following Polchinski, we shall first take the approach of light-cone quantisation (LCQ). In truth, however, there are a few different approaches to quantising the string and each is useful in offering insight into our theory. Much later in this paper we will primarily focus on path integral quantisation and, too, on BRST. But for now, prior to exploring LCQ, it is worth sharing a few comments on another approach called old covariant quantisation (OCQ), or what is also referred to as old canonical quantisation or just simply canonical quantisation.
One will see OCQ employed in many older texts. While this approach is not really of much focus these days, it is still useful to reference as it gives further definition to some contextual points given by Polchinski, particularly about negative norms (pp.27-28). To explain what I mean, let us consider the following.
In short and simple terms, OCQ implements the Virasoro constraints at the quantum level. The success of this procedure is primarily in how it is manifestly Lorentz invariant. Another success is that this manifestly Lorentz invariant theory of the bosonic string predicts the existence of negative-norm states. What happens is that after being forced to generalise the transverse commutator, the preserved Lorentz invariance forces the timelike oscillators to have the wrong sign commutator, which means we then end up with states with odd number timelike excitations that finally result in negative norms.
If it is not clear why this is a problem, it may be pointed out in a rather terse way that negative-norm states are inconsistent with quantum mechanics, considering that the norm is a probability (p.28).
We can conspire to remove the negative-norm physical states from the quantised bosonic string theory. In doing so, in the context of OCQ, the price we pay is putting constraints on the constants $a$ and $c$. Moreover, we are forced to set $a = 1$ and $c = 26$ as primary constraints of the Virasoro algebra.
This approach essentially precedes the one taken in Polchinski. For the enquiring reader I recommend a review of the procedure in Wray, 2009 and Weigand, 2015/16.
What we want to do is explore an alternative to quantising the bosonic string. We shall use the procedure of LCQ, which, at this point in parallel studies of bosonic strings, is common in the literature. The reason it is common in today’s literature refers to the same motivation for Polchinski: LCQ allows us fairly quick and easy access to a study of the physical properties of the string. Later, as I alluded, we will want to study path integral quantisation. But for now, the advantage of LCQ is not only its utility. When we quantise the string in this way, we no longer offer predicted negative-norm states. The price? Our theory is also no longer manifestly Lorentz invariant.
But it is not all bad. It turns out that when we impose the requirement of Lorentz invariance, we are forced to once again constrain our theory. This leads us to enforcing $a = 1$ and $c = 26$.
In the next post we’ll establish our notation, and then following that we’ll look to begin our study with the point particle case.