As done in several text on string theory, you can quantize the worldsheet action to get the quantum string. But first, they start with the quantum theory of a single point particle by quantization of a worldline action theory.

So, I want to draw a parallel between them. For the point particle we promote position and momentum to operators fulfilling ($\hbar=1$):

$$[x^\mu(\tau),p_\mu(\tau)]=i\delta^\mu_\nu$$

Also we have the Klein-Gordon equation which arise as the first-class constraint $p^2+m^2=0$ associated with reparametrization invariance of the worldline, yielding free particle momentum states $|p \rangle$.

As we learned in QFT courses this theory has many problems, such as negative energies, negative probabilities and causality violation (which are the main motivations they state to develop another theory, field theory). Note that with the usual BRST procedure we get a hamiltonian $H=\frac{1}{2}(p^2+m^2)$ and acording to Polchinki's book page 130:

The structure here is analogous to what we will find for the string. The
constraint (the missing equation of motion) is $H = 0$, and the BRST
operator is $c$ times this.

So $p_0<0$ states are in the state space too.

Now, we do the same for the string.
We now have the operators satisfying
$$[x^\mu(\tau,\sigma),p_\mu(\tau,\sigma')]=i\delta(\sigma-\sigma')\delta^\mu_\nu$$
and we can costruct a state space and so on. But the books never talk about the potential problems that could arise such as:

Negative energies

Negative Probabilities

Causality violation

So, do these problems arise in the theory of a single string in the same way as arised in the relativistic single particle?

This post imported from StackExchange Physics at 2017-08-04 22:00 (UTC), posted by SE-user MoYavar