Covariant Quantisation and the Time Operator in String Theory

Covariant quantisation in string theory is accomplished by giving the commutator relations

$[X^\mu(\sigma,\tau),P^\nu(\sigma',\tau)] = i \eta^{\mu\nu} \delta(\sigma - \sigma')$.

Although string theory is not a field theory this approach is identical to the quantisation of fields $X^\mu$ in 1+1 dimensions. From this point of view it is also clear how to proceed from here (e.g. getting rid of the negative norm states due to the "wrong" sign of the zeroth component).

However, there are two issues that I do not quite understand. Physical states are supposed to have positive energy, i.e. $P^0 \geq 0$. Especially when supersymmetry comes into the game, the algebra so to say dictates positive energies. But when the energy is bounded from below it is not possible to have a time operator obeying the commutator relation

$[x^0,p^0] = -i$

where $x$ and $p$ are the above Operators $X$ and $P$ integrated over $\sigma$.

For the case of quantum mechanics this has been proved by Pauli in 1933. The prove goes roughly like this: For an operator $H$ which is bounded from below let there be a Hermitian time operator $T$ with

$[T,H] = i$

It is possible to shift the spectrum of H because of

$e^{-i\epsilon T} H e^{i\epsilon T} = H - \epsilon$

for arbitrary $\epsilon$. That means for an eigenstate $|\psi\rangle$ with

$H|\psi\rangle = E|\psi\rangle$

there is also an eigenstate

$|\phi\rangle = e^{i\epsilon T}|\psi\rangle$

with

$H|\phi\rangle = (E-\epsilon)|\phi\rangle$

For $\epsilon > E$ one has to either conclude that H can not be bounded from below or that T is not a Hermitian operator because it would generate states that lie outside of the Hilbert space.

One possible solution could be to allow negative $p^0$ and to apply some kind of Feynman-Stueckelberg approach by reversing the sign of the world-sheet coordinate $\tau$. In this sense the decay of the vacuum into a positive and negative energy state is simply the propagator of a free state. However, this would not be possible in the supersymmetric case where the states have $p^0>0$.

The other issue related to defining the zeroth component of the commutator is the construction of the Hilbert space of the physical states. In his book String Theory Polchinski starts with a "vacuum" state $|0;k\rangle$ with the norm

$ \langle 0;k|0;k'\rangle \sim \delta^{26}(k-k')$

However, when integrating over $k$ one ends up with a superficial delta function because of the additional mass shell constraint. Polchinski solves the issue by introducing a reduced product

$ \langle 0;k||0;k'\rangle \sim \delta^{25}(k-k')$

This seems to be a logical step because selecting the physical states by

$L_0|\psi\rangle = \alpha'(p^2+m^2)|\psi\rangle = 0$

yields a Hilbert space whose functions depend on 25 momentum components. Still, the full Hilbert space is not normalisable (this differs from what is usually done in QFT during gauge fixing, where the Hilbert space is constructed via an equivalence relation for the gauge and not by selecting a subspace). The question is whether the Hilbert space constructed in such a way does exist at all. Also it seems as if the number of spacetime operators is reduced by one, thus rendering the steps taken for the covariant quantisation needless.

(REMARK: I edited my question to make the point a bit clearer. This is not meant to be a criticism against string theory. It is just that during studying string theory I was confronted by someone with this issue. I just want to understand this detail of the basics because I haven't yet been able to resolve it. I hope somebody can help me.)