States in light cone string theory

Question

Currently I'm trying to understand string theory in the light cone quantization. I just have had a look into Polchinski (Vol. 1, Introduction to the bosonic string), because – as far as I could see – GSW doesn't cover Vertex Operators in the light cone formulation (correction appreciated).

There he constructs (almost) general states on page 21 (eq. 1.3.28) via raising operators acting on Fock space vacuum states $\left|0;k \right\rangle$ with $k = (k^+, k^i)$. A few sentences later he introduces the state $\left| 0;0 \right\rangle$ for the ”ground state of a single string with zero momentum“. I think this state does not even exist as $$p^- = \frac{p^i p^i + m^2}{2 p^+}$$ diverges. This can equivalently be concluded by the fact that $\left|0;k \right\rangle$ (with arbitrary $k$) is a tachyonic state which can never have zero momentum.

This is problematic as this state is heavily used e.g. in chapter 2.8 (eq. 2.8.2 ff.).

Is there anything wrong about my arguments?

Answer 1

Comments to the question (v1):

1. We will not discuss tachyonic states here, because they are pathological and signal an instability of the theory. Then $$\tag{1} p^{\pm}~\equiv~\frac{p^0 \pm p^1}{\sqrt{2}}~\geq~0 $$ is manifestly non-negative, since the energy $p^0\geq |p^1|$.

2. In the light-cone formalism $p^{+}>0$ is strictly positive, since the special case $p^{+}=0$ is by definition regularized away. For the record, Ref. 1 (Ref. 2) mention that $p^{+}$ is positive on the bottom of p. 187 (on the top of p. 20), respectively.

3. A Fock space of definite momentum is built from a ket |$N; p^+, \vec{p}_T \rangle$, where $\vec{p}_T$ is the center-of-mass transverse momentum, and $N$ are oscillator modes. Note that $p^{+}$ is often not written explicitly in the ket state notation $| 0;0\rangle$.

4. The momenta $\hat{p}^{+}$ and $\hat{p}^I_T$ are constants of motion. In first quantized string theory [as opposed to string field theory, where strings can interact], the momentum $p^{+}$ can be assumed to be a fixed (but arbitrary) positive constant.

5. Ref. 1 chooses light-cone gauge $$\tag{2} \hat{X}^{+}~=~2\alpha^{\prime}\hat{p}^{+}\tau,$$ and identifies $$\tag{3}\hat{H}~=~2\alpha^{\prime}\hat{p}^{+}\hat{p}^{-}$$ with the Hamiltonian, see Ref. 1 p. 238.

6. Note that Ref. 2 at first identifies $\hat{p}^{-}$ with the Hamiltonian, cf. e.g. eqs. (1.3.6) and (1.3.30), while Ref. 2. in later chapters (implicitly?) uses the same convention as Ref. 1.

7. Note that light-cone time $\tau$ is a parameter; not an operator. As a consequence, the CCRs
   $$\tag{4}[\tau,\hat{H}]~=~0\quad\text{and}\quad[\hat{X}^{+},\hat{p}^{-}]~=~0.$$ This is similar to ordinary quantum mechanics, cf. Pauli's objection to a time operator, see e.g. Ref. 1 p. 221; and this, this and this Phys.SE posts.

8. On the other hand, the CCR $$\tag{5} [\hat{X}^{-},\hat{p}^{+}]~=~i\hbar {\bf 1}$$ is implemented in the light-cone formalism. Therefore the operator $e^{-ia\hat{X}^{-}/\hbar}$ becomes an intertwining operator between Fock spaces with different values of $p^{+}\to p^{+}+a$. Again, Fock spaces with non-positive $p^{+}\leq 0$ are dismissed in the light-cone formalism.

References:

1. B. Zwiebach, A first course in String Theory, 2nd edition, 2009; p. 187; p. 221; p. 238.

2. J. Polchinski, String Theory, Vol. 1, 1998; p. 21.