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Comments to the question (v3):

  1. The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate.

  2. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, one has to go back to the full string formulation prior to the LC reduction, cf. e.g. my Phys.SE answer herehere. By doing this, one may show that the $\alpha^-_0$ mode term indeed appears in the expansion of $X^-$ as it should.

Comments to the question (v3):

  1. The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate.

  2. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, one has to go back to the full string formulation prior to the LC reduction, cf. e.g. my Phys.SE answer here. By doing this, one may show that the $\alpha^-_0$ mode term indeed appears in the expansion of $X^-$ as it should.

Comments to the question (v3):

  1. The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate.

  2. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, one has to go back to the full string formulation prior to the LC reduction, cf. e.g. my Phys.SE answer here. By doing this, one may show that the $\alpha^-_0$ mode term indeed appears in the expansion of $X^-$ as it should.

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Qmechanic ♦
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Comments to the question (v3):

  1. The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate.

  2. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, one has to go back to the full string formulation prior to the LC reduction, cf. e.g. my Phys.SE answer here. By doing this, one may show that the $\alpha^-_0$ mode term indeed appears in the expansion of $X^-$ as it should.