Facing a problem in Katrin Becker, Melanie Becker, John Schwarz's String Theory.

A state $\left| \psi \right>$ is called spurious if it satisfies the mass-shell condition and is orthogonal to all physical states $$( L_0 - a ) \left| \psi \right> = 0$$ and $$\langle \phi |\psi \rangle=0 \tag{2.106}$$ where $\left| \phi \right>$ represents any physical state in the theory. An example of a spurious state is $$\left| \psi \right> = \sum_{n=1}^\infty L_{-n} \left| \chi_{n} \right> $$ where $$(L_{0} - a + n) \left| \chi_{n} \right>= 0 \tag{2.107}$$

Now, my question is how we arrived to conclude the second part in eq.(2.107)? The second one, is what forces us to write the form of $\left| \chi_{n} \right>$ in this expression, $\left| \chi_{n} \right> = L_{-1} \left| \chi_{1} \right> + L_{-2} \left| \chi_{2} \right>$ ?

put on hold as unclear what you're asking by rob 2 days ago

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    It's not clear what you're asking. The equation (2.107) you write is simply supposed to be an example of a spurious state - there's no need to "conclude" the definitions used in an example. Additionally, there second part of 2.107 you have written makes no sense - $L_0 - a + n$ is an operator, $\lvert \chi_n\rangle$ is a state. They cannot be equal. – ACuriousMind 2 days ago
  • so you want to say that eq. (2.107) as hole is a definition, right? – Student404Mus 2 days ago
  • in my opinion, the answer requires a good reader to the book in order to get rid of these questions, since the authors uses assumptions/predictions to arrive to the desired result. – Student404Mus 2 days ago