# How could we conclude this relationship? [on hold]

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 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