# What is the physical meaning of a descendent state in covariantly quantized string theory?

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In the context of covariantly quantized open string theory, states that are annihilated by positively moded Virasoro operators are called Virasoro primaries, whereas Virasoro descendents of a primary state are obtained as linear combination of products of negatively moded Virasoro operators that act on the primary state.

Concerning the spectrum of the theory, real physical states are then given by primary states that are not descendents and that are annihilated by $(L_0 -1)$ and states that are primary and descendent are pure gauge states.

What is the physical meaning of descendent states (that are not primaries at the same time), what do they correspond to in the spectrum of the theory?

An aside: the notion of primary operators etc appears in CFT too, is this a just coincidence or does the terminology have the same meaning there?

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The answer is in the question: "descendent are pure gauge states" and so have no physical meaning, are completely decoupled of the physical states, and do not appear in the spectrum of the physical theory. The theory living on the string world-sheet is a 2dCFT (coupled to 2d gravity): the notion of primary... are the same.

answered May 11, 2014 by (5,140 points)

Thanks @40227. So it seems I am a bit confused by the additional notion of a null state, which is primary and descendent at the same time. This null state ortogonal to the primary and descendent states and can therefore freely be added to any physical state without effecting the physical expectation value. In my book this null state is called a pure gauge state too, this is why I thought a descendent that is not a primary at the same time is something else ...

Sorry, I read your question a bit too quickly because I read  "descendent are pure gauge states"  instead of "states that are primary and descendent are pure gauge states". Nevertheless, all descendants are decoupled from the physical states and they do not appear in the spectrum. More precisely, the real physical Hilbert space of the theory is the space of physical states divided by the space generated by the null states (i.e. states primary and descendant). It's a bit confusing that what is called a "physical state" is not the same thing as a real physical state: a real physical state is an equivalence class of "physical states" up to the equivalence relation defined by the addition of null states.

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