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"...the possible states of a quantum mechanical system are represented by unit vectors (calledstate vectors). Formally, these reside in a complex separable Hilbert space—variously called thestate spaceor theassociated Hilbert spaceof the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes"

In what sense the states are points in the projective space? Is this because they are only defined up to the phase rotation (i.e. a U(1) gauge redundancy)? Does this mean that when I think of a theory admitting gauge symmetry I can think of the states of this theory as living not in a Hilbert space, but a projective Hilbert space, technically?

Also, what is it meant by the last sentence, that "state space for the spin of a single proton is just the product of two complex planes" ?

Could you comment on this?

Thanks!

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# Formulation of quantum mechanics

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