Is the distintion between vector and rays relevant?

[carllacan]

Hi.

Quantum states are usually represented as vectors, and treated as such, even when distinct states should be represented by rays. Is there any case (in physics) when it is important to take into account this distinction?

Thanks.

 

[Matterwave]

Not that I know of, other than the obvious fact that you have to normalize your probabilities before you can count them as proper probabilities.

 

[atyy]

As Matterwave says, in most treatments there is no difference, because only normalizable states are physical states. However, there is one conceptual point for which rays as states may be better, and that is that multiplication by a global phase is a gauge operation in quantum mechanics. In principle, the states as rays naturally incorporates the global phase as gauge, while it has to be put in "by hand" or inferred from the Born rule if states are considered as unit vectors.

The Born rule has to be adjusted depending on whether rays or unit vectors are considered states. Most statements of the Born rule assume that states are unit vectors.

Another formalism that takes care of the global phase as gauge automatically is density matrix formalism.

 

[bhobba]

The truth is states are not elements of a vector space; they are positive operators of unit trace. The Born Rule is given an operator O the expected value is E(O) = Trace (PO). Pure states are states of the form |x><x|. Mixed states are convex sums of pure states. It can be shown that all states are either mixed or pure. Pure states can be mapped to the underlying vector space, but only up to an arbitrary phase factor c because |cx><cx| = |x><x|. But it must always be borne in mind they are not in general elements of that space, they are really operators.

This is the density matrix formalism Atty referred to. Best to view it that way to avoid confusion.

Thanks
Bill

 

[dextercioby]

Your question is relevant

Hi.

Quantum states are usually represented as vectors, and treated as such, even when distinct states should be represented by rays. Is there any case (in physics) when it is important to take into account this distinction?

Thanks.

The mathematical theory of symmetries in quantum mechanics as a whole (the so-called Wigner-Weyl formulation of quantum mechanics) is built on the distinction between rays and vectors. Without this, there would be nothing in QM, really.

From a statistical standpoint, you treat pure states as rays in the Hilbert space (= points in the projective Hilbert space) and mixed states as density operators. Since mixed states are a generalization of pure states, you can describe pure states as density operators as well. In particular, if a pure state is described as a ray Psi, you can write the density operator for that state as a projector from the Hlbert space onto any (normalized) vector from the ray Psi (such a vector is called state representatives) .