# Is the distintion between vector and rays relevant?

1. May 7, 2014

### 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.

2. May 7, 2014

### 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.

3. May 7, 2014

### 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.

4. May 7, 2014

### Staff: Mentor

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

Last edited: May 7, 2014
5. May 8, 2014