if |psi> is a quantum state vector, we know that exp(ia)* |psi> is also a quantum state of the system, where exp(ia) is the global phase factor, and they are indistinguishable w.r.t the expectation value of any system observable or density matrix representation of the state.
my question is there any new developments in QM formalism that distinguishes |psi> and exp(ia)* |psi> ?
Also, is there any significance of the matrices of the form exp(ia)* |psi>
The phase of a wave function shows up when you do interference.
(between two or more waves) , that come from some comon source.
There are various ways to shift the phase and therefore the interference pattern using electromagnetic fields.
There are other types of Gauge invariance than just global, more interesting.
It is clear that the type you describe preservs the probability density, so this phase in itself has no effect. For example the negative of a wave function, has the same meaning as phi itself.
I'm not sure if this is what you are asking, but you might want to take a look at the article below. It obtains the Born rule by assuming that |psi> and exp(ia)* |psi> are distinct states, even if no experiment can tell them apart.
Preprint Quantum Fractionalism: the Born Rule as a Consequence of the...
Thank you Andre Mandolesi, this was precisely what i was looking for.
André Mandolesi Thanks, I look forward to reading that paper. There's something I'm not understanding about global phases, and maybe you have an idea.
Suppose I have a phase shifter that introduces a phase of exp(i*pi)=-1, so that it takes |ψ> to -|ψ>. The thing is, I can also write this as |ψ>⊗|world> --> -|ψ>⊗|world> = |ψ>⊗-|world>. So it seems we can't tell whether we phase-shifted the particle (and if so, which observable), or the world (or any component therein). I think the standard answer is that sure, we can't tell; global phase factors are irrelevant. But this makes it seem as though a phase shifter doesn't do anything, when it clearly does something (as evidenced by putting one in one of the paths of a Mach-Zehnder interferometer, for example).
I'm not sure how to phrase my question, but something about this confuses me.
Aditya
I think you are on the right path. A given particle can describled by phi or -phi, and you can never know the difference.
The exception is interference phenomena with two or more
wave functions, and then the sum of the two depends on the phase of either one, relative to the other.
But if you apply the same global change to two wave functions,
the relative phase will not change, so it is true that nothing really changes.
Regards, JW
Aditya
The point is that when you say you put a phase shifter in ONE of the paths, you are implying there is another path with no shifter. So the process you have is actually
( |ψ1> + |ψ2> )⊗|world> --> ( |ψ1> - |ψ2> )⊗|world>
and now you can no longer move the minus sign to the |world> , as its not a global phase factor anymore.
On the other hand, if put phase shifters in both paths, you would have a global phase, and all observational results would be the same as if you had not put any shifter.
André Mandolesi Yes, thank you, that much is clear to me. I'm not sure how to phrase my question. If a light beam passes entirely through a waveplate, it will acquire a (relative) polarization phase shift. So we can call a waveplate a "polarization phase shifter." What about the device we simply call a "phase shifter?" Which observable does it phase shift? The answer seems to be position, but this is only by virtue of the fact that we can place the device in some positions and not others. In contrast, there's no way to place it in "some polarizations and not others." Something about this asymmetry is bothering me, and I can't pin down why.
Well, in some experiments you can shift one of two interfering
beams to get a different final result.
A global phase shift is more of a general idea in that if you globally shift all the beams, nothing should be different.
Look up the topic gauge invariance, which can be either local or global.
In optics, if one path is longer than the other, you get a shift away from constructive interference. The idea is as simple as that.
Aditya K. Prasad , I'm not sure I understand your doubt, but I'll try to answer as best as I can. The diference with respect to light polarization is that the quantum phase is not an observable, there is no way to measure the phase of a quantum state. Which is why some physicists consider it as a mathematical artifact with no physical meaning, and prefer to work in a projective Hilbert space to eliminate it. But the relative phase between two components does have observable consequences, and it is this relative phase that is affected by a phase shifter in a single beam. Phase shifters in both beams sort of cancel the effect of one another, leaving the relative phase unchanged.
Thank you Prof. Janusz Milek for the extremely insightful reply and observations on the global phase factor.
Regards
D.Ghosh
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