That is precisely the crux of the measurement problem! From the viewpoint of the friend, the state has "collapsed", which is not a unitary evolution - the friend does not perceive themselves to be in a superposition. But Wigner - until he himself "measures" his friend - is free to look at the system as a normal quantum system where the measurement apparatus (the friend) has simply interacted with the system S and become entangled with it.

Your confusion is understandable, but also not new. This is precisely where all the different interpretations spring up - trying to explain what this "collapse" where we always observe definite results for measurement is.

"the friend does not perceive themselves to be in a superposition" ... I'm not sure if has the capability to percieve this

Wigner's friend, like Schrödinger's cat, is a thought experiment designed to make the idea that the measurement apparatus is entangled with the measured system look absurd: "Of course the cat is either dead or alive, not both!"

Also, my point was the system the friend measures also undergoes untiary evolution (Just a different unitrary operator ... one which includes the interaction as well)

The interaction between the friend and the system (and the rest of the lab) is precisely the decohering measurement interaction that evolves it into the entangled state $\lvert \uparrow \rangle_S \lvert \uparrow \rangle_F + \lvert \downarrow \rangle_F \lvert \downarrow \rangle_F$, where $\lvert \uparrow\rangle_S$ is "the particle has spin up" and $\lvert \uparrow \rangle_F$ is "the friend has observed the particle to have spin up".

From Wigner's viewpoint, this has been a unitary interaction, but no human being has ever reported being in a superposition of having observed contradictory events, i.e. the friend only observes $\lvert \uparrow\rangle_S\lvert \uparrow\rangle_F$ or $\lvert \downarrow \rangle_S\lvert \downarrow\rangle_F$, not a superposition.

The measurement problem is essentially "how can this be", and every interpretation has a different answer to that

@HDE226868 your friend asked Sean Paul what a Kahler metric is in the middle of his talk

It is crucial to realize that Wigner does not observe the friend + system to be in this superposition, he's just predicting them to be. When Wigner makes a measurement himself, he will also only get one of the possible results (and of course you can now imagine another outside friend modelling Wigner + friend + system to be in an entangled state, and so on, ad infinitum)

Ah .. I think I appreciate the whole interpretation-al issue more now ...

But I still think one can make some precise mathematical statements about the whole thing as I do here

@MoreAnonymous You never state what $\lvert \psi_\text{net}\rangle$ is.

This whole discussion reminds me of a Not even wrong's question is QM a probabilistic theory ...

yea ... Because I assume it is in an artbitary state to some extent ...

But you don't even say the state of what it is supposed to be

well its a state of the net Hamiltonian ...

There is no such thing.

Hamiltonians do not have states, systems do.

so it is state of the system 1 +2 + 12 (interaction)

Whether there is interaction or not is irrelevant for the definition of the system, as we already discussed. If you're doing a strong (=standard) measurement, then the resultant state is $\sum_i c_i \lvert \lambda_i\rangle_1 \otimes \lvert \lambda_i\rangle_2$, where $\lvert \lambda_i\rangle_1$ is the eigenstate of the measured system with the corresponding eigenvalue of the observable and $\lvert \lambda_i\rangle_2$ is the corresponding pointer state of the measurement apparatus

and the $c_i$ the probability amplitudes for the eigenstate in the state of 1 that was originally being measured.

So your $\psi_\text{net}$ is already known - this is the definition of what the result of a strong measurement is.

@RyanUnger Sounds about right, I think.

is the saying that there are no stupid questions true given the fact that Wikipedia exists

Note also that the usual approach to decoherence reaches this state by not only considering the system and the apparatus, but also "the environment". If you want to engage with discussions of measurement on such a technical level, you should first read up on the rather large existing literature on decoherence and strong and weak measurements

There are maybe no stupid questions but there are definitely inappropriate times and places to ask them

@ACuriousMind Im confused where in that massive post are u ... And which equations are wrong?

@MoreAnonymous I'm saying that your starting point of acting as if $\lvert \psi_\text{net}\rangle$ were unknown is already questionable, and that neglecting the environment is non-standard. Additionally, a measurement is certainly not well-modeled as a "small perturbation" - the resultant state of a strong measurement is very far from the non-entangled non-interacting state!

It's not the equations that are wrong, it's the approach.

Ah ... about the measurement is certainly not well-modeled as a "small perturbation" ... I agree but I do not specify how far ahead in time $\tilde t^+$ nor how behind $\tilde t^-$ are ... NOte: I think my calcukations are immune if its a dirac delta funciton of some sort

Also, I'm not exactly working from the perspective point of view that Wigner did a measurement ... Right now I assume Wigner has not done a measurment ... I do however try ask about Wigner's perspective later ...

(in the questions)

In order for it to be a perturbation of any kind, you'd have to explain what the parameter $\epsilon$ you're expanding in is. But you can't measure systems "a little" - you either measure them or you don't - so there is no parameter for strong measurements you could smoothly expand in.

wait Im confused again ... Isn't the interaction hamiltonian only important during the measurement

?

at a time $t$

like sure if u wanna spike it up use a delta function is my opinion

Yes

So where's the time-dependence in your post?

which part of the post?

are on again?

You just wrote $ \hat H_{net} = \hat H_{\text{non-int}} + \epsilon \hat H_{\text{int}(1,2)}$ for the perturbed Hamiltonian

But if you wanted to only switch on the interaction for a short time, there should be a time-dependent function in front of the interaction term

Also, you can't use time-independent perturbation theory in that case, which is what you do.

Ah .. right ... fair play ... I think I'm gonna have re-analyse the situation ... whelp!

sorry for wasting ur time ... :/

Mostly we hang out

I am Wigner's friend

Watch movies

Go to the park

Sneak round to Hamilton's place and put his cat in a box.

then hide his quaternions

Schrodinger

Hamilton's a bit of a different era

@Slereah True, but Hamiltonians got mentioned a fair bit in the Wigner's Friend discussion.

FWIW, Schrödinger was a cat lover. He chose a cat for his thought experiment because he wanted people to have some empathy for it.

Cat lovers don't typically try to murder cats

Even 50%

@Slereah You've obviously never had a cat ;)

I mean the experiment would have been identical if the device either gave the cat tuna r chicken

@ACuriousMind i've had many

Would have made for a happier cat

superposition of tuna-happy/chicken-happy