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## Homework Statement

The conjecture implies the eigenstate of the observer and observed given by alpha and beta. The problem i am reaching at the end, is whether the well-known Wigners Friend Paradox is answerable by describing the observer and the observed, and then a second observer in terms of Dual 4-Space modelling.

## Homework Equations

## The Attempt at a Solution

[itex]\alpha (\beta) = \beta[/itex]

where alpha is the observer, beta is the oberved and act as an eigentstate equation above. Beta on the left handside operates as a ''point reference''.

It is found that:

[itex]\alpha (\beta) \in R^3[/itex]

[itex]R^3[/itex] is the flat spacetime metric which has no time description, yet.

[itex]\zeta_{\alpha} (\psi (t))=(\alpha ,t)[/itex]

[itex]\zeta_{\beta} (\psi^{\dagger} (t))=(\beta ,t)[/itex]

These two equations act as scalar reference elements. The first equation for instance is not so much a function of alpha on psi, but depends on psi. The second is its complex conjugate transpose. This can be given a retarded and advanced solution:

[itex]| \psi > \in V[/itex]

Where V is the acting linear space and [itex]\overline{V}[/itex] is its dual space:

[itex]< \psi | \in \overline{V}[/itex]

Elementary one can write:

[itex]\alpha (\beta (t))=\beta(t+1)[/itex]

[itex]\rightarrow \alpha (\beta (\Delta t)=\alpha (\beta (A^{n+1}))[/itex]

where [itex]A^{n+1}= \Delta t[/itex]

I now derive the relationship:

[itex]\alpha \psi(k \beta \psi^{\dagger}( \Delta t))= \int |\alpha , \beta (\psi)|^2[/itex]

For the probability of the measurement in conjecture with the original eigenstate equation we used.

**k**is proportional to the coupling:

[itex]k= \frac{ (t_1 - t_2)(t_2+t_1) }{ \int \alpha, \beta dt}[/itex]

For now we could assume the coupling is where the wave function collapses when professor Wigner observes the particle system. We also assume that the two conjugate psi waves describing both the prof. and the particle are describe under a final state vector [itex]< 0 >[/itex] ~ If Wigners friend enters the room it must indicate that there is a retrocausal event in the wave function which no longer states that the wave state on the professor and the observed particle is all there is. A change occurs and the state vector now covers both the mind of Wigner and his counterpart.

If we allow

[itex]{\alpha_1, \alpha_2, ..., \alpha_n}[/itex] to be the basis of [itex]V[/itex] then let [itex]{\beta_1, \beta_2, ... , \beta_n}[/itex] be its dual basis. Then [itex]{\beta_1*, \beta_2*, ..., \beta_n*}[/itex] is the dual basis of [itex]{\beta_1, \beta_2, ... , \beta_n}[/itex].

I ask if this dual 4-basis approach would suffice an answer to the Wigners Paradox?

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