A Understanding Barandes' microscopic theory of causality

[Morbert]

I will attempt to describe a position that seems reasonable to me. First, transition matrices define a nomological entity that is only relevant to the extent that we have (contingent) information about the outcome of an event. That is, given a division event at $t'$, the standalone probability $p_i(t)$ can be written as $p_i(t) = \sum_j \Gamma_{ij}(t \leftarrow t') p_j(t')$, where $p_j(t')$ are, in Barandes's words, "the contingent ingredient". Then, if we have precise information about the system configuration at $t'$, all of the probabilities $p_j(t')$ become zero, with the exception of one, which corresponds to the observed value. Applying this information to our expression for the standalone probability $p_i(t)$, we obtain $p_i(t) = \Gamma_{ij}(t \leftarrow t')$ where $j$ refers here to the system configuration $q_j(t')$ for which we have information. That's the "collapse" in this stochastic formulation.

Now, how is it possible for all the other branches still exist if the aforementioned standalone probability only includes one of them? In other words, how is that possible for different branches to recohere (as in Wigner's friend thought experiments) if they are not included in this standalone probability? Well, the answer lies in the fact that what is "objective" is the transition matrices, not the standalone probabilities, since the latter also depend on contingent information about the ocurrence of other events. In cases such as Wigner's friend scenarios, the recoherence of different branches is intrinsically related with the loss of information about the system configuration at $t'$. Without this information, we have to use again the $\Gamma(t \leftarrow 0)$ transition matrix because, in addition, the dynamics are no longer divisible at $t'$.

@Morbert, I think you'll agree. Am I right?

Lucas.

The divisible transition map of quasiclassical subsystems are near exact. This means they will effectively never fail. You would need to introduce superobservers capable of suspending irreversibility or specific models of the universe permitting Poincare recurrence (for such a universe the size of our observable universe the recurrence time would be something like 10^(10^122)).

Note that Albert is not considering these speculative scenarios. He is considering timescales of at most a couple centuries. The divisible stochastic map obtained from marginalizing over environmental degrees of freedom is effectively exact for these timescales.

 

[Sambuco]

The divisible transition map of quasiclassical subsystems are near exact. This means they will effectively never fail. You would need to introduce superobservers capable of suspending irreversibility or specific models of the universe permitting Poincare recurrence (for such a universe the size of our observable universe the recurrence time would be something like 10^(10^122)).

I agree, but... The issue I tried to address in my last post is about how it is possible, within Barandes's formulation, to restrict the system configuration for $t > t'$ to only one of the possible branches, while still preserving the other branches for those cases with superobservers, Poincaré recurrence and so on. The only way I see to achieve this within Barandes' formulation is by letting the information about the occurrence of an event at $t'$ play an active role through the (contingent) probabilities $p_j(t')$. As long as we are certain of the outcome of the event at $t'$, the probability of a future event will be determined exactly by the transition matrix $\Gamma_{ij}(t \leftarrow t')$ associated with the branch corresponding to the outcome at $t'$. Interference between different branches is still possible whenever something happens that is equivalent to erasing all information about the outcome of the event at $t'$.

Note that Albert is not considering these speculative scenarios. He is considering timescales of at most a couple centuries. The divisible stochastic map obtained from marginalizing over environmental degrees of freedom is effectively exact for these timescales.

I disagree. Marginalizing over environmental degrees of freedom only causes the probability of a future event to be represented by a statistical mixture, not the system to remain on a single branch.

Lucas.

 

[Morbert]

I agree, but... The issue I tried to address in my last post is about how it is possible, within Barandes's formulation, to restrict the system configuration for $t > t'$ to only one of the possible branches, while still preserving the other branches for those cases with superobservers, Poincaré recurrence and so on. The only way I see to achieve this within Barandes' formulation is by letting the information about the occurrence of an event at $t'$ play an active role through the (contingent) probabilities $p_j(t')$. As long as we are certain of the outcome of the event at $t'$, the probability of a future event will be determined exactly by the transition matrix $\Gamma_{ij}(t \leftarrow t')$ associated with the branch corresponding to the outcome at $t'$. Interference between different branches is still possible whenever something happens that is equivalent to erasing all information about the outcome of the event at $t'$.

The divisible subsystem transition map prevents the configuration of the subsystem from jumping across quasiclassical branches. The divisible transition map is a near exact description of the subsystem, and only fail to apply in the superobserver/recurrence scenarios I mentioned above. You seem to be suggesting the divisible transition map applies even in these speculative scenarios. It does not.

I disagree. Marginalizing over environmental degrees of freedom only causes the probability of a future event to be represented by a statistical mixture, not the system to remain on a single branch.

The transition map obtained by marginalizing over environmental degrees of freedom will indeed prevent the system from jumping across branches.

 

[Sambuco]

You seem to be suggesting the divisible transition map applies even in these speculative scenarios. It does not.

No. What I said (or at least tried to say) is that if there is information (even in the environmental degrees of freedom) about the outcome of the event at $t'$, this makes it a division event, and the standalone probability of a future event is exactly $p_i(t) = \Gamma_{ij}(t \leftarrow t')$. If, on the other hand, that information is erased, there is no longer a division event at $t'$, so the stochastic map is now $p_i(t) = \Gamma_{ik}(t \leftarrow 0) p_k(0)$. I think we both agree on that.

The transition map obtained by marginalizing over environmental degrees of freedom will indeed prevent the system from jumping across branches.

Maybe I'm misunderstanding you. Marginalizing over environmental degrees of freedom produces the transition map $p_i(t) = \sum_j \Gamma_{ij}(t \leftarrow t') p_j(t')$. Are you also saying something like "What this statistical mixture means is that the system is constrained to a single branch, we just don't know which one it is"? If that is the case, I agree.

Lucas.