Consider three kinds of ancestry trees: 1) souls of some odd human mothers, 2) ems and their copies, and 3) splitting quantum worlds. In each kind of tree, agents can ask themselves, “Which future version of me will I become?”
SOULS First, let’s start with some odd human mothers. A single uber-mother can give rise to a large tree of descendants via the mother relation. Each branch in the tree is a single person. The leaves of this tree are branches that lead to no more branches. In this case, leaves are either men, or they are women who never had children. When a mother looks back on her history, she sees a single chain of branches from the uber-mother root of the tree to her. All of those branches are mothers who had at least one child.
Now here is the odd part: imagine that some mothers see their personal historical chain as describing a singular soul being passed down through the generations. They believe that souls can be transferred but not created, and so that when a mother has more than one child, at most one of those children gets a soul.
Yes, this is an odd perspective to have regarding souls, but bear with me. Such an odd mother might wonder which one of her children will inherit her soul. Her beliefs about the answer to this question, and about other facts about this child, might be expressed in a subjective probability distribution. I will call such a distribution a “progeny prob”.
EMS Second, let’s consider ems, the subject of my book The Age of Em: Work, Love, and Life when Robots Rule the Earth. Ems don’t yet exist, but they might in the future. Each em is an emulation of a particular human brain, and it acts just like that human would in the same subjective situation, even though it actually runs on an artificial computer. Each em is part of an ancestry tree that starts with a root that resulted from scanning a particular human brain.
This em tree branches when copies are made of individual ems, and the leaves of this tree are copies that are erased. Ems vary in many ways, such as in how much wealth they own, how fast their minds run relative to humans, and how long they live before they end or next split into copies. Split events also differ, such as re how many copies are made, what social role each copy is planned to fill, and which copies get what part of the original’s wealth or friends.
An em who looks toward its next future split, and foresees a resulting set of copies, may ask themselves “Which one of those copies will I be?” Of course they will actually become all of those copies. But as human minds never evolved to anticipate splitting, ems may find it hard to think that way. The fact that ems remember only one chain of branches in the past can lead them to think in terms of continuing on in only one future branch. Em “progeny prob” beliefs about who they will become can also include predictions about life details of that copy, such as wealth or speed. These beliefs can also be conditional on particular plans made for this split, such as which copies plan to take which jobs.
QUANTUM Third, let’s consider quantum states, as seen from the many worlds perspective. We start with a large system of interest, a system that can include observers like humans and ems. This system begins in some “root” quantum state, and afterward experiences many “decoherence events”, with each such event aligned to a particular key parameter, like the spatial location of a particular atom. Soon after each such decoherence event, the total system state typically becomes closely approximated by a weighted sum of component states. Each component state is associated with a different value of the key parameter. Each subsystem of such a component state, including subsystems that describe the mental states of observers, have states that match this key parameter value. For example, if these observers “measured” the location of an atom, then each observer would have a mental state corresponding to their having observed the same particular location.
These different components of a quantum state sum can thus be seen as different “worlds”, wherein observers have different and diverging mental states. Decoherence events can thus be seen as events at which each quantum world “splits” into many child worlds. The total history starting from a root quantum state can be seen as a tree of states, with each state containing observers. And so a quantum history is in part a tree of observers. Each observer in this tree can look backward and see a chain of branches back to the root, with each branch holding a version of themselves. More versions of themselves live in other branches of this tree.
After a split, different quantum worlds have almost no interaction with each other. Which is why we never notice this quantum splitting process in the world around us. So observers typically never see any concrete evidence of that there exist other versions of themselves, other than their past versions in the chain from them now back in time to the root state. That is, we never see other quantum worlds. As observers see only a sequence of past versions of themselves, they can naturally expect to see that sequence continue into the future.
That is, observers typically ask “In the future, what will be the state of the one world, including the one version of my mind?” Even though in fact there will be many worlds, holding many versions of their minds. (Quantum frameworks other than many worlds struggle to find ways, usually awkward, to make this one future version claim actually true.) Beliefs about this “who will I be?” question are thus “progeny probs”, analogous to the beliefs that an em might have about which future copy they will become, or that an odd human mother might have on which future child inherits her soul.
The standard Born rule in quantum mechanics is usually expressed as such a progeny prob. It says that if the current state splits into a weighted sum of future states, one should expect to find oneself in each component of that sum with a chance proportional to the square of that state’s weight in the sum. This is a remarkably simple and context-independent rule. Technically, quantum states are vectors, and the Born rule uses the L2 norm for relative vector size. And a key question about many worlds quantum theory, perhaps the key question, is: from where comes this rule?
IN GENERAL These three cases, of human souls, em copies, and quantum worlds, all have a similar structure. While the real situation is a branching tree of agents, an agent who looks back to see a sequence of ancestors can be tempted to project that sequence forward, predict that they will become only one next descendant, and wonder what that descendant will be like. This temptation is especially strong in the quantum case, where agents never see any other part of the tree than their ancestor sequence, and so can fail to realize that a larger tree even exists.
An agent’s beliefs about which next descendant will “really” be them can be described by a probability distribution, which I’ve called a “progeny prob”. This gives the chance this agent will “really” become a particular descendant, conditional on the details of a situation. For ems, this chance may be conditional on each copy’s wealth, or speed, or job role. For quantum systems, this chance is often conditional on the value of the key parameter associated with a decoherence event.
In the rest of this (long) post, I make three points about progeny probs.
IS FICTION The first big thing to notice is that, for an agent who is a branch in some tree of agents, there is actually no truth of the matter regarding which future branch that agent will “really” be! They will become all descendant branches in that tree. So one of the most fundamental elements of quantum theory, the Born probability rule, is typically expressed in terms of an incoherent concept. Also incoherent is a big question that ems will often ask, “Who will I be next?”
However, even though progeny probs are in this sense fictional, we usually connect them to some very real data: the past sequence of ancestors we see up until today. Agents who believe that their past history was generated by the same sort of progeny prob that applies to their future should expect this history to be typical of sequences generated by such a progeny prob. This test has in fact been applied to the quantum progeny prob, which passes with high accuracy.
If one has has a detailed enough model of how a certain kind of ancestry tree of observers is generated, then one can use this tree model to predict a probability distribution over possible trees. Each such generated tree comes with a set of ancestor sequences, one for each branch in the tree. So given a distribution over trees, one can generate a distribution over ancestor sequences in these trees.
IS RELATIVE However, in order to take a tree model and generate a distribution over ancestor sequences, one needs to pick some relative branch weights, weights which say how much each branch counts relative to others in that tree. And the progeny prob that best fits this total distribution of ancestry sequences will depend on these relative branch weights.
For example, consider all the ems that descend from some particular human, and consider a late time when there are many such descendants. There are several different ways that one could sample from these late ems to create a distribution of ems. For example, one could repeatedly sample 1) a random memory unit able to store part of the mental state of an em, 2) a random processor able to run part of an em mind, or 3) a random dollar of wealth and pick the em who owns it.
The random processor approach tends to fit better with progeny probs which say that you are more likely to be a descendant who runs faster (and who has descendants who run faster). The random memory approach tends to fit better with progeny probs that count descendants more equally, regardless of speed. And the random dollar of wealth approach tends to fit better with progeny probs that say you are more likely to become the descendants who inherit more wealth from you. Which of their descendants an em should expect to become depends on which of these methods this em thinks makes more sense for weighting future ems.
Each progeny prob predicts the existence a tiny fraction of very weird ancestors sequences, ones quite unlikely to be generated by that progeny prob. But such sequences are only actually rare if this progeny prob fits with the correct distribution. For example, few ems chosen by looking at random memories should have ancestor histories that are weird according to a memory-based progeny prob. But most of them might have ancestors histories that are weird according to processor- or wealth-based progeny probs.
For the quantum case, the standard Born rule progeny prob seems to fit well with distributions that sample from later quantum worlds in proportion to the same L2 norm that the Born rule uses. However, we lack a good widely accepted derivation of this distribution from the basic standard core of quantum mechanics. That is, we can’t explain why we should focus on later quantum worlds in proportion to this L2 norm, and mostly ignore the far larger numbers of quantum worlds that have much smaller values of this norm.
Yes, some try to derive this norm from other axioms, but none of these derivations seems a compelling explanation to me. The L2 norm is so simple that it must be implied by a great many sets of axioms. I’ve proposed a “mangled worlds” approach, and show that the Born rule can result from counting discrete worlds equally, if we ignore worlds below a size threshold that are mangled by larger worlds, and so are not hospitable to observers. But my proposal is so far mostly ignored.
IS COMPLEX Finally, it is worth noting that the implicit assumptions of a progeny prob model are typically violated by the reality of even simple tree models. As a result, the best fit progeny prob for a simple tree model can be quite complex.
The progeny prob framework assumes that one and only one “me” travels along some path in the tree. Conditional on being in one branch, the chances that I become each of the child branches must sum to one. And it may seem natural to have those chances be independent of what would have happened had I instead gone to other branches at earlier times.
But even in simple tree models, there is not a fixed total quantity of branches. So when there is more growth in other branches of the tree, this part of the tree shrinks as a fraction of the whole. And so the sum of the weights for the children of a branch do not usually sum to the weight of that branch. And such weights do typically depend on what happens in distant branches that split off long ago.
It thus seems all the more remarkable that the mysterious Born rule progeny prog for quantum mechanics is so simple and context independent.