I think that the solution to the puzzle of Boltzmann Brains will come out of the interpretation of quantum mechanics via the lens of Formal Computational Realism (FCR). On that view, the universe is sampling every possible quantum observable s.t. (i) the marginal distribution of each observable agrees with the Born rule (ii) the overall amount of computation made is minimal. (Tbc this is a very informal description of a rigorous mathematical framework.) For a time moment in an early history of the universe, if the observable you’re sampling is related to the physical state of brains that might exist at time , sampling this observable requires running some computations which we normally think as running inside brains. For a time moment in the late (thermal equilibrium) history of the universe, you’re just sampling a (quantum) Maxwell-Boltzmann distribution, there’s no need to evaluate any cognitive algorithm.
In fact, given two late moments and and an observable , the most economic strategy of sampling and is assigning the same value to both (since they have equal marginal distributions), something that is counterintuitive to how we usually think of time in QM. This is in contrast to early moments and when it’s more economical to reuse the computation that went into sampling when sampling , making the observables correlated approximately in the way the Copenhagen interpretation would predict (for a toy model, the latter claim is proved in the linked post).
That said, a fully formal analysis of BB in the framework is still pending.
Formal Computational Realism also aims to solve the confusions surrounding computationalism (as the name suggests). The key philosophical insight is that computations are actually more fundamental than “atoms”, rather than emergent from atoms. Instead, physical theories are sort of book-keeping devices for predicting which computations actually occur.
While it is possible, in some sense, to answer which computations occur according to a physical theory (this is what the “bridge transform” operator in FCR is doing), this requires information not contained in the physical theory itself, namely knowledge about mathematics. Notice that when we use physical theories in practice, we invoke our knowledge about mathematics all the time. We might naively imagine that it makes sense to think of a mathematically omniscient mind using the same physical theory to draw similar conclusions; however, it doesn’t really make sense: the existence of such an omniscient mind would require all possible computations to already occur inside the mind (or in the process of creating it).
If we start from some fixed state, the distribution of fluctuates with time in a precise way, even if it approximates a Maxwell-Boltzmann distribution (because it arises from a pure quantum state that is just undergoing unitary evolution). Eg. after a Poincare recurrence time, the state has returned to its original value, and takes a non-equilibrium value. This would seem to imply that we must compute the full unitary evolution, even at late times, in order to get the correct fluctuations. Is this wrong?
Does this theory say that any other kinds of observers besides Boltzmann brains don’t have subjective experiences? Can we engineer a scenario that creates such an observer (eg if we have an AI design that is very useful, but we don’t know if it is conscious or not)?
Sampling from a thermal equilibrium distribution is not necessarily cheaper than time evolution. That kind of sampling is roughly speaking “in NP”, while time evolution is just . How sure are we that thermal equilibrium sampling doesn’t involve computations with subjective experience? Like, if the sampling process goes down certain very improbable paths, it’s difficult to know what the most efficient algorithm is going to do.
Thank you so much for this question! It’s silly of me that I haven’t seriously thought about quantum Poincare recurrences in this context, but now that I did, I finally see a path towards formally testing the “no BB in FCR” claim.
Explanation for readers who are following some of the formal details of FCR:
Consider the same setting of Gergely’s post that I linked, but instead of making time evolution stop after T steps, we can make it go on forever. The agent’s memory tape is still of size T, so it will end up cycling through it and (reversibly) overriding it infinitely many times. My conjecture is that, in this new setting, we still have a version of Theorem 4.19 with a non-trivial lower bound. This would imply no BB in the traditional sense: despite the Poincaré recurrences, the agent does not experience all possible histories.
Why would that be true? Because, if you measure the agent’s memory time at a time in which its memory tape is full of “garbage”, the results conveys only a little information about its policy, and its unlikely to make the agent “experience” too much in the formal sense defined by the bridge transform. Whereas if you measure the agent’s memory at time in which the Poincare recurrence recently reset the tape, the “minimal computations” principle would make it likely for it see the same observations it saw during previous such cycles.
Explanation for readers who are not following the formal details of FCR:
The thing is, whether we “must” compute something or not is not a binary. Rather, the probability we compute something increases with the total variation distance between the distributions we need to distinguish. So, as long as different agent policies produce similar distributions, we only have a low probability of computing the policy (which in this framework is equivalent to the agent “experiencing” something). I think this also addresses your remark about “NP”: it’s likely easy to approximate the thermal equilibrium distribution without simulating brains.
Other “observers” without subjective experiences:
I don’t know, but you can in principle use this theory to predict the experiences of an agent inside something like Wigner’s friend experiment or any other scenario that violates decoherence. Implementing such an agent would require a quantum computer.
Actually, I think things might be even better than this: in semiclassical quantum gravity in asymptotically de Sitter spacetime, your quantum state is necessarily mixed (due to tracing over things outside the cosmological horizon, which is how you get Unruh radiation). So, there are no quantum Poincare recurrences. If you plug a stationary mixed state into the FCR interpretation, all the observables become completely frozen in time. If you’re just converging towards a stationary mixed state, I expect the observables to converge towards becoming frozen.
I think that the solution to the puzzle of Boltzmann Brains will come out of the interpretation of quantum mechanics via the lens of Formal Computational Realism (FCR). On that view, the universe is sampling every possible quantum observable s.t. (i) the marginal distribution of each observable agrees with the Born rule (ii) the overall amount of computation made is minimal. (Tbc this is a very informal description of a rigorous mathematical framework.) For a time moment
In fact, given two late moments
That said, a fully formal analysis of BB in the framework is still pending.
Formal Computational Realism also aims to solve the confusions surrounding computationalism (as the name suggests). The key philosophical insight is that computations are actually more fundamental than “atoms”, rather than emergent from atoms. Instead, physical theories are sort of book-keeping devices for predicting which computations actually occur.
While it is possible, in some sense, to answer which computations occur according to a physical theory (this is what the “bridge transform” operator in FCR is doing), this requires information not contained in the physical theory itself, namely knowledge about mathematics. Notice that when we use physical theories in practice, we invoke our knowledge about mathematics all the time. We might naively imagine that it makes sense to think of a mathematically omniscient mind using the same physical theory to draw similar conclusions; however, it doesn’t really make sense: the existence of such an omniscient mind would require all possible computations to already occur inside the mind (or in the process of creating it).
This sounds very intriguing. Questions:
If we start from some fixed state, the distribution of
Does this theory say that any other kinds of observers besides Boltzmann brains don’t have subjective experiences? Can we engineer a scenario that creates such an observer (eg if we have an AI design that is very useful, but we don’t know if it is conscious or not)?
Sampling from a thermal equilibrium distribution is not necessarily cheaper than time evolution. That kind of sampling is roughly speaking “in NP”, while time evolution is just
Quantum Poincare recurrences:
Thank you so much for this question! It’s silly of me that I haven’t seriously thought about quantum Poincare recurrences in this context, but now that I did, I finally see a path towards formally testing the “no BB in FCR” claim.
Explanation for readers who are following some of the formal details of FCR:
Consider the same setting of Gergely’s post that I linked, but instead of making time evolution stop after T steps, we can make it go on forever. The agent’s memory tape is still of size T, so it will end up cycling through it and (reversibly) overriding it infinitely many times. My conjecture is that, in this new setting, we still have a version of Theorem 4.19 with a non-trivial lower bound. This would imply no BB in the traditional sense: despite the Poincaré recurrences, the agent does not experience all possible histories.
Why would that be true? Because, if you measure the agent’s memory time at a time in which its memory tape is full of “garbage”, the results conveys only a little information about its policy, and its unlikely to make the agent “experience” too much in the formal sense defined by the bridge transform. Whereas if you measure the agent’s memory at time in which the Poincare recurrence recently reset the tape, the “minimal computations” principle would make it likely for it see the same observations it saw during previous such cycles.
Explanation for readers who are not following the formal details of FCR:
The thing is, whether we “must” compute something or not is not a binary. Rather, the probability we compute something increases with the total variation distance between the distributions we need to distinguish. So, as long as different agent policies produce similar distributions, we only have a low probability of computing the policy (which in this framework is equivalent to the agent “experiencing” something). I think this also addresses your remark about “NP”: it’s likely easy to approximate the thermal equilibrium distribution without simulating brains.
Other “observers” without subjective experiences:
I don’t know, but you can in principle use this theory to predict the experiences of an agent inside something like Wigner’s friend experiment or any other scenario that violates decoherence. Implementing such an agent would require a quantum computer.
Actually, I think things might be even better than this: in semiclassical quantum gravity in asymptotically de Sitter spacetime, your quantum state is necessarily mixed (due to tracing over things outside the cosmological horizon, which is how you get Unruh radiation). So, there are no quantum Poincare recurrences. If you plug a stationary mixed state into the FCR interpretation, all the observables become completely frozen in time. If you’re just converging towards a stationary mixed state, I expect the observables to converge towards becoming frozen.