The Giant Measurement at the End of Time

2.5.6.0.1 Kent and Bousso&Susskind have proposed approaches to defining quantum wavefunction branches based on late-time information. The basic idea is to exploit the way that quantum information propagates in our universe in particular: (a) photons(and gravitons) carry away information at the speed of light, (b) the universe is expanding and cooling, meaning that these photons(in some cosmological environments) may propagate uninterruptedly forever.

2.5.6.0.2 What this means is that photons very late in time are a privileged set of degrees of freedom: they plausibly carry information about everything that has classically "happened" previously. This suggests a way of defining the set of branches: just "measure"(apply a fictitious measurement operator, that is) the photon degrees of freedom very late in time!

2.5.6.0.3 That's the basic idea. There are many subtleties of course, and Kent and Bousso/Susskind have different proposals. I'll review each in turn and then talk about the philosophical merit of these approaches in general.

2.5.6.0.3.1 The following discussion uses some elementary definitions of objects from general relativity. If you know what a spacelike surface and lightcone are, that should be enough.

2.5.6.0.3.2* Note that as of the time of writing, QFTs in arbitrary space-times are not totally mathematically well-defined objects -- and what's more, the fully correct theory of the objects we are discussing should be quantum gravitational, where we don't even know what the exact form of the theory should look like. Thus these constructions are a bit sketchily defined, even compared to QFTs which are already mathematically sketchy.

2.5.6.1 Quantum Reality via Late-Time Photodetection

2.5.6.1.1 Kent has proposed a few variants of this idea. I'll review the version in his paper Quantum Reality via Late-Time Photodetection.

2.5.6.1.2 Kent is interested in defining a set of possible histories of what he calls "beables", which are essentially the classical-like variables which we think of as defining a possible world, e.g. the positions of objects.

2.5.6.1.3 For his set of beables Kent considers the expected values of quantum observables at a given time, conditioned on some particular late-time outcome of photodetection measurements across all of space(with an additional subtlety I'll get to momentarily).

2.5.6.1.4 Why does this make sense? The idea is that the late-time photons should contain all the information about possible classical experiments, pinning down a possible quantum state associated with a set of measurement outcomes. The expected values then give us the classical-like variables associated with this set. Taking the expected value is necessary because even in a classical-like world all degrees of freedom will still be "spread out" in a quantum manner to some degree.

2.5.6.1.4.1 For example, if we consider a covalently-bonded molecule, its position might be decohered by a measurement and thus ultimately correlated with late-time photons. However, the internal state of the electrons in the covalent bond may not be. So if we condition on late time photons, the position of the molecule will be approximately fixed, but the positions of the electrons will not. Taking the expected value of the probability of detecting an electron in a given place will give a spread out "blob" around the covalent bond. This gives the quasi-classical state we are looking for.

2.5.6.1.5 Now, however, a problem arises. If we project onto an entire late-time space-like surface, this is correlated with the results of all past experiments. This means that the beables at a given point in time will be conditioned on the results of experiments in that point's future, which may give strange results.

2.5.6.1.5.1 For example, say that some time later we take our molecule from the previous experiment and measure the position of the electrons in the covalent bond. The results of this experiment will be recorded in some late-time photons. So if we condition on all these photons, the electrons' positions will be decohered at earlier points as well! In other words, when we consider the expected value of the electrons' position earlier, we will no longer find them in a "blob" corresponding to a normal covalent bond, but rather a state which will evolve to whatever position measurement we end up sampling.

2.5.6.1.6 Clearly, this is undesirable. Kent's solution is to restrict the set of photodetection results we condition on in the future space-like surface. For a given point in spacetime p, we only condition on points on this surface outside the forward light cone of p.

2.5.6.1.6.1 In the molecule example, if we consider a point before we measure the electrons' position, the future experiment, and hence all photons downstream of it, are in the forward light-cone. So these experimental results would not be conditioned on, restoring the normal covalent bond.

2.5.6.1.6.2 This means that measured systems will "collapse" to a given value slightly after they have been measured. In particular, this will occur once there are outgoing photons spacelike separated from the system.

2.5.6.1.6.3 Note however, that we still sample a single shared set of measurement outcomes for all points in spacetime; it's just that when defining the beables at a particular point, we only condition on the outcomes outside of the forward light cone. This ensures we define a consistent shared reality.

Kent defines beables at the point y by conditioning on photon measurements on the blue portion of the forward surface. From page 19 here.

Kent defines beables at the point y by conditioning on photon measurements on the blue portion of the forward surface. From page 19 here.

2.5.6.1.7 So in summary, Kent's prescription for defining a set of quasiclassical worlds and their probabilities is as follows: given some initial quantum state on a spacelike surface, we evolve it forward until some late-time spacelike surface. We then measure the state of photons on this surface, obtaining a particular set of measurement values defining a quasiclassical world. The quasiclassical "beables" at a given point p in spacetime between these two surfaces are defined to be the expected values of quantum observables(Kent uses the values of the stress-energy tensor) conditioned on the portion of these measurements outside the forward-light cone of p.

2.5.6.1.7.1 To define such a history for the entire universe, and remove the dependence on choice of surface, Kent takes the limit as the spacelike surface is evolved to future infinity. Given our cosmology(an increasingly empty and sparse universe) it seems plausible that this gives rise to a well-defined limit. There are also various other technicalities I won't get into here.

2.5.6.2 The Multiverse Interpretation of Quantum Mechanics

2.5.6.2.1 Bousso and Susskind defined a related construction in their paper The Multiverse Interpretation of Quantum Mechanics, initially motivated by the problem of defining measurement outcomes in an inflationary multiverse.

2.5.6.2.2 They consider both a local and global construction; the global construction is a limit of local constructions so let's talk about the local construction first. The idea is to consider the possible experiments that can be conducted by a local observer, using what's called a causal diamond.

2.5.6.2.3 What's a causal diamond? It's essentially defined by gluing together a forward and backwards lightcone. In more detail, if we have a point p in spacetime and a point q in its future, we consider the future lightcone of p and the backwards lightcone of q. These meet at a spacelike surface S; the causal diamond is the union of the interiors of the forward and backwards lightcone up until S.

Figure: A causal diamond. From here.

Figure: A causal diamond. From here.

2.5.6.2.4 Any finite experiment carried out by an observer lives in some causal diamond. Why is this? If we consider an observer who evolves from a point p in spacetime to a later point q, any effect the observer can have, for instance setting up a measurement device, lies in the future lightcone of p. And any information that they can observe lies in the past lightcone of q. Thus the effects they can observe from a given experiment are determined by the sequence of quantum states inside the diamond.

2.5.6.2.5 This means that the future boundary of the causal diamond(i.e. the part consisting of the backwards lightcone) provides an objective choice of "environment" to condition on when considering the results of experiments performed in the diamond.

2.5.6.2.5.1 Note that conditioning on photon states on the boundary may also decohere events outside of the diamond in its causal past. For example, we can run an experiment on Earth measuring the cosmic microwave background. This is correlated with quantum events in the distant past, but photons leaving our local causal diamond contain enough information to decohere them(due to correlations of our measuring apparatus, brains etc. with the events). A finite diamond will generally not have enough information on its future boundary to decohere all classical-like events in its past, however.

2.5.6.2.6 So this gives us a local objective notion of decoherence. The causal diamond also has the nice feature that it defines a branch-like structure on the set of possible measurement results. We can consider information that has left the future boundary up until a given point. For instance, say we measure a system, wait a bit, then perform further measurements on the same system. The photons associated with the first measurement will leave the causal diamond's future boundary before the photons associated with the second.

Figure: Defining a branching structure from the order in which photons exit the diamond's future boundary. From page 15 here.

Figure: Defining a branching structure from the order in which photons exit the diamond's future boundary. From page 15 here.

2.5.6.2.7 How do we extend this to a global picture? The idea is to consider an indefinitely long-lived observer, and take the limit of diamonds defined by a sequence of points on their future trajectory. This will produce a sequence of diamonds of unbounded area(in asymptotically flat universes). Some photons will escape the sequence of diamonds -- aka, for any diamond, some time evolution of the photon's state will be on the boundary. We may equivalently say that these photons escape to null infinity(which is simply the space of possible unbounded photon trajectories). Conditioning on these photon states provides a limiting notion of decoherence and branching for the past of the limiting causal diamond.

2.5.6.2.7.1 In what sense is this a global picture? For an indefinitely long-lived observer in an asymptotically flat universe, their past light cone eventually includes every location in their universe[1]. So every location in spacetime is eventually included in the volume that can potentially be decohered by the limiting photons.

2.5.6.2.7.2 This construction doesn't apply to non-asymptotically flat spacetimes. In universes with a positive cosmological constant, observers' past lightcones don't eventually include everything, as accelerating expansion means an increasing volume of spacetime is forever out of their reach. In this case the future boundary of the diamond approaches a cosmic horizon with finite entropy.

2.5.6.2.7.3 Our universe appears to be of this type, with a (very tiny) positive cosmological constant. That said, it's not out of the question that the apparent c.c. is actually time-dependent(due to quintessence say) and decaying to zero. And there are ways to try adapting the construction to non-asymptotically-flat universes.

2.5.6.2.8 The use of the order in which photons escape to define a branching order bears some resemblance to Kent's proposal to define beables via conditioning only on future events outside a point's future lightcone. If we consider an observer evolving forward in time through a Kentian multiverse, they condition on an increasingly large portion of the future surface. This defines a tree-like structure on the set of possible worlds, which seems plausibly equivalent to that defined by Bousso & Susskind's construction in the limit of the future surface approaching infinity.

2.5.6.3 General Discussion

2.5.6.3.1 So that's my preferred family of definitions of wavefunction branches. Why do I like them?

2.5.6.3.2 First of all, they are plausibly correct. That is, it's plausible that, given a quantum state corresponding to our early universe, if you applied the time evolution operator followed by a late-time photon measurement operator, you would in fact get the set of states and probabilities corresponding to what we intuitively think of as quasiclassical worlds. I know of no other proposed definition that seems as likely to me to meet this bar.

2.5.6.3.3 Secondly, they are elegant. The set of worlds is defined by a single measurement, using the standard quantum definition of measurement. The branching structure is determined by a sequence of measurements, themselves determined by geometrically natural constructs from relativity. There are no arbitrary constants.

2.5.6.3.4 "Isn't the retroactive nature of the construct kinda wacky?" Any definition of wavefunction branching will have at least a bit of a retroactive character, because isolated systems can only be said to have decohered/branched some time after their initial interaction with the environment has sufficiently propagated records/increased the complexity.

2.5.6.3.5 Here's a general argument against any definition of wavefunction branching that doesn't rely on some sort of asymptotic late-time measurement. Say we have a definition based on (e.g.) number of copies of records, or circuit complexity. Then it would be possible for a sufficiently technologically advanced civilization to "spoof" the definition by conducting an experiment in which they evolve an isolated quantum system to meet the definition, then reverse the evolution. According to the definition, branches would form and then re-merge! A set of branches in a sensible definition should never do this.

2.5.6.3.6 This concludes my account of the problem of defining wavefunction branches/the measurement problem. Now what does this have to do with our original goal of explaining the relation between the laws of physics and the existence of life/sentience?

2.5.6.3.6.1 The full answer is complicated and will occupy much of the rest of the blog. But basically, the idea is that quantum mechanics combines two features: observations are defined by information copying, or Darwinism, or propagation of influence. And yet at the same time, the state of the system being measured/copied is to some degree sensitive to the context/basis in which it is measured. The measurements are chained together to form an irreversible history. I think these features recur in many other contexts in our world.

2.5.6.3.6.2 But first I'll briefly go over some more aspects of physics that strike me as potentially relevant re: life and sentience.

^Although, as Bousso & Susskind discuss in their paper, if the asymptotically flat universe is merely a bubble in an inflationary multiverse, the past light cone will not eventually include everything in this wider universe. It will include everything in the bubble's past lightcone. They discuss a proposal to define observables in the rest of the multiverse via "hat complementarity", in analogy with black hole complementarity.

Although, as Bousso & Susskind discuss in their paper, if the asymptotically flat universe is merely a bubble in an inflationary multiverse, the past light cone will not eventually include everything in this wider universe. It will include everything in the bubble's past lightcone. They discuss a proposal to define observables in the rest of the multiverse via "hat complementarity", in analogy with black hole complementarity.