Hilbert Space

Heroic & Dark Fantasy and Science Fiction Character created by Kevin L. O'Brien

Untitled, © by Boris Vallejo

Untitled, © by Adult Art Network

Hilbert Spacen the essay on Parallel Universes, we described what is called a Level 3 multiverse (ML3). An ML3 is one in which all possible outcomes of every decision, every cause, every interaction permitted by quantum mechanics already exists simultaneously in separate universes. For example, if you role a die, you will get one of six possible outcomes. In any one universe, only one outcome is possible, but in an ML3, all six outcomes occur at once, just in separate universes. Being trapped in our universe, we can see only the one outcome, but if we were somehow able to transcend our universe and observe all the universes in an ML3 at once, we would see all six outcomes at the same time.

The problem with an ML3 is that it really tells us nothing new. That is, an ML3 is really no different from a Level 1 (ML1) or a Level 2 multiverse (ML2). For example, a classical ML1 occurs when a Big Bang creates an infinite number of Hubble volumes and randomly sets the initial arrangements of their particles. A quantum ML1, however, starts off with just one Hubble volume, that then rapidly splits into an infinite number of branching volumes, each with its own set of probable outcomes. Yet the result is exactly the same in both cases: an infinite number of universes with an infinite variety of timelines. Put another way, the distribution of outcomes on different quantum branches in one Hubble volume is identical to the distribution of outcomes in multiple Hubble volumes within a single quantum branch.

A Level 3 MultiverseSo the result of adding an ML3 to a level one is to simply compound to infinity the number of possible parallel Hubble volumes, most of which will be identical to each other since each can only hold a finite number of particles arranged in a finite number of combinations. And the same result will occur at progressively higher levels as well. If through symmetry breaking an ML2 can create an infinite number of ML1s, each with its own cosmological properties, it can also create one that splits off into an infinite number of quantum branches. This phenomenon is known in statistical mechanics as the principle of ergodicity.

Alternate Timelines

One can, however, imagine a way in which an ML3 can add something new to the nature of the cosmos. One aspect of an ML1 is that, because there are many more Hubble volumes than there are particles contained within one, the arrangements of these particles must start to repeat themselves, and the more universes there are, the more duplications there would have to be. Yet even in an ML1 in which the arrangement that constitutes our Hubble volume is duplicated trillions upon trillions of times, the chances of having other Hubble volumes exactly identical to ours except for differences in their timelines is still so exceedingly remote as to be virtually impossible. Only if there were an infinite number of Hubble volumes could that be the case, because in any infinite series all events with non-zero probabilities must occur. If, instead, there are only a near-infinite number of Hubble volumes, then some events would still be so remote as to be unrepresented in any existing Hubble volume.

If, however, we assume that there is an ML3 superimposed upon our ML1, we can allow for a limited number of classical Hubble volumes, only now each will have an infinite number of timelines branching off, each with its own alternate history. In that way, we can have just one volume that looks like ours, but still have others that represent every possible way that our volume can evolve.

Abstract Space

A mathematical representation of Hilbert SpaceBut where are these other Hubble volumes? In an ML1, the other classical volumes lie elsewhere in our "local" spacetime continuum, beyond the boundary of our volume, across trillions upon trillions upon trillions of light years of space. We can conceivably travel to them, if we can ever develop a way to go faster than light, teleport ourselves, or otherwise jump across the enormous distances. But the quantum Hubble volumes in the ML3 lie elsewhere in an abstract, infinite-dimensional realm known as Hilbert space. They occupy the same physical spacetime as our Hubble volume, but along a different timeline that is separated from ours in Hilbert space. At present, there is no conceivable way we could even detect these parallel volumes, much less visit them, even though they technically lie right next door to us, quantumly speaking.

The nature of Hilbert space is bound up in decoherence theory, which was described in the essay on the history and theory of quantum mechanics. Hilbert space is the theoretical region in which a wave function exists and evolves unitarily. The many-worlds interpretation of quantum mechanics predicts that even if you start out with a single, classical universe, its wave function will evolve to the point where it will be transformed into multiple superpositions of many different classical universes, each one corresponding to a different outcome created by the wave function as dictated by Schrödinger's wave equation. As the wave function evolves and creates new outcomes in Hilbert space, new classical universes split off from the others, forming an ML3. This splitting, however, also occurs in Hilbert space, so decoherence prevents any of the inhabitants of these universes from noticing the effect. All they see is a single classical universe with an underlying randomness governed by the laws of probability.

Bird's-Eyes and Frog's-Eyes

Imagine, though, if there was a being or a group of beings that were not affected by decoherence. What if their brains, or whatever passes for their brains, were perfectly coherent at all times, so that they could see superpositions? What would an ML3 look like to them?

This situation is analogous to the different views that a bird and a frog have of a countryside. A bird can take in the whole landscape, whereas a frog can only see what's immediately around it. A bird's-eye view of an ML3 is unaffected by decoherence, but it doesn't see multiple parallel worlds. What it does see is a single quantum world in Hilbert space described by a single wave function, with no splitting or parallelism. As the wave function evolves smoothly, continuously, and deterministically over time, it would see within the universe a vast number of parallel timelines, continuously splitting and merging. A frog's-eye view, however, sees only a tiny fraction of this Infinitely branching timelinesreality, because it is a resident of one of these timelines and is subject to decoherence. What it sees is a single universe that operates by a set of classical laws with a degree of randomness built into it governed by probabilities.

Such beings would exist as a multitude of superpositions, except they would be aware of it and could perceive and interact with all their parallel selves in all parallel universes throughout all the ML1s and ML2s contained in the ML3. This would make them "multidimensional" beings without having to deal with the modern scientific reality of four infinite dimensions and seven subatomic dimensions. This could make them virtual indestructible and immortal, and give them great, almost divine power, since their wave functions would now be isolated from that of the universe and thus would evolve independently. Yet it would still limit them by making them subject to the same quantum processes, such as the Schrödinger equation, that we and our universe are subject to. It might also cause their powers to behave much like magic (especially if they have the ability to manipulate the universal wave function), or to cause them to be subject to cyclical periods of torpor and inactivity (do their wave functions evolve in some kind of cyclical pattern?). This idea may prove to be very fruitful for creating new backgrounds for stories.

Sources / Further Reading

"100 Years of Quantum Mysteries" by Max Tegmark and John Archibald Wheeler, Scientific American, Vol. 284, No. 2, February 2001, pp. 68-75

"Parallel Universes" by Max Tegmark, Scientific American, Vol. 288, No. 5, May 2003, pp. 40-51

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