Nomic necessity: are there other worlds than ours?
Position: Multiverse concretism
It’s a good thing that our theory is neutral towards concrete "other worlds". There are developments in modern physics that suggest these may exist (and perhaps provably so one day), so we don’t want to preclude them; at the same time, assuming their existence as a mandatory feature of our model seems like one step too far. Let’s just mention, with an open mind, the relevant scientific avenues.
Physically possible worlds in cosmology
The universe could be finite, or infinite. The impact on "other worlds" does not stem from any complex physical modeling, but from a simple mathematical fact: infinity is big. Intellectually, this is a truism, but not one that human minds are very good at handling. Yet the implacable fact remains that if (1) the universe is infinite, and (2) it is homogeneous and isotropic (that is, if the physical conditions that we observe locally around the Earth are not exceptional), then everything that could exist, does. It may be one gazillion bowjillion light years away, but under such conditions, there must be another Earth, with another Jonathan that’s exactly identical to myself and living the same life, but is right-handed. Or if there’s any physically viable way they may exist, one with unicorns.
Now, bringing our actual theories back into play: is the universe, in fact, infinite? There is a common misconception that the Big Bang models imply a finite universe (a tiny ball that grew up quickly to become a large object). That is however not the case: following the Big Bang, spacetime expanded, which means that density quickly decreased: what was concentrated into a tiny ball was all the matter in the currently observable universe. But it’s perfectly possible to imagine an infinite, isotropic (and extremely dense) universe, expanding into an infinite, isotropic (and less and less dense) universe.
What really determines its finitude is spacetime curvature. Most recent observations seem to point to a negative curvature, which implies... an infinite universe!
If that were true, what would be the impact on our model? At first sight, it would be quite devastating. Remember that we engineered the concept of ordinary objects to enforce a clear countability. I today am the same person I was in 1992, and the same person as myself-without-hair or myself-minus-one-finger. All these Jonathans add up to one single human being. If the universe is indeed infinite and isotropic, there are alternate-world Jonathans which will be captured as "the same human being" under the relative identity relation we’ve defined. But since lots of them are living a different life, the careful construction of ordinary properties falls apart: I cannot say anymore that Jonathan lives in London, since some alternate-world members of the equivalence class live in Tokyo or on Mars.
The solution isn’t conceptually difficult: we can reframe our model using world-indexing. For that, we need to first assume that the concrete world can be partitioned: \(\aleph = \bigcup \aleph_w\). Our ’actual world’ is only a distinguished index \(w_\mathfrak{a}\). Then we can define \(w\)-restricted sets of macroscopic objects \(\aleph ^{\bigodot} _w\), the fusions or divisions of primitive objects from \(\aleph_w\) only. Lastly we can systematically rework our definition of relative identity conditions by imposing that to be "the same" implies that there is some \(w\) such that the objects in relation are both members of \(\aleph ^{\bigodot} _w\). The effect is double:
this precludes any trans-world object to be "the same" as anything else.
this ensures that two things can be the same only if they’re from the same world.
So Tokyo-Jonathan and Mars-Jonathan are not the same as me, and the ordinary object architecture can again serve its purpose.
That said, we are not done with difficulties. The fact the concrete world can be "naturally" partitioned was a leap of faith, and is far from obvious. Yet, general relativity helps us here. After all, when we commonly talk of "the universe" in astronomy, we typically mean "the observable universe". From our vantage point, it is estimated at around a 100 billion light-years radius. But obviously it’s the 4D picture that counts. We know that more and more of the universe becomes observable, as invisible galaxies will progressively come into view. Yet even if we waited for an arbitrarily long time, we wouldn’t be able to observe at an arbitrary far distance, due to the combination of the reduced rate of expansion and the density of dark energy. Therefore, if the universe as a whole is infinite, it must be possible to partition it into an infinity of spatio-temporally disjointed parts, which we’ll call worlds. To refine what I said earlier: if Tokyo-Jonathan exists in one part of the universe that’s wholly disjointed from ours, it will not be the same as me; but if he happens to exist in the same part, albeith 90 billion light years away, he is the same.
We can remedy that by either reframing the relative identity relation from common discourse with an implicit circumstantial restriction (e.g. to "planet Earth, around our day and age"), or by considering that the odds are so long that it would happen (even with trillions of galaxies in the observable universe!) that it’s an acceptable deviation from our initial intent.
Physically possible worlds in quantum physics
A more contemporary development that could tie in with a concretist approach to modality is the so-called Everettian, or ’many-worlds’ interpretation of quantum theory. There is little space here to attempt a full exposé of the matter, so allow me to propose a brief summary instead.
Quantum mechanics is often seen as an intrinsically probabilistic, indeterministic, nonlocal theory, but what it really does is to merge two things:
Some kind of fundamental wave equation. In non-relativistic Quantum Mechanics (QM), that is Schrödinger’s equation. In relativistic QM that is the Klein-Gordon equation or the Dirac equation. In the standard model of Quantum Field Theory (QFT), that is a more complex system of equations that’s governed by the relevant interactions’ symmetry groups. Point is, none of those is qualitatively different from classical equations by Fresnel or Maxwell. They are differential field equations governing the evolution over time of a wavefunction. That evolution is unitary, and the resulting model is entirely deterministic and local.
A concept of ’wavefunction collapse’. Indeed, the spoiler is that, unlike electromagnetics or optics, quantum mechanics cannot stop there, for we know that electrons, atoms and many more bodies don’t behave like waves at all when we observe them. In comes the key postulate, often called Born rule: it states that
the outcome of any measurement (for instance of position, momentum, energy...) is a random function whose probability distribution is given by the wavefunction.
after such a measurement, the wavefunction has collapsed onto a pure state that matches the outcome.
Let’s summon Schrödinger’s always serviceable cat to try and picture things. Pre-measurement, the cat is assumed to be in a superposition of quantum states: \[\lvert\psi_{cat}\rangle = \alpha \lvert \textrm{live}\rangle + \beta \lvert \textrm{dead}\rangle\] The ’live’ and ’dead’ wavefunctions are called "pure states": they are eigenfunctions derived from the wave equation and correspond to a physical state where we have perfect information about the cat’s status. When we make a measurement by looking at the cat, he could be either live or dead. If he is alive, then the wavefunction collapses, and \[\lvert\psi_{cat}\rangle = \lvert \textrm{live}\rangle\] That process is by definition non-deterministic; it’s also non-unitary and nonlocal.
This combination of two disjointed processes characterizes much of quantum theories’ strangeness, and is at the core of all resulting metaphysical puzzles. Cue the interpretations of quantum mechanics: what they all seek to interpret, above all else, is that process of wavefunction collapse.
I believe it is important to dispel an oft-held misconception that conflates the collapse of the wavefunction with quantum decoherence. Like wavefunction collapse, quantum decoherence has a part in explaining how the appearance of classical physics emerges, at macroscopic scale, from the undeniably quantum foundations of our world. Yet, quantum decoherence is part and parcel of the unitary, deterministic world of wave equations. What it does explain is why, at macroscopic scales, objects behave in particle-like fashion. The result of countless interactions and stimuli on a macroscopic body outside of the lab is that the density matrix does not feature any mixed terms — the kind that generate interference patterns in a double-slit experiment —, and thus that the system’s quantum state converges towards a probabilistic mix of pure states. In other words, wave-like quantum effects are supplanted by a distribution that matches the epistemic uncertainty of an ideal observer.
That is why we can observe interference in a double-slit experiment easily with electrons, less easily with atoms, and not at all with small dust particles. However, no wavefunction collapse has taken place. Decoherence tells us why we can’t observe cat-interferences, but doesn’t tell us why and how we observe a live cat or a dead cat, rather than a superposition. Formally, the density matrix may not have near-zero terms like \[\lvert\textrm{live}\rangle \langle \textrm{dead} \lvert\] but it still looks somewhat like \[\frac{1}{2} ( \lvert\textrm{live}\rangle \langle \textrm{live} \lvert + \lvert\textrm{dead}\rangle \langle \textrm{dead} \lvert)\]
From that point on there are three possible attitudes towards wavefunction collapse:
Ignore the problem altogether. Most Copenhagen-like interpretations don’t treat wavefunction collapse as a physical process at all, because they don’t treat wavefunctions as physical either, instead considering them as a measure of the information available to an observer. A number of other approaches plainly accept the co-existence of the unitary, continuous process and the non-unitary, discontinuous collapse.
Treat wavefunction collapse as a physical process that can be explained by a revamped wave equation. Usually that entails giving up on the unitary nature of the quantum time evolution, while retrieving it as an approximation at small scales. An objective collapse (independent of any measurement or observer) is then part of the physical theory, rather than a primitive that needs to be uncomfortably tacked on. Penrose has expressed a particularly elegant proposal that involves the interface between the quantum superposition principle and general relativity’s equivalence principle.
The last group plainly denies that wavefunction collapse takes place at all. Hugh Everett criticized the usual conception of the observed object as a quantum system and the observer as a classical one. The truth is that upon observing a quantum state, the observer becomes entangled with it. According to Everett, there is no discontinuity at all: the only physical process is the unitary wave equation, that keeps moving the quantum state through time.
Post-measurement, the universal quantum state involves a superposition of several entangled states. Coming back to the cat, there would be an entangled state of "Jonathan-who-saw-a-live-cat" with "live-cat" and "Jonathan-who-saw-a-dead-cat" with "dead-cat". The fact that I saw a live cat is not, in that picture, the outcome of some obscure rolling of the dice. It simply stems from the fact that "I" am described by the former quantum state, not the latter.... but I could have been the latter as well. Of course, this observation is not a punctual event: what I have seen will influence my behavior, and then the whole universe going forward. Each measurement event is then re-framed as the branching out into a "different timeline" although the "real world" can still be described as a superposition of all those timelines.
Here are our "possible worlds"... We have used the term "measurements" but won’t endorse a psychologist or subjective approach: objective events can cause wavefunction collapse (and thus ’timeline forking’) in the absence of consciousness, human or otherwise. Therefore, at every turn, new timelines appear, covering all the possible ways the world could be observed at that particular moment. This is exactly our definition of physically possible worlds.
The metaphysical implications are not easy to parse for the Everettian enthusiast. The same issues as for the infinite-universe case arise, whereby we need to somehow partition the world to avoid utter confusion in our modal logic. The physical world may be equal to the universal quantum wavefunction, a superposition of mixed states, but we want to be able to define a world in the metaphysical sense as one where all past observations are indeed certain.
If the characterization of objective measurement "by the Universe" can be rigorously developed, there is a chance to define those worlds properly. Instead of carving the world into spacetime bits like in the previous proposal, we would thus be carving a Hilbert space (or some more complex superstring/quantum gravity avatar) into sectors.
Nomically possible worlds
Those are also conceivable in the context of our current physical theorizing. This term designates worlds that are possible under "some" laws of physics that are different from ours.
One example of multiverse model harboring universes with different laws is given by eternal inflation. By way of brief reminder, inflationary cosmology was borne out of Guth’s work, which proved fruitful in explaining away major issues with standard Friedmann/Lemaître cosmology (essentially, general relativity plus a Big Bang concept with sui generis expansion of the spacetime metric).
The first puzzle was linked with the isotropy of the observable universe: the energy density in any direction we look is remarkably constant. Even the Cosmic Microwave Background radiation, representing the residual photons that were first able to travel freely when the universe cooled after the Big Bang, is isotropic, with variations of \(10^{-5}\) at the most. It’s not a shocking fact in itself. The early universe must have had high degrees of anisotropy because of quantum fluctuations, but fluids like a cannister of gas or a pool of water reach thermal equilibrium when they cool down, and the universe has had billions of years to do so... But the difference lies in the rate of expansion necessary to explain the universe’s size in traditional Big Bang models. It is so high that many regions would find themselves part of distinct causal patches: in other words, they would "travel away from each other" so fast that they would find themselves causally isolated before thermal equilibrium could have taken place. After that point, since communication is impossible, there is no reason the same temperature would be observed in all regions. Inflation can explain that since, as the name indicates, it describes a mechanism (usually involving a scalar quantum field called inflaton) whereby expansion would be exponential for a brief moment (in some models, expansion by a factor of \(10^{78}\) for a duration of \(10^{-32}\textrm{s}\)), before resuming a ’moderate’ pace.
A second enigma is the flatness of the universe: previous cosmological models of course admit a near-flat universe as part of their solutions, but that’s a highly unlikely outcome that can be best described by fine-tuning via a cosmological constant. Inflation automatically explains this with its scale and suddennness. The universe right after the Big Bang was still in a high-energy state where spacetime curvature was a direct result of quantum fluctuations. Inflation was so intense and fast that our whole observable universe expanded from a spacetime volume tiny enough, that even quantum fluctuations created no major homogeneity.
Lastly, this superfast dilution could explain the complete absence of magnetic monopoles or other heavy particles predicted by most candidate GUTs (Grand Unification Theories): since their abundant production would have preceded the inflation phase, their density would be so low that it’s unlikely to find even one in our observable volume.
So, the basic principles of inflationary cosmology seem quite promising. But its initial version was not fit for purpose: it couldn’t explain how inflation would stop and result in a re-heated universe that matches observations. I won’t get into details about all rival models that attempt to remedy this. Suffice it to say here that many of them, and chiefly eternal inflation, describe an overall inflationary universe with no beginning nor end, but where inflation can stop locally, basically creating, hot "bubbles", each of which is like an independent universe with its own Big Bang. New bubble are constantly appearing, then expanding, though of course by definition at a much slower pace than the inflation multiverse (if we are to consider individual bubbles as universes, this is what we should then call the super-structure). This gives the multiverse a fractal structure with in principle an infinity of bubbles. Bubble-universe are endowed with different initial conditions, since those arise from tiny inhomogeneities from quantum fluctuations (still visible in the material structures of a universe, such as our galaxy filaments). But they can also feature different physical laws. Even though we might assume there would be irreducible "multiversal" laws, the way symmetries break (with an element of randomness) could lead to differences in the way extra dimensions would compactify, or different fundamental interactions. Since an infinity of worlds would exist with physical laws that are distinct from our "universal laws", this would be a realization of the nomically possible worlds concept.