Can a Universe Be a ‘Beable’?

One approach to the problem of induction is to build theories from the ground up. That is, rather than construct theories based on our observations of the world, we could attempt to deduce them from first principles, i.e.

I wish to put an emphasis on the phrase ‘universe of our experience’ here. Whatever our motivations as scientists may be, we’re all ultimately trying to understand the world around us. So when physicists postulate things that are far removed from everyday experience like strings or alternate universes, they are not merely engaging in mental gymnastics. Ostensibly they do so in an effort to better explain the universe of their experience. For example, a cosmologist may spend time studying a de Sitter universe, even though it is quite clear that we do not live in one, in order to better understand the universe we do live in.

The concept of a universe would seem to present a problem for Bell’s theory of local beables. As Bell himself said, “When the ‘system’ in question is the whole world where is the ‘measurer’ to be found?” [5], p.

The problem is that defining a universe turns out to be a trickier proposition than it might initially appear. Colloquially, a universe is defined as the totality of everything that exists [1]. The problems with this definition are numerous. First, it is not clear how a universe would be defined within the context of any theory that admits multiple universes, particularly in such a manner that they could be distinguished in some meaningful way. The nature of what we mean by a universe in such instances remains largely unsettled [2, 26]. Second, it is inherently ambiguous in regard to both ‘totality’ and ‘existence.’ The totality of all that exists to a proponent of an Everett-De Witt multiverse, for example, includes an infinite number of universes. This definition is simply too vague to qualify as a beable for any realistic theory.

Alternatively, an operationalist might define a universe as the totality of all that can be measured. Wheeler’s participatory universe takes this idea to its logical extreme by suggesting that only things that can be measured can exist [28]. This, of course, won’t get us very far in the context of beables. In Bell’s conception, the observables corresponding to measurements are constructed from beables. This implies that full knowledge of certain beables may not be possible.

Eddington noted that physical knowledge takes the form of a description of a ‘world’ and thus defined the universe to be this world [13]. In other words, he defined the universe to be the totality of extent of physical knowledge, i.e. “the theme of a specified body of knowledge” [13], p. 3. At first glance, this would appear to be very similar to the operational definition and would thus pose similar problems in relation to Bell’s concept of beables. But this is only true if physical knowledge is limited by what can be directly measured. It leaves the door open to knowledge that cannot be directly measured but might possibly be reliably inferred. But that, of course, brings us back, once again, to the problem of induction. So while Eddington’s definition of the universe may not pose a direct problem for the concept of beables, it does run into the problem of induction.

One could also attempt to define a universe topologically as some kind of space­time manifold, but this presents at least three problems. First, it assumes that the manifold itself is somehow ‘real’ and not merely a mathematical abstraction, e.g. a universe entirely devoid of anything—matter, fields, et al.—would, by definition, still have a metric. Yet it seems nonsensical to even speak of a metric for a perfectly isolated space devoid of literally anything. What meaning would space and time even have in this case? In any case, debate over the ontological status of spacetime is still ongoing [10, 11]. The second problem here is that a topologically defined universe does not seem to explain emergent spacetimes (for examples of theories that involve an emergent spacetime, see [16, 25, 27]).

Some theories define the universe based on a wavefunction of some kind [23], e.g. as a solution to the Wheeler-DeWitt equation, H ) = 0. One might immediately criticize this definition on the grounds that it involves a wavefunction which carries a great deal of interpretational baggage. However, if one derives the Wheeler-DeWitt equation from something like the ADM formalism [3], it becomes clear that it is more formally a field equation. Solutions to the Wheeler-DeWitt equation are not spatial wavefunctions in the sense implied by non-relativistic quantum mechanics.

Rather they are functionals of field configurations taken on all of space. As such there is no time evolution to the system. Time can be introduced by ordering the set of all solutions, though this implies a preferred foliation. In any case, the Hamiltonian, though still an operator in a Hilbert space that acts on wavefunctions, is not quite the same beast as in non-relativistic quantum mechanics. The solutions to the Wheeler- DeWitt equation contain all the information concerning the matter and geometry of the universe, i.e. the topology of spacetime and the matter therein. So, in that sense, solutions to the Wheeler-DeWitt equation would seem to be a more fundamental definition of a universe than one based solely on a topology.

Wavefunctions and wave functionals do not necessarily pose a problem for Bell’s concept of beables. Though it is commonly thought that his theory was one consisting exclusively of local beables, i.e.

[t]he lattice fermion number are the local beables of the theory, being associated with definite positions in space. The state vector \t) also we consider as a beable, although not a local one [4], p. 176. [my emphasis]

So he grants beable status to the state vector. The state vector \t) evolves in time according to the Schrodinger equation and the usual Hamiltonian operator. Wave­functions can, of course, be constructed from state vectors, though that does not necessarily make them beables. Remember that observables are constructed from beables and can occasionally be promoted to the status of beable, but are not usually beables themselves. In fact, it is worth noting that in [7], p. 53, he explicitly does not grant beable status to the usual, spatial wavefunction due to the nonlocality associ­ated with its instantaneous collapse over all space upon measurement. But the wave functional in the Wheeler-DeWitt equation suffers from no such defect since it does not evolve in time. It simply ‘is’.

It is worth pausing here and briefly reviewing the nature of beables. Bell’s defi­nition of the term actually includes subtle variations over the many publications in which he employs it. Likely these represent an evolution of his thinking on the sub­ject. One initially gets the impression that beables must be classical things such as pointers and knobs and instruments and, perhaps, fields (as long as they are classical or, in his words, ‘physical’). Later, Bell suggests that beables are what ‘exist.’ In his discussion of beables in quantum field theory, he leaves any classical notion behind, granting beable status to the lattice fermion number density and the state vector. Nevertheless, the concept of ‘beable’ is very clearly meant to define the ontology of a theory, i.e. what the theory is about.

So where does that leave us? If we define a universe as a solution or set of solutions to the Wheeler-DeWitt equation, where those solutions are functionals of field configurations, then it seems that a universe can be a beable and can thus serve as an ontology for a theory. But is there something more that can be said?

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