In the previous essay, we asked what information is.
Our investigation led us away from the idea that information is a substance which physical objects possess. Information appeared instead as a structure of distinctions, constraints and correlations within a space of possibilities.
This left us with a question that is especially important for quantum theory.
What is the space of possibilities in which quantum information exists?
The usual answer is: the quantum state, represented by the wavefunction.
But what is the wavefunction?
This question is more difficult than it sounds.
We can calculate with it. We can evolve it. We can take its inner products, derive probabilities from it and use it to predict experimental outcomes with extraordinary precision.
But none of this, by itself, tells us what kind of thing the wavefunction is.
Is it a physical object?
Is it a field?
Is it a real entity existing in some high-dimensional space?
Is it merely a mathematical representation of something else?
Or is the question itself malformed?
Our relational investigation suggests another possibility.
The wavefunction is not a thing. It is a structured field of potential instantiations.
This is a small sentence.
It may have large consequences.
From potential to actual
Let us begin with a distinction that is easily blurred in ordinary language:
possible is not actual.
If I say that it might rain tomorrow, I have not described a second, ghostly form of rain which already exists somewhere.
I have described a possibility.
Nor is the possibility arbitrary. It belongs to a structured space. The climate, season, atmospheric conditions and geographical circumstances constrain what kinds of weather can occur.
Tomorrow's weather will be actual.
The climate is not.
The climate describes a structured field within which actual weathers can occur.
This is the distinction we have already used:
climate : potential weathers :: wavefunction : potential instances.
The analogy is not intended to collapse climate science into quantum mechanics.
It is intended to illuminate a conceptual distinction.
A theory of potential is not itself an instance of what it makes possible.
That distinction is surprisingly easy to lose when we talk about the wavefunction.
The wavefunction as potential
Suppose a quantum state describes a system for which several outcomes are possible.
The conventional language encourages us to say that the system is in a superposition of those states.
This is mathematically precise enough.
But the phrase can encourage an ontological picture that is much less precise.
We may begin to imagine that the particle is somehow physically in all of those states at once.
That picture immediately creates difficulties.
What would it mean for an actual particle to be simultaneously in several mutually exclusive actual states?
We then need increasingly elaborate metaphors to explain how the strange quantum object can somehow be all of these things until measurement makes it become one.
But perhaps we began in the wrong place.
Perhaps the wavefunction does not describe a strange object possessing several incompatible actual properties.
Perhaps it describes a structured field of potential actualisations.
The alternatives are not simultaneously actual.
They are possibilities within the quantum state.
An actual event occurs when one of those possibilities is instantiated.
The distinction is fundamental:
The wavefunction is potential; the particle event is actual.
This does not make the quantum state unreal.
Potentiality is not nothing.
The climate is real even though tomorrow's weather has not yet occurred.
A system of legal possibilities is real even though no particular case has yet instantiated one of them.
A grammatical system is real even though no particular sentence is identical with the grammar.
Reality contains structures that are not themselves particular instances.
Quantum theory may be describing one such structure.
Instantiation
This is where the notion of instantiation becomes useful.
A general form can have particular instances.
The word “tree” can be instantiated by countless particular trees.
A grammatical construction can be instantiated in particular utterances.
A pattern can be instantiated in particular physical configurations.
The instance is actual.
The possibility of instantiation is not itself an additional instance.
The distinction allows us to avoid a familiar philosophical confusion: the confusion between a class of possibilities and an actual member of that class.
Our quantum interpretation uses this distinction deliberately.
The wavefunction specifies a structured space of possible instantiations.
A particle detection is an actual instantiation within that space.
The particle is therefore not a little object carrying a wavefunction around inside itself.
Nor is the wavefunction necessarily a mysterious physical object spread throughout space.
Rather, the wavefunction describes the potential structure within which actual quantum events can occur.
That is what makes the wavefunction informationally significant.
It tells us about distinctions among possible outcomes and the relations among them.
What the wavefunction knows
We should be careful with that last phrase.
The wavefunction does not literally “know” anything.
But it contains, in a formal sense, information about what can happen.
If a quantum state assigns different amplitudes to different possible outcomes, it establishes a structured relation among those possibilities.
Some outcomes are more probable.
Some are less probable.
Some may be impossible.
The state therefore constrains the space of possible actualisations.
This gives us a striking correspondence with our previous account of information.
We said:
Information is the structure of distinctions and constraints within a space of possibility.
Now we can say:
The quantum state describes a structured space of possible physical instantiations and the constraints governing their actualisation.
The two ideas fit together remarkably well.
Quantum information does not have to be conceived as a mysterious substance stored inside a quantum object.
It can instead be understood as information concerning the structure of potential actualisation.
Superposition without metaphysical clutter
This also gives us a different way to think about superposition.
Suppose a system has two possible outcomes, A and B.
The ordinary language of superposition can tempt us to say:
The system is really A and B simultaneously.
But that is not the only way to understand the mathematics.
We might instead say:
The quantum state specifies a structured potential for actualising A or B, together with the amplitudes and phase relations that determine how those possibilities participate in subsequent quantum processes.
There is no need to imagine two fully actual worlds hiding inside the particle.
There is a potential structure.
There will be an actualisation.
The quantum state relates the two.
This interpretation also gives us a reason to take interference seriously.
Interference is not merely a peculiar behaviour of tiny objects.
It is evidence that the possibilities themselves have a structure.
The alternatives are not independent boxes from which nature simply selects one.
They possess relations—encoded mathematically in the quantum state—that affect the probabilities of later actualisations.
Potentiality therefore has structure.
That is the important point.
The possibilities are not ignorance
At this point, an obvious objection arises.
Perhaps we are simply replacing quantum reality with classical ignorance.
If the wavefunction merely describes possibilities, perhaps the particle already has a definite state and we simply do not know which one.
But this is not what is being proposed.
Classical ignorance says:
There is already an actual state; we simply do not know which.
Our relational interpretation says something different:
The quantum state describes a structured space of potential actualisations, and the actualisation is not merely the revelation of a pre-existing classical state.
The distinction matters.
If the quantum possibilities were merely hidden classical alternatives, the interference structure of quantum mechanics would be difficult to understand.
The quantum state does not merely list possible classical states.
It relates possibilities in a specifically quantum way.
So potentiality here is not epistemic ignorance.
It is part of the physical description itself.
The particle as instance
We can now turn the usual picture around.
Instead of beginning with:
There is a particle, and it has a wavefunction.
we might begin with:
There is a quantum potential structure, within which particular particle events can be instantiated.
The particle is therefore not ontologically prior to the wavefunction.
Nor is the wavefunction simply a physical cloud surrounding a little particle.
The relation is more like:
potential → instantiation
rather than:
object → property.
This is precisely the kind of reversal that relational ontology invites.
What appears, from an object-centred perspective, to be a thing possessing a state can instead be understood as an actualisation within a structured field of relations and possibilities.
The actual is not less real because it arises from potential.
But neither is the potential an actual thing.
What, then, is a quantum state?
We can now give a tentative answer.
A quantum state is not necessarily a physical object in the same ontological category as the events it describes.
It is a structure of potential physical instantiation.
It specifies relations among possible actual outcomes.
It constrains what can happen.
It assigns weights to possibilities.
It determines how those possibilities participate in subsequent evolution.
And when an actual event occurs, that event is an instantiation within the potential structure described by the state.
This interpretation has an interesting consequence.
The wavefunction does not need to be “located” in the same way as a particle.
We can ask where a particle is.
We can ask where a detection occurs.
But asking where the space of possible instantiations is located may be asking a different kind of question.
The potential structure is represented mathematically over configuration space or Hilbert space, depending on the formulation.
Its physical significance lies in the possibilities and relations it describes.
We should therefore resist turning the mathematical representation into an additional physical object simply because the representation is extraordinarily successful.
Information revisited
This brings us back to information.
In the previous essay, we argued that information is not a substance.
Now we can see why the wavefunction is such a natural bearer of information without itself being an information-containing object.
The wavefunction specifies distinctions among possible actualisations.
It tells us how possibilities are related.
It constrains the outcomes that can occur.
It therefore contains information in precisely the relational sense we developed earlier.
But the information is not something hidden inside the wavefunction as though the wavefunction were a box.
The information is in the structure represented by the wavefunction.
This may sound like a small distinction.
It is not.
If the wavefunction were a physical object containing information, then we could naturally ask what happens to that object.
If it is a structure of potential, we must instead ask what happens to the potential structure itself and to its relations with actual events.
That is already a different black-hole problem.
Quantum information and identity
There is another consequence.
Suppose a quantum state evolves.
We ordinarily say that the state at time (t_1) evolves into the state at time (t_2).
This is perfectly legitimate mathematically.
But what makes it the same information at the two times?
Under an object-centred ontology, we might imagine that some entity persists and carries its information forward.
Under a relational ontology, continuity can be understood differently.
The later state is related to the earlier state by a lawful transformation.
The structure of distinctions and possibilities is transformed, but not arbitrarily.
In unitary quantum evolution, the transformation preserves the relevant mathematical structure.
The important thing is therefore not that an information-substance travels through time.
It is that the relations constituting the informational structure are preserved under the evolution.
This is a much more relational conception of information preservation.
And it will become crucial when we return to the black hole.
Entanglement makes the point sharper
The idea becomes even more compelling in quantum entanglement.
Suppose two systems are entangled.
It is tempting to imagine that each system possesses its own state and that the two states happen to be correlated.
But quantum mechanics often gives us a more radical situation.
The joint state contains information that cannot be reduced to independently specified states of the two components.
The relation is not merely something added to the objects.
It is part of what the physical state is.
This is one of the places where relational ontology seems particularly at home in quantum theory.
The informational structure belongs to the relation.
The systems are physically significant partly through the ways in which their possible states constrain one another.
And now the black-hole problem becomes more interesting still.
Because if information is fundamentally relational, then information loss cannot simply mean:
a thing containing information disappeared.
It must involve a transformation—or perhaps destruction—of the relevant relations and correlations.
That is a considerably more precise question.
The horizon changes the question
Imagine now that one member of an entangled system crosses a black-hole horizon while another remains outside.
The ordinary language says that information has gone into the black hole.
But our ontology asks a different question.
What happens to the relations that constituted the informational structure?
The horizon changes the causal structure connecting interior and exterior events.
Certain future-directed relations that were previously possible are no longer available to an external observer.
The issue is therefore not simply that a physical object has moved behind a boundary.
The issue is that the space of possible relations has changed.
This is exactly why the wavefunction cannot be treated as a thing.
If the wavefunction were merely an object carried along by the particle, we could imagine following it into the black hole.
But if the wavefunction is a structure of potential instantiations, then gravitational collapse raises a deeper question:
How does the structure of quantum potential relate to a geometry whose causal relations are themselves changing?
That is where quantum theory and general relativity begin to press against one another.
The black hole does not swallow a wavefunction
We can now make the provocative statement explicit.
A black hole does not swallow a wavefunction in the same sense that it swallows a physical object.
It may contain, in an appropriate description, the physical system whose quantum state we represent by a wavefunction.
But the wavefunction is our representation of a structured field of potential actualisations.
When the physical system enters a region from which certain external relations are no longer available, the important question is not:
Where is the wavefunction now?
It is:
How has the relational structure of its possible actualisations been transformed?
This is a much harder question.
But it may also be the right one.
Potential does not disappear merely because an instance does
There is one further distinction we should preserve.
Suppose an actual particle event occurs.
The particular potential that preceded it has not necessarily been “destroyed” in the sense that a physical object has been destroyed.
The actualisation changes the state of the system.
It updates the structure of future possibilities.
Potential and actual are therefore dynamically related.
An actual event becomes part of the conditions determining subsequent potential.
This gives us a kind of recursive structure:
potential makes actualisation possible; actualisation transforms subsequent potential.
The world is not a static inventory of things.
It is an evolving relation between what can happen and what has happened.
That formulation may eventually allow us to think about black-hole evaporation in a new way.
A collapsing star establishes one field of possibilities.
The formation of a horizon changes the relations within that field.
Hawking radiation constitutes further actualisations.
Those actualisations modify the subsequent state of the system.
The black hole evaporates.
The question is whether the final field of possibilities and actual events retains the informational distinctions of the initial one.
That is a relational formulation of the information problem.
What we have gained
We have not solved the black hole information paradox.
Nor should we pretend that we have.
But we have changed the conceptual furniture of the problem.
The wavefunction is no longer a mysterious physical thing.
It is a structured field of potential instantiations.
The particle is not ontologically prior to that potential.
It is an actual instance.
Quantum information is not a substance stored inside a quantum object.
It is expressed in the structure of distinctions, constraints and correlations among possible actualisations.
And information preservation is not the survival of an information-object.
It is the preservation, under transformation, of the relevant relational structure.
This gives us a new way of stating the central problem.
The question is no longer:
What happens to the information carried by matter when matter falls into a black hole?
It is:
What happens to the relational structure of quantum potential when the geometry governing physical relations develops an event horizon?
That question brings us to the boundary between quantum theory and general relativity.
And it is precisely there that our next conceptual move becomes necessary.
For if the wavefunction is not a thing, perhaps spacetime is not a thing either.
Perhaps the two theories have been asking us to think relationally all along.
We simply have not always listened.
That is the question of the next essay.
There is no curved thing called spacetime.
There are relations among intervals.
And once we take that possibility seriously, the geometry of a black hole begins to look rather different.
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