Why A Many-Minds Interpretation of Quantum Theory?

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Matthew J. Donald

The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great Britain.

e-mail: mjd1014@cam.ac.uk

home page: http://people.bss.phy.cam.ac.uk/~mjd1014

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Outline

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Why A Many-Minds Interpretation?

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Quantum theory suggests that some version or generalization of the Schrödinger equation
Ψ(t) = e^-itH Ψ can be used to describe every known physical system; at least over short time intervals,

of course, we don't know how to describe gravitational interactions (yet),

moreover, something seems to go wrong over longer times.

What goes wrong?

Suppose we describe, as we can, a single electron heading for an interference grating by a localized wavefunction.

According to the Schrödinger equation, after passing the grating, the wavefunction by itself constitutes the entire interference pattern.

Nevertheless, the electron always appears to make a well-localized impact on a screen at only one of many possible places.

The Schrödinger's cat experiment involves a similar problem:

We only ever see a single cat being either dead or alive, but there are possible circumstances in which the Schrödinger equation will inevitably to lead us to solutions which describe the cat as both.

More generally, there are some wavefunctions which describe situations more-or-less as they seem to be observed — situations, in other words, which we would be reasonably happy to refer to as describing parts of possible worlds — such as

molecules formed of well- but not perfectly- localized nuclei surrounded by clouds of electrons in appropriate orbitals,

live cats,

brains thinking definite thoughts,

but in time such wavefunctions can come to describe unobserved situations such as

delocalized molecules,

cats which are both dead and alive,

brains observing cats which are both dead and alive.

Of course this is a problem which one will only take seriously when one has learned to take the mathematics of quantum theory seriously.

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The search to understand quantum theory is a difficult one which still involves a lot of speculation. Here these speculations will be introduced in a sequence of “suggestions”.

The Conventional Suggestion:

There are two forms of time propagation.

A) The Schrödinger equation:

Ψ(t) = e^-itH Ψ.

B) “Wavepacket collapse/ reduction”:

There are occasional instaneous “collapse” events, when Ψ is abruptly replaced by a new wavefunction taken from some set of possible replacements
{Φ₁, Φ ₂, . . ., Φ_N}.

The probability of the replacement Ψ → Φ_n is |<Ψ|Φ_n>|².

Problems:

When do collapse events happen?

What determines the set {Φ₁, Φ₂, . . ., Φ_N} of possible outcomes?

What do the probabilities mean?

Suggestion:

Given that it is observations that we find anomalous, perhaps collapse has something to do with “measurement” or “observation”.

So perhaps Ψ(t) = e^-itH Ψ except when we look.

Problem:

What is “measurement” or “observation”?

Everett's Alternative Suggestion:

Perhaps Ψ(t) = e^-itH Ψ always holds, whether we look or not.

In this case, collapse doesn't happen — it just appears (to us) as if it happens.

Problems:

When do collapse events appear to happen?

How does one appearance lead on to the next?

What determines the set of possible apparent outcomes?

What do the probabilities mean?

How do we come into the picture?

What indeed are we?

Aside: It is sometimes claimed that all the problems of quantum theory can be solved if we just accept that quantum states are states of information. This begs the questions of whose information is being considered and of what it is to be a consumer of information.

Motivation for Everett's Suggestion:

Consider a simple quantum model which can be used to describe the possibilities arising when an individual observer observes some particular system.

In this model, the observer wavefunctions Ψ and the observed system wavefunctions Φ are combined to form wavefunctions which are sums of tensor products
∑_n a_nΨ_n Φ_n.

Such sums can be thought of as describing, for each n, correlation between the observer wavefunction Ψ_n and system wavefunction Φ_n occuring with probability |a_n|².

Suppose next that we can find some suitable wavefunctions:

Ψ₀ — a wavefunction describing the observer about to observe the outcome of an experiment,

Φ_n — a wavefunction describing the n^th directly observable individual outcome of the experiment — for example, a screen hit in a definite position, or a dead cat,

Ψ_n — a wavefunction describing the observer observing outcome n,

and a mechanism and a corresponding Hamiltonian H such that, for each n,

e^-itH Ψ₀ Φ_n = Ψ_n Φ_n.

This plausible-enough mechanism simply models the process of the observer looking at directly observable outcome n and observing it.

But then the linearity of the Schrödinger equation requires that
e^-itH Ψ₀ ∑_n a_nΦ_n = ∑_n a_nΨ_n Φ_n.

Analysing the sum ∑_n a_nΨ_n Φ_n into the terms a_nΨ_n Φ_n suggests that, in this situation, the mathematics describes the observer observing each of the possible outcomes separately with outcome n being observed with subjective probability |a_n|².

The cat may be both dead and alive, but, according to this analysis, the observer does not see that combination. He sees dead and, separately, he sees alive. As an individual, he does not see both and nor does he see himself seeing both.

Again, taking Everett seriously involves being led by the mathematics of quantum theory.

Summary:

If Ψ(t) = e^-itH Ψ is always true

then we need to analyse Ψ(t) into appropriate possibilities.

Note: As the Schrödinger equation is an equation for complete systems rather than for subsystems, here and henceforth we take Ψ(t) to be the wavefunction describing the entire universe and everything in it.

Problem:

It isn't obvious how to make such an analysis.

Sophisticated Problem:

Even in the light of decoherence theory.

Decoherence theory is useful in showing that appropriate analyses of Ψ(t) are possible, but it leaves serious ambiguity problems; particularly when faced with the complexities of neural systems.

Consequence of Everett's Suggestion:

If Ψ(t) = e^-itH Ψ and Ψ(t) is a description of reality and if
Ψ(t) = ∑_n a_nΨ_n Φ_n, then all the possibilities Ψ_nΦ_n are real.

So we have a many-worlds theory.

Note that not all possibilities are equal — both because of the probabilistic weightings a_n and because the only possibilities which appear are those Ψ_nΦ_n which form components of Ψ(t).

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The first great problem of the interpretation of quantum mechanics is the Schrödinger cat problem:

Observers see individual possibilities which are described by only part of the total solution to the Schrödinger equation.

The second great problem is the locality problem:

There are apparently situations in which

one observer, Alice, can find out which of two genuinely random events

a distant colleague, Bob, is observing,

simultaneously with his observation,

but without there being anything in Bob's laboratory which predetermines his result,

or any message of any kind passed between the laboratories between Alice's observation and Bob's.

This problem one takes seriously in as far as one takes the theory of special relativity seriously.

Suggestion:

Suppose that Alice and Bob make independent local choices. Then note that it is only when they meet that they can compare their results.

In the context of a global wavefunction with no collapse, this allows local explanations of all observations, but implies that when Alice and Bob eventually meet they may encounter any of several of the other's possibilities.

A typical analysis of models for the Schrödinger equation in this context will then lead to something like
Ψ^A Φ^A Ψ^B Φ^B → ∑_m,n a_mnΨ^A_m Φ^A_mΨ^B_n Φ^B_n

in which the structure of the initial wavefunction determines the pattern of correlations which can subsequently be observed.

In this picture, we cannot say that Alice finds out instantaneously which of two genuinely random events her distant colleague Bob is observing, but rather that she finds out enough about the state in her locality to know, from the correlations of that state, what it is that he will tell her that he observed when they eventually meet.

This then is a many-minds theory:

Observations are relative to individual observers.

“Worlds” are distinguished at the level of the minds of individual observers.

Alice has her observations and Bob has his.

Avoiding solipsism requires that we assign consciousness (or reality) to everyone we could meet who is sufficiently similar to ourselves. So Alice should assign reality to each of Bob's possible futures, and, by symmetry, to each of her own.

Summary:

Ψ(t) = e^-itH Ψ

We need to analyse Ψ(t) into possibilities for individual minds.

We need to find descriptions in Ψ(t) of the structure of an individual mind.

Problem:

How do the components at different instants match up?

Suppose that at time t we analyse Ψ(t) as some complicated sum of many terms
Ψ(t) = ∑_m,n a_mnΨ^A_m Φ^A_mΨ^B_n Φ^B_n . . .

and then at a later time t' we have another analysis
Ψ(t') = ∑_p,q a'_pqΨ^A'_p Φ^A'_pΨ^B'_q Φ^B'_q . . .

Then we will need to decide whether A' and A both describe the same Alice.

Can we do this by using locality or geometry?

How?

Suppose, for example, that Alice and Bob are waltzing together.

Once we have worked out who is who, we will still need to discover which term Ψ^A_m is part of the history of some particular Ψ^A'_p.

This is also not obvious, when one starts from just the complete function Ψ(t).

The technical version of this problem requires an analysis of relative probabilities, involving identifications across state decompositions at different times.

Summary:

We need to analyse Ψ(t) into possibilities for individual minds

and to specify the temporal development of individual minds.

Suggestion:

Minds are described by brains.

Brains are described by wavefunctions.

As Ψ(t) provides a description of phyical reality, it contains components which describe living brains at an instant.

We could use the wavefunctions which describe brains somehow to index the set of individual observed possibilities and their histories.

Problems:

There is considerable ambiguity in what might be meant by THE wavefunctions which describe brains.

Indeed, until we know how to analyse Ψ(t), we won't know how to identify how many brain wavefunctions form components of Ψ(t), or how likely each of those components is.

Wavefunctions can have infinite complexity. Minds are finite.

There are also:

coarse-graining problems,

problems with time,

and problems which arise because of the nature of Ψ(t) and the fact that it is very rich in possibilities.

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Coarse-graining problems.

Consciousness is the existence of the brain for itself.

But at what level of detail?

Is awareness awareness of localization for each individual molecule in the brain, or awareness of neural firing patterns?

Aside: To be more precise, this is not so much a question of “awareness” as of “existence for itself”: Does a brain exist for itself at the level of localization of each individual molecule in the brain, or at the level of definite neural firing patterns?

In quantum theory, different coarse-grainings can be incompatible.

This is expressed especially clearly in the ambiguity problems of the consistent histories formalism.

Incompatible coarse-grainings can lead to diverging probabilities and futures. Such divergencies are magnified because brains are vast communities of metastable information-carrying fluid systems with the timing of each neural firing linked to that of many others.

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Problems with time.

How are the instants at which Ψ(t) is decomposed to be chosen in a theory supposed to be compatible with special relativity theory?

How are instantaneous decompositions matched?

Does personal identity depend only on what we are at the present moment?

What moment is that?

More generally, what should be taken to be determined and what not determined in an indeterministic physical theory compatible with relativity theory?

What is it like to live in a world governed by an indeterministic theory?

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What is Ψ(t)?

By assumption, Ψ(t) never collapses.

So it never has collapsed.

For radiation and matter, this means that the part of Ψ(t) of which we have some knowledge, can be modelled by the Schrödinger equation time propagation of the mathematically quite simple quantum state which emerges at decoupling shortly after the big bang. This state is close to thermal equilibrium save for the density inhomogeneities which will initiate the highly random and indeterminate process of star formation.

Individual stars and planets, let alone individual humans, exist only as multi-potent possibilities within that state, not as definite objects.

Possible humans exist within Ψ(t), but it is hard to decide what makes a component of Ψ(t) a description of a unique human possibility.

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The mathematics of quantum theory thus leads us well beyond the simple models with which we began.

In order to turn the many-minds idea into a fully-developed formalism for the interpretation of quantum theory, it is necessary to step back from those simple models, and look afresh at the relationship between mind and brain.

In the classical mind-body problem, body seems sufficient to explain all behaviour. Mind therefore seems epiphenomenal, or even superfluous.

However quantum theory calls into question the definiteness of every physical object.

So perhaps after all mind is fundamental and it is the mind which determines the body.

But what is mind?

In particular, if the apparent, or classical, brain is an adequate model of the mind, then what aspects of that brain are required to form the model?

In other words, what characterizes the physical structure of observers?

Or, to put it yet another way, if there is something which it is like to be a bat, then what physical structure does a bat have to have to be what it is that makes it something which there is something which it is like to be?

Suggestion

We begin an analysis of brains whose existence is classically-speaking uncertain by trying to understand when it might be sensible to suppose that two clasical brains which are physically sufficiently similar, necessarily give rise to the same mental phenomena.

Consider, as examples, two brains which differ only by a change in temperature of less than 0.001 K,

or in pH by less than 0.001 units,

or by small redistribution of the atomic weights of the carbon atoms,

or by a tree falling in a forest on another planet,

or by a complete alteration in future,

or by a complete alteration in history.

Consider the infamous brain in a vat:

the body other than the brain has been removed,

the external world has been replaced.

Can anything else be removed or replaced?

Have we taken too much?

A brain magically and instantaneously constructed by bringing atoms together in a vat would function but does not contain or imply its past.

The same might be true of the result of a teleportation process.

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The conventional idea is that a brain functions through a pattern of neural firing.

Suggestion:

Find a minimal finite abstract model of some aspect of the neural functioning of an individual, sufficient to identify EVERYTHING about the physical structure of that individual which is relevant to his or her mental life.

What would be “sufficient” mean in this context?

Let's consider some possibilities:

Suppose that just a name and a time is sufficient. In this case we have the idea of the Cartesian theatre, in which the physical structure of the brain is irrelevant. Mind is a little person, or soul, inside the head. All we need to know is which head.

A second possibility corresponds to some variety of computational functionalism, in which we could suppose that a Gödel number is sufficient, and that mind is a little computer programmer with a large but simple computer. All we need to know is the computer program.

In this case, meaning arise from behaviour under different counterfactual circumstances, from how the program would behave given different inputs, not from actual experience.

On the other hand, if mere function is not sufficient, then it is possible that something could behave as if conscious, without being conscious.

As a third possibility, one might suppose that the physical structure of the brain at an instant identifies everything about the physical structure of an individual which is relevant to his or her instantaneous mental life. But in this case, it seems to me hard to understand how that physical structure is directly experienced for itself, without supposing that mind is a little neurophysiologist with access to a variety of suitable possible worlds.

Although the instantaneous structure of the brain is sufficient for its instantaneous functioning and behaviour, I do not see it as being sufficient to provide the meaning of that functioning.

As a final possibility, I therefore propose, that some finite abstract model of the total history of the neural functioning of an individual is required to identify everything about the physical structure of that individual which is relevant to his or her mental life.

In this case it is possible that mind can give meaning to a simple structure:

a pattern of two-state elementary events abstracted from the pattern of neural firings.

Such a structure can be expressed in simple terms in the mathematics of quantum theory.

Mind does not analyse that structure as a homunculus.

It is that structure.

Summary:

We need to analyse Ψ(t), considered as a function of time rather than at a fixed instant, into possibilities for individual mental histories.

These possibilities need not be directly identified with wavefunction components.

Minds are described by brains.

Brains produce developing abstract patterns of information over time.

These patterns define individual mental histories.

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We have now reached the starting point for the interpretation of quantum theory presented in the core papers on my web site and reviewed in the summary on that site.

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Solvable Problems:

How might abstract patterns of information be defined?

How might those patterns be expressed within the function Ψ(t) and used to analyse it?

Could human brains correspond to such expressions?

How can probabilities be defined?

Are the definitions compatible with observation?

More Problems:

Is the theory too complicated?

Is the theory too speculative?

Consequences of Solving the Solvable Problems:

An interpretation of quantum theory.

An idealist theory in which minds are fundamental and Ψ(t) defines the probabilities of possible mental histories.

Minds are defined by their histories.

Time is an aspect of our structure as individual observers and need not be universal.

Probabilities are objective numbers explicitly definable for each individual observer at each appropriate moment, and providing, for an observer at a moment, the relative likelihoods of seeing elements of a finite set of possible immediate outcomes.

Thus the experience of an observer is that of observing a particular, identifiable, discrete stochastic process.

Any person you choose to consider is leading one of the lives that you might have led had a finite sequence of stochastic events come out differently.

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Matthew J. Donald

The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great Britain.

e-mail : mjd1014@cam.ac.uk

home page: http://people.bss.phy.cam.ac.uk/~mjd1014