IIIE. The Mind Interpretation.

We have seen in sections IIIA through D that none of the proposed interpretations of quantum mechanics is satisfactory.  So we will propose a different one here.  We start by laying out the state of affairs as we see it. 

1. Summary of Relevant Points.

S1. Only the wave function physically exists.  (IIIA, IIIB3).

S2. The usual noncollapse time evolution of each branch of the wave function always holds, so all branches continue forever.  (IIIB4, IIIB5)  Thus, a little more explicitly than S1,  physical existence consists solely of the noncollapsed wave function, with all its versions of reality.

S3. The Everett approach, in which all branches are equally valid, is flawed. (IIIC)

S4. Because that which is perceived agrees numerically with the predictions of quantum mechanics, we assume it is the wave function that is perceived. More specifically, each of us is presumed to be “consciously” aware of one specific version of the wave function of his or her own brain.

S5. Quantum mechanics, by itself, cannot explain why one branch is singled out for “conscious” perception while the others are not. (IIIB5, subsection 2). An interpretation must explain why one branch is singled out.

S6.  The equations of motion of quantum mechanics do not imply the probability law (although they are consistent with it).  So an interpretation must also explain why the probability law holds.

The only one of these points I am not certain of is collapse.  From the strength of the arguments in Section IIIB4, I would estimate is that there is a 95% chance there is no collapse.

2. Basic Points of the “Mind” Interpretation.

All that physically exists, in our view, is the wave function of quantum mechanics.  But conventional quantum mechanics—which we assume holds—cannot explain why or how one branch of the wave function is singled out for “conscious” perception.  It is thus inevitable (if we agree on S1 through S5) that the process of perception involves something outside of quantum mechanics.  In the Mind interpretation, we implement this idea in the following way:

M1. In addition to our physical bodies, we each possess our own, individual, perceiving Mind (separate from the physical brain).

M2. That Mind is not subject to the laws of quantum mechanics (or any other mathematical laws).  For that reason, we call it nonphysical (because those laws presumablly govern all of physical existence).

M3. Our individual, nonphysical Mind perceives only the wave function of our own individual brain (or perhaps brain-body).  The nonphysical Mind knows of the external physical world only through our sensory perceptions, which change the state of our brain. 

M4. When there are several branches of the wave function, and hence several quantum mechanical versions of our brain, our individual, nonphysical Mind can concentrate its awareness on one particular version.  This particular version is what we are aware of in the normal sense of awareness.  Note that the Mind does not collapse the wave function; it only focuses on one branch.

In this way, we arrive at a scheme that leaves the mathematics of quantum mechanics untouched while explaining why we are conscious of one particular version of reality.  The price paid is that we must postulate the existence something—the Mind—outside the laws of the physical universe.

3. Mental Picture of Perception by the Mind.

Perception by the Mind takes place in the mathematical Hilbert space appropriate to quantum mechanics (see section IV).  It is not possible (at least for me) to visualize in this many-dimensional space.  But to give the idea, we will suppose the space is only three-dimensional.  Imagine the Mind to be a person standing at the center of a sphere.  And imagine each of the allowed versions of reality to correspond to a ray of light that is directed towards the person.  Each ray will come from a different direction.  The analog of the Mind concentrating its awareness on just one version is to suppose the person in the center of the sphere looks in the direction of just one ray of light.

There is a second way of visualizing the focus of the Mind on just one version.  Each branch of the wave function, and hence each version of the brain wave function, exists in a different space, a different universe.  We could imagine that the Mind can “exist” in only one of those universes at a time.  Or we could imagine that the Mind can perceive all the universes but chooses just one as the one to concentrate on and therefore physically perceive.  The awareness belongs to the Mind, but the mechanism is the wave function of the brain.

4. Choice in the Mind Interpretation.

The Mind interpretation presents the possibility that we each have the freedom to choose our thoughts and actions.  To understand this process, we must first make a distinction between internal and external branchings of the wave function.  Internal branchings are branchings that take place only in the wave function of the brain.  External branchings involve parts of the physical universe outside the brain.  So, for example, the Schrödinger’s cat example involves an external branching because there are two versions of the cat in addition to the two versions of the brain.  In this nonphysical Mind interpretation, the individual Mind has no choice of which branch is perceived in an external branching.  That is, no human Mind chooses whether the cat is dead or alive.  We do not know how the choice of cat alive or cat dead is made.

An internal branching would be a superposition of wave functions corresponding to different thoughts within the brain (section IIH, subsection 7).  Our assumption here is that the brain is not effectively a classical device.  The quantum branchings presumably come at the synapses, where the relevant distances are small enough so that quantum uncertainty implies a quantum superposition of states.  The nonphysical Mind then has a free (not mathematically or mechanically constrained) choice of which one of the superposition of states, each corresponding to a particular thought or action, to bring into effective physical consciousness. 

5. Agreement among Observers

We found in section IIIA3, subsection 10 that quantum mechanics implies observers cannot disagree.  However, the Mind interpretation complicates this result.  How do we ensure that all observers put their “conscious” awareness on the same branch?  We suggest two possible ways.

The first is to suppose that the Mind of each observer is a part or fragment or facet of a single overarching MIND.  And because of this connection to a single MIND, we must all perceive the same branch.  The second possibility is to suppose that the individual Minds interact and have an “agreement” that they will all put their consciousness on the same branch.

Neither of these possibilities is attractive to the purely rational physicist because they are unsupported metaphysical assumptions.  But the first might be palatable to the mystically inclined physicist because it aligns well with the views of several major mystical (spiritual) systems.

6. Probability in the Mind Interpretation.

Probability Is Not Classical in
Noncollapse, Wave-Function-Only Interpretations

In addition to explaining why our awareness is on one particular version of reality, an interpretation should also explain why the probability law holds.  When we look at noncollapse, wave-function-only interpretations, however, we discover a most interesting problem with respect to probability.  We will illustrate it with the half-silvered mirror experiment (sections IID and IIE).

In that experiment, the wave function of light impinging on a half-silvered mirror splits into two parts, one of which travels on a vertical path and the other on a horizontal path.  In an objectively real model (such as Bohm’s particle model) or a collapse model, each time the experiment is run, there is a probability of a definite, classical outcome.  The particle of light will travel on the vertical path, or the wave function will collapse to the vertical alternative, with probability ; or the particle of light will travel on the horizontal path, or the wave function will collapse to the horizontal alternative, with probability .

However, in a noncollapse, wave-function-only interpretation, when an experiment is run many times without looking at individual results, there is no definite, classical outcome—V path or H path—for each individual run.  Instead, there is simply the full wave function, with all the "classical" options present.  Thus there can be no definition of the probability of a specific outcome per run in this type of interpretation because there is no specific outcome!

Derivation of the Probability Law

In spite of this problem, we can still show that, under certain assumptions, the Mind interpretation can yield the probability law for a large number of runs of the experiment.  The derivation of this result is based on Everett’s flawed derivation in his many-worlds interpretation (section IIIC). 

We make the following assumptions:

P1. From standard quantum mechanics, we know that each branch of the wave function breaks into a product with three parts: the state of the brain, of length 1 by definition; the state of the external world, again of length 1 by definition; and the norm, , of the state.  The assumption is that when the Mind looks at the various states, the probability of its concentrating on (and thus bringing into our everyday awareness) a particular state of the brain depends to some extent on the norm.  More specifically, the Mind is more likely to bring into our awareness a state in which the norm of the state is relatively large.  It doesn’t matter what the actual numerical relation is between the norm and the probability of perception, or even that it is constant over time.

P2. We run the half-silvered mirror experiment N times, with N large.  Suppose there is a dial that keeps track of the total number, , of runs that have V as a result.  There will be many different sequences of V’s and H’s that lead to the same . For a given , the norm squared, , of the associated vector will, according to the rules of quantum mechanics, be the sum of the squares of the norms of all the different sequences of states that give .  The numerical result is

(IIIE-1)  

 

P3. When the Mind perceives the N+1 ( ) states of the brain, the norm squared of the state associated with a particular  will be that of equation (IIF-1).

We can now apply the reasoning that was used in the Everett interpretation (where it was not possible to justify it because there was no “external” observer that might be sensitive to the norm of the state vector).  The norm squared of the state has a very sharp maximum about .  Thus, independent of the actual numerical “probability law” for the Mind, if there is one, the Mind will only perceive the state which has (relatively) a very large associated norm.  This implies that the Mind will always perceive an very close to .

This is our proposed derivation of the probability law.  It has the interesting property that it does not make each measurement a random event with probability .  Instead it only specifies that a large number of measurements, considered as a single observation, will give the probability law.

7. Experimental Test of the Probability Law.

The above derivation of the probability law depends on N being large and on perceiving only the final result.  So we might expect deviations from the  law if intermediate results with small N are observed.  (Remember, since there is no actual event associated with each run, we are not dealing with standard probability theory.)  How do we test this?  We use a procedure akin to the experimental test of the probability law in section IIIB4, but we observe intermediate results.  This will be illustrated with the half-silvered mirror experiment. 

Conventional Analysis of the Half-Silvered Mirror Experiment.

Suppose we run the half-silvered experiment mN times, with m small, say between 2 and 10,  and N large.  We group the data into N sets of m measurements each.  The conventional probability for obtaining j  V measurements and  H measurements in a given set is

(IIIE-2)  

If the m-group experiment is run N times, with  V readings on the kth run, then we conventionally expect the number of results, N( j ), for which  to be

(IIIE-3)  

And we expect the deviation squared from the average for each j to be

(IIIE-4)  

If we take the sum over j of all these deviations, remembering that the sum of the ’s is 1, we see that

(IIIE-5)  

Experimental Test.

As in the conventional analysis, we do the half-silvered mirror experiment m times and observe the number, , of V results.  Then we do the experiment again and observe , and so on, up to , with N large.  We next use these   , individually observed, to calculate , and , which, through the , is a function of .  We then minimize  with respect to  .  If conventional probability concepts apply, then the minimum  should be on the order of N.  If it is significantly larger, then we know that observation of intermediate results has made a difference and conventional probability theory does not apply. 

Note 1:Instead of calculating by the minimizing process, we could calculate it from the experiment itself, through .

Note 2: We emphasize that the results each m-set must be individually observed.

Note 3: We are assuming that the effeciency of detection is near 1.  If it is not, then the analysis must be redone.

Discussion.

Because of our familiarity with conventional probability, this test appears ridiculous.  How could observation of intermediate results affect probabilities?  The point is that conventional probability theory is implicitly based on the idea that there is a specific outcome in each individual case.  But there is no specific outcome (that is, no single, classical version of reality) in noncollapse, wave-function-only interpretations, and so conventional probability concepts do not apply.

Note that this argument has nothing to do with paranormal phenomena.  We are not claiming that observation can influence the outcome of a specific, objectively occurring event.  We are only saying that observation of intermediate events in noncollapse, wave-function-only interpretations produces results that are almost certainly at variance with conventional probability theory.

Finally, how trustworthy is this simple test for objective reality?  I must admit that I don't fully trust my reasoning here.  It needs to be checked and verified.

8. Experimental Test of Noncollapse, Wave-Function-Only Interpretations.

This experimental test of the probability law also provides a specific, relatively simple experimental method of determining whether or not there is an objective reality (composed of particles or the collapsed wave function or ...).  Suppose first that the intermediate-observation results agree with conventional probability theory.  As far as I can see, the only way this could happen in a noncollapse, wave-function-only theory is if the Mind of the observer perceived branch i with probability .for each run of the experiment.  The necessity for such a strong assumption would make a noncollapse, wave-function-only interpretation untenable.  The task then would be to determine whether the objective reality consisted of a collapsed wave function, or of hidden variables (particles).

If, on the other hand, the intermediate-observation results do not agree with conventional probability theory, then that is a certain sign there is no objective reality.  Instead, only the wave function exists and there is no collapse.