An outline of my thoughts on quantum mechanics is as follows:

(1) Quantum mechanics gives many simultaneously existing versions of the physical world, including many versions of our brain.

(2) If there are particles, or if there is collapse of the wave function, then one version of the brain is singled out and that is the conscious version.  So if either the particle or the collapse interpretation is correct, then conscious awareness is almost certainly based in the physical, objectively existing brain.

(3) But there is no experimental evidence for particles or collapse.  Further, theoretical arguments imply these two interpretations are most likely not valid.  My subjective estimate is that there is no chance that particles exist and less than a 10% chance that collapse occurs.  See sections IIIA and IIIB for details. 

(4) If neither the particle nor the collapse interpretation is correct—if only the wave function, with all its versions of reality, exists—then one can prove that conscious awareness cannot be based in the physical brain.  Its origin must be outside physical reality (in the sense that it cannot be subject to the mathematics of quantum mechanics).  The proof is given below.

Proof that Conscious Awareness Cannot Be Based

in the Physical Brain Wave Function

Assume that only the wave function, with all its branches, exists; there are no particles, there is no collapse, and there is no “being” outside physical reality that is the source of conscious awareness.  Under these circumstances, only the wave function exists, so conscious awareness can only reside in the wave function of the brain.  We will show this is not possible because it violates the probability law. 

Proof that Every Version of the Observer is Conscious

To illustrate the problem, we do a modified Schrödinger’s cat experiment (with no cat).  We have a source of radiation and a nearby detector with a dial that gives the number of particles detected.  The detector is turned on for one second and, to make matters simple, we assume the only possible readings are 0, 1 and 2.  Before starting the experiment, we cover the dial with a piece of paper with an x on it.  The state of the dial-conscious observer system is, schematically,

(1)                           [dial:0][obs brain state x; conscious of x]

where the brain state refers to the wave function of the neurons, which will be in a configuration corresponding to seeing x.  What does conscious awareness correspond to?  In this scheme where there is only the wave function, conscious awareness can only correspond to some property of the wave function—say the quantum configuration of the prefrontal lobe.

Now we turn on the detector for one second but at first do not lift the paper with the x on it.  If we indicate this time by a 0 subscript, then the state of the system is (ignoring the ai coefficients)

      ([dial: 0]0 and [dial:1]0and [dial:2] 0)[obs brain state x; conscious of x] 0=

                                  [dial:0] 0 [obs brain state x; conscious of x] 0

                                                                and

(2)                             [dial:1] 0 [obs brain state x; conscious of x] 0

                                                                and

                                  [dial:2] 0 [obs brain state x; conscious of x] 0

The three brain states are identical and are therefore all conscious.

We now remove the paper with the x on it, exposing the reading on the dial.  We know how the wave functions of the brain evolve; the associated neural firing pattern will now correspond to the dial reading.  But at this point, we don’t know how the state of conscious awareness of each version evolves.  So we write the state of the system as

                                  [dial:0] [obs brain state 0; conscious(?) of 0]

                                                                 and

(3)                             [dial:1] [obs brain state 1; conscious(?) of 1]

                                                                 and

                                  [dial:2] [obs brain state 2; conscious(?) of 2]

One might think that because conscious awareness is such an ephemeral quality, we could not determine in quantum mechanics how the conscious awareness of each version of the brain evolves.  But we can!  The method hinges on the property that the time evolution of any version of reality in the wave function is independent of all the other versions that might be present in the total wave function.  So we do a separate, auxiliary “experiment.”  We simply take the paper with the x off the dial, which has been preset at j (j=0, 1, 2).  Then we know from our experience—first we consciously see x then we consciously see j—that, no matter what the technical definition of consciousness, the time evolution is always conscious to conscious:

(4)         [dial: j]0 [obs brain state x: consc of x]0   [dial: j] [obs brain state j: conscious of j]

Therefore, since we know that each version of the brain evolves from conscious to conscious, no matter what other versions are present, state (3) must be

                                  [dial:0] [obs brain state 0; conscious of 0]

                                                                and

(3a)                            [dial:1] [obs brain state 1; conscious of 1]

                                                                and

                                  [dial:2] [obs brain state 2; conscious of 2]

So in unamended quantum mechanics (with no “sentient being” outside the physical-mathematical system), we end up with many consciously aware versions of the observer.  That is, there are many consciously aware versions of each of us!  This is one potential interpretation of quantum mechanics—basically the Everett many-minds interpretation.  We can show, however, that this bizarre interpretation is not acceptable.

Problem with the Many-Conscious-Versions

Interpretation of Quantum Mechanics

This interpretation is unacceptable because it contradicts the |ai|2 probability law of quantum mechanics.  To see this, suppose we do the above experiment, except that the only possible outcomes are 0 and 1.  And we repeat the experiment N times, with N very large.  There will be possible outcomes to the experiment, one for each sequence of 0’s and 1’s.  If the observer perceives the whole sequence, then according to the above, there will be 2N consciously aware versions of the observer.  N!/m!(N-m)! of the versions will perceive m 0 readings and N-m 1 readings.  This distribution has a sharp peak about m=N/2, so “almost all” versions of the observer will perceive m close to N/2.  If we are the observer, almost all versions of us will perceive N/2, so “we” would, in effect, always see m near N/2.  But if the coefficient on the 0 version is a0, the probability law of quantum mechanics says that we will almost certainly perceive m close to N|a0|2 rather than N/2.  Thus there is an irreconcilable conflict between the probability law and the many-conscious-versions interpretation of quantum mechanics.

Therefore unamended (no particles, no collapse) quantum mechanics cannot properly account for conscious awareness.  So if there actually are no particles or collapse, then that-which-is-consciously-aware must not be subject to the laws of quantum mechanics.  In particular, conscious awareness cannot reside in the (wave function of the) physical brain. 

Note: The “nonphysical perceiving aspect” does not necessarily collapse the wave function.

Note on weighting factors:  Everett attempted to get around the above criticism, and at the same time derive the probability law, by introducing a weighting factor for the various versions.  It was composed of two parts; the first was the norm squared of the individual states and the second was the number of states which corresponded to a given value of m.  He then said, in effect, that the only states which count were those in which the product of these two factors was relatively large.  But putting a ranking system on the different versions of consciousness seems like a very odd idea to me, so odd that it is hard to criticize.  Nevertheless I will try.

• Why should some conscious versions count more than others.  More what?  It cannot be more aware.  And it cannot be more likely to be observed because there is no one to observe the many versions of the observer.  What physical process determines how much each one counts?  It seems to me the idea of weighting the various aware versions has an unclear physical origin, an unclear meaning, and unclear implications

• Suppose |a0|2=.25.  Then the relative number of (conscious) versions that perceive m near N/4 is down by a huge factor (on the order of exp(N/4)).  Why do the vast majority of conscious versions “not count?”  Why should N/4 be what our consciousness is aware of when almost all versions of our “self” perceive N/2?

• From our experience, we know there is nothing in our perception (of a single run of an experiment) that corresponds to the value of ai; we simply perceive 0 or 1 (plus our surroundings). So why should the value of ai be of relevance in a weighting?

• Each conscious version perceives only its own sequence of 0’s and 1’s.  There is no version—and no “outside being”—that perceives all the states with the same value of m.  So if it is unobserved, why would such a weight be relevant to our perceptions?

• One cannot say that “the observer” is more likely to be in one state or the other because there is no “the observer;” there are 2N simultaneously aware versions of the observer.

Everett's Many-Minds Interpretation

Everett’s "many-worlds" interpretation is an attempt to show that noncollapse quantum mechanics—with no amendments or presumption of consciousness outside quantum mechanics—is sufficient, by itself, to explain all observations.  It assumes there is a different universe for each branch of the wave function and that each universe is equally valid.  Its veracity hinges on whether or not it can explain the probability law.  We have basically shown above that it cannot, but will give a slightly different argument here.

1. The Probability Law. 

The probability law says that the probability of perceiving a given state, i,  is equal to the amplitude squared, , of that state.  This perception-related law appears to be separate from the time-evolution laws of quantum mechanics.  Everett, however, claims it is not separate because it can be derived from the other mathematical principles of conventional, no-collapse quantum theory.  This claim, if true, would further reduce the mysteries presented by quantum mechanics.  We will consider Everett's derivation and then point out its problems.

2. Everett’s Derivation and Criticisms of It. 

I must admit that the longer I study the Everett derivation of probability, the more confusing it becomes, so I may not get his logic entirely correct.  (This, however, won't stop us from giving valid criticisms of it.)  The ground rules are that there are no particles, so that only the wave function exists, and there is no collapse of the wave function (as we argued in sections IIIA and B). To illustrate Everett's ideas, we do a half-silvered mirror light wave experiment (IIA5) N times, with N large.  After the mirror, each light wave has wave function , , where V indicates the vertical path and H the horizontal path.   There is a dial that registers the outcome of each run, and there is an observer who perceives the readings on the dial at the end of the experiment.   In the mathematics of quantum mechanics, there will be many different versions of the observer.  The quantum state of the N particles plus the observer at the end of the experiment is

(IIIC-1)  

where the D indicates the readings on the dial of the detector.  The state of Eq. (IIIC-1) is a superposition of states, with different observer states.  That is, if you are the observer, there are different versions of you!

Now Everett introduces a "measure."  In his words (in the RMP article),

In order to establish quantitative results, we must put some sort of quantitative measure (weighting) on the elements of a final superposition.  This is necessary to make assertions which hold for almost all of the observer states described by elements of a superposition.  We wish to make quantitative statements about the relative frequencies of the different possible results of observation - which are recorded in the memory - for a typical observer state; but to accomplish this we must have a method for selecting a typical element from a superposition of orthogonal states. (Italics added.)

Everett points out that the measure or weighting must satisfy certain criteria if it is to be consistent with quantum mechanics.  Not surprisingly, this leads to the measure  .

First criticism:  There is no real justification provided by Everett (or apparent to me) for assuming there is a measure.  If there is no "being" looking in from outside, if there is only the wave function, what is it that "requires" a measure (that is consistent with quantum mechanics)?

Everett's Logic.  The pivotal step in Everett's logic is this: All versions of the observer are presumed to exist, with each one being counted as a valid version of the observer.  If "almost all" versions perceive results in agreement with the probability law, and if "we," as a typical version of an observer, belong to this "almost all" group, then we will (almost) always observe that the probability law is satisfied.

So far so good (except for the first criticism).  But I am not able to follow Everett's logic any further.  So I will guess at what he is implying.  Suppose we change the dial so that it only tells the observer the final numer of V states, ,so that Eq. (IIIC-1) becomes

(IIIC-2)  

Everett then notes that the length squared of the vector associated with observer state

 is

                                             ,

(IIIC-3)

where the binomial coefficient gives the number of different states (different ordering of detected V’s and H’s) associated with a given value of .  This length squared has a very sharp maximum about .  Everett then apparently supposed that the only "valid" observers were those for which the norm of the state was (relatively) very large.  He says

Therefore all predictions of the usual theory will appear to be valid to the observer in almost all observer states.

But as far as I can tell, what he meant was

Therefore all predictions of the usual theory will appear to be valid to the observer in almost all valid observer states (that is, states with large norms).

However, I can see no persuasive reason why the observer states should be limited to those with a large norm.  This leads to the

Second Criticism (originally pointed out by Ballantine). Suppose we go back to Everett's logic and to Eq. (IIIC-1).  There are 2^N  versions of the observer.  of those versions will perceive a result in which there are m V’s and N – m H’s.  This binomial coefficient has a sharp peak about m=N/2.  Thus “almost all” versions of the observer will perceive a value of m very near N/2.  This implies that each of us, as a typical version of the observer, will “always” perceive a value of m very near N/2, independent of the coefficients.  This contradicts the probability result that we will always perceive a value of m near .

Conclusion. Our conclusion is that the probability law cannot be derived from quantum mechanics in the manner suggested by Everett, and that his "many-worlds" interpretation is therefore not justifiable.  Further, because the argument is general - for almost all versions of the observer no matter what - we also conclude that there can be no way to derive the probability law from the mathematics of quantum mechanics alone. 

The crucial missing piece in Everett’s reasoning is a justification of the assumption that perception of a particular state by the observer depends on the norm of the vector associated with that state.  This assumption can be more readily justified in the model of section IIIE.

References.

H. Everett III, Reviews of Modern Physics 29, 454 (1957). 

L. E. Ballantine, Foundations of Physics 3, 229 (1973).