A Brief History of the Density Matrix

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What I like about the physics sections of Sir Penrose’s two books, The Emperor’s New Mind and Shadows of the Mind, is that they are not the standard layman’s or “popular” versions of the discussion. Not that these are by any means less interesting [two of my very favorites are Michio Kaku’s Hyperspace and John Gribbin’s Schrodinger’s Kittens] but I was looking for something new and something more and I found it in these two books. I especially like that Penrose is not afraid of the quantum superposition; rather, he bravely, brazenly faces the two-peaked amplitude without blinking and calculates the extremely reasonable squared modulus of the amplitude to obtain the probability. He uses the unit circle in complex space to show why it is the square of the modulus that is the “real” probability, but we all know that doesn’t really explain anything.

There is such a richness and depth of mathematical knowledge in these two books that I know I will be returning to them again in the future and I cannot hope to cover every detail here. Instead, I will try to carry a thread through some of the more relevant concepts, at least as I see them at this point. The point about Sir Penrose’s discussion of the deterministic wave equation (U) is that the reduction (R) of this equation appears arbitrary and cannot be derived from (U). There is no law of physics and no mathematical rule for when to apply R. This is the impetus for his assertion that the wave equation is an incomplete description of reality and that a new, greater mathematics, a GUT, is necessary especially in order to derive R.

The importance of this problem is fundamental. If R is not applied, quantum entities remain in superposition…indefinitely, according to ψ (Psi is the symbol denoting the quantum state, or “state vector” of a particle, whether in superposition or not). What this means is hotly debated in the field. No overall consensus exists regarding the meaning of a particle in superposition [aside from the significant fact that unless a waveform resolves, there is no particle, no ‘reality,’ for waves are intangible; this places objective reduction at the centre of the question of universal reality and how it is involved in productions for perception], even though the mathematics provide an unparalleled predictive accuracy demonstrated by endless experimental evidence. What the wave function (ψ) shows is that a measurable, “real” entity seems to exist in two (or more) separate, distinct places at exactly the same time, but only until its state is “reduced” (R) or, as Penrose sees it, “magnified to the classical level” by a measurement (R), by observation. This is ridiculous from the point of view of a mathematician. A valid mathematical description of reality should provide a specific location (not a probable one) of a particle for every time (τ), independent of observation. ψ does not give us this information and it is only upon the arbitrarily applied magnification by measurement, or observation, that the operation R takes place and a specific location at a specific time, τ, is obtained.

According to the Copenhagen interpretation, it is said that at this point the wave function “collapses” and all probabilities are cancelled except the one that is measured. The implication is that of wave-particle duality, and for many physicists and philosophers, this is good enough. Not so for Sir Penrose, whose reasons are encompassed by his theory of consciousness, which involves the necessity of an objective reduction of the wave function, independent of measurement or observation. It is this objective reduction that produces consciousness. Later we will see the biological evidence, discovered by Hameroff, et. al. For now, let us explore the problem of duality and quantum superposition as they relate to basic ontology and objective reality.

According to Victor Lenzen, “Einstein’s Theory of Knowledge,” Albert Einstein, Philosopher -Scientist, Volume II (Paul Arthur Schilpp, ed., Harper & Brothers, New York, 1959):

“Einstein in an essay on Maxwell appears to have accepted the realist doctrine, for he says, “The belief in an external world independent of the percipient subject is the foundation of all science. But since sense-perceptions inform us only indirectly of this external world, of Physical Reality, it is only by speculation that it can become comprehensible to us.” In his essay on the method of theoretical physics he expresses the conviction that pure mathematical construction is the method of discovering the concepts and laws for the comprehension of nature…[Einstein says]”Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity.””(p.363)

The implications of the quantum superposition in ψ, as I said, have been hotly debated among physicists, philosophers, mathematicians…let’s just say it is the foremost scientific mystery of our time. What the theory seems to say is that the processes which underlie and produce productions for reality are themselves not real, for if they were, we are left with the embarrassing problem of explaining why there are “classical” states of reality. Here is what Sir Penrose says:

“I do not see how reality can transform itself from a complex (or real) linear superposition of two alternatives into one or the other of these alternatives, on the basis merely of the evolution of U….”(The Emperor’s New Mind, p.380).

“I have made no bones of the fact that I believe that the resolution of the puzzles of quantum theory must lie in our finding an improved theory…But even if one believes that the theory is somehow to be modified, the constraints on how one might do this are enormous. Perhaps some kind of “hidden variable” viewpoint will eventually turn out to be acceptable. But the non-locality that is exhibited by the EPR-type experiments severely challenges any ‘realistic’ description of the world that can comfortably occur within an ordinary space-time of the particular type that has been given to us to accord with the principles of relativity – so I believe that a much more radical change is needed. Moreover, no discrepancy of any kind between quantum theory and experiment has ever been found – unless, of course, one regards the evident absence of linearly superposed cricket balls as contrary evidence. In my own view, the non-existence of linearly superposed cricket balls actually is contrary evidence!…Somewhere in between, I would maintain, we need to understand the new law, in order to see how the quantum world merges with the classical. I believe, also, that we shall need this new law if we are ever to understand minds!” (ENM, p.385-6).

So what did Einstein think of the quantum superposition? In his “Reply to Criticisms: Remarks Concerning the Essays Brought Together in this Volume,” (op. cit., Albert Einstein, pp. 665-688), Einstein emphatically states:

“…I reject the basic idea of contemporary statistical quantum theory, insofar as I do not believe that this fundamental concept will provide a useful basis for the whole of physics…I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory], sic., operates with an incomplete description of physical systems.”

To emphasize his repugnance toward the metaphysical implications of ψ, Einstein says,

“Whenever the positivistically inclined modern physicist hears such a formulation his reaction is that of a pitying smile. He says to himself: “there we have the naked formulation of a metaphysical prejudice, moreover, the conquest of which constitutes the major epistemological achievement of physicists within the last quarter-century. Has any man ever perceived a ‘real physical situation’? How is it possible that a reasonable person could today still believe that he can refute our essential knowledge and understanding by drawing up such a bloodless ghost?” (p.667)

Well, I have no response to that.

Einstein rails against quantum uncertainty. There must be a real, objective universe for me to be conscious of, not a universe created by the observation of consciousness. Yet, he admits that any construct of reality, including mathematical ones, can emerge only from pure consciousness and not from empirical evidence. Here is where Sir Penrose has come to rescue us! And from his scabbard he draws, of all things, quantum gravity! We have come full circle.

Now, I don’t know about you, but I remember thinking that quantum entanglement seemed strangely like consciousness. The uncanny awareness exhibited by the particles themselves, the instantaneous nature of this awareness over timespace, the strange irreverence for the laws of physics demonstrated by this phenomenon, the time symmetry, all seemed unreal in the illogical, intuitive, surreal way that my own consciousness seems unreal, able to defy the universe by, say, imagining purple cows. When I wrote my essay on the Star Trek robot, Data, I was feeling around in the dark, looking for something in the quantum material structure of his brain that utilized or organized this quantum awareness [see my blog entitled, Still Don’t Want To Talk About It]. Was I astounded or what when I stumbled upon Stuart Hameroff’s YouTube video discussions (A New Marriage of Brain and Computer) about quantum consciousness?

Stuart Hameroff identifies a biological quantum process of neurotransmission, gamma synchrony, that is associated directly with consciousness in that, consciousness is present with gamma synchrony and not present without it (i.e., under anesthesia, of all brain activity it is only the gamma synchronies, “coherent 40 Hz” high frequency activity, that disappear). These are produced at body temperature in dendritic microtubule components, called tubulin. Inside each tubulin dimer there are hydrophobic pockets, which are insulated regions produced by folding proteins. These protein chains are induced to fold around dipolar aromatic rings, forming hydrophobic pockets in which the quantum dipoles, produced by van der Waals forces, combine in concert to produce high frequency potentials. It is found that these high frequency wave potentials – gamma synchronies, about 40-80 per second – maintain coherency and transmit non-local, instantaneous signals via gap junctions in the dendrites.

Gap junctions are connections between dendrites that are not involved in the sequential chemically induced potentials at dendritic junctions. Rather, gamma synchronies transmit simultaneously, instantaneously across many thousands of dendrites at a time through the gap junctions, thus strengthening the coherence of the high frequency waves. This is achieved by quantum superposition of the magnetic dipoles inside the tubulin dimers, so that they become quantum bits. These individual bits in concert create interference patterns over the cylindrical tubules, thought to be possible code for quantum computational activity. These potentials maintain coherence over several thousand neurons simultaneously and each, single potential is thought to be a conscious moment.

So here we have brain activity on the grain level of quantum processes. Hameroff theorizes that the collapse of each of these gamma potentials is a conscious moment. This process is described by the deterministic wave function (U) for the rise of the potential, and the probabilistic collapse of the wave (R) for each potential. The potential itself involves coherent quantum superposition across many individual tubulin dimer proteins. The problem addressed by Penrose is to explain the collapse of the wave function objectively – why does the gamma potential reduce all by itself, without an observer? And, what has this got to do with consciousness?

Enter, the Density Matrix.


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