A Note on R and the 2nd Law of Thermodynamics

Photo taken from NASA video, Solar Dynamics Observatory

What is really cool is that Penrose challenges the time-symmetry of quantum mechanics from two directions, the non-reversibility of the reduction of the Schrodinger wave and the irreversibility of entropy according to the second law of thermodynamics. He shows that while the linear superposition exists going forward in time for a particle in the classic, simple half-silvered mirror scenario, when one attempts to reverse the process it becomes obvious that once R has occurred, absurdities, such as emission of a photon from a non-light source which absorbed it, will occur. R seems, therefore, to be related to entropy, another irreversible process.

Penrose discusses how entropy is related to gravity using the idea of light cones. On the theory of general relativity, he proposes that the gravitational field is a kind of all-pervasive refracting medium, tilting light cones in space, thereby influencing the direction of light (by curving it), and thus, influencing the relations of cause and effect. He reminds us that cause and effect are only relevant under general relativity and not under Newtonian physics, because in relativity theory, light has a speed limit beyond which paradoxes occur in time-space. This is crucial when one is considering quantum superpositions, as there are also time-space paradoxes involved in their current mathematical description.

In The Emperor’s New Mind, Penrose states:

“…there was a huge gain in entropy due to gravitational contraction…all the remarkable lowness of entropy that we find about us – and which provides this most puzzling aspect of the second law – must be attributed to the fact that vast amounts of entropy can be gained through the gravitational contraction of diffuse gas into stars.” (417)

In fact, what is really bizarre is the assertion that such a low entropy state as the original singularity could have existed spontaneously, as the “natural” state of matter is a high-entropy state of thermal equilibrium. In discussing why entropy is not time-symmetric, Penrose notes that, on the basis of the phase-space model of entropy:

“Our phase-space argument gave us completely the wrong answer when we tried to apply it in the reverse direction of time!…What that argument actually showed was that for a given low-entropy state (say for a gas tucked in a corner of a box), then, in the absence of any other factors constraining the system, the entropy would be expected to increase in both directions in time away from the given state…The argument has not worked in the past direction in time precisely because there were such factors. There was indeed something constraining the system in the past. The tendency towards high entropy in the future is no surprise. The high-entropy states are, in a sense, the ‘natural’ states which do not need further explanation. But the low-entropy states in the past are a puzzle. What constrained the entropy of our world to be so low in the past? The common presence of states in which the entropy is absurdly low is an amazing fact of the actual universe that we inhabit – though such states are so commonplace and familiar to us that we do not normally tend to regard them as amazing. We ourselves are configurations of ridiculously low entropy!” (410)

So, entropy and gravity are related.

Penrose notes that in the space-time of general relativity there is an ‘obstruction,’ called the WEYL tensor, which is the conformal part of the relativistic equations. This obstruction prevents uniformity of space-time in terms of his illustration using light cones; that is, the light cones cannot be aligned perfectly with one another because of WEYL:

“The tensor WEYL describes just half of the information – the ‘conformal’ half – that is contained in the full Reimann curvature tensor of space-time…Only if WEYL is zero can we rotate all the light cones into the Minkowskian arrangement [i.e. perfectly aligned with one another]. The tensor WEYL measures the gravitational field – in the sense of the gravitational tidal distortion – so it is precisely the gravitational field, in this sense, that provides the obstruction…”(Shadows of the Mind, 224)

The point about light cone tilting is that this action, this character of gravity had gone unnoticed in classical physics and was only identified in Einstein’s theory. Recent observations of gravitational lensing have provided good evidence for this hitherto “invisible” aspect of gravity. Analogously, perhaps there is some unseen, non-computational aspect of physical matter that is organized in biological design for the purpose of producing consciousness.

Penrose argues that instances of quantum superposition in nature are rare and unstable. The occurrence of coherent, prolonged quantum superposition in a biological organism must be the result of design and constitutes a novel use, in nature, of such special properties of matter that are not well understood. Stuart Hameroff alludes to this when he mentions that the objective reduction time of an individual particle in space, which avoided decoherence, would be in the order of 10 million years, and that it would be of low frequency, low intensity. The implication is that for consciousness to occur on the basis of the objective reduction of coherent quantum superpositions, a special design which organizes this phenomenon is necessary.


The Density Matrix


“…mathematical understanding is something different from computation and cannot be completely supplanted by it. Computation can supply extremely valuable aid to understanding, but it never supplies actual understanding itself.” (Penrose, Shadows of the Mind, 199)

Basically, mathematics is a descriptive language, like any other language, and as such is not a generator of conscious perception.

Penrose’s discussions of quantum physics explore its mathematical “ability” to describe reality. This line of inquiry appears to be motivated by his conjecture that there must be some property of physical reality that is related to the production of consciousness which science has either overlooked or not discovered yet:

“…the phenomenon of consciousness can arise only in the presence of some non-computational physical process taking place in the brain. One must presume, however, that such (putative) non-computational processes would also have to be inherent in the action of inanimate matter…First, why is it that the phenomenon of consciousness appears to occur, as far as we know, only in (or in relation to) brains…Second, we must ask how it could be that such a seemingly (putative) ingredient as non-computational behaviour, presumed to be inherent – potentially at least – in the actions of all material things, so far has entirely escaped the notice of physicists?” (SotM, 216)

Penrose finds this ingredient in the diaphanous vicissitudes of the gravitational field because, “gravity actually influences the causal relationships between space-time events, and it is the only physical quantity that has this effect” (SotM, 219). Gravity really alters the geometry of space-time and of all particulate matter found within it. Because particles (or “lumps” of matter with specific mass-energies) in superposition also have a gravitational field which also must be part of the superposition, “the state involves a superposition of two different gravitational fields. According to Einstein’s theory, this implies that we have two different space-time geometries superposed!” (SotM, 337)

“The point is that we really have no conception of how to consider linear superpositions of states when the states themselves involve different space-time geometries. A fundamental difficulty with ‘standard theory’ is that when the geometries become significantly different from each other, we have no absolute means of identifying a point in one geometry with any particular point in the other – the two geometries are strictly separate spaces – so the very idea that one could form a superposition of the matter states within these two separate spaces becomes profoundly obscure.” (SotM, 337)

This is where the rubber really hits the road for the brilliant Sir. It is stunning, awesome and totally amazing to witness the invention, right before our very eyes, of a beginning of a new mathematical description of reality.

The density matrix becomes important at this point because it is the mathematics of the density matrix, rather than simply the state vector, ψ, that is involved in the state vector Reduction or “measurement” process. The density matrix is a deliberately fuzzy description of multiple state vectors, a “probability mixture”:

“…with a density matrix, there is a (deliberate) confusion, in this description, between these classical probabilities, occurring in this probability-weighted mixture and the quantum-mechanical probabilities that would result from the R-procedure. The idea is that one cannot operationally distinguish between the two, so a mathematical description – the density matrix – which does not distinguish between them is operationally appropriate. (SotM, 317)

As a description, Penrose calls the density matrix “elegant” and useful “for all practical purposes” (FAPP); however, as a complete description of reality, it will not do:

“The fact that the physicist considers that the state of his detector is described by the density matrix D does not in any way explain why he always finds that the detector is either in a YES state…or else in a NO state…For precisely the same density matrix would be given if the state were an equal-probability-weighted combination of classical absurdities…(…the quantum linear absurdities ‘YES plus NO’ and ‘YES minus NO’)!…The upshot of all this is that merely knowing that the density matrix is some D does not tell us that the system is a probability mixture of some particular set of states that give rise to a particular D. There are always numerous completely different ways of getting the same D, most of which would be ‘absurd’ from the common-sense point of view. Moreover, this kind of ambiguity holds for any density matrix whatsoever.” (SotM, 326-7)

What is being said, here, is that there is no reason whatsoever given by the current, state-of-the-art mathematical description why a quantum system assumes some particular real, observable, even in principle, classical answer to the experimental question, Where is the particle now? Even more bizarrely, one cannot ascertain why, on the basis of the density matrix, one ever finds a real answer, a real position, a real particle, at all!

What this really means, argues Sir Penrose, is that the R procedure cannot and does not follow from the unitary evolution of the wave equation and seems to represent a completely independent and as yet not understood process of which R is only an approximation. R must be some kind of gravitational or gravitationally-related process; in fact, it must be a quantum gravitational process:

“[in a quantum superposition] when are two geometries to be considered as actually ‘significantly different’ from one another? It is here, in effect, that the Planck scale of 10^-33 cm comes in. The argument would roughly be that the scale of the difference between these geometries has to be, in an appropriate sense, something like 10^-33 cm or more for reduction to take place. We might, for example, attempt to imagine that these two geometries are trying to be forced into coincidence, but when the measure of the difference becomes too large, on this kind of scale, reduction R takes place – so, rather than the superposition involved in U being maintained, Nature must choose one geometry or the other.” (SotM, 337)

The reasoning, it seems, is that we don’t have a mathematics of quantum gravity and this must be why scientists have not found a non-computable physical process as described above. So, Penrose sets out to develop one for us!

Hence, the need for The New Criterion. Penrose simply and elegantly surmises that the reduction of a quantum superposition is analogous to the spontaneous decay of atomic nuclei in that it is unstable. He calculates the simple gravitational displacement, in absolute units (see 338-9) and:

“…we ask that there be a rate of state-vector reduction determined by such a difference measure. The greater the difference, the faster would be the rate at which reduction takes place…In general, when we consider an object in a superposition of two spatially displaced states, we simply ask for the energy that it would take to effect this displacement, considering only the gravitational interaction between the two. The reciprocal of this energy measures a kind of ‘half-life’ for the superposed state. The larger this energy, the shorter would be the time that the superposed state could persist.” (SotM, 339, 341)

Penrose goes on to explain that the numbers at the Planck scale for this descriptive equation, E= h/t, correspond well with observations of nature, “It is reassuring that this provides very ‘reasonable’ answers in certain simple situations.” (340) (cf Diósi) In terms of biological systems,

“A biological system, being very much entangled with its environment…would have its own state continually reduced because of the continual reduction of its environment. We may imagine, on the other hand, that for some reason it might be favourable to a biological system that its state remain unreduced for a long time, in appropriate circumstances. In such cases it would be necessary for the system to be, in some way, very effectively insulated from its surroundings.” (SotM, 343)

Here we have the rudiments of a mathematical language for consciousness. We will now have to wait till I finish the book to see how this all ties in with brain science. For a preview, see Stuart Hameroff’s YouTube video, A New Marriage of Brain and Computer.

A Brief History of the Density Matrix

NASA Cassini Image

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.