The Density Matrix

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“…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.

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