Commenting on my thoughts about decision analysis and Schroedinger’s cat (see here for my clarifications), Dave Krantz writes,

I’d first like to comment on the cat example, and then turn to the relationship to probabilistic modelling of choice.

I think one can gain clarity by thinking about simpler analogs to Schroedinger’s cat. Instead of poison gas being released, killing the cat, let’s suppose that a single radioactive decay just releases one molecule of hydrogen (H2) into an otherwise empty (hard vacuum) cat box. Now an H2 molecule is something that, in principle, one can describe pretty well by a rather complicated wave function. The wave function for an H2 molecule confined to a small volume, however, is different from the wave function for an H2 molecule confined to a much larger cat box. At any point in time, our best description (vis-a-vis potential measurements we could make that would interact with the H2 molecule) is a superposition of these two wave functions, narrowly or broadly confined. As long as we don’t know whether the radioactive decay has taken place, and we make no observation that directly or indirectly interacts with the H2 molecule, the superposition continues to be the best physical model.

This example points up the fact that Schroedinger’s cat involves two different puzzles. The first is epistemological: we are used to thinking of a cat as alive or dead, but equally used to thinking of a H2 molecule as confined narrowly or broadly. How can it be both? But this way of thinking just won’t work in QM. The point of the double-slit experiments is to show clearly that an unobserved photon does NOT go through one slit or the other, it goes through both, in the sense of its wave function giving rise to coherent circularly symmetric waves emanating from each slit and interfering. It is equally wrong to think that a H2 molecule is either confined narrowly or broadly. Observations are going to be accounted for by assuming a superposition.

The second puzzle arises because a cat cannot in practice be described by a single wave function at all. That’s at least true of an ordinary cat, subject to many sorts of observation. But in practice, even an unobserved cat is not describable by a wave function. There are wave functions for each molecule, but the best descriptions do not collapse these into a single wave function. Coherence fails. To take an analogy, one can get monochromatic light by passing a beam through an interference filter; though the frequencies of the different photons are all alike, the phases still vary randomly. This is very different from the coherent light of a laser, where everything is in phase.

There is a real problem of understanding when incoherent wave functions collapse into a single coherent one. This has been dramatized, in recent years, by studies of Bose-Einstein condensates. Rubidium atoms can be very near one another, yet still incoherent; but at low temperatures, they become a single molecular system, with a condensed wave function. The study of conditions for coherence is on-going, as I understand it. A cat is outside the boundaries of coherence.

Epistemologically, the introduction of probabilities as fundamental terms in choice modelling is rather analogous to the introduction of probabilities in QM measurement. It has always struck me as curious that the two happened in the same year, 1927: Born developed the probabilistic interpretation of QM measurement and Thurstone formulated the law of comparative judgment.

Where the analogy breaks down, however, is that there isn’t any analog to a wave function in choice models. Thurstone actually tried to introduce something like it, with his discriminal processes, but from the start, discriminal processes were postulated to be independent rather than coherent random variables. Thus, I don’t see much point in pushing the analogy of any DM problem with the Schroedinger cat problem, where the essence is superposition rather than independence.

My thoughts

OK, that was Dave talking. To address his last point, yes, I don’t see where the complex wave function would come in. (Dsquared makes the same point in the comments to this entry. In probability theory we’re all happy to use Boltzmann statistics (i.e., classical probability theory). I’ve never seen anyone make a convincing case (or even try to make a case) that, for example, Fermi-Dirac statistics should be used for making business decisions.)

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That being said, Dave’s point above about “coherence” is exactly what I was talking about. Also there’s the bit about the collapse of the wave function (or of the decision tree). But I suppose Dave would say that, without complex wavefunctions, there’s no paradox there. With classical Boltzmann statistics, the cat really is just alive or dead all along, with no need for superposition of states.

Jim Thompson’s cat

Hmmm…my feeling is that the act of deliberation, or even just of keeping a decision “open” or “alive,” creates a superposition of states. If I’m deciding whether or not to flip the switch, then I would’t say that the cat is “either alive or dead.” I haven’t decided yet! In The Killer Inside Me, Jim Thompson writes, “How can you hurt someone that’s already dead?”, but I don’t take such a fatalistic position.

Roger Penrose’s consciousness

But hey, let’s take this one step further. In my experiment (as opposed to Schroedinger’s), the cat is alive or dead based on my decision of whether to flip a switch (and, in turn, this decision is ultimately coupled with other outcomes of interest; e.g., the switch also turns off the light in the next room, which encourages the lab assistant to go home for the day, and then he might bump into someone on the subway, etc., etc.). If it is true, as Penrose claims in The Emperor’s New Mind, that consciousness is inherently quantum-mechanical and non-algorithmic, then my decision of whether to flip the switch indeed must be modeled as a superposition of wave functions. Although then I’m not quite sure how deliberation fits in to all this.

Anyway, to get more positivistic for a moment, maybe the next research step is to formulate some actual decision problems (or realistic-seeming fake problems) in terms of coherence, and see if anything useful comes of it.

P.S. Dave is very modest on his webpage but he’s actually the deepest thinker I know of in decision analysis.

P.P.S. It’s funny that Dave has a cat living in a “cat box,” which I always thought was equivalent to the litterbox (so I recall from my catful days). Maybe “cat container” would be a better phrase?