Schrödinger’s Cat

Erwin Schrödinger describes the context for his thought experiment with a cat:

The other alternative consists of granting reality only to the momentarily sharp determining parts – or in more general terms to each variable a sort of realization just corresponding to the quantum mechanical statistics of this variable at the relevant moment.

That it is in fact not impossible to express the degree and kind of blurring of all variables in one perfectly clear concept follows at once from the fact that Q.M. as a matter of fact has and uses such an instrument, the so-called wave function or psi-function, also called system vector. Much more is to be said about it further on. That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message. At all events it is an imagined entity that images the blurring of all variables at every moment just as clearly and faithfully as does the classical model its sharp numerical values. Its equation of motion too, the law of its time variation, so long as the system is left undisturbed, lags not one iota, in clarity and determinacy, behind the equations of motion of the classical model. So the latter could be straight-forwardly replaced by the psi-function, so long as the blurring is confined to atomic scale, not open to direct control. In fact the function has provided quite intuitive and convenient ideas, for instance the “cloud of negative electricity” around the nucleus, etc. But serious misgivings arise if one notices that the uncertainty affects macroscopically tangible and visible things, for which the term “blurring” seems simply wrong. The state of a radioactive nucleus is presumably blurred in such a degree and fashion that neither the instant of decay nor the direction, in which the emitted alpha-particle leaves the nucleus, is well-established. Inside the nucleus, blurring doesn’t bother us. The emerging particle is described, if one wants to explain intuitively, as a spherical wave that continuously emanates in all directions and that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however does not show a more or less constant uniform glow, but rather lights up at one instant at one spot – or, to honor the truth, it lights up now here, now there, for it is impossible to do the experiment with only a single radioactive atom. If in place of the luminescent screen one uses a spatially extended detector, perhaps a gas that is ionised by the alpha-particles, one finds the ion pairs arranged along rectilinear columns, that project backwards on to the bit of radioactive matter from which the alpha-radiation comes (C.T.R. Wilson’s cloud chamber tracks, made visible by drops of moisture condensed on the ions).

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a “blurred model” for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

We see here the two elements described at the end of this earlier post. The psi-function is deterministic, but there seems to be an element of randomness when someone comes to check on the cat.

Hugh Everett amusingly describes a similar experiment performed on human beings (but without killing anyone):

Isolated somewhere out in space is a room containing an observer, A, who is about to perform a measurement upon a system S. After performing his measurement he will record the result in his notebook. We assume that he knows the state function of S (perhaps as a result of previous measurement), and that it is not an eigenstate of the measurement he is about to perform. A, being an orthodox quantum theorist, then believes that the outcome of his measurement is undetermined and that the process is correctly described by Process 1 [namely a random determination caused by measurement].

In the meantime, however, there is another observer, B, outside the room, who is in possession of the state function of the entire room, including S, the measuring apparatus, and A, just prior to the measurement. B is only interested in what will be found in the notebook one week hence, so he computes the state function of the room for one week in the future according to Process 2 [namely the deterministic  wave function]. One week passes, and we find B still in possession of the state function of the room, which this equally orthodox quantum theorist believes to be a complete description of the room and its contents. If B’s state function calculation tells beforehand exactly what is going to be in the notebook, then A is incorrect in his belief about the indeterminacy of the outcome of his measurement. We therefore assume that B’s state function contains non-zero amplitudes over several of the notebook entries.

At this point, B opens the door to the room and looks at the notebook (performs his observation.) Having observed the notebook entry, he turns to A and informs him in a patronizing manner that since his (B’s) wave function just prior to his entry into the room, which he knows to have been a complete description of the room and its contents, had non-zero amplitude over other than the present result of the measurement, the result must have been decided only when B entered the room, so that A, his notebook entry, and his memory about what occurred one week ago had no independent objective existence until the intervention by B. In short, B implies that A owes his present objective existence to B’s generous nature which compelled him to intervene on his behalf. However, to B’s consternation, A does not react with anything like the respect and gratitude he should exhibit towards B, and at the end of a somewhat heated reply, in which A conveys in a colorful manner his opinion of B and his beliefs, he rudely punctures B’s ego by observing that if B’s view is correct, then he has no reason to feel complacent, since the whole present situation may have no objective existence, but may depend upon the future actions of yet another observer.

Schrödinger’s problem was that the wave equation seems to describe something “blurred,” but if we assume that is because something blurred exists, it seems to contradict our experience which is of something quite distinct: a live cat or a dead cat, but not something in between.

Everett proposes that his interpretation of quantum mechanics is able to resolve this difficulty. After presenting other interpretations, he proposes his own (“Alternative 5”):

Alternative 5: To assume the universal validity of the quantum description, by the complete abandonment of Process 1 [again, this was the apparently random measurement process]. The general validity of pure wave mechanics, without any statistical assertions, is assumed for all physical systems, including observers and measuring apparata. Observation processes are to be described completely by the state function of the composite system which includes the observer and his object-system, and which at all times obeys the wave equation (Process 2).

It is evident that Alternative 5 is a theory of many advantages. It has the virtue of logical simplicity and it is complete in the sense that it is applicable to the entire universe. All processes are considered equally (there are no “measurement processes” which play any preferred role), and the principle of psycho-physical parallelism is fully maintained. Since the universal validity of the state function is asserted, one can regard the state functions themselves as the fundamental entities, and one can even consider the state function of the whole universe. In this sense this theory can be called the theory of the “universal wave function,” since all of physics is presumed to follow from this function alone. There remains, however, the question whether or not such a theory can be put into correspondence with our experience.

This present thesis is devoted to showing that this concept of a universal wave mechanics, together with the necessary correlation machinery for its interpretation, forms a logically self consistent description of a universe in which several observers are at work.

Ultimately, Everett’s response to Schrödinger is that the cat is indeed “blurred,” and that this never goes away. When someone checks on the cat, the person checking is also “blurred,” becoming a composite of someone seeing a dead cat and someone seeing a live cat. However, these are in effect two entirely separate worlds, one in which someone sees a live cat, and one in which someone sees a dead cat.

Everett mentions “the necessary correlation machinery for its interpretation,” because a mathematical theory of physics as such does not necessarily say that anyone should see anything in particular. So for example when Newton when says that there is a gravitational attraction between masses inversely proportional to the square of their distance, what exactly should we expect to see, given that? Obviously there is no way to answer this without adding something, and ultimately we need to add something non-mathematical, namely something about the way our experiences work.

I will not pretend to judge whether or not Everett does a good job defending his position. There is an interesting point here, whether or not his defense is ultimately a good one. “Orthodox” quantum mechanics, as Everett calls it, only gives statistical predictions about the future, and as long as nothing is added to the theory, it implies that deterministic predictions are impossible. It follows that if the position in our last post, on an open future, was correct, it must be possible to explain the results of quantum mechanics in terms of many worlds or multiple timelines. And I do not merely mean that we can give the same predictions with a one-world account or with a many world account. I mean that there must be a many-world account such that its contents are metaphysically identical to the contents of a one-world account with an open future.

This would nonetheless leave undetermined the question of what sort of account would be most useful to us in practice.

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