The great scandal of physics

Featuring Schrödinger’s cat

In The Ashgate Companion to Contemporary Philosophy of Physics,¹ David Wallace introduces the chapter on “Philosophy of Quantum Mechanics” with the following words:

By some measures, quantum mechanics is the great success story of modern physics: no other physical theory has come close to the range and accuracy of its predictions and explanations. By other measures, it is instead the great scandal of physics: despite these amazing successes, we have no satisfactory physical theory at all — only an ill-defined heuristic which makes unacceptable reference to primitives such as “measurement”, “observer” and even “consciousness”. This is the measurement problem, and it dominates philosophy of quantum mechanics.

Wallace draws a distinction between (i) a “bare quantum formalism,” which he describes as “an elegant piece of mathematics” that comes “prior to any notion of probability, measurement etc.,” and (ii) a “quantum algorithm,” which he describes as “an ill-defined and unattractive mess.” And he insists that measurement “is a physical process, not an unanalyzable primitive.”

It does indeed seem perfectly reasonable to hold that a measurement involves an interaction between an apparatus and a physical system on which the measurement is performed. Yet if this interaction is modeled as a physical process, and if this process is described in quantum-mechanical terms, what follows is that no measurement ever has an outcome (which is self-contradictory) or that a measurement involves something beyond the realm of the physical, something like the consciousness of an observer (which to most philosophers of science is worse).

Let’s consider what it means, to most physicists and philosophers of science, to describe the interaction between a measurement apparatus and a physical system as a quantum-mechanical process:

|A₀⟩⊗|Ψ⟩ → Σₖ cₖ |Aₖ⟩⊗|qₖ⟩ → |A(q)⟩⊗|q⟩

I will make this as simple as possible (but not simpler). To the left of the first arrow we have a mathematical operation represented by the symbol ⊗, which combines |A₀⟩ (a mathematical thingummy said to represent the apparatus in its neutral state, prior to the measurement interaction) and |Ψ⟩ (a mathematical thingummy said to represent the initial state of the quantum system). The first arrow represents the period of time during which the interaction takes place, and the expression between the two arrows represents the result of the interaction. As indicated by the symbol Σₖ, it is a sum of as many terms as there are possible outcomes. cₖ is a complex number that depends on |‍‍Ψ⟩. |qₖ⟩ is said to represent the system as having the property qₖ (one of the possible outcomes), and |Aₖ⟩ is said to represent the apparatus as indicating that the system has the property qₖ.

Because the interaction results in a sum of as many terms as there are possible outcomes, the measurement is as yet unfinished. The first stage of the purported measurement process is therefore known as pre-measurement. The second arrow represents the transition to a state in which the system has the property q (one of the possible outcomes) and the apparatus indicates that this is the case. This stage is known as objectification, and it is what quantum mechanics cannot explain. For this reason the measurement problem is also known as “the disaster of objectification”.²

How, then, is the connection made between the “bare quantum formalism,” according to which measurements lack definite outcomes, and human experience, in which measurements have definite outcomes? It is made by stipulating that the squared magnitude |cₖ|² of cₖ is the probability with which the measurement yields the experienced outcome qₖ.

Wallace is right that probability cannot emerge from a “bare quantum formalism” which has no bearing on probability. Probability is the kind of something that you cannot get from nothing. Precisely for this reason there is no such thing as a bare quantum formalism “prior to any notion of probability, measurement etc.” Every single axiom of every axiomatization of the quantum theory is, and only makes sense as, a feature of a calculus that serves to assign probabilities to measurement outcomes and on the basis of measurement outcomes.³

If the expression Σₖ cₖ |Aₖ⟩⊗|qₖ⟩ would actually represent the combined state of system and apparatus, one would have to admit that the interpretation of the squared magnitude of cₖ as a probability is inacceptable and may even be considered scandalous. But what is actually inacceptable is the set of notions which render the interpretation of |cₖ|² as a probability inacceptable. Foremost among these are the notions that |Ψ⟩ and |A₀⟩ represent the respective physical states of system and apparatus prior to the measurement, that |qₖ⟩ represents the system as having the property qₖ, and that |Aₖ⟩ represents the apparatus as indicating this outcome. For it is self-contradictory to interpret the sum Σₖ cₖ |Aₖ⟩⊗|qₖ⟩ as the combined physical state of system and apparatus and also to interpret the respective terms |qₖ⟩ and |Aₖ⟩ as representing the system in possession of the property qₖ and the apparatus as indicating this outcome. This is exactly the befuddled thinking which leads to the notorious Schrödinger-cat state

|S-cat⟩ = c₁ |A₁⟩⊗|cat(alive)⟩ + c₂ |A₂⟩⊗|cat(dead)⟩,

where |cat(alive)⟩ is supposed to represent the cat as being alive, |A₁⟩ is supposed to represent the apparatus as indicating this, and ditto for |cat(dead)⟩ and |A₂⟩. The crowning absurdity of this way of thinking about the expression |S-cat⟩ is that, as the sum of the two terms on the right-hand side, it implies that the cat is both dead and alive, or else neither dead nor alive.

Note: “Copenhagen interpretation” is often used as a derogatory term by those who believe that each measurement interaction splits the world into as many worlds as there are measurement outcomes, so that in each world the cat is either dead or alive. And of course it should be “poison.”

Here is the scenario as originally thought up by Schrödinger⁴:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical 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 one 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 first atomic decay would have poisoned it. The ψ-function of the entire system would express this by having in it the living and the 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.

The fact of the matter is that the symbol |x⟩ denotes a vector in a certain mathematical space. While by itself it bears no relation to either physical reality or human experience, it is often used (and should in fact only be used) as a convenient shorthand for another mathematical animal, which is designated by the symbol |x⟩⟨x|. This represents a measurement outcome — either that to which a probability is assigned or that on the basis of which probabilities are assigned. But it does this only in the context of a probability calculus, where it serves either as the assignment basis or as the possibility to which a probability is assigned.

From quantum theory’s early days, the goal of making physical sense of the theory’s mathematical formalism has been pursued along two divergent lines, one fundamentally philosophical, the other essentially mathematical; one spearheaded by Niels Bohr, the other set in motion by the mathematician and computer scientist John von Neumann. One of von Neumann’s claims to fame is that he was the first to work out the mathematical formalism of quantum mechanics in a systematic and mathematically precise way. Many published discussions of the meaning of quantum mechanics present von Neumann as viewing the “quantum state” |Ψ⟩ as a representation of a physical state that is capable of changing (“evolving”) in two distinct ways: continuously between measurements (as also during the so-called pre-measurement stage), and discontinuously at the objectification stage, when a measurement is completed and the system’s state is said to “collapse.”

It is less widely known that, soon after the publication of his celebrated 1932 book,⁵ von Neumann rejected the central role he had assigned to |Ψ⟩ in favor of |Ψ⟩⟨Ψ|. He thereby abandoned the notion of a physical state with two distinct modes of change, and instead espoused as the physically relevant core of quantum mechanics the conditional probabilities defined by a “trace operator,” which has input slots for two measurement outcomes — one for the outcome to which a probability is assigned, and one for the outcome on the basis of which this probability is assigned.⁶ While Bohr and von Neumann (after the publication of his book) were thus on convergent tracks, too many physicists and philosophers of science still see the issue of making sense quantum mechanics as a choice between a view that Bohr never held (instrumentalism) and a view that von Neumann soon abandoned (quantum state realism).

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There are actually two measurement problems. One, sometimes called the “big” measurement problem, is the problem of explaining how measurement outcomes come about “dynamically,” i.e., as a result of a single continuous mode of change, without invoking a second, discontinuous mode of change. This problem arises from the false premise that |Ψ⟩ represents a physical state, and that its dependence on time is the continuous time dependence of a physical state.

While the passage of time between the first measurement (on the basis of whose outcome probabilities are assigned) and the second measurement (to the possible outcomes of which probabilities are assigned) is taken care of by an operator that depends on the respective times of the two measurements, this operator does not represent a physical process that brings about a physical change. Any story purporting to relate what happens between the two measurements is (in Wolfgang Pauli’s felicitous phrase) “not even wrong,” inasmuch as it can be neither proved nor disproved.

The second measurement problem, sometimes referred to as the “small” one, is the question why certain projection operators |x⟩⟨x| (or the subspaces into which they project) represent possible measurement outcomes, while others do not. The reason we never experience a measurement pointer as simultaneously pointing in two different directions is that measurement outcomes are experiences, and experiences conform to Kant’s principle of thoroughgoing determination. Assuming that every thing was accessible to direct sensory experience, Kant⁷ concluded that

every thing, as to its possibility, stands under the principle of thoroughgoing determination, according to which, among all possible predicates of things, insofar as they are compared with their opposites, one must apply to it.

Because Kant’s principle applies to everything that is accessible to sensory experience, it applies to every outcome-indicating property, and therefore implies the definiteness of every measurement outcome.

As long as the cat is directly accessible to sensory experience, it can serve as a measurement pointer: if after an hour the cat is alive, it indicates that as yet no atom has decayed, and if after an hour the cat is dead, it indicates that at least one atom has decayed. Is it possible to make the cat inaccessible to direct sensory experience by, say, penning it up in a steel chamber? Suppose that it is. While we are then ignorant of the state of the cat (alive or dead), we are by no means cognizant of the cat’s being neither dead nor alive. To be cognizant of such a state, we must have evidence that such a state obtains. Is it possible to have such evidence?

Let’s start with the simplest possible situation. It is perfectly feasible to prepare a particle in such a way that its spin, if measured with respect to the vertical axis, is certain to be found up, and that its spin is equally likely to be found up or down if measured with respect to a horizontal axis. In this case, finding the spin up with respect to the vertical axis implies that the spin is neither up nor down with respect to any horizontal axis. It is also possible (if technically more challenging) to perform an experiment that has a possible outcome which implies that neither of the following is the case: no atom has decayed and at least one atom has decayed. And it would also be possible to obtain evidence that the cat is neither alive nor dead if it were possible to perform an experiment that has a possible outcome which implies that the cat is neither alive nor dead.

Let us furthermore take into account that the properties of quantum systems are contextual: they are defined by the experimental conditions under which they are observed, and they only exist if their presence is indicated. Since presently we are treating the cat as a quantum system, we need to ask: is it possible to conceive of experimental conditions that define a property whose existence would, if indicated, imply that the cat is neither alive nor dead?

Let’s take the craziness a step further. If such experimental conditions could be created, it would be possible to transform a living cat into one which is neither dead not alive. It would then also be possible to transform a dead cat into one which is neither dead nor alive. And it would then be possible to determine, by a subsequent measurement, whether the cat is dead or alive, and it would be possible to find that the cat is alive. In other words, it would be possible to resurrect a dead cat. I am not making this up. Luigi Picasso, for one, writes in his Lectures in Quantum Mechanics (Springer, 2016, p. 341) that “tomorrow, when the observables that today do not exist will become available, we will be able, by means of two measurements, to resurrect dead cats...”

1. Edited by D. Rickles (Ashgate, 2008).

2. B.C. van Fraassen, The problem of measurement in quantum mechanics, in P. Lahti and P. Mittelstaedt (Eds.), Symposium on the Foundations of Modern Physics, pp. 497‒503 (World Scientific, 1990).

3. See my paper Quantum mechanics explained, International Journal of Quantum Information 7, 435‒458 (2009).

4. J.D. Trimmer, The Present Situation in Quantum Mechanics: A Translation of Schrödinger's “Cat Paradox” Paper, Proceedings of the American Philosophical Society 124, 323–338 (Oct. 10, 1980).

5. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, 1932); Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1955).

6. M. Rédei, Why John von Neumann did not Like the Hilbert Space Formalism of Quantum Mechanics (and What he Liked Instead), Studies in History and Philosophy of Modern Physics 21, 493–510 (1996).

7. I. Kant, Critique of Pure Reason, p. 553 (Cambridge University, 1998).