Many words or many worlds?

“We use mathematics in physics so that we won’t have to think” — Bryce DeWitt

Rudolf Peierls¹ once said: “The Copenhagen interpretation is quantum mechanics!” These days it is more usual to claim, as Ted Bunn did,² that henceforth the many-worlds interpretation of quantum mechanics (MWI) should be known simply as “quantum mechanics.” To David Deutsch, the MWI isn’t an interpretation of quantum mechanics, any more than dinosaurs are an interpretation of the fossil record. “The only astonishing thing is that that’s still controversial,” he adds.³

These are examples of one of the more common errors that physicists tend to make, which is to mistake the mathematical formalism of a physical theory for its interpretation. The purpose of an interpretation is to spell out how the mathematical formalism of the theory relates to the physical world, to human experience, or to some aspect thereof. Here is how philosopher of science Michael Redhead put it: to give an interpretation of quantum mechanics is to provide “some account of the nature of the external world and/or our epistemological relation to it that serves to explain how it is that the statistical regularities predicted by the formalism ... come out the way they do”.⁴ While it should be obvious that this cannot be done without introducing background knowledge and/or assumptions unwarranted by the formalism itself, this is precisely what the original proponents⁵ of the MWI have claimed: “The mathematical formalism of the quantum theory is capable of yielding its own interpretation.” In other words, the MWI delegates the formidable task of making physical sense of quantum theory’s mathematical formalism to the formalism itself, absolving its proponents from the need to think. Many worlds instead of many words!

There is also the small matter of which mathematical formulation of the quantum theory is (allegedly) capable of yielding its own interpretation. For there are not only different interpretations of the theory, there are also different formulations.⁶

The first coherent formulation of (non-relativistic) quantum theory was developed in 1925 by Werner Heisenberg, Max Born, and Pascual Jordan. Because it was the first, it was then known simply as “quantum mechanics.” Heisenberg’s leading idea at the time was that only those quantities that are in principle observable should play a role in the theory, and that all attempts to form a picture of what goes on inside the atom should be abandoned. Why, then, did his article⁷ of 1927, in which he introduced the uncertainty principle, bear the title “On the visualizable content of quantum kinematics and mechanics”? Why has visualizability suddenly become a concern to him?

The reason was that meanwhile, in 1926, Erwin Schrödinger had presented his formulation of the theory, then called “wave mechanics.” When it was shown that quantum mechanics and wave mechanics were different mathematical formulations of the same physical theory, this physical theory came to be called “quantum mechanics” and Heisenberg’s formulation came to be known as “matrix mechanics.” While matrices are indisputably mathematical constructs, for which physical visualizations are hard to come by, waves are visualizable, and this made Schrödinger’s formulation more attractive to many leading physicists.

Heisenberg’s based his claim to the visualizability of matrix mechanics on (i) our ability to picture the theory’s experimental consequences and (ii) the fact that “all concepts that are used to describe a mechanical system in the classical theory can also be exactly defined for atomic processes in analogy to the classical concepts.” That claim was, however, undermined by (a) the uncertainty principle and (b) the need for experiments to define those concepts: “the experiments which provide such a definition contain in themselves an indeterminacy when we require of them the simultaneous determination of two canonically conjugate quantities,” such as position and momentum.

As regards Schrödinger’s wave mechanics, there are at least two obvious reasons why its centerpiece — the wave function Ψ — should not be taken to represent a physical wave. The first is that Ψ is a complex function (i.e., a function that outputs a complex number), and the second is that the wave function associated with a physical system made up of as few as two particles “exists” or “propagates” in a six-dimensional abstract space, rather than in physics space.

Of the remaining formulations (see Note 6) suffice it to name but one — Feynman’s path integral formulation,⁸ a.k.a. the sum-over-histories formulation — for this gives explicit expression to what is common to all formulations of quantum mechanics, namely that they provide us with tools for calculating conditional probabilities. Insofar as all these empirically equivalent formulations differ, they differ in what some may consider idle metaphysical baggage — claims that are “not even wrong” (to use Wolfgang Pauli’s famous phrase) since it is impossible to prove them wrong. But humans are, as Schopenhauer put it, metaphysical animals,⁹ to the extent that even such paradigmatically calculational tools as Feynman diagrams have been touted as “literally pictures of what happened,.... pictures in spacetime of the antics of particles, coming together, colliding and producing other particles, and so on”.¹⁰ (This kind of metaphysics is known as realism about calculational devices.)

Why among all the empirically equivalent mathematical formalisms should Schrödinger’s be chosen as the one that is “capable of yielding its own interpretation”? The answer is that no calculational tool is easier to reify than a time-dependent “quantum state.” All one has to do is (i) misconstrue the wave function’s dependence on the time of a measurement (to the possible outcomes of which it serves to assign probabilities) as the time dependence of an evolving physical state of some kind, and (ii) posit the chimera of a universal wave function. Since there are neither apparatuses nor observers outside the Universe, a universal wave function cannot depend on measurement outcomes. It is what it is, never mind that nobody knows what it is nor on what it depends (since it cannot depend on the outcomes of measurements).

Those who believe that a universal wave function is all there is, will have some tough question to answer. For one, how is it that there are observers? And if observations involve neurons, how is it that there are neurons? In fact, how is it that there are things like molecules, or atoms, or subatomic particles? A few years ago, Sheldon Goldstein¹¹ raised this very question:

It seems clear that quantum mechanics is fundamentally about atoms and electrons, quarks and strings, not those particular macroscopic regularities associated with what we call measurements of the properties of these things. But if these entities are not somehow identified with the wave function itself — and if talk of them is not merely shorthand for elaborate statements about measurements — then where are they to be found in the quantum description?

Much the same question was raised by Wojciech Zurek¹² in the context of the decoherence program, where it is assumed that the universe can be divided into at least three parts: an object system, a measurement apparatus, and an environment:

one issue which has often been taken for granted is looming large, as a foundation of the whole decoherence programme. It is the question of what the “systems” which play such a crucial role in all the discussions of the emergent classicality are. ...the problem of measurement cannot even be stated without a recognition of the existence of systems. ...a compelling explanation of what the systems are ... would undoubtedly be most useful.

Microscopic objects exist because they are individuated by macroscopic objects, and macroscopic objects exist because experience tells us that they do. But if we are forbidden to appeal to experience because the goal is to account for the experience of a classical world (“classicality”), then this hurdle remains to be overcome. Assuming that it can or has been overcome, we are faced with the preferred basis problem, which the decoherence program was designed to solve.

If there is a universal wave function, and if this evolves deterministically at all times (as it must since there is nothing outside the universe which could cause it to “collapse”), and if there are physical systems that interact with each other, then these systems will get entangled. If the mathematical formalism of quantum mechanics is understood as a probability calculus, then saying that two or more quantum systems are entangled is the same as saying that the outcomes of measurements performed on these systems are statistically correlated (in ways that generally defy causal explanation). But what does it mean in the context of a universal wave function which evolves deterministically at all times?

To find out, let’s consider how the measurement interaction between an apparatus and an object system is usually discussed. Symbolically:

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

Put in words: Initially, the object on which the measurement is made is assumed to be in the state |Ψ⟩, and the apparatus by which the measurement is made is assumed to be in the ready state |A₀⟩. The measurement has a certain number of possible outcomes, labeled by an index k = 1,2,3,.... After the measurement interaction has taken place but before there is an outcome, the combined system is assumed to be in a superposition, which is the sum of the states |A(qₖ)⟩⊗|qₖ⟩ each multiplied by a complex number cₖ. Each individual expression |A(qₖ)⟩⊗|qₖ⟩ signifies that the measured quantity has the value qₖ, and that the apparatus indicates that this is the case.

This story is patently absurd: if each individual expression |A(qₖ)⟩⊗|qₖ⟩ signifies that the measured quantity has the specific value qₖ, then the combined system cannot be in a state that is a sums over these expressions. Quantum-state realists, however, cannot be deterred by such trifles. They take the superposition to signify, either that the measured quantity has all of its possible values at once, or that it has none of them.

(You may also ponder what it means to say that a physical system is “in” a quantum state if a quantum state is a probability algorithm. The only way a quantum state describes a system is by assigning probabilities to the possible outcomes of the measurements to which the system may be subjected.)

The desired result of a measurement process is — one should think — a single outcome. Yet this desire is one that the “elegant piece of mathematics,” which comes “prior to any notion of probability, measurement etc.,” cannot satisfy. To somehow account for the coming into being of an outcome, one needs to supplement the elegant piece of mathematics with an “ill-defined and unattractive mess,” which postulates that a single outcome does in fact come into being, and that the probability for this to happen is the squared absolute value of cₖ.

Hugh Everett III

Hugh Everett the Third’s stroke of genius¹³ was to view the measurement process as terminating, not in the state |A(q⟩⊗|q⟩, but in the superposition of states |A(qₖ)⟩⊗|qₖ⟩. His original intention was to think in terms of relative states — i.e., relative to the apparatus state |A(qₖ)⟩ the object state is |qₖ⟩ — which could be interpreted in conditional terms — i.e., if the apparatus state is |A(qₖ)⟩ then the object state is |qₖ⟩ (or vice versa) — but which, for want of a reality criterion for the antecedent (the “if” part), soon devolved into the MWI, according to which each measurement interaction produces as many worlds as there are possible outcomes. In each of these worlds, the measured observable has one of its possible values, the apparatus indicates that this is the case, and the observer learns that this is the case. So a measurement with N possible outcomes yields not only N actual outcomes but also N hitherto numerically identical observers.

A determination whether Schrödinger’s cat is dead or alive yields two worlds, one in which it is dead and another in which it is alive. Image by Christian Schirm.

If we have two interacting systems one of which is a measurement apparatus, then the apparatus defines a basis. If it is designed to measure the object’s position, then the basis will consist of states in which the object has a definite position, and if it is designed to measure the object’s momentum, then the basis will consist of states in which the object has a definite momentum. But if we have no idea of the criteria that a physical system has to satisfy to qualify as a measurement apparatus (or as an observer-cum-measurement-apparatus), then the preferred basis problem rears its ugly head.

In keeping with Everett’s idea that observers are physical systems within the quantum universe, Murray Gell-Mann and Jim Hartle¹⁴ have postulated the existence of “information gathering and utilization systems” (IGUSes). IGUSes are subsystems of the Universe characterized by the ability to acquire information about their environments, to use the regularities in the acquired information to create and update models of their environments called schemata, and to act on the predictions of their schemata, typically acquiring new information in the process. According to Hartle,¹⁵ an IGUS’s Reality “consists of those features of its schema that it can rely on in the calculation of productive behavior.” It isn’t something “out there independent of human cognition” but more like “what’s in there as a consequence of human cognition and observation.”

Well, duh. We’ve learned this from Kant, we’ve learned it again from Piaget,¹⁶ and it has been shouted from the rooftops by many an evolutionary biologist, neuroscientist, cognitive scientist, psychologist, and philosopher — and all without the benefit of the quantum theory. Where it gets weird is when certain features of our collective schema — specifically, the statistical correlations that hold between the results of certain productive behaviors (those we call “measurements”) — get elevated to an independent reality from which we human IGUSes have emerged, along with our individual and collective schemata and the realities they define: because IGUSes “are emergent features of our quantum Universe along with their schemata that define Realities, a construction or postulate of what’s real in the Universe is not necessary. It’s already there in the schemata of its IGUSes.”

According to Hartle, human IGUSes thus bear a twofold relation to the quantum-mechanical wave function. On the one hand, the wave function encapsulates statistical correlations between certain events within their shared reality. And on the other, the wave function of the universe is the underlying reality from which human IGUSes (along with their shared reality) have emerged. This would be an intriguing circularity of the kind suggested by John Wheeler,¹⁷ were there any intelligible way of cashing out the notion that IGUSes have emerged from the universal wavefunction. But there isn’t.

Wheeler’s vision of the universe viewed as a self-excited circuit.

The wave function formulation of quantum mechanics is actually a special application (the so-called position representation) of a more general formulation of the theory. This more general formulation “knows” nothing of space, time, position, or duration. To make it applicable to the contents of human experience, we must choose between the position representation (suitable for assigning probabilities to the possible outcomes of position measurements) and the momentum representation (suitable for assigning probabilities to the possible outcomes of momentum measurements). Note that both “position” and “momentum” derive their meanings from our experience of space.

So what kind of reality does space possess? Could it be that space enjoys the same measurement-independent existence as the universal wave function supposedly does, notwithstanding the measurement-dependence of the positions and momenta of quantum objects (i.e., the contextuality of these quantities)? While the existence of space, considered as an intrinsically undivided expanse, may not depend on measurements, the same cannot be said of any partition of space into distinct parts. For the extent to which a partitioning of space can be considered objective is limited by the uncertainty principle, as we have seen in this post, and the uncertainty principle involves measurements.

If we partition space into smaller and smaller regions, then, so David Wallace informs us,¹⁸ “we will eventually reach a point where interference between branches ceases to be negligible, but there is no precise point where this occurs.” Saying that interference between branches ceases to be negligible is the same as saying that decoherence ceases to be effective. While the multiplication of worlds in the many-worlds interpretation is often compared to the bifurcation of futures in Jorge Luis Borges’ story “The Garden of Forking Paths,” a better analogy (suggested by Philip Ball¹⁹) “might therefore be something like the gradual separation of shaken salad dressing into layers of oil and vinegar.”

In human experience decoherence is in effect. A preferred basis exists. The instrument pointer points either left or right, and if one looks again, it continues to point in the same direction. On the other hand, if it were possible to put the pointer into a coherent superposition of “left” and “right,” it would almost instantly decohere: looking again, one would find it pointing left or right. Decoherence theory explains this by invoking the environment: the pointer acts on the environment in such a way that we can know the direction in which it points by observing the environment rather than the pointer, and thus without disturbing the pointer.

So here is the question. In human experience the pointer points in a definite direction (either left or right). We know it, and decoherence theory was designed to explain it. How far can the definiteness of the pointer be extended into the quantum domain, which, unlike the pointer, is not directly accessible to our senses? To what extent can it be objectivized? And here is the answer: not “all the way down.” We will eventually reach a point where decoherence ceases to be effective.

If decoherence were effective all the way down, it would be coherent to assume that one of the branches spawned by a measurement-like interaction is the actual branch, while the other branches lost out in the competition for reality. But it is not effective all the way down, though to us it may seem that way. So if a universal wave function is what is ultimately real, and if in addition (and in spite of this) there are physical system, and if for some of them something can seem to be some way, then this seeming to be some way must have emerged, and an essential aspect of emergent items is that they are not definable in the language of the underlying reality. End of story.

But we aren’t finished yet. There is also the small matter that in any relativistic theory simultaneity is relative. Hence there is no unique answer to the question when the splitting of the universal wave function into different branches occurs.

Again, if every possible outcome of a measurement is obtained in some world, where do probabilities come in? Where does “possible” come in? If every possible outcome is the actual outcome in some world, every possible outcome has a probability equal to 1. The concept of probability is meaningful in relation to a single-world interpretation of quantum mechanics. There is a huge, inconclusive literature on the meaning of “probability” in the context of many worlds, and some researchers see it as the issue on which the idea stands or falls. (Trust me, it falls.)

And again, decoherence is predicated on possibility and probability, and therefore it cannot account for the emergence of possibility and probability.. Decoherence theory concerns the apparent or effective loss of coherence of (or of interference between) certain elements of a density operator, and everyone agrees that a density operator is a tool for assigning probabilities to possible measurement outcomes.

Chris Fuchs²⁰ once wrote that

physics as a subject of thought is a dynamic interplay between storytelling and equation writing. Neither one stands alone, not even at the end of the day. But which has the more fatherly role? If you ask me, it’s the storytelling.... An interpretation is powerful if it gives guidance, and I would say the very best interpretation is the one whose story is so powerful it gives rise to the mathematical formalism itself.... The “interpretation” should come first; the mathematics ... should be secondary.

For Yours Truly, the story is about the manifestation of the world. The manifestation of the world is a transition from the One to the Many, in which there are stages, which are typified, respectively, by formless particles, non-visualizable atoms, and partly visualizable molecules. Instead of being considered constituent parts of the manifested world, these quantum objects are best thought of as instrumental in the manifestation of the world.

In the context of this story, how are we to describe the stages intermediate between the One and the Many, the stages at which the distinguishability of objects and the definiteness of properties is as yet incompletely realized? The answer is that whatever is not intrinsically distinct or definite can only be described in terms of probability distributions over objects or properties that are intrinsically distinct or definite, to wit: property-defining instruments and their outcome-indicating properties. What is instrumental in the manifestation of the world can only be described in terms of correlations between events that happen (or could happen) in the manifested world. This goes a long way towards explaining why the general theoretical framework of contemporary physics is a probability calculus, and why the events to which it serves to assign probabilities are measurement outcomes.²¹

1. Quoted by Chris Fuchs in M. Schlosshauer, Elegance and enigma: The Quantum Interviews, p. 71 (Springer, 2011).

2. E.F. Bunn, Many worlds, blog post, June 30, 2014.

3. Quoted by Philip Ball, Why the Many-Worlds Interpretation Has Many Problems, Quanta Magazine, October 18, 2018.

4. M. Redhead, Incompleteness, Nonlocality and Realism, p. 44 (Clarendon, 1987).

5. B.S. DeWitt, Quantum mechanics and reality, Physics Today 23 (9) (1970); reprinted in B.S. DeWitt and N. Graham, The Many-Worlds Interpretation of Quantum Mechanics, pp. 155–165 (Princeton University Press, 1973).

6. D.F. Styer et al., Nine formulations of quantum mechanics, American Journal of Physics 70, 288–297 (2002).

7. W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43, 172–198 (1927); English translation in J.A. Wheeler and W.H. Zurek (Eds.), Quantum Theory and Measurement, pp. 62–84 (Princeton University Press, 1983). “Anschaulich” is one of those German words that defy unambiguous translation. While the article’s title has been variously translated using the expressions “the physical/actual/perceptible content,” the most appropriate translation surely is “the visualizable content.”

8. R.P. Feynman, A.R. Hibbs, and D.F. Styer, Quantum Mechanics and Path Integrals (Dover, 2005).

9. A. Schopenhauer, The World as Will and Idea, Chapter XVII: On Man’s Need Of Metaphysics (Delphi Classics, 2017).

10. A. Zee, Quantum Field Theory in a Nutshell, pp. 53, 57 (Princeton University Press, 2003).

11. S. Goldstein, Bohmian mechanics, in E.N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2017 Edition).

12. W.H. Zurek, Decoherence, einselection and the existential interpretation (the rough guide), Philosophical Transactions of the Royal Society (London) A 356, 1793–1821 (1998).

13. H. Everett III, “The theory of the universal wavefunction,” Princeton University Dissertation, reprinted in B.S. DeWitt and N. Graham, The Many-Worlds Interpretation of Quantum Mechanics, pp. 3–140.

14. M. Gell-Man, and J.B. Hartle, Quantum Mechanics in the Light of Quantum Cosmology, in W. Zurek (Ed.), Complexity, Entropy, and the Physics of Information (Addison Wesley, 1990).

15. J.B. Hartle, What are the Realities, arxiv.org (2021).

16. J. Piaget, La construction du réel chez l’enfant (Delachaux et Niestlé, 1937); The construction of reality in the child (Basic Books, 1971). Jean Piaget’s central aim has been “the analysis of that higher form of adaptation which is scientific thought” (in H.E. Gruber and J.J. Vonèche (Eds.), The Essential Piaget, p. xii (Routledge and Kegan Paul, 1977).

17. J.A. Wheeler, Law without law, in J.A. Wheeler and W.H. Zurek (Eds.), Quantum Theory and Measurement, pp. 182–213 (Princeton University Press, 1983).

18. D. Wallace, Decoherence and ontology (or: How I learned to stop worrying and love FAPP), in S. Saunders, J. Barrett, A. Kent, and D. Wallace (Eds.), Many Worlds? Everett, Quantum Theory, and Reality, pp. 53–72 (Oxford University Press, 2010).

19. P. Ball, Why the Many-Worlds Interpretation Has Many Problems, Quanta Magazine, October 18, 2018.

20. In M. Schlosshauer, Elegance and enigma: The Quantum Interviews, p. 71 (Springer, 2011).

21. As for why the laws of physics have the particular form that they do, see U. Mohrhoff, Quantum mechanics explained, International Journal of Quantum Information 7, 435‒458 (2009); U. Mohrhoff, The World According to Quantum Mechanics: Why the Laws of Physics Make Perfect Sense After All, 2nd Edition, Sec. 20.2 (World Scientific, 2018).