Hyperion and the Decoherence of Worlds - (ChatGPT)

Hyperion and the Decoherence of Worlds

In a previous post, we explored how the Saturnian moon Hyperion—thanks to its chaotic rotation—provides a vivid case study for how classical unpredictability collides with quantum indeterminacy. But what stops us from encountering Hyperion in a state where it's simultaneously in multiple orientations? Why does it always appear to us as a moon tumbling this way or that, but never in some bewildering quantum blur?

The answer lies in decoherence. And in the context of the Many Worlds Interpretation (MWI), decoherence is not a marginal side effect—it’s the mechanism that gives structure and observational content to the branching wavefunction. Without decoherence, the wavefunction evolves but remains unstructured. With decoherence, the wavefunction evolves into effectively distinct classical histories.

1. Superposition: What It Is and Isn’t

A quantum system is said to be in a superposition when its state vector is a linear combination of eigenstates of some observable. For instance:

|ψ⟩ = α |A⟩ + β |B⟩

This is a statement about the system’s state in configuration space, not a claim about what one sees in any individual measurement. Upon observation, the system yields a single outcome—|A⟩ or |B⟩—with probabilities determined by the squared moduli of the coefficients. There is no such thing as “observing a superposition” in a single event.

Superposition is not a visual phenomenon, nor does it correspond to a macroscopic body appearing in multiple classical states at once. Rather, it is a mathematical descriptor of how the system's amplitudes are distributed across its configuration space.

2. Interference and Its Prerequisites

Interference is a physical phenomenon, not a formal one. It arises when different components of a quantum superposition recombine in such a way that their relative phases affect the probabilities of measurement outcomes. Interference can only be detected through statistical regularities in ensembles of measurements—such as the classic fringe patterns in a double-slit experiment with electrons.

A single electron does not interfere. Only an ensemble of identically prepared electrons, evolving through identical dynamics, can display interference effects.

For a system like Hyperion, there is no physical possibility of preparing an ensemble of identically initialised moons in the same quantum state. We cannot rerun the universe multiple times with Hyperion in the same chaotic quantum configuration. Consequently, even if Hyperion's quantum state becomes a superposition of orientations, there is no operational method by which interference effects between those orientations could be revealed.

3. Decoherence and the Emergence of Classicality

Hyperion is constantly interacting with its environment—sunlight, cosmic radiation, thermal photons, gravitational tides. These interactions cause decoherence: a process whereby phase coherence between components of the system’s wavefunction (expressed in a specific basis) is effectively destroyed by entanglement with environmental degrees of freedom.

(Here's a tutorial on the concept of density matrix, before you read further.)

This doesn’t collapse the wavefunction. Instead, it causes the system’s reduced density matrix to become diagonal in a preferred basis—typically one aligned with classical observables such as position or orientation. The states in this basis, called pointer states, are those that remain stable under environmental interactions. Because most interactions are local, the environment couples most strongly to position, making position eigenstates the natural classical basis.

In configuration space terms: decoherence suppresses the off-diagonal elements of the density matrix in the basis of classical configurations. This renders the quantum state a statistical mixture of distinguishable macroscopic states—each evolving independently. Hyperion’s rotational state, initially a quantum object, becomes a set of non-interfering classical alternatives, each entangled with a different environment.

4. Many Worlds Needs Decoherence

In the Many Worlds Interpretation, the universal wavefunction never collapses. All components persist. But without decoherence, there is no way to carve that wavefunction into meaningful branches—no structure of “worlds” that match the experienced classical order.

Decoherence solves this by defining the dynamical conditions under which different components of the wavefunction become mutually inaccessible. It ensures that each branch contains a consistent classical history, unpolluted by phase interference from others. In short: decoherence provides the effective disconnection that makes classical-looking worlds emerge from a fundamentally quantum substrate.

5. What We Don’t See

So when we say “we don’t see macroscopic superpositions,” we do not mean that there are such things which evade our detection, nor that a superposition is something you might glimpse like a ghost. What we mean—more precisely—is this:

We never observe interference effects between macroscopically distinct configurations, because we cannot prepare ensembles of systems like Hyperion in the same quantum state, and because environmental decoherence renders such interference physically inaccessible, even in principle.

The wavefunction may formally contain many such components—different orientations of Hyperion, for instance—but those components no longer interact. Their relative phases are scrambled into environmental degrees of freedom, never to return. In effect, the branches of the wavefunction become autonomous classical narratives.

6. Final Reflection

Hyperion is not both “this way” and “that way” until observed. Its quantum state evolves as a linear combination of possibilities, but decoherence ensures that these possibilities become mutually opaque long before any observation is made. There is no mystery in why we observe it in a single orientation. The mystery lies in how classicality emerges at all from the linear formalism of quantum mechanics—and decoherence is the key that makes that transition intelligible.

So the moon tumbles on, a chaotic fragment of ice and rock, participating silently in the cosmic branching of possibility. And yet, each time we train a telescope on it, we find a world that has—reliably and quietly—made up its mind.

The Chaotic Delocalisation of Hyperion - (ChatGPT)

The Quantum Weirdness of Hyperion

The Saturnian moon Hyperion is often cited as a natural example of quantum chaos, and it has played an interesting role in debates about the quantum-classical boundary—especially regarding decoherence and the role of the observer.

1. Classical Chaos in Hyperion's Rotation

Hyperion is a small, potato-shaped moon of Saturn. It rotates chaotically: that is, its axis of rotation wobbles so much that its orientation in space becomes unpredictable, because:

• It is non-spherical, so torques from Saturn’s gravity are complex and time-dependent.
• It is in an eccentric orbit, and experiences perturbations from other moons (notably Titan, with which it is in a 3:4 orbital resonance).

These features lead to chaotic tumbling: Hyperion’s rotational phase changes in a way that is exponentially sensitive to initial conditions—the hallmark of classical chaos.

This is a classic example of a "three-body problem" in celestial mechanics, where the interactions between Saturn, Titan, and Hyperion make it impossible to predict Hyperion's orientation more than a few months in advance. The system's "Lyapunov time" (the timescale over which small uncertainties in initial conditions become large, unpredictable differences) for Hyperion's rotation is about 30 days.

So far, all of this is standard Newtonian mechanics.

2. Quantum Analogue: Quantum Chaos

In quantum mechanics, the classical concept of chaos doesn’t apply straightforwardly. Schrödinger’s equation is linear and unitary; it doesn’t permit the divergence of trajectories in the classical sense. Nevertheless, we can still ask: what happens to the quantum state—the wavefunction—of a classically chaotic object like Hyperion?

In 1995, physicists Wojciech Zurek and Don Paz explored this question by modelling Hyperion as an isolated quantum system—neglecting environmental interactions for the sake of analysis.

3. The Argument: Delocalisation of the Wavefunction

In quantum mechanics, a system is described by a wavefunction that evolves deterministically over time. For macroscopic bodies like Hyperion, this wavefunction is typically sharply peaked in configuration space (that is, over classical variables like orientation), which allows us to approximate the system as behaving classically.

But for a chaotic system, Zurek showed that the wavefunction becomes delocalised in configuration space—though the underlying spreading is best understood via the system’s evolution in classical phase space.

Specifically:

• The wavefunction describing Hyperion’s rotational degree of freedom spreads exponentially over time.
• Within roughly 20 years (some estimates suggest even less), the quantum state becomes widely spread over many possible orientations.
• However, this spreading doesn’t lead to any observable interference, because the orientations become effectively non-overlapping in configuration space.

This is counterintuitive: the quantum state of Hyperion does not converge toward a classical trajectory but becomes a superposed, delocalised object. Yet we always observe Hyperion in a definite orientation.

4. The Role of Decoherence

The missing element is decoherence.

In reality, Hyperion is not isolated. It constantly interacts with its environment—sunlight, cosmic rays, thermal radiation, and so on. These interactions entangle the moon’s quantum state with vast numbers of environmental degrees of freedom.

Before continuing, you may wish to review this tutorial on the density matrix.

Decoherence causes the off-diagonal terms in Hyperion’s reduced density matrix (in the orientation basis) to rapidly vanish. In effect, the quantum coherence between different orientations becomes inaccessible—even in principle—because the environment has recorded “which orientation is which.”

This does not collapse the wavefunction or select a unique outcome. The total system—Hyperion plus environment—remains in a superposition of different orientation branches, each entangled with a corresponding environmental state. But from the perspective of any internal observer, each branch evolves as if the others do not exist.

So decoherence explains why Hyperion behaves as if it were in a definite orientation, even though its quantum state remains a superposition. It renders the alternatives mutually non-interfering.

5. Philosophical Bite: Many Worlds or Collapse?

What we make of this depends on how we interpret quantum mechanics:

• In the Many Worlds Interpretation (MWI), each decohered orientation corresponds to a separate branch of the universal wavefunction. All still exist. Decoherence tells us where the branches are.
• In collapse models, decoherence helps explain why collapse appears to occur in a specific basis—typically position or orientation—but collapse itself must still be postulated separately.

Either way, decoherence does not explain why one result occurs, but it does explain why we see stable classical behaviour, and why interference between macroscopically distinct outcomes never appears.

6. So, Is Hyperion in a Superposition?

Yes—prior to any observation, Hyperion remains in a quantum superposition of rotational states. Decoherence does not change this. It only ensures that these components are entangled with distinct environmental records and that their interference effects (if any) are physically unobservable.

This is not a superposition that could be revealed by measurement—because there’s no way to prepare an ensemble of Hyperions in the same quantum state, nor to recombine the branches. The superposition is real, but inaccessible.

7. Final Thought: Quantum and Classical Chaos

Hyperion’s case illustrates that:

• Classical chaos amplifies quantum uncertainty by exponentially spreading the wavefunction in configuration space.
• Decoherence suppresses quantum interference, producing the appearance of classical spacetime behaviour.
• The interplay of these two effects is central to understanding how the classical world emerges from quantum mechanics.

It remains one of the clearest examples of where quantum theory touches the macroscopic world—not through spooky paradoxes, but through the subtle mathematics of entanglement, entropy, and environmental indifference. A tumbling moon becomes a case study in the fragility of classicality itself.

In the next post we dig deeper into the role of decoherence in all of this.

What is an Electron Really?

From ChatGPT

This summary, the result of a little prompt-dialogue between ChatGPT and myself, seems as good an account as any other, and is consistent with Matt Strassler's book, "Waves in an Impossible Sea" .

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In the realm of classical physics, an electron is often conceived as a tiny, negatively charged particle orbiting the nucleus of an atom. However, modern physics, particularly Quantum Field Theory (QFT), offers a radically different perspective.

In QFT, an electron is not a point-like particle but rather a quantum excitation of an underlying entity known as the electron field. This field pervades all of space, and what we perceive as an electron is simply a localized disturbance or excitation within this field. This excitation carries certain properties—such as charge, spin, and mass—that we associate with the electron.

The electron's position is not definite until it is measured. Instead, the electron exists in a superposition of possible states, each corresponding to a different location. This superposition means that the electron doesn’t have a single, well-defined position but rather a range of probabilities (quantum amplitudes really) of being found in various locations.

When we incorporate the Many-Worlds Interpretation (MWI) of quantum mechanics, this notion takes on an even more intriguing aspect. According to the MWI, each possible position of the electron corresponds to a different “world” or branch of reality. 

In one world, the electron might be detected at a particular point, while in another world, it appears somewhere else. These different worlds coexist in a vast multiverse which constitutes reality, and the electron’s delocalization can be understood as it existing in multiple worlds simultaneously, with each world realizing a different outcome of its position. Note that in each specific world, the electron is still an 'excitation of the electron field', a 'wavicle' according to Strassler.

This is a difficult ontology to imagine or believe, but it's the best we can do today.

"Something Deeply Hidden" - some thoughts

Amazon link

The title is a quote from Einstein, revealed at the end of the book. Carroll is on Einstein's side in the great Einstein-Bohr debates on the meaning and completeness of quantum theory. Sort of.

Carroll writes well, mostly. He's a fluent author and has the surpassing virtue of conceptual clarity. This is a book about concepts: conceptual analysis and what the equations are telling us. This works well at the beginning where he covers the material in a typical undergraduate course, falters later when he discusses quantum field theory, and stutters at the end, where he sketches the research program of deriving spacetime from Hilbert space.

Who is this book for? Not the layperson - it's too unfamiliar, too conceptually abstract and dense. It's for people who know quantum mechanics - the mathematics and the calculations - people who understand the machinery but, like everyone else, struggle to understand what it's telling us about the universe itself. The math he doesn't mention underpins the concepts he's keen to articulate and talk around.

He's persuasive on the many-worlds interpretation, mostly because it seems plausible to start with the wave-function of the universe-as-a-holistic-entity. He's at pains to point out that the MWI is really no more than that: austere quantum theory.

He's excellent on how decoherence works, giving a clear conceptual overview. You have to have covered superpositions and entangled states in your QM course - and to have thought about it - to really grasp what he's saying, though. He writes like it's pretty clear but it isn't.

Should one walk away from this book an Everettian?  Carroll makes a very strong case for this over all the other interpretations - his critiques sometimes feel like shooting fish in a barrel. He's convincing that we should conceptualise QM as if the MWI were true.

But is the universe really a state vector in a high-dimensional Hilbert space? With our familiar classical-looking spacetime something emergent? A reality emergent from entropic-entanglement (hence locality and metric) and then local sampling of that so-structured Hilbert space?

No-one knows. It might be nice .. but the research isn't in.

And then there are the lacunae. The genesis of the standard model is nowhere mentioned. It may all be quantum fields - but how did we get separate quantum fields for all the different fermions and bosons?

My conclusion: every physics undergraduate should read this book. All the questions they have about how quantum theory is put together (the map of the territory in fact) and how it all relates to the universe we experience are honestly discussed here. They won't find those issues addressed in class or in their textbooks.

They will also appreciate how much we still don't understand about the fundamentals of the theory and about reality itself. Quantum gravity is still, most likely, the holy grail. But in the absence of meaningful experiments (the collider plans don't really help) solid progress is likely to remain stalled.

Reality and the MWI

From "Many Worlds? An Introduction" by Simon Saunders.

“As Popper once said, physics has always been in crisis, but there was a special kind of crisis that set in with quantum mechanics. For despite all its obvious empirical success and fecundity, the theory was based on rules or prescriptions that seemed inherently contradictory. There never was any real agreement on these matters among the founding fathers of the theory.
...
“In what sense are the rules of quantum mechanics contradictory? They break down into two parts. One is the unitary formalism, notably the Schrödinger equation, governing the evolution of the quantum state. It is deterministic and encodes spacetime and dynamical symmetries.

“Whether for a particle system or a system of fields, the Schrödinger equation is linear: the sum of two solutions to the equation is also a solution (the superposition principle). This gives the solution space of the Schrödinger equation the structure of a vector space (Hilbert space).

“However, there are also rules for another kind of dynamical evolution for the state, which is - well, none of the above. These rules govern the collapse of the wavefunction. They are indeterministic and non-linear, respecting none of the spacetime or dynamical symmetries. And unlike the unitary evolution, there is no obvious route to investigating the collapse process empirically.

“Understanding state collapse, and its relationship to the unitary formalism, is the measurement problem of quantum mechanics. There are other conceptual questions in physics, but few if any of them are genuinely paradoxical. None, for their depth, breadth, and longevity, can hold a candle to the measurement problem.

“Why not say that the collapse is simply irreducible, ‘the quantum jump’, something primitive, inevitable in a theory which is fundamentally a theory of chance? Because it isn’t only the collapse process itself that is under-specified: the time of the collapse, within relatively wide limits, is undefined, and the criteria for the kind of collapse, linking the set of possible outcomes of the experiment to the wavefunction, are strange.

“They either refer to another theory entirely - classical mechanics - or worse, they refer to our ‘intentions’, to the ‘purpose’ of the experiment.

“They are the measurement postulates - (‘probability postulates’ would be better, as this is the only place where probabilities enter into quantum mechanics). One is the Born rule, assigning probabilities (as determined by the quantum state) to macroscopic outcomes; the other is the projection postulate, assigning a new microscopic state to the system measured, depending on the macroscopic outcome.

“True, the latter is only needed when the measurement apparatus is functioning as a state-preparation device, but there is no doubt that something happens to the microscopic system on triggering a macroscopic outcome.

“Whether or not the projection postulate is needed in a particular experiment, the Born rule is essential. It provides the link between the possible macroscopic outcomes and the antecedent state of the microscopic system. As such it is usually specified by giving a choice of vector basis - a set of orthogonal unit vectors in the state space - whereupon the state is written as a superposition of these. The modulus square of the amplitude of each term in the superposition, thus defined, is the probability of the associated macroscopic outcome.

“But what dictates the choice of basis? What determines the time at which this outcome happens? How does the measurement apparatus interact with the microscopic system to produce these effects? From the point of view of the realist the answer seems obvious. The apparatus itself should be modelled in quantum mechanics, then its interaction with the microscopic system can be studied dynamically. But if this description is entirely quantum mechanical, if the dynamics is unitary, it is deterministic. Probabilities only enter the conventional theory explicitly with the measurement postulates. The straightforwardly physicalistic strategy seems bound to fail. How are realists to make sense of this?

“The various solutions that have been proposed down the years run into scores, but they fall into two broadly recognizable classes. One concludes that the wavefunction describes not the microscopic system itself, but our knowledge of it, or the information we have available of it (perhaps ‘ideal’ or ‘maximal’ knowledge or information). No wonder modelling the apparatus in the wavefunction is no solution: that only shifts the problem further back, ultimately to ‘the observer’ and to questions about the mind, or consciousness, or information - all ultimately philosophical questions.

“Anti-realists welcome this conclusion; according to them, we neglect our special status as the knowing subject at our peril. But from a realist point of view this just leaves open the question of what the goings-on at the microscopic level, thus revealed, actually are. By all means constrain the spatiotemporal description (by the uncertainty relations or information-theoretic analogues), but still some spatiotemporal description must be found, down to the length-scales of cells and complex molecules at least, even if not all the way to atomic processes.

“That leads to the demand for equations for variables that do not involve the wavefunction, or, if none is to be had in quantum mechanics, to something entirely new, glimpsed hitherto only with regard to its statistical behaviour. This was essentially Einstein’s settled view on the matter.

“The only other serious alternative (to realists) is quantum state realism, the view that the quantum state is physically real, changing in time according to the unitary equations and, somehow, also in accordance with the measurement postulates.

“How so? Here differences in views set in. Some advocate that the Schrödinger equation itself must be changed (so as to give, in the right circumstances, collapse as a fundamental process). They are for a collapse theory.

“Others argue that the Schrödinger equation can be left alone if only it is supplemented by additional equations, governing ‘hidden’ variables. These, despite their name, constitute the real ontology, the stuff of tables and chairs and so forth, but their behaviour is governed by the wavefunction. This is the pilot-wave theory.

“Collapse in a theory like this is only ‘effective’, as reflecting the sudden irrelevance (in the right circumstances) of some part of the wavefunction in its influence on these variables. And once irrelevant in this way, always irrelevant: such parts of the wavefunction can simply be discarded. This explains the appearance of collapse.

“But for others again, no such additional variables are needed. The collapse is indeed only ‘effective’, but that reflects, not a change in the influence of one part of the quantum state on some hidden or ‘real’ ontology, but rather the change in dynamical influence of one part of the wavefunction over another - the decoherence of one part from the other.

“The result is a branching structure to the wavefunction, and again, collapse only in a phenomenological, effective sense. But then, if our world is just one of these branches, all these branches must be worlds. Thus the many worlds theory - worlds not spatially, but dynamically separated.”

Saunders' introductory chapter from the book, "Many Worlds?" underlines the central puzzle of quantum mechanics. What would reality have to be like to make the theory of quantum mechanics so incredibly accurate?

Realists driven to the 'Many Worlds Interpretation' can still make no sense of it (Sean Carroll is a consistent defender, though). As Saunders observes on page 20,

“How does talk of macroscopic objects so much as get off the ground? What is the deep-down ontology in the Everett interpretation? It can’t just be wavefunction [...]; it is simply unintelligible to hold that a function on a high-dimensional space represents something physically real, unless and until we are told what it is a function of  - of what inhabits that space, what the elements of the function’s domain are.

“If they are particle configurations, then there had better be particle configurations, in which case not only the wavefunction is real.”

And so I have bought "The Many Worlds of Hugh Everett III: Multiple Universes, Mutual Assured Destruction, and the Meltdown of a Nuclear Family" by Peter Byrne.

Some really good-bad books

Fay Weldon writes one of my favourite classifications:

"In the nineties, moved by the new GOO (‘GOOd read’) classification in Camden public libraries, I devised my own classification scheme as follows.

Good-good books – the best of contemporary literary novels, plus classics which were best-sellers in their day and have withstood the passage of time: all engaging with the intellect, if ‘difficult’.

Bad-good books – pretentious and dreadfully boring, yet taken seriously by the occasional reviewer (usually a friend of the author) and funded by the Arts Council.

Good-bad books – intensely readable, unpretentious and seldom reviewed.

Bad-bad books – worthy only to be hurled into the corner or dropped in the bath."

Sometimes you just need to switch off your intellect and enjoy a piece of page-turning escapism. If you are a science-fiction fan, you could do considerably worse than turn to B. V. Larson, for example his Undying Mercenaries series of exceedingly good-bad books.

The first volume ...

Larson also writes insightfully about the craft of successful, self-published writing ("Advice Concerning the Self-Publishing Game ").

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Steve Hsu has an interesting post on the interpretation of quantum mechanics (PDF here). Basically he does a good job of explaining where the many worlds interpretation comes from, and how decoherence plays a critical role in 'separating branches'. He then sheds a really clear light on the issue of 'where do the Born Rule probabilities come from?' concluding this is still an unresolved problem.

As a bonus, the comments feature the infamous Luboš Motl who is coaxed to clearly explain his own views on quantum reality. As a hard-line 'Copenhagenist', Motl appears to believe that reality is both ill-defined and lacks objective (classical-ish) reality in the absence of sentient observers - a highly counter-intuitive view for sure! Here is how he explains things:

"The conceptually right to describe a world without sentient beings is that an unspecified and unknown initial wave function evolves unitarily according to Schrödinger's equation and never collapses because it's only measurements that may collapse and there are none in your theory. The complete "diffusion" of the wave function (into the linear superposition of dead and alive cats and all objects, small and big, in the most general superpositions of all conceivable states) may be said to be a problem - but another problem is that the initial state is totally unknown, too.

"It makes no sense to say that the initial wave function is a particular thing because one may only say that the wave function is a particular thing [if] something is [a] measurement - if a sentient being becomes aware of the result of some measurement. This is not happening in a universe without sentient beings. So there's no specific science to discuss in a universe without sentient beings at all. The laws may still be the same as they are in our world but they won't be applied in any particular situation because there are no particular situations or particular special wave functions in a world where no one ever measures anything.

"Einstein asked whether there is any Moon over there if no one looks. In practice, classical physics is a good enough approximation, so one may assume that the Moon is pretty much there even before observers look etc. But conceptually, if you care about similar objects for which the quantum effects are strong, the right answer is that the Moon just isn't at any particular location and has no other particular properties if no one looks. The wave function isn't a real object of any type. Its amplitudes can't be measured in a single repetition of the situation. It is only a template storing information allowing to predict probabilities of things that actually can be measured - the observables."

This is all accessible to anyone who has taken and understood QM at an undergraduate level.

Many Worlds: the "loss of coherence" and decoherence

What problem does the "many worlds interpretation" try to solve? The core idea is captured by the infamous Schrödinger's cat thought-experiment. The cat ends up in a superposition:

α |cat-alive> + β |cat-dead>           (where α, β are complex amplitudes)

but in reality we never observe such a superposition. What we see is either that the Geiger counter has triggered, releasing the poison and the cat is dead or there was no particle-emission and the cat remains alive. Making the act of measurement explicit, the observer becomes entangled with the cat-box on observation and joins the superposition thus (with updated amplitudes α' and β') :

α'∣Live cat> ⊗ ∣Observer sees live cat> + β'∣Dead cat>  ⊗ ∣Observer sees dead cat>.

The overall situation is still a superposition, but an Everettian would say that since the observer has 'split',  each 'copy' doesn't see a superposition but just one outcome. If world-splitting is not your thing, then you have to postulate (as an extra axiom) that 'measurement' somehow causes the superposition to collapse to one or the other outcome according to the probabilities |α'|² and |β'|².

We now focus more precisely on how the process of 'measurement' destroys superpositions - a topic called 'decoherence'. The formal treatment of decoherence involves graduate-level concepts and is formidably inaccessible, while analogies such as 'phase information leaking into the environment' are unhelpful at best. I will say more about a 'simplest possible model' another day.

However, something can be said even at undergraduate level, and for this we are indebted to non-Everettian Eric L. Michelsen and his excellent "Quirky Quantum Concepts: Physical, Conceptual, Geometric, and Pictorial Physics that Didn't Fit in Your Textbook (Undergraduate Lecture Notes in Physics)". This is available as a PDF but in a fit of guilt I bought it. An extract, [somewhat annotated] follows (page 51 and following).

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"In theoretical QM, we usually focus on perfect systems, and pure states. We frequently say that a measurement “collapses” the quantum state vector to one agreeing with the measurement, and this is often a useful simplification of the measurement process. However, in practice, the measurement process is more complicated than that, because most measuring equipment, and all observers, are macroscopic. The “decohered” state is the norm; you must work hard to achieve even an approximately pure entangled state.

We show here that elementary QM can explain some of the features of real measurements, however, the full explanation of decoherence is beyond our scope. (The term “decoherence” has a specific meaning: the process of a system becoming entangled with its environment in irreversible ways, resulting in the loss of a consistent phase relationship between components of the system state. We therefore use the more general term “loss of coherence” for both decoherence and other processes.)

Most macroscopic measurements do not show quantum interference [as in the two-slit experiment]. Why not? One reason is that macroscopic bodies suffer unknowable, and unrepeatable energy interactions, i.e. they gain or lose an unknowable amount of energy due to uncontrollable interactions with their environments. In other words, they are subject to simple “noise.” This results in the loss of a consistent phase relationship between components of a superposition state. We discuss below how such a loss of consistent phase leads to classical probabilities.

Let us walk through a plausible measurement, and consider the elementary quantum mechanics involved. [The system pictured below shows the famous Stern-Gerlach experiment, which first demonstrated the quantization of angular momentum.]

Suppose we start with a particle which can be in either of two states, |s₁> or |s₂>, such as polarization (horizontal or vertical), or spin (up or down). A general particle state is then:

|ψ> = a|s₁> + b|s₂>     where a,b are complex coefficients and |s₁>,  |s₂>  are basis states.

This is called a coherent superposition, because a and b have definite phases. (This is in contrast to a mixed state or incoherent mixture, where a and b have unknown phases.) All that is required for loss of coherence is for the relative phases of a and b to become unknown. For simplicity, we take |s₁> and |s₂> to be energy eigenstates, and the particle is spread throughout our measurement system [i.e. it is in a spatial superposition].

According to the Schrödinger equation, every state time-evolves with a complex phase determined by its energy, then our 2-state system time evolves according to:

|ψ(t)> = ae^-iE₁t/ℏ|s₁> + be^-iE₂t/ℏ|s₂>.

Since the energies E₁ and E₂ are quantized, the complex phases multiplying |s₁> and |s₂> maintain a precise (aka coherent) relationship, though the relative phase varies with time.

When we measure the particle state, the state of the measuring device becomes entangled with the measured particle. Let |M1> and |M2> be states of the whole measuring system in which either detector 1 detected the particle, or detector 2. If we look directly at the indicator lights, we will observe only state 1 or state 2, but never both. This means |M1> and |M2> are orthogonal. As the measuring system first detects the particle, the combined state of the particle/measuring-device starts out as a coherent superposition: [this is the same as the Schrödinger's cat case above]

|Ψ> = c|M1>|s₁> + d|M2>|s₂>       where c, d are complex coefficients.

The combined system time evolves according to its new energies:

|Ψ(t)> = ce^{-i(E₁ + E_M1)t/ℏ}|M1>|s₁> + de^{-i(E₂ + E_M2)t/ℏ}|M2>|s₂>

If the energies of the two measuring device states fluctuate even a tiny bit, the two components of the superposition will rapidly wander into an unknown phase relation. They will lose coherence.

Every macroscopic system suffers unrepeatable and unknowable energy fluctuations due to its environment.

We estimate a typical coherence loss rate shortly.

[So what does this loss of phase coherence mean in practice?]. Let us examine the effects of various kinds of energy transfers between a system and its environment. In our two-path experiment, [I think he means that this is a variant experiment where we don't 'look at' (i.e. measure) the indicator lights 1 and 2 so allowing the interference pattern to emerge on the screen in the figure to the right] the interference pattern is built up over many trials, by recording detections on film. Now suppose one path suffers an energy transfer to/from its environment before recombining and interfering. There are four possibilities:

1. The energy transfer is knowable and repeatable. Then one can predict and see an interference pattern in the usual way.
2. The energy transfer is unknowable, but repeatable. Then we can record an interference pattern, and from it, determine the relative phases of the two paths (mod 2π), and therefore the relative energies (mod 2πħ/t) from (ΔE/ℏ)t.
3. The energy transfer is knowable for each trial, but not repeatable. Essentially, each trial has its own position for the interference pattern. One can then divide the detection region into intervals of probability calculated for each trial, and then show consistency with QM predictions, but contrary to classical probability.
4. The energy transfer is unknowable and unrepeatable. Then there will be no interference pattern, and repeated trials do not allow us to measure any quantum effects, since the phase is unknown on each trial. Therefore, the measurements are equivalent to classical probabilities: it is as if a single path was chosen randomly, and we simply don’t know which path it was.

This fourth condition, of unknowable and unrepeatable energy transfer, causes loss of coherence, the randomization of phase of components of a superposition. Loss of coherence makes measurements look like the system behaves according to classical probabilities, with no “wave” effects. Loss of coherence destroys the interference pattern when we try to measure through “which slit” a particle passes. Full loss of coherence leads to classical probabilities.

Our example process leading to loss of coherence follows directly from the Schrödinger equation and unknown energy transfers. There is no need to invoke any “spooky” quantum effects.

Note that even accounting for loss of coherence, quantum theory still requires the axiom of collapse of the wave-function upon observation. When a particle’s wave splits, then passes through both detector 1 and detector 2, and then loses coherence because of entanglement with a macroscopic measuring device, the system is still left in a superposition of both slits:

|Ψ(t_after)> = f|M1>|s₁> + g|M2>|s₂>

we just don’t know f or g. We can’t generate an interference pattern from multiple trials, because each trial has a different phase relation between f and g, putting the peaks and valleys of any hoped-for interference pattern in a random place on each trial. These shifts average over many trials to a uniform distribution. Nonetheless, each trial evolves in time by the Schrödinger equation, which still leaves the system in a superposition. Once we “see” the result, however, the unobserved component of the wave-function disappears, i.e. the wave-function collapses.

Collapse of the wave-function is outside the scope of the Schrödinger equation, but within the scope of QM, because collapse is a part of QM theory. It is one of our axioms. Some references confuse this issue: they try to avoid assuming such a collapse as an axiom, but cannot derive it from other axioms. From this, they conclude that QM is “incomplete.” In fact, what they have shown is that the axiom of collapse completes QM.

Note that once the measuring system fully loses coherence, we could just as well say that the wavefunction has then collapsed, because from then on the system follows classical probabilities (equivalent to a collapsed, but unknown, wave-function). However, we now show that a binary model of “collapse or not” cannot explain partial coherence.

Partial coherence: What if we start with a microscopic system but replace our microscopic atoms with mesoscopic things: bigger than microscopic, but smaller than macroscopic? Mesoscopic things might be a few hundred atoms. These are big enough to lose coherence much faster than single atoms, but still slowly enough that some amount of interference is observed. However, the interference pattern is weaker: the troughs are not as low, and the peaks are not as high. A superposition leading to a weak interference pattern is called partially coherent. We describe partial coherence in more detail in section 8.4. The simple model that the wave-function either collapsed or didn’t cannot describe the phenomenon of partial coherence.

The larger the mesoscopic system, the more uncontrollable interactions it has with its environment, the faster it loses coherence, and the less visible is any resulting interference pattern. We can estimate the time-scale of coherence loss from our example energy fluctuations as follows: a single 10 μm infrared photon is often radiated at room temperature. It has an energy of ~0.1 eV = 1.6 x 10^–20 J. This corresponds to ω = E/ħ ~ 2 x 10¹⁴ rad/s. When the phase of the resulting system has shifted by an unknowable amount > ~2π, we can say the system has completely lost coherence. At this ω, that takes ~ 4 x 10^–14 s. In other words, thermal radiation of a single IR photon causes complete loss of coherence in about 40 femtoseconds. In practice, other effects cause macroscopic systems to lose coherence in dramatically shorter times.

Summary: A measurement entangles a measuring device with the measured system. The entangled state of device and system time-evolves according to the Schrödinger equation. Macroscopic devices lose coherence, due to interactions with the environment. Lack of coherence prevents any interference pattern within the system. Therefore, measurement by a macroscopic device produces subsequent results that are classical, as if the system collapsed into a definite state upon measurement, but observers only “see” which state when they look at the measuring device. Any observation by a person is necessarily macroscopic, because people are big. Such an observation collapses the (incoherent) device/system/world state to that observed. Quantum interference can only be seen if it occurs before any entanglement with a macroscopic system (and therefore before any loss of coherence in the system).

The model of “collapse of the wave-function” is a binary concept: either the wave-function collapses or it doesn't. Such a model cannot account for the phenomenon of partial coherence. Loss of coherence is a continuous process, taking a fully coherent state through less and less partially coherent states and eventually to incoherent (aka “mixed”) states. Continuous loss of coherence fully explains partial coherence and the varying visibility of interference patterns.

Some quantum effects, such as the spectrum of atoms, do not rely on interference, and are therefore macroscopically observable. In fact, measurement of such effects led to the development of QM."