The (Measurement Problem) Problem

The measurement problem is perhaps the best-known puzzle in quantum mechanics. A vast literature addresses this problem without ever resolving it. The measurement problem is also responsible for one of the most recognisable symbols of quantum mechanics, i.e. Schrödinger's cat.

The proposed solutions to this problem—the so-called "interpretations of quantum mechanics"—keep the mathematics as it is, and redefine reality to make sense of the maths. This procedure is problematic for several reasons: it inverts the scientific method of describing reality with mathematics (and it reifies mathematics), it's not testable, the different realities proposed are all mutually exclusive, and in the final analysis, changing reality to validate mathematics has no explanatory power.

Worse, the competing "interpretations" act more like ideologies in that they attract adherents who take sides and proselytise. I refer to these as "ideological" or "programmatic" interpretations because they are accompanied by a theoretical agenda. New ideological interpretations appear all the time, but none ever disappears. My intention here is not to spend time debating the merits of the ideological interpretations, all of which seem to me to be fundamentally unscientific, but to show that they are merely the froth on a much deeper set of philosophical issues.

The problem I wish to focus on is that the measurement problem does not, in fact, arise in the context of making measurements. Rather, it is a theoretical problem that arises from attempting to physically interpret a highly abstract mathematical procedure with no physical analogue.

In this essay, I will argue that the mathematical tail is wagging the metaphysical dog in quantum physics. Nature is not a mathematician. A problem arising from a mathematical procedure is being foisted onto reality rather than being dealt with in the mathematics.

Like many paradoxes, the associated difficulties seem to reside in how the problem is framed. So I want to begin this essay by attempting to accurately frame the situation in which the problem occurs.

What is the "Measurement Problem"?

The measurement problem hinges on precise distinctions between:

• What the formalism says,
• What the interpretations claim, and
• What the observations show.

So we need to be as clear as possible about each of these.

The Formalism.

By the 1920s, a combination of empirical results and theoretical breakthroughs—such as Max Planck's description of blackbody radiation and Albert Einstein's description of the photoelectric effect—had shown that existing theories were insufficient to explain atoms.

Light had started off as vague "rays" and, under the influence of James Clerk Maxwell, had become electromagnetic waves propagating in space. Then light was observed to be quantised and to have particle-like properties. In 1923, Louis de Broglie proposed that all matter particles are also wave-like and have a "wavelength". At the time, this crystallised as the wave-particle duality postulate: matter is both a particle and a wave. As we will see, the tension implied by this duality continued to be influential, even after the advent of quantum field theory (QFT), which proposed that all matter is fundamentally excitations in fields.

Atomic spectra were known to consist of a handful of frequencies rather than a continuous rainbow, and some progress had been made on clarifying the relationships between the frequencies. Both Werner Heisenberg and Erwin Schrödinger were trying to explain atomic spectra in the wake of Planck and Einstein. Schrödinger aimed at a realist explanation in terms of wave mechanics using a Hamiltonian, rather than a Newtonian, formulation. He was inspired by the Hamilton-Jacobi equation because it draws a formal analogy between classical mechanics and wave optics. This suggested that classical trajectories might arise from some underlying wave phenomenon.

The result was the famous Schrödinger equation, which describes a "wavefunction", 𝜓. The equation can be written in many ways, though the most familiar forms came later. Where Schrödinger's formalism relied on differential equations, Heisenberg used matrices to describe virtual oscillations. Schrödinger himself proved that the two approaches were equivalent. Max Born and Pascual Jordan developed the matrix formulation into a kind of algebra with "functions" and "operators". An operator is a rule that can be applied to a function and returns another function.

There's a disconnect between what Schrödinger set out to do and how quantum mechanics turned out, which he was never reconciled with. Schrödinger was seeking a realist theory and wanted to treat 𝜓 as a physical wave in space. Heisenberg was influenced by positivism and wanted to explain phenomena purely in terms of what could be observed. His approach treated atomic spectra lines as resulting from a virtual oscillator. These early ideas were soon replaced by the idea that 𝜓 is a vector in Hilbert space (named after the mathematician David Hilbert).

The modern formalism of quantum mechanics has two modes: evolution over time and extracting information. To make for a simpler narrative, I'm going to focus on what it tells us about the position of a particle.

Other things being equal, the wavefunction 𝜓 evolves smoothly and continuously in time according to the time-dependent Schrödinger equation. We can rewrite the Schrödinger equation to give us 𝜓 as a function of position: 𝜓(x). For Schrödinger, x was simply a coordinate and 𝜓(x) represented the amplitude of the wave at that coordinate. In his view, the modulus squared of the wavefunction at that point |𝜓(x)|² reflected the charge density.

However, Max Born had other ideas. As described by John Earman (2022):

Born rejected Schrödinger's proposal because the spreading of the wave function seemed incompatible with the corpuscular [i.e. particle] nature of the electron, of which Born was convinced by the results of scattering experiments.

Born introduced a further level of abstraction. In Born’s view, 𝜓 does not describe a physical wave. It is an abstract mathematical function. The quantity |𝜓(x)|² gives the probability density for the electron, which tells us how probability is distributed over space. To find the probability of detecting the particle in a given range of x, one must integrate |𝜓(x)|² over that range. This is called "applying the Born rule". (Note that the procedure is slightly different for discrete quantities like spin).

Paul Dirac and John von Neumann added yet more layers of abstraction. Dirac invented his own operator algebra. Von Neumann recast 𝜓 in terms of vectors in an infinite-dimensional space—a Hilbert space. The Dirac-von Neumann formalism is how quantum mechanics is taught.

We can also retrieve another kind of position information, which is the expectation value for x. This is the long-term average position of the particle. This involves applying the position operator x̂ to the 𝜓, producing another function.

In the standard textbook formulation of quantum mechanics, 𝜓 represents a vector in an abstract vector space, and 𝜓(x) is a function representing that vector in terms of positions in space. It evolves smoothly and continuously; then we apply the Born rule, ∫ |𝜓(x)|² dx, to extract probabilities of particles appearing in defined regions of space. The physical interpretation of this formalism presents seemingly intractable problems.

Interpretations

Having attempted to present the formalism in neutral terms, we could simply stop at this point. In practice, however, we are compelled by inclination and convention to say what the theory means. And this involves saying what the mathematics means in physical terms. As noted above, by "interpretations" I'm not referring to ideological views like "Copenhagen" or "Many Worlds". Rather, I'm trying to draw attention to the contradictory assumptions that underlie these ideological views.

Traditionally, scientists aim to explain some phenomenon P in terms of its causes. If A causes P, then A explains P. In the abstract mathematics of vectors in Hilbert space, causality is not defined. However, rather than this fact shutting down speculation, it has enabled a proliferation of competing interpretive frameworks, many of which are treated as axiomatic.

There are numerous ideas about what, if anything, 𝜓 represents, and these include some very fundamental tensions: particle versus wave, realist versus anti-realist; ontic versus epistemic, observer-independent versus observer-dependent.

As noted, the wave-particle duality postulate remains at the heart of quantum mechanics. Waves and particles are not simply different kinds of entities. The mathematics used to describe them is also fundamentally different. Particles have qualities like position and momentum, while waves have properties like displacement, wavelength, and period. Quantum mechanics purports to replace waves and particles with a single unified mathematics, but what this means in physical terms remains unclear. And in the end, we still use the wavefunction to extract information on position and momentum (though not at the same time).

Quantum field theory reformulates particles as excitations of fields, but the legacy of dualistic terminology and beliefs persists, especially in the use of the wavefunction to compute particle positions (which is partly why I chose the example of position above).

The realist view is that reality extends to the atomic world, even though we cannot directly experience it (by "directly" here, I mean without apparatus). In this view, atoms, electrons, and photons are assumed to be real entities with real properties that we can either measure or infer. We are so used to science being committed to realism that it may seem strange to insist on this. However, the considered opinion is that classical physics fails to describe the atomic and subatomic scales, and this opened the door to antirealism.

The antirealist view is that if we cannot see or measure something, then it makes little sense to assert that it is real. Non-observable entities, such as electrons, are merely mathematical objects that we use for making calculations to account for experimental observations. Quantum theory emerged in the early twentieth century when positivism was having its brief moment in the sun. Positivists were against metaphysics on principle (based on a rather naive reading of Kant). Where something could not be observed, they argued that it should not have a place in our descriptions of reality. And in the 1920s, this included atoms, electrons, and protons. Heisenberg initially set out to eliminate anything unobservable from his formulation of quantum mechanics.

As noted Schrödinger was a realist. As were Planck and Einstein, although Einstein had changed what "real" meant. Historian Mara Beller (1996) notes that Heisenberg, Pauli, Born, and Bohr all denied being positivists, but were given to making strongly positivist pronouncements in their work.

We can also note that, on the whole, scientific realism is strongly associated with a commitment to metaphysical reductionism. Reductionism holds that only substances are real. Substance is described as "fundamental". In this view, structures and systems are simply adventitious phenomena. Thus, physicists tend to view the "particles" of the standard model (or the fields of QFT) to be the ultimate building blocks, the foundations of reality. Although, ironically, the fields of QFT are not observable, even in principle.

Another level of debate occurs over the ontological status of the wavefunction 𝜓. How one thinks about this issue is naturally influenced by one's pre-existing view about realism versus antirealism. For example, anti-realists are not inclined to see 𝜓 as real; realists are so inclined.

A major issue with such views is that assumptions can concatenate: if A, then B; if B, then C; if C, then D; and so on. For example, if I assume a realist stance, then it may seem natural to conclude that the wave function is real. And if the wavefunction is real, then something real must happen to it when we measure the location of a particle. And so on.

Yet realism is not a given; it is a metaphysical stance—one that rests on its own chain of assumptions.

Most of us fall one way or the other in the realism versus antirealism divide, and this limits what views are available in other contexts. Views in which the wavefunction is a real entity are called 𝜓-ontic, and views in which it is only concerned with knowledge are called 𝜓-epistemic (I mentioned this in my essay about Probability).

A 𝜓-ontic view holds that the wavefunction is a real entity, which is obviously also a realist view. Some high-profile physicists, notably Sean Carroll, now routinely state "the wavefunction is real" without qualification. Such views are colloquially referred to as 𝜓-ontologies. For Carroll, this move is a prelude to introducing Hugh Everett's Many Worlds ideology, in which the reality of 𝜓 is axiomatic.

A 𝜓-epistemic view holds that the wave function only represents our knowledge of the system. The issue of what real process creates atomic phenomena is incidental. The problem here is that, e.g. 𝜓(x) on its own provides us no knowledge of position in terms of space or time. To get knowledge out, we have to apply the Born rule, and even then, we don't obtain knowledge about position, but only about probabilities.

None of these views resolves all the issues. For example, 𝜓(x) appears to represent all possible locations without distinguishing between them. And since this situation arises prior to applying Born’s rule, it is typically interpreted as describing the system "before a measurement is made". However, if all positions are represented and none is distinguished—let alone selected—then the interpretation itself seems to require further interpretation.

If each interpretation needs another to explain it, we’re not solving the problem; we’re just piling words on top of mathematics. And this process can lead to infinite regress. Adding interpretive layers does nothing to resolve more fundamental ambiguities and contradictions, even when it obscures them.

The next dichotomy brings us to the heart of the measurement problem.

On one view, by far the dominant view, the observer plays a central role in causation at the atomic level. Readers are likely familiar with the phrase that observation collapses the wavefunction, even if they are vague about what it means. In my essay Observations and Superpositions, I refuted the idea that observation can occur prior to the outcome of the events we are observing. Still, this issue continues to arise and cause confusion.

However, the term "observer" is a misnomer. What "observation" means in this context is that a particle is detected by a particle detector. The "observer" is never a human being—we cannot see particles. The "observer" is a Geiger counter, a photomultiplier, a photographic plate, etc.

Eugene Wigner went one step further and postulated that "consciousness" collapses the wavefunction, and this idea has proved irresistible to mystics. "Consciousness" is in scare quotes because it is undefined by Wigner, and decades later, there is still no standard definition of this often reified abstraction. Physicists, by and large, view Wigner's theory as an embarrassing moment in an otherwise distinguished career and sweep it under the rug. The idea that "consciousness" is involved in deciding outcomes is simply nonsensical.

It is also possible to take the view that quantum phenomena are not affected by observation. That the "collapse of the wave function" is, for example, random.

The Measurement Problem

The "measurement problem" can be viewed from a plurality of philosophical standpoints, and thus, there are many ideas about what it connotes. However, the basic problem is that to extract a probability, we have to switch from evolving the wavefunction in time to applying the Born rule.

This leads to a number of unresolved ontological questions, some of which are artificial. The ontological status of the wavefunction is generally decided prior to these questions, but there is an ongoing dissensus on this issue. A constant question for physics is how descriptions, especially mathematical descriptions, relate to reality (presuming our metaphysical commitments admit a reality). This leads to a number of deep questions:

• What does it mean for the wavefunction to "collapse"?
• If there is a change in mathematical procedure between time evolution and making a prediction, does this amount to a change in reality?
• If we cannot make any distinction in terms of position using time evolution, what is the ontological status of position prior to applying the Born rule?
• What is the ontological status of probability?
• Is making a prediction using the Born rule equivalent to making a measurement?

Each question has multiple serious answers, and is surrounded by a halo of versions for non-specialists that are more or less inaccurate. The idea of wavefunction collapse, in terms of position, is an interpretation of three facts:

1. During time evolution of the wavefunction 𝜓, we obtain no definite information about position.
2. When we apply the Born rule to 𝜓(x), we get the probability of a particle appearing in a range.
3. When we measure position, we get definite position information.

An obvious question is, given how definitely we can measure position, why is the formalism that supposedly predicts this so vague and indefinite?

During time evolution, an ontic interpretation is that the particle has no definite position. An epistemic interpretation is that we don't know anything about the position prior to measuring it. A popular interpretation is that the particle is in all positions simultaneously.

Quantum physicists adopted the term "superposition" from wave mechanics to describe this situation. In wave mechanics, superposition refers to the simple mathematical fact that if two waves are described by functions f(x) and g(x), then their combined displacement at x is just f(x) + g(x). There is nothing remarkable about this; it is commonly observed and well understood in classical terms.

Superposition of waves

Quantum superposition, in the case of a single particle, can be read as "no position", "unknown position", or "all positions". We often read about electrons being "smeared out" in space. Or forming a cloud around the nucleus. But these images raise more questions than they answer (not least questions about how such an atom can be stable). My impression is that most scientists now lean towards "no position".

The epistemic position is more or less intuitive, but this does not make it right. As we have seen, physical intuition can be a poor guide to the physics of things we can't physically experience (like galaxies or atoms). Ignorance is our natural state. Still, what can it mean for a particle to have no definite position in an ontological sense? Is it extended in space? Does it not exist? No one can say.

How we view superposition determines how we view applying the Born rule. An epistemic view says that applying the Born rule changes our state from ignorant to knowledgeable (again, this is intuitive, but so what?). But if we take the wavefunction to be something real, then we are forced to seek a realist interpretation of applying the Born rule. Somehow, and no one knows how, the Born rule causes the wavefunction to do or be something different. If we start with the idea that all positions exist simultaneously, then applying the Born rule appears to do something dramatic, which has been called "collapsing the wavefunction". Which of these views actually applies is unclear.

And finally, we have the fact that when we measure the positions of particles, they appear in one definite place. Again, depending on how we interpret the wavefunction to start with, and how we interpret the outcome of applying the Born rule, there are several possible positions one can take on this.

It's seldom explicit, but it seems to be common to assume that making a measurement is functionally equivalent to applying the Born rule. This is why people say "measurement collapses the wavefunction". However, the result of the former is a single location in space, while the result of the latter is a probability of occurring within a range of space. In fact, it's not at all clear how taking a measurement relates to the quantum formalism.

The idea that the wavefunction "collapses" is a rather dramatic way of framing the issue. The idea of "collapse" requires that we view the wavefunction as real, in a realist framework. The time evolution is viewed as pregnant with possibilities, and applying the Born rule gives birth to a single possibility. And how we frame this in physical terms depends on our prior philosophical commitments as sketched out above.

Conclusion

One way to view science is that it seeks to replace guesses and superstition with objective knowledge that is accurate and precise to certain limits. Belief is merely a feeling about an idea, and thus highly subjective. The antipathy between belief and science is never far from the news.

And yet, beliefs play a central role in the story of quantum mechanics, even if it is merely the belief in instrumentalism. Some physicists believe that the wavefunction is real; others believe it is not real. Some believe that the wavefunction represents reality; others believe it represents knowledge about reality. Some believe that measurement collapses the wavefunction; others believe that wavefunction collapse is an incoherent idea. After a century of trying, physics alone seems incapable of resolving such issues.

Part of the reason that quantum theory seems so complicated is that it involves multiple layers of abstraction, each of which is subject to multiple competing interpretations. Some interpretations are accepted as axiomatic, and others are treated as postulates. Just when you think you've understood one aspect, you discover another that contradicts it. Commentators often take philosophical stances without sufficient justification. And at the popular level, explanations of quantum mechanics often do more harm than good.

It's almost guaranteed that any given statement about quantum beyond the bare mathematics will be contradicted by another point of view. It is ridiculously difficult to give a concise account of quantum mechanics that won't be shot down by someone.

However, given the choices on offer, a thoughtful person might well choose none of the above. My sense is that the ideological/programmatic interpretations, which keep the maths and redefine reality, are all incoherent. Notably, as I have pointed out many times now, all of these interpretations are mutually exclusive. For example, the ontology of Many Worlds is completely unrelated to the ontology of, say, Pilot Wave theory, which is unrelated to spontaneous collapse interpretations, which are unrelated to information-theoretic interpretations. Given the plurality of choices on offer and the lack of any coherent ideas that can distinguish between them, I can see how it makes sense to focus on what works. In the absence of a viable ontology, adopting an instrumentalist stance—in which we have a functioning mechanism for obtaining probabilities—makes a lot of sense. However, it abandons the attempt at explanation, which I find unsatisfactory.

For realism, at least, reality itself is a point of reference. A good theory is not only consistent with reality, it is consistent with other theories about reality. When two theories are diametrically opposed, like realism and anti-realism, then only one of them can reflect reality. One of them must be a bad theory. But when it comes to quantum mechanics, we still cannot tell which is which.

~~Φ~~

Bibliography

Beller, Mara. (1996). "The Rhetoric of Antirealism and the Copenhagen Spirit". Philosophy of Science 63(2): 183-204.

Earman, John. (2022) “The Status of the Born Rule and the Role of Gleason's Theorem and Its Generalizations: How the Leopard Got Its Spots and Other Just-So Stories.” [2022 preprint]

Quanta, Quantization, and the Myth of Quantised Gravity.

In 1905, Einstein famously published four papers, each of which caused a revolution in physics. Although he is better known for his theory of relativity, it was his 1905 explanation of the photoelectric effect that won him a Nobel Prize.

In the photoelectric effect, UV radiation falling on the surface of a metal can make the metal emit electrons. This turned out to be difficult to explain based on the then-current theory of light as a wave. Experimenters had observed that the kinetic energy of the emitted electrons depends on the frequency of the radiation, not the intensity. The wave theory of light predicts that intensity will be the controlling variable. Moreover, the effect has a threshold frequency, below which no effect is seen.

Max Planck had already postulated that light also had a particle nature. Einstein applied this idea to explain the photoelectric effect, arguing that light consisted of packets of energy and that an electron absorbed all of it or none of it. We now call such a packet a quantum (Latin for "How much?".) We call a quantum of electromagnetic radiation a photon.

In the first essay in this series, I suggested that, given the role of atoms in energy quantisation, we should see quantisation as a structural (aka "emergent") property, rather than a substantive (aka fundamental) property. In this essay, I will pursue this argument a little further and then comment on the myth of quantised gravity.

What is and is not Quantised?

Quantisation is simply a fact of nature. Some quantities are naturally quantised. These include

• Franck-Hertz experiment (1914): Showed electrons in gases absorb energy in discrete steps.
• Stern-Gerlach (1922): Silver atoms deflect in discrete trajectories, proving spin quantization.
• Zeeman effect (1896): Magnetic fields split spectral lines into discrete frequencies.
• Neutron interferometry (1970s): Confirms half-integer spin for fermions.
• Millikan’s oil drop (1911): Measured charge strictly as ne.
• Quantum Hall effect (1980): Conductance steps at e²/h

However, we know of several quantities for which there is no evidence of quantisation.

• Space and Time: No experiment has ever detected quantized spacetime.
• Mass: There is no evidence of a discrete unit of mass. And again, no observation requires it.
• Gravitational waves show no signs of discreteness. No quantum theory of gravity successfully describes observation. And no explanation of an observation so far made requires quantum gravity.
• Fluids/Collective Phenomena: Sound waves, ocean waves, etc., are not discrete.

At present, it is still unfortunately the case that quantum theory has not offered us any coherent explanation of the atom or subatomic world. There is certainly no reason to believe that quantum physics explains anything at all about the world in its current state. It's merely a mathematical formalism that accurately predicts the probabilities of all outcomes for any observation.

Attempts to find the physical meaning of this have amounted to reifying the probability distribution. I will cover the metaphysics of probability in my next essay.

For now, we can simply note that no attempt to reify the wavefunction has resulted in a coherent metaphysics. We are left with a number of mutually exclusive metaphysical speculations that manage to "explain" quantum mechanics at the cost of having to abandon Realism and locality.

99% of physicists are apparently willing to pay this toll. While a dogged band of eccentric tenured professors (who can't be sacked for heresy) and a ragged collection of outsiders continue to try to do better and rescue Realism in the process. This essay, and the previous two, are my reflections on how we could do better.

Particles and Properties

In my view, the term "particle" is confusing. For physicists, it continues to conjure the idea of a point mass, which is to say a particle that has mass but no length dimensions. The problem with point masses is that density is mass divided by length. If an electron had finite mass but zero length (radius or volume), it would have infinite density since x/0 = ∞; for all values of x. That is to say, every electron would be a tiny blackhole. Observation tells us this is not the case. So we can simply, logically, rule out any talk of "point masses" or "point charges". They are mathematical fictions. 

We often assume, for the purposes of calculation that all the mass of an object is concentrated at the centre of mass. This massively simplifies mechanics. But it is precisely in atomic physics that such idealised models run into trouble.

In our universe, there are no points, no singularities, and no infinities (though some quantities may be uncountable). The same observation means that general relativity is incomplete, since it currently predicts singularities in black holes and at the Big Bang. Indeed, at the Big Bang, all energy was supposedly contained within a tiny volume, creating near-infinite spacetime curvature. If the universe started out that way, then it was a black hole, and nothing ever escapes from a black hole. There are dozens of ideas about how to prevent singularities in GR, but as far as I can see, there is no consensus on the horizon.

Wave-particle duality made enough sense in the 1920s that no one seriously questioned it. In the 2020s, however, it has become clear that "particle-like" behaviour is an artefact of detector design and, as I have explain, related to the standing wave structure of the atom. Matter is waves (likely soliton waves in spacetime). However, this realisation did not lead professional physicists to go back to the original formulations and root out all mention of particles. Rather, wave-particle duality is still retained as a fundamental postulate even in quantum field theory. 

To put it another way, one of the quantities that physicists still use the maths for is to find the "position" of the electron in an atom. And one way of talking about superposition is that the electron in an atom is in all possible locations at once (hence the idea of electron "clouds". But note that the mass of all these superpositions is still just the mass of the electron. I'm not sure how this can be explained. Can an electron in a superposition have fractional mass? When you ask physicists this is the kind of question that they seem to struggle. I assume that this is because the question is not canonical. The answer is not in the textbooks. 

That said, the electron-as-wave has a very clear physical reality. The electron has mass, charge, and angular momentum. These are all real quantities that can be measured. Moreover, electron interact in causal ways. So the electron behaves like a real object with objective properties that can be measured precisely. So why the formalism is so vague I do not understand. But since "particle" is inaccurate and carries baggage, I propose to use the term wavicle for such objects.

I discussed the roles of substance and structure in reality at length in a series of three essays in 2016. In the meantime, I've decided to drop the terms "fundamental" and "emergent", which are legacies of metaphysical reductionism. Reductionism holds that only the irreducible (i.e. "fundamental") features of the world are real. In this view, "real" and "fundamental" are synonyms. The corollary is that structure is not real. I reject this notion.

I take structure to be every bit as real as substance. For example, structure can persist over time, we can interact with structures as objects (over and above any interaction with their parts), and structure is capable of participating in sequences of events that appear causal. 

Note, I need to go back over the issue of causality. Since the last time I wrote about it, I've learned that causality is, in fact, built into relativity. Relativity takes as axiomatic the idea that causality is independent of frame of reference. Time and space are relative to one's frame of reference, but causality is not.

While reductionism is the optimal approach for studying substances, it gives us no leverage on understanding structures. Most of the significant properties of water, for example, are due to the arrangement of the three atoms that make up the molecule. For example, while both hydrogen and oxygen atoms are electrically neutral, the water molecule is an electric dipole. As such, electrostatic forces created by the dipole play a significant role in how water behaves. This is why, for example, water has such a high surface tension and a high specific heat compared to similar liquids.

In my view (influenced by Jones 2013), there are two kinds of properties.

1. Substantial Properties are the result of substance. Substantial properties are unaffected by the imposition of structure (no "downward" causation).
2. Structural Properties are the result of structure. Structural properties are (at least to some extent) independent of substance.

This is a static view. If we need to talk about a dynamic structure, it can be helpful to switch to the language of systems and speak of systemic properties. A "system" is simply a persistent dynamic structure. And a structure is a static system. Keep in mind that "static" and "dynamic" are idealisations. In the real world, things are always in motion, even those that persist over time.

How we perceive such systems is affected by the scale on which we observe them. On the millimetre scale, for example, few objects are as stable and unchanging as diamonds, which were formed in the mantle of the Earth millions of years ago. On the picometer scale, the diamond resolves into a vibrating lattice of vibrating atoms, with everything in constant motion. Matter is vibrant.

Whether we experience it as a static structure (a diamond) or a dynamic system (vibrating lattice) depends on scale. There's inevitably a transition zone in which making the distinction is difficult. We simply choose whichever viewpoint suits our aims.

That said, I think the substance/structure gestalt is helpful for anyone wishing to move beyond the limits of reductionism and into a more coherent view of reality. After all, almost everything that we can interact with is a structure or system. And our language about the world is based on how we physically interact with such objects.

Quantisation of Energy

In my first attempt at a description of the hydrogen atom, I described the electron as a (real) spherical standing wave (I leave open for the time being the question of what is waving, but I'm leaning towards the answer being "spacetime"). The basic idea is not new; both Heisenberg and Schrodinger started off with a similar approach. The difference is that they start off assuming wave-particle duality, which we now know to be a false step. Let's look again at the concept of a standing wave.

Standing Waves on a String

A guitar string is anchored at the ends by the "bridge" and the "nut". No matter how much a string is vibrating, the amplitude of vibration at the ends is always zero. We call these points nodes. If I tune a guitar string to a pitch and strike it, other things being equal, it will always vibrate at that precise pitch (pitch is determined by the density of the material, the radius of the string, temperature, and the tension it is under). 

Note that because the ends of the string are anchored and cannot vibrate, a guitar string is only capable of vibrating in standing waves. Once you impose boundary conditions such as anchoring the ends, the oscillations can only be standing waves.

Standing waves form on a guitar string when we strike it. Striking the string causes it to be locally stretched out of shape and experience a restoring force so that the stretched part accelerates back towards the centre but overshoots. The restoring force always acts towards where the resting state would be. This perturbation travels along the string in both directions. When it reaches the bridge or nut, the perturbation is reflected back along the string in the opposite direction.

If you think about how wavelength is defined in terms of sine waves (crest to crest), you can see that nodes correspond to where a cycle of the sine wave crosses the x-axis, i.e. at 0, π, and 2π radians. Thus, other things being equal, the fundamental mode (= "ground state" for an electron) corresponds to a wavelength of twice the string length.

Other things being equal, the string can also vibrate at higher energies, with pitches corresponding to the string length (L) divided by a positive integer.

For example, I can make a guitar string vibrate in the first harmonic mode. This is done by lightly placing a finger on the string at the halfway point (the 12th fret) to prevent it vibrating there, and striking the string with the other hand. This forces the string to vibrate at twice the frequency, with zero amplitude at the ends and in the middle. Now there is a node (zero amplitude) in the middle of the string as well as at the ends, corresponding to a whole wavelength. If the wavelength is halved, the frequency doubles. In music, we call this first harmonic the octave (referencing the fact that there are seven unevenly spaced notes in diatonic scales).

One can make the string vibrate at L/3 by damping the oscillations at 1/3 of the string length. In the case of a guitar, this corresponds to having nodes over the 7th and 19th frets. The second harmonic has a frequency of 3/2 of the fundamental. In music terms, this is an octave and a fifth above the fundamental.

Note that for n > 3, the nodes are no longer precisely aligned to the frets because the spacing between them is based on an equal temperament scale, to allow the guitar to sound in tune across different keys. 

Also, in real life, guitar strings vibrate in two dimensions, and/or the plane of vibration may precess around the axis of the string. Strings also vibrate in many modes at once, and, in practice, the first harmonic is often more prominent than the fundamental.

There are many videos online explaining this, so if my explanation is not clear to you, I recommend looking up "harmonics". This video gives you a decent brief intro plus some basic maths.

Note. Some people refer to the fundamental as the "first harmonic". As a musician, I think of harmonics as additional modes of vibration. So the "first harmonic" in my view is the octave.

Standing Waves in a Sphere

As noted, in order to balance the spherically symmetric electric field of the proton, the electron in a H atom must be configured as a sphere. Any other configuration leaves the electrostatic field unbalanced, meaning that the atom would not be electrically neutral. However, all atoms are electrically neutral.

Ergo, electrons in atoms are spherical with the nucleus at the centre of the sphere. The nuclear reaction of an electron and a proton to form a neutron (and a neutrino) is endothermic, requiring energy to make it happen. This means that it is not thermodynamically favoured despite the strong electromagnetic attraction. And this means that electrons seldom combine with protons except in places like the core of stars.

Keep in mind here that I'm describing an idealised situation in which no external forces are acting on the atom. In reality, everything influences everything else via the four forces.

Still, the electrostatic attraction between electron and proton is powerful and draws them together, pulling the spherical electron towards the centre. However, the electron wave has to accommodate a minimum number of wavelengths, so, despite the electrostatic attraction, it cannot get any smaller than the size needed to accommodate that number. Attracted inwards, the electron hits the minimum size limit, and it rebounds outwards. Just like the perturbation on a guitar string reflecting off the bridge and nut. The outgoing wave is prevented from escaping by the electrostatic attraction, which provides the restoring force. The outgoing wave stalls and accelerates back towards the centre again. With two waves of the same wavelength travelling in opposite directions, we have the conditions for the formation of a standing wave. 

Energy radiates away from the guitar string in the form of sound waves and heat. And it gradually loses amplitude but stays at the same frequency. But there is no equivalent of this process in an atom, so the oscillation of the electron goes on indefinitely. Each atom is a kind of perpetual motion machine.

A logical consequence of this is that energy absorbed from or emitted by atoms will be quantised. The only way a resting state (fundamental) standing wave can absorb energy is for it to jump to a harmonic mode of vibration. The actual energies can be predicted from first principles using the maths of spherical standing waves developed by Pierre-Simon Laplace, ca 1782 (long before quantum mechanics was envisaged). Similarly, energy can only be emitted when an electron falls from a higher harmonic to a lower harmonic or to the fundamental.

For similar reasons, if the electron itself is emitted from a hydrogen atom, then it has a very specific energy corresponding to the ground state energy (looked at another way, this is the ionisation potential of the atom). An atom cannot capture an electron with less than the ground state energy. If it captures an electron with more than the ground state, then the excess will be emitted as a photon of that energy.

Outside of the atom, the energy of electrons is not quantised. A free electron can have any energy. Similarly, free photons can have any energy. We can design systems so that they emit photons of any arbitrary wavelength as long as the associated energy is greater than the ground state energy (ca. -13 eV). Any colour of light is possible, and for any two colours, we can always find an intermediate wavelength (even if the human eye can't resolve the difference).

However, note that when we need "free" electrons to study, we have to get them from atoms. So all the electrons we get to study always come pre-quantised. This may have blinded physicists to the role of the standing wave in energy quantisation and caused them to see quantisation as an independent idea.

This account explains the photoelectric effect and the ionisation energy of an atom as natural consequences of the standing wave configuration of the atom. And without invoking quantum mechanics.

Quantisation of "Spin"

A more complete treatment of the angular momenta of wavicles using a theory proposed by Jackson & Minkowski (2021) and similar to Macken's (2024) is in the pipeline. Today, I will just outline the basic idea. Spin is quantised for quite different reasons than energy. In this view, we think of the spherical electron as rotating. As I have shown, simple logic dictates that, in a hydrogen atom, an electron can only be a sphere. Any other configuration leads to unbalanced electrostatic forces and unstable atoms, because the electric field of the proton is spherically symmetrical.

It's less clear that a free electron has a shape, but we can account for at least some of its characteristics by modelling it as a tiny, rotating spherical soliton wave. This approach also comes with some non-trivial problems that I will address in a future essay.

We can illustrate the general principle using the planet Earth as an analogy.

Looked at from above the north pole, the Earth is rotating anticlockwise around the y-axis. Moreover, if I hovered above Cambridge at ~52° north, the Earth is still spinning anticlockwise. And so for any position that is not directly above the equator.

If I look at exactly the same planet from above the south pole, I see it rotating clockwise. Similarly, as I move from the pole (90° S) down towards the equator (0° S), I see the rotation as going clockwise all the way.

Now, if I measure the angular momentum of the Earth on the y-axis, from any angle, the vector will point along the y-axis in the positive direction (i.e. north). If it were spinning clockwise on the y-axis, the vector would point in the negative y-axis. And this is precisely analogous to "quantum spin" having just two values: "up" and "down" (with respect to the axis of measurement). If I measure the y-axis angular momentum of a sphere rotating on some (unknown) arbitrary axis, I will still see it as either pointing to +y or -y.

The analogy of rotation has been historically rejected for two main reasons. One of which was the strange behaviour of the angular momentum of spin ½ wavicles. If we impose a 360° rotation on a spin 1 wavicle, it behaves as expected: if we see it as spin up initially, following a 180° rotation we see it as spin down, and a further 180° rotation (360° in total), we see the spin as up again.

Spin ½ wavicles do not behave like this. If we rotate a spin ½ wavicle, such as an electron, it comes back to spin "up" in just 180° (though this state is not quite identical to the starting position). The hypothetical spin 2 wavicle only returns to spin up in 720°.

As a result, physicists generally abandoned the idea that angular momentum is related to rotation. Instead, they invented the idea of "intrinsic angular momentum", which doesn't involve any actual rotation. This is very dubious from a philosophical point of view. It's an ad hoc assumption that is only justified because it works mathematically, but it cannot be tested. And, in the process, it destroys any hope of a realist description of wavicles. Jackson and Minkowski have shown that we have to allow for gyroscopic effects of magnets in magnetic fields.

The gyroscopic effect is traditionally demonstrated in physics classes by an instructor sitting on a rotating chair holding a bicycle wheel. In this video for example, the wheel is pitching (rotating around the x-axis), the instructor attempts to make it roll (rotation around the z-axis), and the result is torque creating a yawing motion (rotation around the y-axis).

Something analogous seems to be happening to a spin ½ wavicles because they are magnetic dipoles (aka magnets). If a spherical magnet is rotating around the y-axis and I try to flip it around the z-axis using a magnetic field, then there is a torque around the x-axis. The result is that for a spin ½ wavicle that is rotating around the y-axis, a 180° rotation in the z-axis is accompanied by a 180° rotation in the x-axis. The direction of the angular momentum goes from up and back to up. 

But it's not exactly the same, because the 180° rotation in the x-axis leaves the wavicle with reversed orientation in the z-axis. Another 180° rotation in z does bring it back to the starting position.

So for a sphere rotating about the y-axis, subjected to a 180° rotation about the z-axis:

• spin 0 wavicles cannot be rotated by magnetic fields.
• spin 1 involves a 180° rotation in z + 0° rotation in x.
• spin 2 involves a 180° rotation in z + 90° rotation in x.
• spin ½ involves a 180° rotation in z + 180° rotation in x.

This reproduces the observed properties of wavicle angular momentum without abandoning realism. And to date, spin 2 is purely hypothetical. Once again, there is a "classical" description of the system that predicts the behaviour we observe. And it is of a piece with the other classical physics that I have invoked in this series of essays. 

The other, more serious problem with modelling the electron as a sphere is that, based on what was known a century ago, it was calculated that the surface speed of a point on such a sphere would be moving faster than the speed of light. There are ways around this, but it needs a lengthy explanation that I will work through in a forthcoming essay.

Conclusions and Gravity

To be clear, I'm not trying to explain (or justify) the postulates of quantum physics. Indeed, I think that postulates such as a wave-particle duality or fundamental quantisation are false. I am an old-fashioned kind of scientist, and I am only trying to explain observations. Philosophically, I remain committed to pragmatic Realism and the substance/structure gestalt (in place of reductionism).

As far as I can see, quantisation of energy (in experiments like the photoelectric effect) is driven by the structure of atoms, i.e. it results from standing waves, all of which are inherently quantised. It is the structure of the atom rather than the substance of the electron that imposes this configuration on matter. And the quantisation of angular momentum is because the spherical wavicle is rotating, with the proviso that spin ½ wavicles have a magnetic moment (they are magnets), so that using a magnet to rotate the angular momentum in one direction causes an orthogonal torque. This means that we can say:

Quantisation of energy is a structural property of atoms, rather than a substantial property of wavicles.

Or, for those who prefer the less precise legacy terminology:  

Quantisation is an emergent property of atoms, rather than a fundamental property of particles.

And none of this is, in any way, "weird". Rather, we fully expect this behaviour from what we know about the (classical) mechanics of standing waves and spinning magnets. 

An important corollary of this is that, where there is no standing wave involved, we should not expect energy to be quantized because there is no reason for it to be quantised. Where a wavicle is not a rotating sphere, we don't expect angular momentum to be quantised either.

A case in point is the gravitational wave. We have been measuring gravitational waves using the LIGO apparatus for a few years now. Gravitational waves are unequivocally real. The waves we detect are mostly from binaries of black holes or neutron stars colliding. It seems clear that such systems are not stable, and that (therefore) they do not form standing waves. Ergo, we would not expect gravity waves to be quantised.

Despite a century of strenuous efforts by a cadre of maths geniuses, there is still no way to meaningfully quantify gravity using the standard methods of quantum physics. The attempts to quantise gravity have not worked. And we should not expect this to ever work.  

The quest for quantised gravity is not driven by any practical consideration or observation. Quantum gravity is not required to explain any existing experimental observations. Rather, the search for quantum gravity is being driven by ideological and aesthetic concerns. 

Worse, physicists appear to believe that GR and QM ought to be reconcilable while at the same time acknowledging that neither theory is yet complete. My thought is that we cannot expect two incomplete theories to be reconcilable anyway. In all likelihood, completing them will solve the apparent problems. 

Atoms, protons, and electrons are real. They persist over time and have measurable properties such as mass, charge, and angular momentum. In sequences of events, these entities play a causal role. In which case, I cannot see the logic in modelling atoms, protons, or electrons in terms of abstract probabilities. I believe this category error explains why the metaphysics that scientists have drawn from the formalism are so divergent and mutually exclusive: they are not anchored in reality. 

The title of my next essay will be: "Ψ-ontologies and the Nature of Probability."

~~Φ~~

Bibliography

Jackson, Peter A. & Minkowski, John S. (2021). "The Measurement Problem, an Ontological Solution." SpringerNature, Foundations of Physics 51 (article 77).

———. (2014). "Quasi-classical Entanglement, Superposition and Bell Inequalities." Unpublished essay. https://www.academia.edu/9216615/

Jones, Richard H. (2013). Analysis & the Fullness of Reality: An Introduction to Reductionism & Emergence. Jackson Square Books.

Macken, John. (2024). "Oscillating Spacetime: The Foundation of the Universe". Journal of Modern Physics 15 (8): 1097-1143. DOI: 10.4236/jmp.2024.158047

Why Quantum Mechanics is Currently Wrong and How to Fix It.

It is now almost a century since "quantum mechanics" became established as the dominant paradigm for thinking about the structure and motion of matter on the nanoscale. And yet the one thing quantum mechanics cannot do is explain what it purports to describe. Sure, quantum mechanics can predict the probability of measurements. However, no one knows how it does this. 

Presently, no one understands the foundations of quantum mechanics. 

Feynman's quote to this effect is still accurate. It has recently been restated by David Deutsch, for example:

"So, I think that quantum theory is definitely false. I think that general relativity is definitely false." (t = 1:16:13)

"Certainly, both relativity and quantum theory are extremely good approximations in the situations where we want to apply them... So, yes, certainly, good approximations for practical purposes, but so is Newton's theory. That's also false." (t = 1:28:35)

—David Deutsch on Sean Carroll's podcast.

I listened to these striking comments again recently. This time around, I realised that my conception of quantum field theory (QFT) was entirely wrong. I have a realistic picture in my head, i.e. when I talk about "waves", something is waving. This is not what GFT says at all. The "fields" in question are entirely abstract. What is waving in quantum mechanics is the notion of the probability of a particle appearing at a certain location within the atom. Below I will show that this thinking is incoherent. 

There have been numerous attempts to reify the quantum wavefunction. And they all lead to ridiculous metaphysics. Some of the most hilarious metaphysics that quantum mechanics has produced are:

1. The universe behaves one way when we look at it, and a completely different way when we don't.
   
2. The entire universe is constantly, and instantaneously, splitting into multiple copies of itself, each located in exactly the same physical space, but with no connections between the copies.
   
3. Electrons are made of waves of probability that randomly collapse to make electrons into real particles for a moment.

None of these ideas is remotely compatible with any of the others. And far from there being a consensus, the gaps between "interpretations" are still widening. Anyone familiar with my work on the Heart Sutra will recognise this statement. It's exactly what I said about interpretations of the Heart Sutra.

Physics has lost its grip on reality. It has a schizoid ("splitting") disorder. I believe I know why.

What Went Wrong?

The standard quantum model embraces wave-particle duality as a fundamental postulate. In the 1920s, experiments seemed to confirm this. This is where the problems start.

Schiff's (1968) graduate-level textbook, Quantum Mechanics, discusses the idea that particles might be considered "wave packets":

The relation (1.2) between momentum and wavelength, which is known experimentally to be valid for both photons and particles, suggests that it might be possible to use concentrated bunches of waves to describe localized particles of matter and quanta of radiation. To fix our ideas, we shall consider a wave amplitude or wave function that depends on the space coordinates x, y, z and the time t. This quantity is assumed to have three basic properties. First, it can interfere with itself, so that it can account for the results of diffraction experiments. Second, it is large in magnitude where the particle or photon is likely to be and small elsewhere. And third, will be regarded as describing the behavior of a single particle or photon, not the statistical distribution of a number of such quanta. (Schiff 1968: 14-15. Emphasis added)

I think this statement exemplifies the schizoid nature of quantum mechanics. The Schrödinger model begins with a particle, described as a "wave packet", using the mathematics of waves. The problem is that physicists still want to use the wave equation to recover the "position" or "momentum" of the electron in the atom, as though it is a particle. I have seen people dispute that this was Schrödinger's intention, but it's certainly how Schiff saw it, and his text was widely respected in its day.

The obvious problem is that, having modelled the electron as a wave, how do we then extract from it information about particles, such as position and momentum? Mathematically, the two ideas are not compatible. Wave-talk and particle-talk cannot really co-exist. 

In fact, Schrödinger was at a loss to explain this. It was Max Born who pointed out that if you take the modulus squared value of the wave function (which outputs complex-numbered vectors), you get a probability distribution that allows you to predict measurements. As I understand it, Schrödinger did not like this at all. In an attempt to discredit this approach, he formulated his classic thought experiment of the cat in the box. A polemic that failed so badly, that the Copenhagen crowd adopted Schrödinger's cat as their mascot. I'll come back to this.

However, there is a caveat here. No one has ever measured the position of an electron in an atom, and no one ever will. It's not possible. We have probes that can map out forces around atoms, but we don't have a probe that we, say, can stick into an atom and wait for the electron to run into it. This is not how things work on this scale.

Can We Do Better? (Yes We Can!)

Electric charge is thought to be a fundamental property of matter. We visualise the electric charge of a proton as a field of electric potentials with a value at every point in space, whose amplitude drops off as the square of the distance. The electric field around a proton is observed to be symmetrical in three dimensions. In two dimensions, a proton looks something like this with radiating, evenly spaced field lines:

An electron looks the same, but the arrows point inwards (the directionality of charge is purely conventional). So if the electron were a point charge, an atom would be an electric dipole, like this:

This diagram shows that if the electron were a point mass/charge, the hydrogen atom would be subject to unbalanced forces. Such an atom would be unstable. Moreover, a moving electric dipole causes fluctuations in the magnetic field that would rapidly bleed energy away from the atom, so if it didn't collapse instantaneously, it would collapse rapidly. 

Observation shows atoms to be quite stable. So, at least in an atom, an electron cannot be a point mass/charge. And therefore, in an atom, an electron is not a point mass/charge.

Observation also shows that hydrogen atoms are electrically neutral. Given that the electric field of the proton is symmetrical in three dimensions, there is only one shape the electron could be and balance the electric charge. A sphere with the charge distributed evenly over it.

The average radius of the sphere would be the estimated value of the atomic radius. Around 53 picometers (0.053 nanometers) for hydrogen. The radius of a proton is estimated to be on the order of 1 femtometer.

Niels Bohr had a similar idea. He proposed that the electron formed a "cloud" around the nucleus. And this cloud was later identified as "a cloud of probability". Which is completely meaningless. The emperor is not wearing any clothes. As David Albert says on Sean Carroll's podcast:

“… there was just this long string of brilliant people who would spend an hour with Bohr, their entire lives would be changed. And one of the ways in which their lives were changed is that they were spouting gibberish that was completely beneath them about the foundations of quantum mechanics for the rest of their lives…” (emphasis added)

We can do better, with some simple logic. We begin by postulating, along with GFT, that the electron is some kind of wave. 

If the electron is a wave, AND the electron is a sphere, AND the atom is stable, AND the atom is electrically neutral, then the electron can only be a spherical standing wave.

Now, some people may say, "But this is exactly what Schrödinger said". Almost. There is a crucial difference. In this model, the spherical standing wave is the electron. Or, looked at from the other direction, an electron (in a hydrogen atom) is a physical sphere with an average radius of ~53 pm. There is no particle, we've logically ruled out particles.

What does observation tell us about the shape of atoms? We have some quite recent data on this. For example, as reported by Lisa Grossman (2013) for New Scientist, here are some pictures of a hydrogen atom recently created by experimenters.

The original paper was in Physical Review.

Sadly, the commentary provided by Grossman is the usual nonsense. But just look at these pictures. The atom is clearly a sphere in reality, just as I predicted using simple logic. Many crafty experiments, have reported the same result. It's not just that the probability function is spherical. Atoms are spheres. Not solid spheres, by any means, but spheres nonetheless.

We begin to part ways with the old boys. And we are instantly in almost virgin territory. To the best of my knowledge, no one has ever considered this scenario before (I've been searching the literature).

The standard line is that the last input classical physics had was Rutherford's planetary model proposed in 1911, after he successfully identified that atoms have a nucleus, which contains most of the mass of the atom. This model was debunked by Bohr in 1913. And classical physics has nothing more to say. As far as any seems to know, "classical physics says the electron is a point mass". No one has ever modelled the electron in an atom as a real wave. At least no one I can find.

This means that there are no existing mathematical models I can adapt to my purpose. I have to start with the general wave equation and customise it to fit. Here is the generalised wave equation of a spherical standing wave:

Where r is the radius of the sphere, θ and φ are angles, and t = time. Notice that it is a second-order partial differential equation, and that the rates of change in each quantity are interdependent. It can be solved, but it is not easy.

The fact is that, while this approach is not identical to existing quantum formalism, it is isomorphic (i.e. has the same form). Once we clarify the concept and what we are trying to do with it, the existing formalism ought to be able to be adapted. So we don't have to abandon quantum mechanics, we just have to alter our starting assumptions and allow that to work through what we have to date. 

An important question arises: What about the whole idea of wave-particle duality?

In my view, any particle-like behaviour is a consequence of experimental design. Sticking with electrons, we may say that every electron detector relies on atoms in the detector absorbing electrons. And there are no fractional electrons. Each electron is absorbed by one and only one atom. It is this phenomenon that causes the appearance of discrete "particle-like" behaviour. At the nano-scale, any scientific apparatus is inevitably an active part of the system.

The electron is a wave. It is not a particle. 

Given the wild success of quantum mechanics (electronics, lasers, and so on), why would anyone want to debunk it? For me, it is because it doesn't explain anything. I didn't get into science so I could predict measurements, by solving abstract maths problems. I got into it so I could understand the world. Inj physics maths is supposed to represent the world and to have a physical interpretation. I'm not ready to give up on that.

The Advantages of Modelling the Electron as a (Real) Wave.

While they are sometimes reported as special features of quantum systems, the fact is that all standing waves have some characteristic features.

In all standing waves, energy is quantised. This is because a standing wave only allows whole numbers of wavelengths. We may use the example of a guitar string that vibrates in one dimension*.

*Note that if you look at a real guitar string, you will see that it vibrates in two dimensions: perpendicular to the face of the guitar and parallel to it.

The ends of the string are anchored. So the amplitude of any wave is always zero at the ends; they cannot move at all. The lowest possible frequency is when the wavelength equals the string length.

The next lowest possible frequency is when the wavelength equals half the string length. And so on.

This generalises. All standing waves are quantised in this way. This is "the music of the spheres". 

Now, spherical standing waves, with a central attractive force exist and were described ca 1782 by Pierre-Simon Laplace. These entities are mathematically very much more complicated than a string vibrating in one dimension. Modelling this is a huge challenge. 

For the purposes of this essay, we can skip to the end and show you what the general case of harmonics of a spherical standing wave looks like when the equations are solved and plotted on a graph.

Anyone familiar with physical chemistry will find these generalised shapes familiar. These are the theoretical shapes of electron orbitals for hydrogen. And this is without any attempt to account for the particular situation of an electron in an atom (the coulomb potential, the electric field interfering with itself, etc).

So not only is the sphere representing the electron naturally quantised, but the harmonics give us electron "orbitals". And, if we drop the idea of the electron as a particle, this all comes from within a classical framework (though not Rutherford's classical framework). 

Why Does Attempting to Reify Probability Lead to Chaos?

As already noted, Schrödinger tried and failed to relate his equation back to reality. Max Born discovered that the modulus squared of the wavefunction vector at a given point could be interpreted as the probability of finding the "the electron" (qua particle) at that point. This accurately predicts the probable behaviour of an electron, though not its actual behaviour. But all this requires electrons to be both waves and point-mass particles. 

Since the real oscillations I'm describing are isomorphic with the notional oscillations predicted by Schrödinger, we can intuit that if we were to try to quantify the probability of the amplitude of the (real) spherical standing wave at a certain point around the sphere, then any probability distribution we created from this would also be isomorphic with application of the Born rule to Schrödinger's equation.

What I've just done, in case it wasn't obvious, is explain the fundamentals of quantum mechanics (in philosophical terms at least) in one sentence. The predicted probabilities take the form that they do because of a physical mechanism: a spherical standing wave. And I have not done any violence to the notion of "reality" in the process. To my knowledge, this has not been done before, although I'm certainly eager to learn if it has.

However, the isomorphism is only causal in one direction. You can never get from a probability distribution to a physical description. Let me explain why by using a simple analogy that can be generalised.

Let's take the very familiar and simple case of a system in which I toss a coin in the air and, when it lands, I note which face is up. The two possible outcomes are heads H and tails T. The probabilities are well-known:

P(H) = 0.5 and P(T) = 0.5.

And as always, the sum of the probabilities of all the outcomes is 1.0. So:

P(H) + P(T) = 1.0

No matter what values we assign to P(H) and P(T), they have to add up to 1.

In physical terms, this means that if we toss 100 coins, we expect to observe heads 50 times and tails 50 times. In practice, we will most likely not get exactly 50 of each because probabilities do not determine outcomes. Still, the more times we toss the coins, the closer our actual distribution will come to the expected value.

Now imagine that I have tossed a coin, it has landed, but I have not yet observed it (call this the one-dimensional Schrödinger's cat, if you like). The standard rhetoric is to say that the coin is in a superposition of two "states". One has to be very wary of the term "state" in this context. Quantum physicists do not use it in the normal way, and it can be very confusing. But I am going to use "state" in a completely naturalistic way. The "state" of the tossed coin refers to which face is up. And it has to be in one of two possible states: H or T.  

Now let's ask what I know and think about what I can know about the coin at this moment before I observe the state of the coin.

I know that the outcome must be H or T. And I know that the odds are 50:50 that it is either one. What else can I know? Nothing. Despite knowing to 100 decimal places what the probability is, I cannot use that information to know what state the coin is in before I observe it. If I start with probabilities, I can say nothing about the fact of the matter (using a phrase David Albert uses a lot). If I reify this concept, I might be tempted to say that there is no fact of the matter. 

Note also that it doesn't matter if P(H) and P(T) are changing. Let us say that the probabilities change over time and that the change can be precisely described by a function of the coin: Ψ(coin). Are we any better off? Clearly not.

This analogy generalises. No matter how complex my statistical model, no matter how accurately and precisely I know the probability distribution, I still cannot tell you which side up the coin is without looking. There is undoubtedly a physical fact of the matter, but as the old joke goes, you cannot get there from here.

There are an infinite number of reasons why a coin toss will have P(H) = P(T) = 0.5. We can speculate endlessly. This is why the "interpretations" of quantum mechanics are so wildly variable and the resulting metaphysics so counter-intuitive. Such speculations are not bound by the laws of nature. In fact, all such speculations propose radical new laws of nature, like splitting the entire universe in two every time a quantum event happens. 

So the whole project of trying to extract meaningful metaphysics from a probability distribution was wrong-headed from the start. It cannot work, and it does not work. A century of effort by very smart people has not produced any workable ideas. Or any consensus on how to find a workable idea. 

Superposition and the Measurement Problem

The infamous cat experiment, in all its varieties, involves a logical error. As much as Schrödinger resisted the idea, because of his assumption about wave-particle duality, his equation only tells us about the probabilities of states; it does not and cannot tell us which state happens to be the fact of the matter. The information we get from the current formalism is a probability distribution. So the superposition in question is only a superposition of probabilities; it's emphatically not a superposition of states (in my sense). A coin cannot ever be both H and T. That state is not a possible state. 

Is the superposition of probabilities in any way weird? Nope.

The fact that P(H) = 0.5 or P(H) = Ψ(coin) and that P(T) = 0.5 or P(T) = Ψ(coin) are not weird facts. Nor is the fact that P(H) + P(T) = 1. These are common or garden facts, with no mystical implications.

If we grant that the propositions P(H) = 0.5 and P(T) = 0.5 are logically true, then it must also be logically (and mathematically) true to say that P(H) + P(T) = 1. Prior to observations all probabilities coexist at the same time.

For all systems we might meet, all the probabilities for all the outcomes always coexist prior to observing the state of the system. And the probabilities for all but one outcome collapse to zero at the moment we observe the actual state. This is true for any system: coins, cats, electrons, and everything. 

Note also that this is not a collapse of anything physical. No attempt to reify this "collapse" should be made. Probability is an idea we can quantify, but it's not an entity. No existing thing collapses when we observe an event. 

Moreover, Buddhists and hippies take note, our observing an event cannot influence the outcome. Light from the event can only enter our eye after the event has occurred, i.e. only after the probabilities have collapsed. And it takes the brain an appreciable amount of time to register the incoming nerve signal, make sense of it, and present it to the first-person perspective. Observation is always retrospective. So no, observation cannot possibly play any role in determining outcomes. 

One has to remember that probability is abstract. It's an idea about how to quantify uncertainty. Probability is not inherent in nature; it comes from our side of the subject-object divide. Unlike, say, mass or charge, probability is not what a reductionist would call "fundamental". We discover probabilities through observation of long-term trends. At the risk of flogging a dead horse, you cannot start with an abstraction and extract from it a credible metaphysics. Not in the world that we live in. And after a century of trying, the best minds in physics have signally failed in this quixotic endeavour. There is not even a working theory of how to make metaphysics from probabilities. 

The superposition or collapse of probabilities is in no way weird. And this is the only superposition predicted by quantum mechanics. 

In my model, the electron is a wave, and the wave equation that describes it applies at all times. Before, during, and after observation. 

In my model, probabilities superpose when we don't know the facts of the matter, in a completely normal way. It's just that I admit the abstract nature of probability distributions. And I don't try to break reality so that I can reify an abstraction.

On the other hand, my approach is technically classical. A classical approach that ought to predict all the important observations of quantum mechanics, but which can also explain them in physical terms. As such, there is no separation between classical and quantum in my model. It's all classical. And I believe that the implications of this will turn out to be far-reaching and will allow many other inexplicable phenomena to be easily explained.

The so-called measurement problem can be seen as a product of misguided attempts to hypostatise and reify the quantum wavefunction, which only predicts probabilities. It was only ever a problem caused by a faulty conceptualisation of the problem in terms of wave-particle duality. If we drop this obviously false axiom, things will go a lot more smoothly (though the maths is still quite fiendish).

No one ever has or ever will observe a physical superposition. I'm saying that this is because no such thing exists or could exist. It's just nonsense, and we should be brave enough to stand up and say so.

There is no "measurement problem". There's measurement and there is ill-advised metaphysical speculation based on reified abstractions.

What about other quantum weirdness?

I want to keep this essay to a manageable length, so my answer to this question must wait. But I believe that Peter Jackson's (2013) free electron model as a vortex rotating on three axes is perfectly consistent with what I outlined here. And it explains spin very elegantly. If the electron is a sphere in an atom, why not allow it to always be a sphere?

Jackson also elegantly explains why the polarised filter set-up to test Bell's inequalities is not quantum weirdness, but a result of the photon interacting with, and thus being changed by, the filter. At the nano-scale and below, there are no neutral experimental apparatus.

What about interference and the double-slit experiment? Yep, I have some ideas on this as well.

Tunnelling? I confess that I have not tried to account for tunneling just yet. At face value, I think it is likely to turn out to be a case of absorption and re-emission (like Newton's cradle) rather than Star Trek-style teleporting. Again, there is no such thing as a neutral apparatus on the nano-scale or below. If your scientific apparatus is made of matter, it is an active participant in the experiment and at the nano-scale, it changes the outcomes. 

It's time to call bullshit on quantum mechanics and rescue physicists from themselves. After a century of bad metaphysics, let's put the phys back into physics!

~~Φ~~

P.S. My book on the Heart Sutra is coming along. I have a (non-committal) expression of interest from my publisher of choice. I hope to have news to share before the end of 2025.

PPS. I'd quite like to meet a maths genius with some time on their hands...

PPPS (16 Apr). I now have an answer to the question "What is waving?". An essay on this is in progress but may take a while. 

Bibliography

Grossman, Lisa. (2013). "Smile, hydrogen atom, you're on quantum camera." New Scientist. https://www.newscientist.com/article/mg21829194-900-smile-hydrogen-atom-youre-on-quantum-camera/

Jackson, Peter. (2009). "Ridiculous Simplicity". FQXi. What is Fundamental? https://forums.fqxi.org/d/495-perfect-symmetry-by-peter-a-jackson

Schiff, Leonard I. (1968). Quantum Mechanics. 3rd Ed. McGraw-Hill.

Quantum Bullshit

I was appalled recently to see that a senior professor of Buddhism Studies—whose work on Chinese Buddhist texts I much admire—had fallen into the trap of trying to compare some concept from Buddhist philosophy to what he calls "quantum mechanics". Unfortunately, as seems almost inevitable in these cases, the account the Professor gives of quantum mechanics is a hippy version of the Copenhagen interpretation proposed by Werner Heisenberg back in the 1920s. In a further irony, this same Professor has been a vocal critic of the secularisation and commercialisation of Buddhist mindfulness practices. The same problems that he identifies in that case would seem to apply to his own misappropriation of quantum mechanics.

As I've said many times, whenever someone connected with Buddhism uses the word "quantum" we can safely substitute the word "bullshit". My use of the term "bullshit" is technical and based on the work of Princeton philosopher Harry Frankfurt (image left). I use "bullshit" to refer to a particular rhetorical phenomenon. Here is the anonymous summary from Wikipedia, which I think sums up Frankfurt's arguments about bullshit precisely and concisely:

“Bullshit is rhetoric without regard for truth. The liar cares about the truth and attempts to hide it; the bullshitter doesn't care if what they say is true or false; only whether or not their listener is persuaded.”

What I am suggesting is that Buddhists who refer to quantum mechanics are not, in fact, concerned with truth, at all. A liar knows the truth and deliberately misleads. The bullshitter may or may not know or tell the truth, but they don't care either way. Their assertions about quantum mechanics may even be true, but this is incidental. The idea is to persuade you of a proposition which may take several forms but roughly speaking it amounts to:

If you sit still and withdraw attention from your sensorium, another more real world is revealed to you.

Certain Buddhists argue that a specific man sitting under a specific tree ca 450 BCE, while ignoring his sensorium, saw such a reality (Though he neglected to mention this). And then this thesis is extended with the proposition:

The reality that one "sees" when one's eyes are closed is very like the descriptions (though not the mathematics) of quantum mechanics.

I imagine that these statements strike most scientists as obviously false. The first hint we had of a quantum world was in 1905 when Einstein formalised the observation that energy associated with atoms comes in discrete packets, which he called "quanta" (from the Latin with the sense "a portion"; though, literally, "how much?"). Even this nanoscale world, which we struggle to imagine, is established by observation, not by non-observation. Equally, there is no sign in early Buddhist texts that the authors had any interest in reality, let alone ultimate reality. They didn't even have a word that corresponds to "reality". They did talk a lot about the psychology of perception and about the cessation of perception in meditation, within the context of a lot of Iron Age mythology. Given that there is no prima facie resemblance between science and Buddhism whatever, we might well ask why the subject keeps coming up.

I think this desire to positively compare Buddhism to quantum mechanics is a form of "virtue signalling". By attempting to align Buddhist with science, the highest form of knowledge in the modern world, we hope to take a ride on the coat-tails of scientists. This is still the Victorian project of presenting the religion of Buddhism as a "rational" alternative to Christianity. Generally speaking, Buddhists are as irrational as any other religieux, it's just that one of the irrational things Buddhists believe is that they are super-rational.

Had it merely been another misguided Buddhism Studies professor, I might have let it go with some pointed comments on social media. Around the same time, I happened to watch a 2016 lecture by Sean Carroll on YouTube called, Extracting the Universe from the Wave Function. Then I watched a more recent version of the same lecture from 2018 delivered at the Ehrenfest Colloquium. The emphasis is different in the two forums and I found that watching both was useful. Both lectures address the philosophy of quantum mechanics, but in a more rigorous way than is popular amongst Buddhists. Sean thinks the Copenhagen interpretation is "terrible" and he convinced me that he is right about this. The value of the lectures is that one can get the outlines of an alternative philosophy of quantum mechanics and with it some decisive critiques of the Copenhagen interpretation. Sean is one of the leading science communicators of our time and does a very good job of explaining this complex subject at the philosophical level.

What is Quantum Mechanics?

It is perhaps easiest to contrast quantum mechanics with classical mechanics. Classical mechanics involves a state in phase space (described by the position and momentum of all the elements) and then some equations of motion, such as Newton's laws, which describe how the system evolves over time (in which the concept of causation plays no part). Phase space has 6n dimensions, where n is the number of elements in the state. Laplace pointed out that given perfect knowledge of such a state at a given time, one could apply the equations of motion to know the state of the system at any time (past or future).

Quantum mechanics also minimally involves two things. A state is described by a Hilbert Space, the set of all possible quantum states, i.e., the set of all wave functions, Ψ(x). It is not yet agreed whether the Hilbert Space for our universe has an infinite or merely a very large number of dimensions.

For the STEM people, there's a useful brief summary of Hilbert spaces here. If you want an image of what a Hilbert Space is like, then it might be compared to the library in the short story The Library of Babel, by Jorge Luis Borges. (Hat-tip to my friend Amṛtasukha for this comparison).

Mathematically, a Hilbert Space is a generalisation of vector spaces which satisfy certain conditions, so that they can be used to describe a geometry (more on this later). One thing to watch out for is that mathematicians describe Hilbert Spaces (plural). Physicists only ever deal with the quantum Hilbert Space of all possible wavefunctions and have slipped into the habit talking about "Hilbert Space" in the singular. Sean Carroll frequently reifies "Hilbert Space" in this way. Once we agree that we are talking about the space defined by all possible wave functions, then it is a useful shorthand. We don't have to consider any other Hilbert Spaces.

The second requirement is an equation that tells us how the wave functions in Hilbert Space evolve over time. And this is Schrödinger's wave equation. There are different ways of writing this equation. Here is one of the common ways:

The equation is a distillation of some much more complex formulas and concepts that take a few years of study to understand. Here, i is the imaginary unit (defined as i² = -1), ħ is the reduced Planck constant (h/2π). The expression δ/δt represents change over time. Ψ represents the state of the system as a vector in Hilbert Space -- specifying a vector in a space with infinite dimensions presents some interesting problems. Ĥ is the all important Hamiltonian operator which represents the total energy of the system. And note that this is a non-relativistic formulation.

We owe this formalisation of quantum theory to the fact that John von Neumann studied mathematics with David Hilbert in the early 20th Century. Hilbert was, at the time, trying to provide physics with a more rigorous approach to mathematics. In 1915, he invited Einstein to lecture on Relativity at Göttingen University and the two of them, in parallel, recast gravity in terms of field equations (Hilbert credited Einstein so no dispute arose between them). In 1926, Von Neumann showed that the two most promising approaches to quantum mechanics—Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation—could be better understood in relation to a Hilbert Space.

[I'm not sure, but this may the first time a Buddhist has ever given even an overview of the maths in an essay about Buddhism and quantum mechanics.]

By applying the Born Rule (i.e., finding the square of the Wave Function) we can find the probability that any given particle will be found in some location at any given time. A common solution to the wave equation is a map of probabilities. For example, the probability plot for an electron in a resting state hydrogen atom looks like this (where shading represents the range probability and the black in the middle is the nucleus). And btw this is a 2D representation of what in 3D is a hollow sphere.

If we give the electron more energy, the probably map changes in predictable ways. An electron bound to an atom behaves a bit like a harmonic oscillator. A good example of a harmonic oscillator is a guitar string. If you pluck a guitar string you get a complex waveform made from the fundamental mode plus harmonics. The fundamental mode gives a note its perceived pitch, while the particular mixture of harmonics is experienced as the timbre of the note. The fundamental mode has two fixed points at the ends where there is zero vibration, and a maximum in the centre. The next mode, the 2nd harmonic takes more energy to produce and the string vibrates with three minima and two maxima - the pitch is an octave above the fundamental.

Using the fleshy parts of the fingers placed at minima points, it is possible to dampen extraneous vibrations on a guitar string and pick out the harmonics. Such notes have a very different timbre to regular notes. An electron bound to an atom also has "harmonics", though the vibrational modes are three dimensional. One of the striking experimental confirmations of this comes if we split sunlight up into a rainbow, we observe dark patches corresponding to electrons absorbing photons of a precise energy and becoming "excited". One of the first confirmations of quantum mechanics was that Schrödinger was able to accurately predict the absorption lines for a hydrogen atom using it.

And on the other hand, after we excite electrons in, say, a sodium atom, they return to their resting state by emitting photons of a precise frequency (in the yellow part of the visible spectrum) giving sodium lamps their characteristic monochromatic quality. The colour of light absorbed or emitted by atoms allows us to use light to detect them in spectral analysis or spectroscopy. For example, infrared light is good for highlighting molecular bonds; while green-blue visible and ultraviolet light are good for identifying individual elements (and note there are more dark patches towards the blue end of the spectrum).

The wave function applied to the electron in an atom gives us a map of probabilities for finding the electron at some point. We don't know where the electron is at any time unless it undergoes some kind of physical interaction that conveys location information (some interactions won't convey any location information). This is one way of defining the so-called the Measurement Problem.

rugby ball

I have a new analogy for this. Imagine a black rugby ball on a black field, in the dark. You are walking around on the field, and you know where you are from a GPS app on your phone, but you cannot see anything. The only way to find the ball is to run around blindly until you kick it. At the moment you kick the ball the GPS app tells you precisely where the ball was at that moment. But kicking the ball also sends it careering off and you don't know where it ends up.

Now, Buddhists get hung up on the idea that somehow the observer has to be conscious, that somehow consciousness (whatever that word means!) is involved in determining how the world evolves in some real sense. As Sean Carroll, says in his recent book The Big Picture:

“...almost no modern physicists think that 'consciousness' has anything whatsoever to do with quantum mechanics. There are an iconoclastic few who do, but it's a tiny minority, unrepresentative of the mainstream” (p.166).

The likes of Fritjof Capra have misled some into thinking that the very vague notion of consciousness plays a role in the measurement problem. As far as the mainstream of quantum mechanics is concerned, consciousness plays no part whatsoever in quantum mechanics. And even those who think it does have provided no formalism for this. There is no mathematical expression for "consciousness", "observer", or "observation". All of these concepts are completely nebulous and out of place around the wave equation, which predicts the behaviour of electrons at a level of accuracy that exceeds the accuracy of our measurements. In practice, our experiments produce data that matches prediction to 10 decimal places or more. Quantum mechanics is the most accurate and precise theory ever produced. "Consciousness" is the least well-defined concept in the history of concepts. "Observation" is not even defined.

In the image of the black rugby ball on a black field in the dark, we don't know where the ball is until we kick it. However, a ball and a field are classical. In the maths of quantum mechanics, we have no information about the location of the ball until we physically interact with it. Indeed, it appears from the maths that it's not physically in one place until information about location is extracted from the system through a physical interaction. And by this we mean, not a conscious observer, but something like bouncing some radiation off the electron. It's as though every time you take a step there is a possibility of the ball being there and you kicking it, and at some point, it is there and you kick it. But until that moment, the ball is (somehow) smeared across the whole field all at once.

Put another way, every time we take a step there is some probability that the ball is there and we kick it, and there is some probability that the ball is not there and we do not kick it. But as we step around, we don't experience a probability, and we never experience a ball spread out over all locations. Whenever we interact with the system we experience the ball as being at our location or at some specific other location. Accounting for this is at the heart of different interpretations of quantum mechanics.

Copenhagen

What every undergraduate physics student learns is the Copenhagen Interpretation of the measurement problem. In this view, the ball is literally (i.e., in reality) everywhere at once and only adopts a location at the time of "measurement" (although measurement is never defined). This is called superposition - literally "one thing on top of another". Superposition is a natural outcome of the Wave Equation; there are huge problems with the Copenhagen interpretation of how mathematical superposition relates to reality.

Firstly, as Schrödinger pointed out with his famous gedanken (thought) experiment involving a cat, this leads to some very counterintuitive conclusions. In my analogy, just before we take a step, the rugby ball is both present and absent. In this view, somehow by stepping into the space, we make the ball "choose" to be present or absent.

Worse, the Copenhagen Interpretation assumes that the observer is somehow outside the system, then interacts with it, extracting information, and then at the end is once again separate from the system. In other words, the observer behaves like a classic object while the system being observed is quantum, then classical, then quantum. Hugh Everett pointed out that this assumption of Copenhagen is simply false.

In fact, when we pick up the cat to put it in the box, we cannot avoid becoming entangled with it. What does this mean? Using the ball analogy if we kick the ball and know its location at one point in time then we become linked to the ball, even though in my analogy we don't know where it is now. If someone else now kicks it, then we instantaneously know where the ball was when it was kicked a second time, wherever we happen to be on the field. It's as though we get a GPS reading from the other person sent directly to our phone. If there are two entangled electrons on either side of the universe and we measure one of them and find that it has spin "up", then we also know with 100% certainty that at that same moment in time, the other electron has spin "down". This effect has been experimentally demonstrated so we are forced to accept it until a better explanation comes along. Thus, in Schrödinger's gedanken experiment, we always know from instant to instant what state the cat is in (this is also counter-intuitive, but strictly in keeping with the metaphor as Schrödinger outlined it).

As you move about the world during your day, you become quantum entangled with every object you physically interact with. Or electrons in atoms that make up your body become entangled with electrons in the objects you see, taste, touch, etc. Although Copenhagen assumes a cut off (sometimes called Heisenberg's cut) between the quantum world and the classical world, Hugh Everett pointed out that this assumption is nonsense. There may well be a scale on which classical descriptions are more efficient ways of describing the world, but if one atom is quantum, and two atoms are, and three, then there is, in fact, no number of atoms that are not quantum, even if their bulk behaviour is different than their individual behaviour. In other words, the emergent behaviour of macro objects notwithstanding, all the individual atoms in our bodies are obeying quantum mechanics at all times. There is no, and can be no, ontological cut off between quantum and classical, even if there is an epistemological cutoff.

In terms of Copenhagen, the argument is that wave function describes a probability of the ball being somewhere on the field and that before it is kicked it is literally everywhere at once. At the time of kicking the ball (i.e., measurement) the wave function "collapses" and the ball manifests at a single definite location and you kick it. But the collapse of the wave function is a mathematical fudge. In fact, it says that before you look at an electron it is quantum, but when you look at it, it becomes classical. Then when you stop looking it becomes quantum again. This is nonsense.

In Schrödinger's cat-in-the-box analogy, as we put the cat in the box, we become entangled with the cat; the cat interacts with the box becoming entangled with it; and so on. How does an observer ever stand outside a system in ignorance and then interact with it to gain knowledge? The answer is that, where quantum mechanics applies, we cannot. The system is cat, box, and observer. There is no such thing as an observer outside the system. But it is even worse because we cannot stop at the observer. The observer interacts with their environment over a period of years before placing the cat in the box. And both cat and box have histories as well. So the system is the cat, the box, the observer, and the entire universe. And there is no way to get outside this system. It's not a matter of whether we (as macro objects) are quantum entangled, but to what degree we are quantum entangled.

This is a non-trivial objection because entanglement is ubiquitous. We can, in theory, speak of a single electron orbiting a single nucleus, but in reality all particles are interacting with all other particles. One can give a good approximation, and some interactions will be very weak and therefore can be neglected for most purposes but, in general, the parts of quantum systems are quantum entangled. Carroll argues that there are no such things as classical objects. There are scale thresholds above which classical descriptions start to be more efficient computationally than quantum descriptions, but the world itself is never classical; it is always quantum. There is no other option. We are made of atoms and atoms are not classical objects.

Carroll and his group have been working on trying to extract spacetime from the wave function. And this is based on an idea related to entanglement. Since 99.99% of spacetime is "empty" they ignore matter and energy for the moment. The apparently empty spacetime is, in fact, just the quantum fields in a resting state. There is never nothing. But let's call it empty spacetime. One can define a region of spacetime in terms of a subset of Hilbert Space. And if you take any region of empty spacetime, then it can be shown to experience some degree of entanglement with all the other regions nearby. In fact, the degree of entanglement is proportional to the distance. What Carroll has suggested is that we turn this on its head and define distance as a function of quantum entanglement between regions of spacetime. Spacetime would then be an emergent property of the wave function. They have not got a mathematical solution to the wave equation which achieves this, but it is an elegant philosophical overview and shows early promise. Indeed, in a much simplified theoretical universe (with its own specific Hilbert Space, but in which Schrödinger's wave equation applies), they managed to show that the degree of entanglement of a region of spacetime determined its geometry in a way that was consistent with general relativity. In other words, if the maths works out they have shown how to extract quantum gravity from just Hilbert Space and the wavefunction.

Other questions arise from this critique of Copenhagen. What is an "event"? What is an "observation"? The problem for Buddhists is that we assume that it has something to do with "consciousness" and that "consciousness" has something to do with Buddhism. The first is certainly not true, while the second is almost certainly not true depending on how we define consciousness. And defining consciousness is something that is even less consensual than interpreting the measurement problem. There are as many definitions as there are philosophers of mind. How can something so ill-defined be central to a science that is all about well-defined concepts?

More on Interpretations

In 2013, some researchers quizzed physicists at a conference about their preferred interpretation of the measurement problem. This gave rise to what Sean Carroll called The Most Embarrassing Graph in Modern Physics:

Sean Carroll comments:

 

I’ll go out on a limb to suggest that the results of this poll should be very embarrassing to physicists. Not, I hasten to add, because Copenhagen came in first, although that’s also a perspective I might want to defend (I think Copenhagen is completely ill-defined, and shouldn’t be the favorite anything of any thoughtful person). The embarrassing thing is that we don’t have agreement.

Just 42% of those surveyed preferred Copenhagen - the account of quantum mechanics they all learned as undergraduates. Mind you, Carroll's preferred interpretation, Everett, got even less at 18%. However, it may be more embarrassing than it looks, because there are multiple Everettian interpretations. And note that several existing interpretations had no supporters amongst those surveyed (the survey was not representative of the field).

In Carroll's account, Copenhagen has fatal flaws because it makes unsupportable assumptions. So what about the alternatives? I found Carroll's explanation of the Everett interpretation in this lecture quite interesting and compelling. It has the virtue of being parsimonious.

Just like other interpretations, Everett began with Hilbert Space and the Wave Equation. But he stopped there. There are no special rules for observers as classical objects because there are no classical objects (just classical descriptions). In this view, the rugby ball still both exists and does not exist, but instead of the wave function collapsing, the interaction between the ball, the field, the observer, and the world cause "decoherence". If there are two possible outcomes — ball present at this location, ball somewhere else — then both happen, but decoherence means that we only ever see one of them . The other possibility also occurs, but it is as though the world has branched into two worlds: one in which the ball is present and we kick it, and one in which it is somewhere else and we do not kick it. And it turns out that having split in this way there is no way for the two worlds to interact ever again. The two outcomes are orthogonal in Hilbert Space.

While this sounds counterintuitive, Carroll argues that the many worlds are already present in the Hilbert Space and all the other interpretations have to introduce extra rules to make those other worlds disappear. And in the case of Copenhagen, the extra rules are incoherent. Everett sounds plausible enough in itself, but given the number of particles in the universe and how many interactions there are over time, the number of worlds must be vast beyond imagining. And that is deeply counter-intuitive. However, being counter-intuitive is not an argument against a theory of quantum mechanics. Physics at this scale is always going to be counterintuitive because it's not like the world on the scale we can sense. And at this point, it will be useful to review some of the problems associated with differences in scale.

Scale (again)

I've written about scale before. It is such an important idea and so many of our misconceptions about the world at scales beyond those our senses register are because we cannot imagine very small or very large scales.

We understand our world as classical. That's what we evolved for. Modern humans have been around for roughly between 400,000 and 200,000 years. But we discovered that there are scales much smaller than we can experience with our senses only about 400 years ago with the development of the microscope. As our understanding progressed we began to see evidence of the world on smaller and smaller scales. Each time we had to adjust our notions of the universe. At the same time telescopes revealed a very much larger universe than we had ever imagined.

Quantum mechanics developed from Einstein's articles in 1905 and was formalised mathematically in the 1920s. It has never been intuitive and it is so very far from our experience that is unlikely ever to be intuitive.

Humans with good eyesight can see objects at around 0.1 mm or 100 µm. A human hair is about 20-200 µm. A small human cell like a sperm might be 10 µm, and not visible; while a large fat cell might be 100 µm and be visible (just). A water molecule is about 0.0003 µm or 0.3 nanometres (nm = 10^-9 m). But at this level, the physical dimensions of an object become problematic because the location in space is governed by quantum mechanics and is a probability. Indeed, the idea of the water molecule as an "object" is problematic. The classical description of the world breaks down at this scale. The average radius of a hydrogen atom at rest is calculated to be about 25 picometres or 25x10^-12 m, but we've already seen that the location of the electron circling the hydrogen nucleus is a probability distribution. We define the radius in terms of an arbitrary cut off in probability. The estimated radius of an electron is less than 10⁻¹⁸ m (though estimates vary wildly). And we have to specify a resting state atom, because in a state of excitation the electron probability map is a different shape. It hardly makes sense to think of the electron as having a fixed radius or even as being an object at all. An electron might best be thought of as a perturbation in the electromagnetic field.

The thing is that, as we scale down, we still think of things in terms of classical descriptions and we don't understand when classical stops applying. We cannot help but think in terms of objects, when, in fact, below the micron scale this gradually makes less and less sense. Given that everything we experience is on the macro scale, nothing beyond this scale will ever be intuitive.

As Sean Carroll says, the many worlds are inherent in Hilbert Space. Other theories have to work out how to eliminate all of the others in order to leave the one that we observe. Copenhagen argues for something called "collapse of the wave function". Why would a wave function collapse when you looked at it? Why would looking at something cause it to behave differently? What happened in the universe before there were observers? Everett argued that this is an artefact of thinking of the world in classical terms. He argued that, in effect, there is no classical world, there is only a quantum world. Subatomic particles are just manifestations of Hilbert Space and the Wave Equation. The world might appear to be classical on some scales, but this is just an appearance. The world is fundamentally quantum, all the time, and on all scales.

Thinking in these terms leads to new approaches to old problems. For example, most physicists are convinced that gravity must be quantised like other forces. Traditional approaches have followed the methods of Einstein. Einstein took the Newtonian formulation of physical laws and transformed them into relativity. Many physicists take a classical expression of gravity and attempt to reformulate it in quantum terms - leading to string theory and other problematic approaches. Carroll argues that this is unlikely to work because it is unlikely that nature begins with a classical world and then quantises it. Nature has to be quantum from the outset and thus Everett was on right track. And, if this is true, then the only approach that will succeed in describing quantum gravity will need to start with quantum theory and show how gravity emerges from it. As I say, Carroll and his team have an elegant philosophical framework for this and some promising preliminary results. The mathematics is still difficult, but they don't have the horrendous and possibly insurmountable problems of, say, string theory.

Note: for an interesting visualisation the range of scales, see The Scale of the Universe.


Conclusion

Quantum mechanics is a theory of how subatomic particles behave. It minimally involves a Hilbert Space of all possible wave functions and the Schrödinger wave equation describing how these evolve over time. Buddhism is a complex socio-religious phenomenon in which people behave in a wide variety of ways that have yet to be described with any accuracy. It's possible that there is a Hilbert Space of all possible social functions and an equation which describes how it evolves over time, but we don't have it yet!

Buddhists try to adopt quantum mechanics, or to talk about quantum mechanics, as a form of virtue signalling -- "we really are rational despite appearances", or legitimising. They either claim actual consistency between Buddhism and quantum mechanics; or they claim some kind of metaphorical similarity, usually based on the fallacy that the measurement problem requires a conscious observer. And this is patently false in both cases. It's not even that Buddhists have a superficial grasp of quantum mechanics, but that they have a wrong grasp of it or, in fact, that they have grasped something masquerading as quantum mechanics that is not quantum mechanics. None of the Buddhists I've seen talking or writing about quantum mechanics mention Hilbert Spaces, for example. I'm guessing that none of them could even begin to explain what a vector is let alone a Hilbert Space.

I've yet to see a Buddhist write about anything other than the Copenhagen interpretation. I presume because it is only the Copenhagen interpretation that is capable of being shoehorned into a narrative that suits our rhetorical purposes; I don't see any advantage to Buddhists in the Everett interpretation, for example. Buddhists read — in whacky books for whacky people — that the "observer" must be a conscious mind. Since this suits their rhetorical purposes they do not follow up and thus never discover that the idea is discredited. No one ever stops to wonder what the statement means, because if they did they'd see that it's meaningless.

Thus, Buddhists who use quantum mechanics to make Buddhism look more interesting are not concerned with the truth. They do not read widely on the subject, but simply adopt the minority view that chimes with their preconceptions and use this as a lever. For example, I cannot ever recall such rhetoric ever making clear that the cat-in-the-box thought experiment was proposed by Schrödinger to discredit the Copenhagen interpretation. It is presented as the opposite. Again, there is a lack of regard for the truth. Nor do Buddhists ever present criticisms of the Copenhagen interpretations such as those that emerge from Everett's interpretation. Other criticisms are available.

And this disregard for the truth combined with a concerted attempt to persuade an audience of some arbitrary argument is classic bullshit (as described by Harry Frankfurt). Buddhists who write about quantum mechanics are, on the whole, bullshitters. They are not concerned with the nature of reality, they are concerned with status, especially the kind of status derived from being a keeper of secret knowledge. It's past time to call out the bullshitters. They only hurt Buddhism by continuing to peddle bullshit. The irony is that the truth of Buddhism is far more interesting than the bullshit; it's just much harder to leverage for status or wealth.

~~oOo~~

Frankfurt, Harry G. On Bullshit. Princeton University Press.

For those concerned about the flood of bullshit there is an online University of Washington course Calling Bullshit.

If you have a urge to learn some real physics (as opposed to the bullshit Buddhist physics) then see Leonard Susskind's lecture series The Theoretical Minimum. This aims to teach you only what you need to know to understand and even do physics (no extraneous mathematics or concepts).