How can a particle be in two places at once? (Superposition, Again)

Image of atom

A common question for lay people confronted with counterintuitive popular narrative accounts of quantum physics is:

 

How can a particle be in two places at once?

The idea that a particle can be in two places at once is a common enough interpretation of the idea of quantum superposition, but this is not the only possible interpretation. Some physicists suggest that superposition means that we simply don't know the position, and some say that it means that the "position" is in fact smeared out into a kind of "cloud" (not an objective cloud). However, being in two places at once is an interpretation that lay people routinely encounter, and it has become firmly established in the popular imagination.

Note that while the idea is profoundly counterintuitive, physicists often scoff at intuition. Richard Feynman once said, "The universe is under no obligation to make sense to you." I suppose this is true enough, but it lets scientists off the hook to easily. The universe might be obligation-free, but science is not. I would argue precisely that science is obligated to make sense. For the first 350 years or so, science was all about making sense of empirical data. This approach was consciously rejected by people like Werner Heisenberg, Max Born, and Niels Bohr before arriving at their anti-realist conclusions.

But here's the thing. Atoms are unambiguously and unequivocally objective (their existence and properties are independent of the observer). We even have images of individual atoms now (above right). Electrons, protons, neutrons, and neutrinos are all objective entities. They exist, they persist, they take part in causal relations, and we can measure their physical properties such as mass, spin, and charge. The spectral absorption/emission lines associated with each atom are also objective.

It was the existence of emission lines, along with the photoelectric effect, that led Planck and Einstein to propose the first quantum theory of the atom. And if these lines are objective, then we expect them to have an objective cause. And since they obviously form a harmonic series we ought to associate the lines with objective standing waves. The mathematics used to describe and predict the lines does describe a standing wave, but for reasons that are still not clear to me, physicists deny that an objective standing wave is involved. The standing wave is merely a mathematical calculation tool. Quantum mechanics is an antirealist scientific theory, which is an oxymoron. 

However, we may say that if an entity like the atom in the image above has mass, then that mass has to be somewhere at all times It may be relatively concentrated or distributed with respect to the centre of mass, but it is always somewhere. Mass is not abstract. Mass is physical and objective. Mass can definitely not be in two places at once. Similarly, electrical charge is a fundamental physical property. It also has to be somewhere. If we deny these objective facts then all of physics goes down the toilet. 

Moreover, if that entity with mass and charge is not at absolute zero, then it has kinetic energy: it is moving. If it is moving, that movement has a speed and a direction (i.e. velocity). At the nanoscale, there is built-in uncertainty regarding knowing both position and velocity at the same time, but we can, for example, know precisely where an electron is when it hits a detector (at the cost of not knowing its speed and direction at that moment).

Quantum theory treats such objective physical entities as abstractions. Bohr convinced his colleagues that we cannot have a realist theory of the subatomic. It's not something anyone can describe because it's beyond our ability to sense. This was long before images of atoms were available. 

The story of how we came to have an anti-realist theory of these objective entities and their objective behaviour would take me too far from my purpose in this essay, but it's something to contemplate. Mara Beller's book Quantum Dialogue goes into this issue in detail. Specifically, she points to the covert influence of logical positivism on the entire Copenhagen group.

The proposition that a particle can be in two places at once is not only wildly counterintuitive, but it breaks one of Aristotle's principles of reasoning: the principle of noncontradiction. Which leaves logic in tatters and reduces knowledge to trivia. Lay people can only be confused by this, but I think that, secretly, many physicists are also confused.

To be clear:

• No particle has ever been observed to be in different locations at the same time. When we observe particles, they are always in one place and (for example, in a cloud chamber) appear to follow a trajectory. Neither the location nor the trajectory is described by quantum physics.
• No particle has ever been predicted to be in different locations at the same time. The Schrödinger equation simply cannot give us information about where a particle is.

So the question is, why do scientists like to say that quantum physics means that a particle can be in two places, or in two "states"*, at one time? To answer this, we need to look at the procedures that are employed in quantum mechanics and note a rather strange conclusion.

* One has to be cautious of the word "state" in this context, since it refers only to the mathematical description, not to the physical state of a system. And the distinction is seldom, if ever, noted in popular accounts.

What follows will involve some high school-level maths and physics.

The Schrödinger Equation

Heisenberg and Schrödinger developed their mathematical models to try to explain why the photons emitted by atoms have a specific quantum of energy (the spectral emission lines) rather than an arbitrary energy. Heisenberg used matrices and Schrödinger used differential equations, but the two approaches amount to the same thing. Even when discussing Schrödinger's differential equation, physicists still use matrix jargon like "eigenfunctions" indiscriminately.

The Schrödinger equation can take many forms, which does not help the layperson. However, the exact form doesn't matter for my purposes. What does matter is that they all include a Greek letter psi 𝜓. Here, 𝜓 is not a variable of the type we encounter in classical physics; it is a mathematical function. Physicists call 𝜓 the wavefunction. Let's dig into what this means.

Functions

A function, often denoted by f, is a mathematical rule. In high school mathematics, we all learn about simple algebraic functions of the type:

f(x) = x + 1

This rule says: whatever the current value of x is, take that value and add 1 to it.

So if x = 1 and we apply the rule, then f(x) = 2. If x = 2.5, then f(x) = 3.5. And so on.

A function can involve any valid mathematical operation or combinations of them. And there is no theoretical limit on how complex a function can be. I've seen functions that take up whole pages of books.

We often meet this formalism in the context of a Cartesian graph. For example, if the height of a line on a graph is proportional to its length along the x-axis, then we can express this mathematically by saying that y is a function of x. In maths notation.

y = f(x); where f (x) = x + 1.

Or simply: y = x + 1.

This particular function describes a line at +45° that crosses the y-axis at y = 1. Note also that if the height (y) and length (x) are treated as the two orthogonal sides of a right-triangle, then we can begin to use trigonometry to describe how they change in relation to each other. Additionally, we can treat (x,y) as a matrix or as the description of a vector.

In physics, we would physically interpret an expression like y = x + 1 as showing how the value of y is proportional to the value of x. We also use calculus to show how one variable changes over time with respect to another, but I needn't to go into this.

Wavefunctions and Hilbert Spaces

The wavefunction 𝜓 is a mathematical rule (where 𝜓 is the Greek letter psi, pronounced like "sigh"). If we specify it in terms of location on the x-axis, 𝜓(x) gives us one complex number (ai + b; where i = √-1) for every possible value of x. And unless otherwise specified, x can be any real number, which we write as x ∈ ℝ (which we read as "x is a member of the set of real numbers"). In practice, we usually specify a limited range of values for x.

All the values of 𝜓(x), taken together, can be considered to define a vector in an abstract notional "space" we call a Hilbert space, after the mathematician David Hilbert. The quantum Hilbert space has as many dimensions as there are values of x, and since x ∈ ℝ, this means it has infinitely many dimensions. While this seems insane at first glance, since a "space" with infinitely many dimensions would be totally unwieldy, in fact, it allows physicists to treat 𝜓(x) as a single mathematical object and do maths with it. It is this property that allows us to talk about operations like adding two wavefunctions (which becomes important below).

We have to be careful here. In quantum mechanics, 𝜓 does not describe a objective, physical wave in space. Hilbert space is not an objective space. This is all just abstract mathematics. Moreover, there isn’t an a priori universal Hilbert space containing every possible 𝜓. Every system produces a distinct abstract space. 

That said, Sean Carroll and other proponents of the so-called "Many Worlds" interpretation first take the step of defining the system of interest as "the entire universe" and notionally assign this system a wavefunction 𝜓_universe. However, there is no way to write down an actual mathematical function for such an entity since it would have infinitely many variables. Even if we could write it down, there is no way to compute any results from such a function: it has no practical value. In gaining a realist ontology, we lose all ability to get information without introducing massive simplifications. Formally, you can define a universal 𝜓. But in practice, to get predictions, you always reduce to a local system, which is nothing other than ordinary quantum mechanics without the Many Worlds metaphysical overlay. So in practice, Many Worlds offers no advantage over "shut up and calculate". And since the Many Worlds ontology is extremely bizarre, I fail to see the attraction.

It is axiomatic for the standard textbook approach to quantum mechanics—deriving from the so-called "Copenhagen interpretation"—that there is no objective interpretation of 𝜓. Neutrally, we may say that the maths needn't correspond to anything in the world, it just happens to give the right answers. The maths itself is agnostic; it doesn't require any physical interpretation. Bohr and co positivistically insisted that it's not possible to have a physical interpretation because we cannot know the world on that scale.

As readers likely know, the physics community is deeply divided over (a) the possibility of realist interpretations, i.e. the issue of 𝜓-ontology and (b) which, if any, realist interpretation of 𝜓 is the right one. There is a vast amount of confusion and disagreement amongst physicists themselves over what the maths represents, which does not help the layperson at all. But again, we can skip over this and stay focussed on the goal.

The Schrödinger equation in Practice

To make use of the Schrödinger equation, a physicist must carefully consider what kind of system they are interested in and define 𝜓 so that it describes that system. Obviously, this selection is crucial for getting accurate results. And this is a point we have to come back to.

When we set out to model an electron in a hydrogen atom, for example, we have to choose an expression for 𝜓 whose outputs correspond to the abstract mathematical "state" of that electron. There's no point in choosing some other expression, because it won't give accurate results. Ideally, there is one and only one expression that perfectly describes the system, but in practice, there may be many others that approximate it.

For the sake of this essay, I will discuss the case in which 𝜓 is a function of location. In one dimension, we can state this as: 𝜓(x). When working in three spatial and one time dimensions, for technical reasons, we use spherical spatial coordinates, which are two angles and a length, as well as time: 𝜓(φ,θ,x,t). The three-dimensional maths is challenging, and physicists are not generally required to be able prove the theorem. They only need to know how to apply the end results.

Schrödinger himself began by describing an electron trapped in a one-dimensional box, as perhaps the simplest example of a quantum system (this is an example of a spherical cow approximation). This is very often the first actual calculation that students of quantum mechanics perform. How do we choose the correct expression for this system? In practice, this (somewhat ironically) can involve using approximations derived from classical physics, as well as some trial and error.

We know the the electron is a wave and so we expect it to oscillate with something like harmonic motion. In simple harmonic motion, the height of the wave on the y-axis changes as the sine of the position of the particle on the x-axis.

One of the simplest equations that satisfies our requirements, therefore, would be 𝜓(x) = sin x, though we must specify lower and upper limits for x reflecting the scale of the box.

However, it is not enough to specify the wavefunction and solve it as we might do in wave mechanics. Rather, we first need to do another procedure. We apply an operator to the wavefunction.

Just as a function is a rule applied to a number to produce another number, an operator is a rule applied to a function that produces another function. In this method, we identify operators by giving them a "hat".

So, if p is momentum (for historical reasons), then the operator that we apply to the wavefunction so that it gives us information about momentum is p̂. And we can express this application as p̂𝜓. For my purposes, further details on operators (including Dirac notation) don't matter. However, we may say that this is a powerful mathematical approach that allows us to extract information about any measurable property for which an operator can be defined, from just one underlying function. It's actually pretty cool.

There is one more step, which is applying the Born rule. Again, for the purposes of this essay, we don't need to say more about this, except that when we solve p̂ψ, the result is a vector (a quantity + a direction). The length of this vector is proportional to the probability that, when we make a measurement at x, we will find momentum p. And applying the Born rule gives us the actual probability.

So the procedure for using the Schrödinger equation has several steps. Using the example of 𝜓(x), and finding the momentum p at some location x, we get something like this:

• Identify an appropriate mathematical expression for the wavefunction 𝜓(x).
• Apply the momentum operator p̂𝜓(x).
• Solve the resulting function (which gives us a vector).
• Apply the Born Rule to obtain a probability.

So far so good (I hope).

To address the question—How can a particle be in two places at once?—we need to go back to step one.

Superposition is Neither Super nor Related to Position.

It is de rigueur to portray superposition as a description of a physical situation, but this is not what was intended. For example, Dirac's famous quantum mechanics textbook presents superposition as an a priori requirement of the theory, not a consequence of it. Any wavefunction 𝜓 must, by definition, be capable of being written as a combination of two or more other wavefunctions: 𝜓 = 𝜓₁ + 𝜓₂. Dirac simply stated this as an axiom. He offers no proof, no evidence, no argument, and no rationale.

We might do this with a problem where using one 𝜓 results in overly complicated maths. For example it's common to treat the double-slit experiment as two distinct systems involving slit 1 and slit 2. For example, we might say that 𝜓₁ describes a particle going only through slit 1, and 𝜓₂ describes a particle going through slit 2. The standard defence in this context looks like this:

• The interference pattern is real.
• The calculation that predicts it more or less requires 𝜓 = 𝜓₁ + 𝜓₂.
• Therefore, the physical state of the system before measurement must somehow correspond to 𝜓₁ + 𝜓₂.

But the last step is exactly the kind of logic that quantum mechanics itself has forbidden. We cannot say what the state of the system is prior to measuring it. Ergo, we cannot say where the particle is before we measure it and we definitely cannot say its in two places at once.

To be clear, 𝜓 = 𝜓₁ + 𝜓₂ is a purely mathematical exercise that has no physical objective counterpart. According to the formalism, 𝜓 is not an objective wave. So how can 𝜓₁ + 𝜓₂ have any objective meaning? It cannot. Anything said about a particle "being in multiple states at once", or "taking both/many paths", or "being in two places at once" is all just interpretive speculation. We don't know. And the historically dominant paradigm tells us that we cannot know and we should not even ask.

To be clear, the Schrödinger does not and cannot tell us what happens during the double slit experiment. It can only tell us the probable outcome. The fact that the objective effect appears to be caused by interference and the mathematical formalism involves 𝜓₁ + 𝜓₂ is entirely coincidental (according to the dominant paradigm).

Dirac has fully embraced the idea that quantum mechanics is purely about calculating probabilities and that it is not any kind of physical description. A physical description of matter on the sub-atomic scale is not possible in this view. And his goal does not involve providing any such thing. His goal is only to perfect and canonise the mathematics which Heisenberg and Born had presented as a fait accompli in 1927:

“We regard quantum mechanics as a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification.”—Report delivered at the 1927 Solvay Conference.

I noted above that we have to specify some expression for 𝜓 that makes sense for the system of interest. If the expression is for some kind of harmonic motion, then we must specify things like the amplitude, frequency, direction of travel, and phase. Our choices here are not, and cannot be, derived from first principles. Rather, they must be arbitrarily specified by the physicist.

Now, there are an almost infinite number of expressions of the type 𝜓(x) = sin (x). We can specify amplitude, etc., to any arbitrary level of detail.

• The function 𝜓(x) = 2 sin (x) will have twice the amplitude.
• The function 𝜓(x) = sin (2x) will have twice the frequency.
• The function 𝜓(x) = sin (-x) will travel in the opposite direction.

And so on.

A physicist may use general knowledge and a variety of rules of thumb to decide which exact function suits their purposes. As noted, this may involve using approximations derived from classical physics. We need to be clear that nothing in the quantum mechanical formalism can tell us where a particle is at a given time or when it will arrive at a given location. Whoever is doing the calculation has to supply this information.

Obviously, there are very many expressions that could be used. But in the final analysis, we need to decide which expression is ideal, or most nearly so. 

For a function like 𝜓(x) = sin (x), for example, we can add some variables: 𝜓(x) = A sin (kx). Where A can be understood as a scaling factor for amplitude, and k as a scaling factor for frequency. Both A and k can be any real number (A ∈ ℝ and k ∈ ℝ).

Even this very simple example clearly has an infinite number of possible variations since ℝ is an infinite set. There are infinitely many possible functions 𝜓₁, 𝜓₂, 𝜓₃, ... 𝜓_∞. Moreover, because of the nature of the mathematics involved, if 𝜓₁ and 𝜓₂ are both valid functions, then 𝜓₁ + 𝜓₂ is also a valid function. It was this property of linear differential equations that Dirac sought to canonise as superposition.

To my mind, there is an epistemic problem in that we have to identify the ideal expression from amongst the infinite possibilities. And having chosen one expression, we then perform a calculation, and it outputs probabilities for measurable quantities.

The 𝜓-ontologists try to turn this into a metaphysical problem. Sean Carroll likes to say "the wavefunction is real". 𝜓-ontologists then make the move that causes all the problems, i.e. they speculatively assert that the system is in all of these states until we specify (or measure) one. And thus "superposition" goes from being a mathematical abstraction to being an objective phenomena, and its only one more step to saying things like "a particle can be in two places at once". 

I hope I've shown that such statements are incoherent at face value. But I hope I've also made clear that such claims are incoherent in terms of quantum theory itself, since the Schrödinger equation can never under any circumstances tell us where a particle is, only the probability of finding it in some volume of space that we have to specify in advance. 

Conclusion

The idea that a particle can be in two places at once is clearly nonsense even by the criteria of the quantum mechanics formalism itself. The whole point of denying the relevance of realism was to avoid making definite statements about what is physically happening on a scale that we can neither see nor imagine (according to the logical positivists).

So coming up with a definite, objective interpretation—like particles that are in two places at once—flies in the face of the whole enterprise of quantum mechanics. The fact that the conclusion is bizarre is incidental since it is incoherent to begin with.

The problem is that while particles are objective; our theory is entirely abstract. Particles have mass. Mass is not an abstraction; mass has to be somewhere. So we need an objective theory to describe this. Quantum mechanics is simply not that theory. And nor is quantum field theory. 

I'm told that mathematically, Dirac's canonisation of superposition was a necessary move. And to be fair, the calculations do work as advertised. One can accurately and precisely calculate probabilities with this method. But no one has any idea what this means in physical terms, no one knows why it works or what causes the phenomena it is supposed to describe. When Richard Feynman said "No one understands quantum mechanics", this is what he mean. And nothing has changed since he said it.

It would help if scientists themselves could stop saying stupid things like "particles can be in two places at once". No, particles cannot be in two places at once, and nothing about quantum mechanics makes this true. There is simply no way for quantum mathematics, as we currently understand it, to tell us anything at all about where a particle is. The location of interest is something that the physicist doing the calculation has to supply for the Schrödinger equation, not something the equation can tell us (unlike in classical mechanics).

And if the equation cannot tell us the location of the particle, under any circumstances, then it certainly cannot tell us that it is in two places or many places. Simple logic alone tells us this much.

The Schrödinger equation can only provide us with probabilities. While there are a number of possible mathematical "states" the particle can be in, we do not know which one it is in until we measure it.

If we take Dirac and co at face value, then stating any pre-measurement physical fact is simply a contradiction in terms. Pretending that this is not problematic is itself a major problem. Had we been making steady progress towards some kind of resolution, it might be less ridiculous. But the fact is that a century has passed since quantum mechanics was proposed and physicists still have no idea how or why it works but still accept that "the fundamental physical and mathematical hypotheses are no longer susceptible of modification."

Feynman might have been right when he said that the universe is not obligated to make sense. But the fact is that, science is obligated to make sense. That used to be the whole point of science, and still is in every other branch of science other than quantum mechanics. No one says of evolutionary theory, for example, that it is all a mysterious blackbox that we cannot possibly understand. And no one would accept this as an answer. Indeed, a famous cartoon by Sydney Harris gently mocks this attitude...

The many metaphysical speculations that are termed "interpretations of quantum mechanics" all take the mathematical formalism that explicitly divorces quantum mechanics from realism as canonical and inviolable. And then they all fail miserably to say anything at all about reality. And this is where we are.

It is disappointing, to say the least.

~~Φ~~

Observations and Superpositions

The role of observation in events has been a staple of quantum physics for decades and is closely associated with "the Copenhagen interpretation". On closer inspection, it turns out that everyone connected with Bohr's lab in Copenhagen had a slightly different view on how to interpret the Schrödinger equation. Worse, those who go back and look at Bohr's publications nowadays tend to confess that they cannot tell what Bohr's view was. For example, Adam Becker speaking to Sean Carroll (time index 21:21; emphasis added):

I don't think that there is any single Copenhagen interpretation. And while Niels Bohr and Max Born and Pauli, and Heisenberg and the others may have each had their own individual positions. I don't think that you can combine all of those to make something coherent...

...Speaking of people being mad at me, this is something that some people are mad at me for, they say, "But you said the Niels Bohr had this position?" I'm like, "No, I didn't, I didn't say that Niels Bohr had any position. I don't know what position he had and neither does anybody else."

So we should be cautious about claims made for "the Copenhagen interpretation", which seem to imply a consensus that never existed at Bohr's lab in Copenhagen.

That said, the idea that observation causes the wavefunction to collapse is still a staple of quantum physics. Despite playing a central role in quantum physics, "observation" is seldom precisely defined in scientific terms, or when it is defined, it doesn't involve any actual observation (I'll come back to this). The situation was made considerably worse when (Nobel laureate) Eugene Wigner speculated that it is "consciousness" that collapses the wave function. "Consciousness" is even less well-defined than "observation". While most academic physicists instantly rejected the role of consciousness in events, outside of physics it became a popular element of science folklore and New Ageism.

The idea that "observation" or "consciousness" are involved in "collapsing the wave function" is also an attachment point for Buddhists who wish to bolster their shaky faith by aligning it with science. The result of such legitimisation strategies is rather pathetic hand waving. Many Buddhists want reality to be reductive and idealist: they want "mind" to be the fundamental substance of the universe. This would align with some modern interpretations of traditional Buddhist beliefs about mind. But the idea is also to find some rational justification for Buddhist superstitions like karma and rebirth. As I showed at length in my book Karma and Rebirth Reconsidered, it simply does not work.

In this essay, I will show that it is trivially impossible for observation to play any role in causation at any level. I'm going to start by defining observation with respect to a person and exploring the implications of this, particularly with respect to Schrödinger's cat. I will also consider the post hoc rationalisation of observation qua "interaction" (sans any actual observation).

What is "An Observation"?

We may say that an observer, Alice, observes a process P giving rise to an event E, with an outcome O, when they become aware of P, E and O. It is possible to be aware of each part individually, but in order to understand and explain what has happened, we really need to have some idea of what processes were involved, what kinds of events it engendered, and the specific outcomes of those events. 

It's instructive to ask, "How does Alice become aware of external events?" Information from the process, event, and/or outcome of interest first has to reach her in some form. The fastest way that this can happen is for light from the process, event, and/or outcome to reach Alice's eyes. It always takes a finite amount of time for the light to reach her eye.

But light reaching Alice's eye alone does not create awareness. Rather, cells in the eye convert the energy of light into electrochemical energy (a nerve impulse). That pulse of energy travels along the optic nerve to the brain and is incorporated into our virtual world model and then, finally, presented to the first person perspective. Only then we become aware of it. And this part also takes a finite amount of time. Indeed, this part takes a lot more time than the light travelling.

Therefore, the time at which Alice becomes aware of P, E, and O, is some appreciable amount of time after E happens and O is already fixed. There is no alternative definition of "observation" that avoids this limitation, since information cannot travel faster than the speed of light and the brain is always involved. The only other possibilities are, if anything, slower. Therefore:

Alice can only observe processes, events, and outcomes after the fact.

If observation is always after the fact, then observation can never play any causal role in the sequence of events because causes must precede effects, in all frames of reference. Therefore:

Observation can play no causal role
in processes, events, or outcomes.

This means that there is no way that "observation" (or "consciousness") can cause the collapse the wavefunction. Rather, the collapse of the wavefunction has to occur first, then the light from that event has to travel to Alice's eye. There is no way around this physical limitation in our universe. And given the nature of wavefunctions—the outputs if which are vectors in a complex plane—this can hardly be surprising. 

Observation is never instantaneous let alone precognitive. And this means that all talk of observation causing "wavefunctions to collapse" is trivially false.

We could simply leave it at that, but it will be instructive to re-examine the best known application of "observation".

Schrödinger's cat

Schrödinger's cat is only ever alive or dead. It is never both alive and dead. This was the point that Schrödinger attempted to make. Aristotle's law of noncontradiction applies: an object cannot both exist and not exist at the same time. We cannot prove this axiom from first principles, but if we don't accept it as an axiom, it renders all communication pointless. No matter what true statement I may state, anyone can assert that the opposite is also true.

Schrödinger proposed his thought experiment as a reductio ad absurdum argument against Bohr and the others in Copenhagen. He was trying to show that belief in quantum superpositions leads to absurd, illogical consequences. He was right, in my opinion, but he did not win the argument (and nor will I).

This argument is broadly misunderstood outside of academic physics. This is because Schrödinger's criticism was taken up by physicists as an exemplification of the very effect it was intended to debunk. "Yes," cried the fans of Copenhagen type explanations, "this idea of both-alive-and-dead at the same time is exactly what we mean. Thanks." And so we got stuck with the idea that the cat is both alive and dead at the same time (which is nonsense). Poor old Schrödinger, he hated this idea (and didn't like cats) and now it is indelibly associated with him.

The general set up of the Schrödinger's cat thought experiment is that a cat is placed in a box. Inside the box, a random event may occur. If it occurs, the event triggers the death of the cat via a nefarious contraption. Once the cat is in the box, Alice doesn't know whether the cat is alive or dead. The cat is a metaphor for subatomic particles. We are supposed to believe that they adopt a physical superposition of states: say, "spin up" and "spin down", or "position x" and "position y" at the same time before we measure them, then at the point of measurement, they randomly adopt one or the other of the superposed states.

Here's the thing. The cat goes into the box alive. If the event happens, the cat dies. If it doesn't happen the cat lives. And Alice doesn't know which until she opens the box. The uncertainty here is not metaphysical, it's epistemic. It's not that a cat can even be in a state of both-alive-and-dead, it cannot; it's only that we don't know whether it is alive or dead. So this is a bad analogy.

Moreover, even when Alice opens the box, the light from the cat still takes some time to reach her eyes. Observation always trails behind events, it cannot anticipate or participate in events. Apart from reflected light, nothing is coming out from Alice that could participate in the sequence of events happening outside her body, let alone change the outcome.

Also, the cat has eyes and a brain. It is itself an "observer". 

Epistemic uncertainty cannot be mapped back to metaphysical uncertainty without doing violence to reason. A statement, "I don't know whether the cat is alive or dead," cannot be taken to imply that the cat is both alive and dead. This is definitely a category error for cats. Schrödinger's view was that it is also a category error for electrons and photons. And again, I agree with Schrödinger (and Einstein).

In that case, why do physics textbooks still insist on the nonsensical both-alive-and-dead scenario? It seems to be related to a built-in feature of the mathematics of spherical standing waves, which are at the heart of Schrödinger's equation (and many other features of modern science). The mathematics of standing waves was developed in the 18th century (i.e. it is thoroughly classical). Below, I quote from the Mathworld article on Laplace's equation (for a spherical standing wave) by Eric Weisstein (2025. Emphasis added)

A function psi which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

The last sentence of this passage is similar to a frequently encountered claim in quantum physics. That is to say, the fact that solutions for individual quantum states can be added together and produce another valid solution for the wave equation. This is made out to be a special feature of quantum mechanics that defines the superposition of "particles".

Superposition of waves is nothing remarkable or "weird". Any time two water waves meet, for example, they superpose.

In this image, two wave fronts travel towards the viewer obliquely from the left and right at the same time (the appear to meet almost at right angles). The two waves create an interference pattern (the cross in the foreground) where the two waves are superposed. Waves routinely superpose. And this is known as the superposition principle.

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually."
The Penguin Dictionary of Physics.

For this type of linear function, we can define superposition precisely: f(x) + f(y) = f(x+y)

In mathematical terms, each actual wave can be thought of as a solution to a wave equation. The sum of the waves must also be a solution because of the situation we see in the image, i.e. two waves physically adding together where they overlap, while at the same time retaining their identity.

I've now identified three universal properties of spherical standing waves that are frequently presented as special features of quantum physics:

• quantisation of energy
• harmonics = higher energy states (aka orbitals)
• superposition (of waves)

These structural properties of standing waves are not "secret", but they are almost always left out of narrative accounts of quantum physics. And yet, these are important intuitions to bring to bear when applying wave mechanics to describing real systems.

Something else to keep in mind is that "quantisation" is an ad hoc assumption in quantum physics. It's postulated to be a fundamental feature of all quantum fields. The only problem is that all of the physical fields we know of—which is to say the fields we can actually measure—are smooth and continuous across spacetime: including gravitational fields and electromagnetic fields. Scientists have imagined discontinuous or quantized fields, but they have never actually seen one.

Moreover, as far as I know, the only physical mechanism in our universe that is known to quantize energy, create harmonics, and allow for superposition is the standing wave. The logical deduction from these facts is that it is the standing wave structure of the atom that quantizes the energy of electrons and photons and creates electron orbitals. 

Quantization is a structural property of atoms, not a substantial property of fields. (Or more conventionally and less precisely, quantization is an emergent property, not a fundamental property). 

Also, as I have already explained, the coexistence of probabilities always occurs before any event, and those probabilities always collapse at the point when an event has a definite outcome. There is nothing "weird" about this; it's not a "problem". What is weird, is the idea that hypostatizing and reifying probabilities leads to some meaningful metaphysics. It has not, and it will not.

While the superposition of waves or probabilities is an everyday occurrence. The superposition of physical objects is another story. Physical objects occupy space in an exclusive way: if one object is in that location, no other physical object can also be in that location. Physical objects cannot superpose and they are never observed to be superposed. And yet, the superposition of point particles is how physicists continue to explain the electron in an atom.

The electric field has been measured and it is found to be smooth and continuous in spacetime. Just as predicted by Maxwell. Given this, simple logic and basic geometry dictates that if—

1. the electrostatic field of the proton has spherical symmetry, and
2. a hydrogen atom is electrostatically neutral, and
3. the neutrality is assumed to be the result of the electron's electrostatic field,

—then the electron can only be in one configuration: it must be a sphere (or a close approximation of a sphere) completely surrounding the proton. This is the only way to ensure that all the field lines emerging from the proton terminate at the electron. Otherwise there are unbalanced forces - a net charge rather than neutrality. And a changing electric field dissipates energy, which electrons do not. 

Unbalanced forces

Now, if the electron is both a wave and a sphere, then the electron can only be a spherical standing wave. The Bohr model of the atom was incorrect and it surprises me greatly that this problem was not identified at the time. 

And if the electron is a spherical standing wave then, because these are universal features of standing waves, we expect:

1. The energy of the electron in the H atom will be quantised.
2. The electron will form harmonics corresponding to higher energy states and it will jump between them when it absorbs or emits photons.
3. When two electron waves intersect, the sum of their amplitudes is also a solution to the wave equation.

Moreover, we can now take pictures of atoms using electron microscopes. Atoms are physical objects. In every single picture, atoms appear to be approximately spherical.

And yet mainstream quantum models do not quite treat atoms as real. Quantum physics is nowadays all about probabilities. The problem is that, as I established in an earlier essay, a probability cannot possibly balance an electrostatic field to create a neutral atom. Only a real electric field can do this. Schrödinger was right to be unconvinced by the probability interpretation, even if it works. But he was wrong about modelling a particle as a wave. 

Waves are observed to superpose all the time. Solid objects are never observed to do so. The only reason we even consider superposition for "particles" is the wave-particle duality postulate, which we now know to be inaccurate. "Particles" are waves.

As I understand it, the idea that our universe consists of 17 fields in which particles are "excitations" is a widely accepted postulate. And as such, one might have expected scientists to go back over the physics predicated on wave-particle duality and recast it in terms of only waves. Having the wave equation describe a wave would be a start.

I digress. Clearly the idea that observers influence outcomes is trivially false. So now we must turn to the common fudge of removing the observer from the observation.

Interaction as Observation

One way around the problems with observation, is to redefine "observation" so that it excludes actual observations and observers. The move is to redefine "observation" to mean "some physical interaction". I'm sure I've mentioned this before because I used to think this was a good idea.

While we teach quantum physics in terms of isolated "particles" in empty, flat space, the fact is that the universe is crammed with matter and energy, especially in our part of the universe. Everything is interacting with everything that it can interact with, simultaneously in all the ways that it can interact, at every moment that it is possible to interact. Nothing in reality is ever simple.

In classical physics, we are used to being able to isolate experiments and exclude variables. This cannot ever happen at the nanoscale and below. An electron, for example, is surrounded by an electrostatic field which interacts with the fields around all other wavicles, near and far.

Electrons, for example, are all constantly pushing against each other via the electromagnetic force. If your apparatus contains electrons, their fields invariably interact with the electron you wish to study. This includes mirrors, beam-splitters, prisms, diffraction gratings, and double slits. The apparatus is not "classical", it's part of the quantum system you study. At the nanoscale and below, there are no neutral apparatus. 

Therefore, the idea that interaction causes the wavefunction to "collapse" is also untenable because in the real world wavicles are always interacting. In an H atom, for example, the electron and the proton are constantly and intensely interacting via the electromagnetic force. So the electron in an H atom could never be in a superposition.

Conclusions

Observation can only occur after the fact and is limited by the speed of light (or speed of causality).

Neither "observation" nor "consciousness" can play any role in the sequence of events, let alone a causal role.

Schrödinger's cat is never both alive and dead. And observation makes no difference to this (because observation can only ever be post hoc and acausal).

It is always the case, no matter what kind of system we are talking about, that probabilities for all possibilities coexist prior to an event and collapse as the event produces a specific outcome. But this is in no way analogous to waves superposing and should not be called "superposition".

All (linear) waves can superpose. All standing waves are quantised. All standing waves have harmonics.

Defining observation so as to eliminate the observer doesn't help as much as physicists might wish.

"Observation" is irrelevant to how we formulate physics.

The wave-particle duality postulate is still built into quantum mechanics, despite being known to be false.

For the last century, quantum physicists have been trying to change reality to fit their theory. Many different kinds of reality have been proposed to account for quantum theory: Copenhagen, Many Worlds, Qbism, etc. I submit that proposing a wholly different reality to account for your theory is tantamount to insanity. The success in predicting probabilities seems to have causes physicists to abandon science. I don't get it, and I don't like it. 

~~Φ~~

Bibliography

Weisstein, Eric W. (2025) "Laplace's Equation." MathWorld. https://mathworld.wolfram.com/LaplacesEquation.html