You can't test if quantum uses complex numbers
algassert.com
algassert.com
If you exclude non-real operations and states you are removing part of the system such that it becomes impossible to work with certain cases -- like handling non-real roots of ax^2 + bx + c polynomials.
It is possible to represent complex numbers as 2x2 matrices as those can encode 2D rotations. With the matrix formulation you are not dealing with imaginary numbers -- or you are, but they are not encoded with i = sqrt(-1) but as a 45deg rotation. IIRC, there is a formulation of Dirac's QED (Quantum ElectroDynamics) using matrices.
If you don't realize this, then you can draw conclusions that don't make sense in the space you're working with. Take a simple equation like y = -x^2 - 5, representing a thrown ball's trajectory. It never crosses zero, there are no solutions. You can't "pop into the complex numbers and find a solution" because the thing it represents is confined to the reals.
So if you find yourself reaching for complex numbers, you have to understand that the thing you are working with is no longer one-dimensional, even if that second dimension collapses back to 0 at the end.
Quantum is an odd one, as the name indicates that it deals in quantums. Minimum values that can't be divided. The difficult parts seems more to be in systems that have a probability space more than an analytical model that describes them. Which, fair, it is not a number system.
Classical probability works with real numbers (probabilities between 0 and 1). Quantum probability involves amplitudes represented by complex numbers. These amplitudes can wave-interfere with each other, leading to superposition and entanglement
The post implies flaws in the original paper, then at the very end seems to concede it was technically fine just should’ve been up front about entanglement.
The post should be edited to be more upfront about what he realized after writing it.
By assuming no entanglement, it's also not a paper about quantum. (Entanglement <=> quantum)
It should not be in Nature. It is... a homework problem, at best? To teach students not to get too excited about publishing.
Without the technical footnote, it could still have been an interesting paper. But wrong
As for homework problems, the blogpost is a better starting point for one. Maybe the blog author should be a prof, and the paper authors should become adjuncts
I think the article makes a good point when it says that nobody else he spoke to about it had noticed this qualifier either. There is a long history of papers in physics making grand claims that lead to decades of mistaken beliefs. For instance, von Neumann's mistaken proof ruling out any kind of hidden variable theory from reproducing QM. Some people still cite this incorrect proof to this day to dismiss hidden variables, 70 years later.
And once you make the caveat obvious, the paper is kind of a nothingburger that got published in Nature.
The author even admits this "is better than doing no test at all".
“ (…) you can just mimic the behavior of complex numbers using pairs of real numbers (and appropriately tweaked definitions of operations). (…) What Renou et al are actually claiming is that if you start with quantum mechanics, and then remove all operations and states involving non-real numbers, and then try to emulate what was lost using what remains, you will fail in an experimentally detectable way”
Meaning it’s actually totally possible to only use reals to encode the complex Numbers, but not to also remove all operators which do the same things as the complex numbers would.
"actually you can replace with something equivalent"
"i said just remove the math, not replacing it"
so, what's the news?
It's also a matter of reputation. Everyone knows everyone and have read a few of the previous papers (or papers of the advisor/coworker/whatever). If the previous papers were good, it's a good signal to take a look at the new result. If the previous papers were dubious, you skim it in case there is something interesting, but may just ignore it.
[1] Perhaps the exception are medical trials, but they have a lot of rules and paperwork to avoid lying, error, misrepresentations and other nasty stuff. Anyway, after reading a lot of ivermectin preprints during peak pandemic, I'm not so sure.
I'm not sure how to solve it, that avoids wasting money and time on flat earth research.
(I think the researcher time is more critical. If 10% of the money is wasted, nobody would care, some research flops anyway. If someone wasted 5 years doing flat earth research, it's difficult to get back. (I'd ask x5 of the money in exchange of the reputation damage.))
I think i saw such a warning on a casino door in LV.
Here is my viewpoint, which somehow some people find controversial: quantum theory is first and foremost a description of individual particles. To describe their time evolution, we use the Schrodinger equation:
i d_t Psi = H Psi
What is that "i" there? Oh right, the imaginary unit. So... quantum theory uses complex numbers.
Now you are free to search for another theory without the "i", and perhaps even find something that is somehow mathematically consistent. But that theory either describes experiments just as well as ordinary quantum theory, in which case it is physically equivalent and of no advantage (except to those with strong allergies to complex numbers), or it does not, and then it is wrong.
Of course the last logical possibility is that your theory might do better than quantum theory... but that is the dream only of those who do not known quantum field theory.
/rant, with apologies
As you venture further into the universe of QFT you find that you need even more exotic number like objects like spinors with their own peculiar structures, but the essence is the same: they must serve the purpose of representing probabilities and symmetries. The complex numbers in QM mean nothing at all except in that they serve these purposes.
If we wish to speak informally and wave our hands a bit we can say that it isn't so surprising that we find the complex numbers and related number like objects because the complex numbers are a promise to square something at a later date and recover a real number, which is what we need to satisfy the requirement to represent probabilities.
In fact, we can formulate classical probabilistic mechanics with complex numbers (the Koopman von Neuman operator theory) and again, they appear because we want to operate on objects living in a nice Hilbert space which also square to probabilities. In only took me 20 years to understand this, so I can sympathize with confusion.
It's a really nice book, very self-contained. I think anyone with a basic mathematical education (A-Level or equivalent) could get through it without having to read other things to acquire prerequisites, though they should be prepared to think quite hard.
1. The resemblance to the titles of Gerald Jay Sussman's "Structure and Interpretation" books appears to be coincidental. The title is meant literally: the book is split into two sections, one on the (mathematical) structure of QM and one on its (philosophical) interpretation. There are no similarities in style, pedagogy or subject matter to Sussmann's books and no use of, or reference to, programming. The author was a professor of philosophy at the University of South Carolina.
2. He actually lists a collection of alternatives for that extra ingredient, any one of which has the same effect when added.
The discussion of the EPR paradox and the Kochen-Specker Theorem was really very illuminating.
Except, most complex numbers don't square to a real number. Only those lying along the complex or real axes square to a real number; everything else just squares to another (non-real) complex number. In what way do complex numbers represent a "promise" to square it later and recover a real number? Who is making this promise? I feel like this is falling into the same trap of believing that complex numbers are not allowed to simply exist on their own merit.
I think it's quite serendipitous that the number system designed to algebraically close the reals to include roots of polynomials like x^4 + 1 happens to also cleanly describe so much of physics. There happens to be a lot of physics that boils down to "magnitude and phase" where those quantities interact in the same way complex numbers do, but it's not a-priori obvious that electromagnetism shouldn't need some third quantity as well, nor that we shouldn't be using quaternions instead, nor some other algebraic structure defined over 2D or 3D or 4D vectors.
Indeed, as you point out, there are plenty of more complicated mathematical structures that are best for describing other parts of physics, like spinors, Lie groups, and special unitary groups. It's not a-priori obvious that Lie groups should be so important to physics either. But neither should anyone protest their use as somehow not "really existing". It is true that complex numbers do not physically exist -- neither do Lie groups, and neither does the number 7. We got lucky that mathematicians had already explored an algebra that turned out to be perfect for "magnitude-and-phase" physics, but it doesn't seem like "squaring to a real number" had anything to do with why they are useful. Real numbers have no stronger claim to truly representing physics than complex numbers, spinors, or Lie groups do.
Eh, call me when your detector gives you back a complex number. Measurements return real numbers. I've never known one to return a complex valued one. Probabilities are real numbers. I feel this puts real numbers in a privileged position. If you ever wrote a theory that suggested that you lay a ruler against an object and measure a complex value, you'd be in trouble.
There are an uncountably infinite number of real numbers. 100% of them (but not all) are not computable, and cannot be written down or described. Measurements do not return "true" real numbers. Measurements return whatever the detector is designed to return. Digital measurements return binary floating-point, fixed-point, or integer numbers. Some measurements return "red" vs. "blue". Pregnancy detectors return "1 line" or "2 lines". All it would take for a detector to give a complex number is to design one that measures something that can be described as a complex number, and return it as a complex number. For example, a phasor measurement unit:
https://en.wikipedia.org/wiki/Phasor_measurement_unit
Should I call you?
When I say measurements are real I mean that displacements between objects in space are represented with real numbers.
You make a good and interesting point as to whether the actual structure of the reals, which is, as you say, pretty strange, meaningfully corresponds to relative displacements in space, but this is a separate point. If we wished to be finitist, we could argue that we measure over a sparse subset of the reals or something like that or we could define various methods of putting the rational numbers to use for this purpose. But my larger point is that in the end the physical world appears to be entirely sensible to us only as relative displacements of objects in space and these appear to map to something very like the real numbers.
In fact, physics at its most basic is encoded in these terms as well, where any system is conceptualized as being encoded by its q's, which correspond to relative displacements, and the generators of their motion, roughly speaking either the time derivatives of the q's or their conjugate momenta. But the q's are what we have to work with. At any instant in time we must lay our rulers out, one way or another, and then construct any other physical property of interest in terms of those displacements.
https://plato.stanford.edu/entries/measurement-science/
Note that Bertrand Russel is on "team Nathan" in the sense that he thinks measurements relate to (at most) the reals.
I don't think anyone ever really measures the elecromagnetic field. We might, for example, measure the displacement of a charged object attached to a spring in an electromagnetic field. Or, if the field is changing in time we measure that displacement as a function of the position of the hands on a clock. But it is very hard for me to think of a situation where we measure something other than a distance at its most basic level. Even in a DAC we measure voltage relative to some calibrated voltage which we measured using a voltmeter which shows us our answer as a deflection in a meter.
This is particularly relevant in QM because in fact all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.
I don't think anyone ever really measures the displacement of a charged object attached to a spring in an electromagnetic field. We might, for example, measure the difference in strength of neuronal activation in our brain from neurotransmitters emitted in a chain traveling from the photoreceptor cells in our fovea, which are responding to differing quantities of photoisomers that have had their shape altered by the absorption of different frequencies of photons reflected differently off the needle of a gauge in the display of an instrument which is measuring the displacement of a charged object attached to a spring in an electromagnetic field.
This is what I mean when I say it's overly reductive to say a measurement is necessarily a displacement. A measurement is lots of things, and not all of them can be represented as a spatial displacement unless you really shoehorn it.
> all the values we might measure are the eigenvalues of hermitian operators and they are, in fact, restricted to the real numbers.
Is that even true? Spin is a measurable quantity, and you cannot possibly get a spin of 0.2. Most measurable quantum numbers are essentially integers (integer multiples of some conventional base). Remember this discussion is about whether complex numbers are "lower-class citizens" than real numbers in physics. If you're going for measurable quantum numbers, these are almost all counterexamples to the idea that real numbers are special, and instead hint that it's simply integers that are the only first-class citizens in physics. I also fundamentally disagree that the possible eigenvalues of hermitian operators are the sole criteria we should be using to "rank" the truthiness or realness of mathematical structures.
I deeply do not care what philosophy has to say about this, either. Philosophy as a discipline is completely incapable of determining the truth, because it is unwilling to ever reject a single idea; all it is is a giant collection of shower thoughts. Every philosophy page, including the one you linked, somewhere includes "According to Aristotle ...". If you're trying to learn evolutionary biology and you read "according to Lamarck ...", then you are not learning science, you're reading science history. Yet this is all philosophy ever can say about anything. Science curates ideas; philosophy hoards them.
Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way.
I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.
Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.
Quantum mechanics likewise is just an approximation of quantum field theories.
You can reconstruct our modern understanding of the motion of the planets in the reference frame of a static earth and produce a mathematically equivalent path that draws out epicycles which predict the positions of planets with exactly the same accuracy as our regular formulations. You can rework the representation of the laws of gravity such that they spit out positions in this reference frame. It is an equally valid model of the cosmos, with exactly the same number of starting assumptions, it's just remarkably more complex.
Today with vastly more data and more accurate measurements you’d need effectively infinite terms, which makes it more obvious but you don’t need that level of absurdity to render judgment.
Ptolemy rejects Aristotle's cosmology which relied on perfect spherical motion. Ptolemy really did believe that the planets moved according to his model (ie it wasn't just a pure computational tool) but he was very clear that his model was based purely on mathematics. Not only did he not give a reason for why the cosmos should take this form, he openly speculates that the answer is unknowable, and works under the assumption "maybe they can move wherever they want and they just like moving this way."
Further, cycles were not added over time [1]. On day one there were 31 cycles and circles, and these were exactly the same ones being used at the time of Copernicus. You also don't need many epicycles to accurately produce a path identical to keplerian orbits. Completely arbitrary orbits can be described with finite epicycles. [2] Indeed the problem was that Ptolemy didn't fit the data by adding more epicycles, but instead through the Equant, which moved the positions of the centers of the epicycles, which meant adding more epicycles would not make it more accurate. The story of ever more epicycles being added to a bloated old theory that was streamlined by heliocentrism is a modern myth.
[1] https://diagonalargument.com/2025/05/20/from-kepler-to-ptole...
[2] https://web.math.princeton.edu/~eprywes/F22FRS/hanson_epicyc...
That’s a count of the total need to describe the motion of multiple celestial bodies.
I’m referring to the number of cycles needed to describe the motion of a single celestial body. There wasn’t enough data at high enough precision to need 17 cycles to describe the motion of a single celestial body until much later. At the time lesser precision was more common, but that someone really did go to such an extreme to create the best fit.
> Completely arbitrary orbits can be described with finite epicycles.
The number of points isn’t fixed with continuous observations. Your best fit for past data keeps needing new cycles over time unless you’re working backwards from a much better model. Even then you run into issues with earthquakes changing the length of the day etc. The basic assumptions they where working from don’t actually hold up.
Also, I’m reasonably sure you couldn’t actually write out an infinite decimal representation of the irrational number e using a finite number of epicycles. Not something I’ve really considered deeply, but it seems like an obvious counter example.
The first is overlooking the issue of overfitting using hand calculation and imperfect observations. The calculated “best fit” for the data available did involved adding a bunch of epicycles and there was no theoretical reason to avoid doing so.
The second is playing fast and loose with a fat line drawn over a squiggly line based on a better model. It’s being mathematically rigorous but intentionally deceptive. You can fairly trivially construct a set of epicycles to fit some desired shape, but working backwards from observation there’s nothing guiding you to the most elegant possible solution for a given situation.
If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.
To make the continuous case interesting as a compilation problem, you'd need some alternate formulation of the Schrodinger equation, e.g. based on the limit of small powers of unitaries rather than on the matrix exponential, so that deleting i didn't delete literally all processes. Or you could arbitrarily declare real-only hamiltonians are permitted, despite the Schrodinger equation saying "i". But that'd be kinda lame, imo.
(Note: am author of post)
Huge fan of your work!
I just started my PhD in distributed quantum computing, and my Masters was applying that framework to the QFT.
I came across a number of papers you authored in the process, as well as your blog. In particular, big fan of Kahanamoku-Meyer et al.'s optimistic QFT circuit.
Anyway, keep up the great work!