[prof-dr-ir]

Frankly I am so tired of this whole branch of research where people try to be foundational about "quantum theory" but at the same time boil it down to qubits, gates, bell tests and, well, two-by-two matrices.

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

[nathan_compton]

There is really nothing to the appearance of complex numbers in QM. In QM we must design wave functions which do the double duty of representing the probability of measurement outcomes AND capture the symmetries implicit in the system related to the fact that there are degrees of freedom between preparation of a state and measurement (for example, we may rotate our detector any way we wish before we make a measurement of a particle in a given prepared spin state). To accomplish this we need some number-like objects to denote our wave function in that square to real numbers but have enough structure to represent (in this case) the rotations.

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.

[omnicognate]

It's a long time since I read it, but there's a book called "The Structure and Interpretation of Quantum Mechanics" [1] by R. I. G. Hughes. The "Structure" part of it begins by building up most of the mathematical framework (including use of complex numbers, Hilbert spaces, operators, etc), motivated only by the desire to build a physical theory that is probabilistic in nature. It then shows how you can add one extra ingredient that turns the framework into that used for quantum mechanics [2]. I assume that everything discussed up to that point applies equally to Koopman-von Neumann.

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.

[antognini]

It's nice to see this reference. I'm currently reading it and about halfway through (making my way through the chapter on Quantum Logic).

The discussion of the EPR paradox and the Kochen-Specker Theorem was really very illuminating.

[nathan_compton]

It is one of my favorites.

[feoren]

> complex numbers are a promise to square something at a later date and recover a real number

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.

[rsfern]

I think this is just loose terminology, instead of squaring they should have said “multiply by the complex conjugate”, which is what you do to quantum mechanical wavefunctions to obtain real-valued probability amplitudes

[nathan_compton]

> 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.

[feoren]

> Eh, call me when your detector gives you back a complex number. Measurements return real numbers.

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?

[nathan_compton]

I think there is a credible case to be made that all we ever actually measure is relative displacements in space. We design objects (physical or mathematical) to convert these displacements into quantities or units of interest and might even decorate such with some additional structure beyond the reals, but in the end, we are measuring distances relative to a standard. This account becomes somewhat tricky when digital and/or electronic measurements are taken into account, but goes through, I believe.

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.

[feoren]

A "measurement" in the general sense (not quantum) is about amplifying or isolating some signal out of all the noise. I simply don't agree that all measurements boil down to relative displacements. Did a chemical reaction occur or not? Is it red or is it blue? Sure, in many of these measurements, something tangentially related to distance (like light wavelength) is involved, but lots of things are involved: measurements are inherently "macro" phenomena, which means many processes will be involved, not the least of which is electrochemical signaling in the brain of the measurer. It's just far too reductive to say that every quantity we measure is actually relative displacement. I could just as justifiably (or more so) say that every measurement is actually just measuring the strength of an electromagnetic field. Why is this helping? What does this have to do with which numbers "truly exist" or not?

[jjk166]

The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness.

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.

[ogogmad]

The phase space formulation of QM uses less complex numbers than the Schrodinger one: it models states using quasi-probability distributions, where the "probabilities" behave in all the usual ways except they can go negative. Interestingly, the classical limit of this (that is, when h goes to zero) still has negative probabilities in it.

[Retric]

The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data.

Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.

[jjk166]

And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity.

Quantum mechanics likewise is just an approximation of quantum field theories.

[Retric]

It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.

[jjk166]

Epicycles wasn't a theory, it was a model. It did not try to explain why the planets moved in the sky as they did, it only predicted where they'd be. Neither, for that matter, were copernican or keplerian mechanics theories. They too required unending tweaking because they also were only approximations of what was actually happening. For the first few centuries after heliocentrism was proposed, it gave worse results, and demanded more tweaking. What really won people over was that the phases of the moons of jupiter were accurately predicted by the model as well. The only way to achieve that result with epicycles was to rearrange everything to be mathematically equivalent to a heliocentric model.

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.

[Retric]

It started from an actual theory based around the assumption that spherical motion was perfect. They needed 2 which did actually work for a while, eventually the most accurate model needed ~17 with people giving up on the underlying theory as the number of terms destroyed the initial idea.

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.

[IcyWindows]

More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were.

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.

[Strilanc]

Yes, the post is focusing on the overall effect of operations (unitaries) rather than their continuous trajectories (hamiltonians acting on system via Schrodinger equation) (analogous to working with impulses rather than forces).

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)

[wasabi991011]

Gidney, that's you?

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!

[prof-dr-ir]

I made a related comment about two years ago:

https://news.ycombinator.com/item?id=38255476