[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?

[nathan_compton]

I guess we should both read the SEP page on measurement.

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.