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Comment by voldacar | Hacker News Reader

>the underlying quantum states and governing equations involve imaginary numbers, and there’s no simple way to remove them

Sure you can. Just replace any complex number a+bi with the 2x2 real matrix of the form [[a,-b],[b,a]]. It is easy to show that these matrices form a field under + and * that behaves identically to the complex numbers.* There you go, no more scary square roots of negative numbers.

I honestly don't understand why complex numbers make people so angry. They are just a way of writing a linear map / element of a particular vector space

* You can actually do this for all sort of algebraic objects, most notably groups. I.e. you can "simulate" group operations by doing matrix multiplication on a specific set of matrices, each of which "represents" an element of the group or algebraic structure. This is known as representation theory

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>> I honestly don't understand why complex numbers make people so angry.

I think the reason is very simple: the name. Every time you talk about "real" numbers and "imaginary" numbers, you are telling people implicitly that there is something less legitimate about the imaginary numbers. To put it another way, you're saying that they're not real, when all you really mean is that they're not Real.

Some people just get it quickly: names are just labels, and they don't matter. Others take a long time to realize that, and some never do. Some of us are hard wired to read into the names we give things.

To be fair, they aren't ordered, which some people consider to be the most essential property of a number.

The transfinite numbers, hyperreal numbers, and surreal numbers each have a total ordering -- a transitive antisymmetric relation. But the complex "numbers" do not have this.

The "complex field" would be a better name.

https://en.m.wikipedia.org/wiki/Surreal_number

https://en.m.wikipedia.org/wiki/Transfinite_number

https://en.m.wikipedia.org/wiki/Hyperreal_number

These are not scary enough.

https://en.wikipedia.org/wiki/Schizophrenic_number

Imaginary numbers have an order. 1i is less than 2i for example. You are thinking of complex numbers.

Yeah the shitty terminology can be blamed on Descartes. I think Gauss wanted to rename them "lateral numbers" but I guess that never stuck

Wow, genius. This would have personally helped immensely

This is a strange train of thought, isn't the entire field of mathematics a conceptual framework? i.e it's entirely imaginary.

Not entirely.

Some are searching for signs of ET or God in irrational numbers: https://en.wikipedia.org/wiki/Schizophrenic_number

Your evidence that it isn’t completely made up is people are using it to search for other things that are completely made up?

Their paper and this article both explain why this naive approach fails. Read at least one of them before assuming your simplistic and completely incorrect assessment satisfies the requirements of the underlying physics. This idea and any such idea fails - that is why their work is worth understanding. Simply mapping one vector space to another of a different dimension doesn't satisfy other requirements that the physics requires.

These authors aren't complete idiots.

The article takes a pretty simple view of these approaches. E.g. in the “too many dimensions” example, the problem is that they’re taking tensor products over R, when they should be taking tensor products over this matrix algebra.

More generally, you can always obscure the language of complex numbers in real algebra terms (I.e. replace every instance of “C” with “R[x]/(x^2+1)”), so whatever they proved must make some assumptions about what exactly a “real-valued” formulation of quantum mechanics is.

Look, I've a PhD in algebraic geometry, and a healthy dose of research level physics. I'm fully aware of the math side of such claims. I don't see this thread having any understanding of physics here, which is what this paper is about. When you write "when they should be" as if this naive (mis) understanding of the issues is somehow the correct formulation of the problem, it's not relevant. They address this in the paper. They take a standard set of rules for QM, replace a complex Hilbert space with a real Hilbert space, and ask can it reproduce QM. This includes any real valued structure underlying that space, including any of the field isomorphisms. They derive an experiment that can detect in our particular reality which case we live in. So far experiments rule out real valued QM, just like Bell's inequality allowed experimenters to rule out all possible hidden variable theories.

So apparently there is more to the physics than this naive representation you claim is equivalent.

So please stop with the "mathematically isomorphic as fields" argument. It's ruled out experimentally. This is much deeper, and appears fundamentally interwoven in reality, which is much cooler.

Thank you for reading the paper.

So, they’re asking if you can use real Hilbert spaces instead of complex. No cheating by using “real algebra”-valued Hilbert spaces. Okay, fine, but that’s assuming a whole lot about your model, so the question is fairly limited in scope, right?

I dunno, to be honest I just don’t see what’s interesting about this pursuit at all, since complex numbers are far from the most exotic construction in theoretical physics.

> No cheating by using “real algebra”-valued Hilbert spaces

No, read what I wrote, or read the paper.

They showed that no "real algebra"-valued Hilbert space works.

>I just don’t see what’s interesting about this pursuit at all

Yes, because you don't care to see that your assumptions using naive math constructions are not applicable here. They didn't cheat and use circular arguments, they are not unaware of these naive math constructions, they didn't make assumptions that are not validated by physics.

Doing something as trivial as you assume their work is would not get published in Nature. They far surpassed anything you've put up as some workaround or dumb loophole in their arguments. If you're not willing to follow an explanation, or to read and understand their paper, then please stop poo-pooing work you don't understand.

Their work is actually a solid answer to a question that has been poked at by the best minds in physics since QM was founded over 100 years ago. It's a really deep and interesting result. If you want to see some of the work on this, and there is a lot, simply google "does quantum mechanincs require complex numbers," and you'll find a long history of trying to answer this question one way or another.

And no, the answer is not simply replace C with R[z]/(z^2+1) and everything works.

They showed that no "real algebra"-valued Hilbert space works.

That is literally not true. The abstract clearly states that they only compare Real and Complex-valued Hilbert Spaces; they say nothing about general R-algebras. And of course they couldn’t prove what you claim because C itself is an R-algebra!

I’m not claiming that they’re cheating or that what they did is trivial, just that

1) I don’t think they’ve answered the very broad question “Does quantum mechanics need imaginary numbers?” because as stated it’s sort of ill-posed and unfalsifiable. What they have answered is “Can this Hilbert space formulation of quantum mechanics with Postulates 1-4 work with real coefficients?” and the answer is no. The article itself points out non-Hilbert formulations that do in fact avoid complex numbers. Maybe this narrow question is interesting to physicists, I don’t know!

2) I don’t personally think it’s an interesting question, because I don’t think complex numbers are anywhere near the top of the list of unintuitive abstractions needed for modern physics or mathematics. Square root of -1 is a lot easier to swallow than even Cauchy Sequences or Dedekind cuts, much less I dunno, topoi of presheaves on the category of commutative von Neumann algebras of bounded operators (I wish I were making that up: https://aip.scitation.org/doi/abs/10.1063/1.4898185).

> behaves identically to the complex numbers

Are you sure? For example, is the name "parabola" just a relabeling for ax^2 + bx + c, or does 'parabola' point to deeper unity among conic sections starting in 3D?

Based upon a matrix formulation in place of complex analysis, would anyone have come up with insights and use around all of residuals, cuts, poles, zeros, and so forth? How many entries in books of integrals had solutions found using complex integrals?

> I honestly don't understand why complex numbers make people so angry. They are just a way of writing a linear map / element of a particular vector space

I agree. The jump between rational numbers and real numbers is enormous on many different levels. The jump between reals and complex is about as small as can be.

Try forgetting a lot of what you know about numbers then, and maybe then you might understand better.

Real numbers are quite easy for people to "understand"--you can point out the ratio between a circle's circumference and its diameter will always be constant, and name it, while not being exactly expressible as a ratio between two integers, and people will understand that as a concept. Yes, real numbers have all sorts of weird properties, are challenging to build in a set-theoretic principle, etc., but most people aren't going to reach a level of mathematics where those issues come up.

By contrast, complex numbers are invariably introduced as a mathematical gimmick. The imaginary number 'i' is sqrt(-1)... what does that even mean? What would an i-meter long rod look like? What can you point to as a physical construct that encapsulates complex numbers? Especially as is often taught, complex numbers just feel like someone arbitrarily decides that it means something, and you're then manipulating something without any justification for what it actually is or why you should care.

I'll note that negative numbers are similar to complex numbers: a rod of -1 meters doesn't make sense as a physical construct. Indeed, historically, negative numbers aren't in much use until about the same time complex numbers are invented. However, the value of negative numbers is readily apparent: you can use it to keep track of directionality; a credit of -$100 is clearly the same as a debit of $100, for example. With complex numbers, examples of why you might want to do the work are rarely forthcoming. But to give the example no one gave me in school: complex numbers encode magnitude and phase--it will tell if you adding two things that are equally strong will cancel each other out completely, intensify each other, or something in between. (Of course, this is not readily apparent from the usual representation of complex numbers as a + bi, which may be why it's not really covered).

> The imaginary number 'i' is sqrt(-1)... what does that even mean?

Relating to phase is also hard for some people, so I try to keep it even simpler.

In the same way that negative numbers provide directionality on a line, the complex numbers give directionality in rotation about the plane. The unit `i` is a counter-clockwise quarter-turn about the plane. -1 is a half-turn. -i is a a quarter-turn clockwise (or 3/4 turn counter-clockwise).

`exp(i * tau) = 1` or `exp(i * pi) = -1` are now simple identities, referring to the fact that a full turn doesn't change orientation, and a half-turn inverts it.

I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem.

Our "comfort" with mathematics isn't because some concepts are better and some aren't. It's probably more how often we encounter it in everyday life. How often people need to apply complex numbers is WAY less than postive/negative/real numbers.

ALL of the concepts are made up though. There isn't a set of math that isn't entirely made up, with no relation the physical world except for the relationships we also make up.

> I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem.

The point is that, when negative numbers are introduced, they're invariably done so in a way that helps people get over that no-physical-meaning hump. Complex numbers generally aren't, and often, if you try to push for examples, the people teaching themselves don't give anything that helps.

Compare to the article on "Negative number" on Wikipedia to "Complex number". For the former, you are immediately greeted in the first paragraph three separate example use cases for negative number. Turn to the latter, and... it's basically all just mathy justification: complex numbers make this bit of math work. Oh, there's a sentence on how they are "fundamental in many aspects of the scientific description of the natural world"--but there's no examples! Even scrolling down to the applications section, the descriptions tend more towards "complex numbers are used in this field" rather than "this is what a complex number represents."

Put differently, if someone asks a question "what is a negative number? why would I use it?", you'd get a reasonable response. But swap "negative" for "complex", and the answer usually becomes "just shut up and do your math!"

While you probably need to follow some links to grok all of it, in the introduction on Wikipedia they talk about the geometric interpretation which is probably one of the most easily graspable answers to the question "what does a complex number represent?":

> This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

I went to a middling high school, and even our algebra teacher covered the geometric interpretation of complex numbers.

That’s just a familiarity thing. Negative numbers are typically taught in primary school, complex numbers are perhaps only encountered in university.

If you want to see some interesting applications of them, check out some explanations of the Riemann Hypothesis on YouTube.

I don’t understand how “i” could be called a gimmick, if “sqrt(2)” is not. Or pi. Or as you point out “-1” for that matter.

>magnitude and phase

Can you go a bit further? Magnitude or strength is not just the value (+5 or -5) itself?

Maybe in a rigorous mathematical sense that's true but for most people rational numbers as fractions OR fixed point decimal makes enough sense, and the jump to real numbers is just: more decimal places? I might have to round. You don't even really need algebra first just arithmetic.

The jump to complex numbers suddenly introduces a whole bunch of things: letters AND numbers? i*i = -1? Thinking in 2 dimensions and rotations?

I don't know anything about quantum mechanics, but if I'm reading the article right the experiments are meant to disprove this?

> Perhaps it’s most natural to represent each complex dimension by two real dimensions, but real-valued theories need not be so limited. There could be three real dimensions per complex dimension—or four, or even infinitely many.

> Navascués and colleagues found a function of measurement correlations for the entanglement-swapping experiment that could reach 62‾√=8.49 in standard quantum theory but that could never exceed 7.66 in a real-valued formulation.

edit: ok I think they're talking about something different entirely, reading more just made me confused. It seems to me that complex numbers are easier to work with than 2d vectors with operations defined to make them behave the same as complex numbers though.

It's not anger. It's just an objection to treating complex numbers as somehow "special" or as "our jewel".

You can iterate the Cayley-Dickson construction any number of times. Zero times gives you the reals, once gives you the complex numbers, twice gives you the octoneons, etc. Each time you gain closure under more operations but lose other properties (for example, the complex numbers aren't an ordered field).

https://en.m.wikipedia.org/wiki/Cayley–Dickson_construction

Complex numbers are just an arbitrarily-chosen point in a much larger space. If you find them useful, use them. Just don't try to claim that they are some kind of divine gift from the heavens.

Then you can use a helpful little notational shorthand and write matrices of the form [[a,-b,],[b,a]] as a+bi.

You can define i to be

  0 -1
  1  0

and then the two become identical. (A cute detail is that this is a 90 degree rotation matrix)

Those are complex numbers, just expressed differently. Complex numbers are not tied to a notation.

It generates clicks. It's something that most of people in the developed world have seen in high school and it has nice applications on higher mathematics. The name "imaginary" echoes on some deeper, more abstract mathematics that the layman can somewhat relate to.

"Matrices" sound booooorrriiiing next to "COMPLEX IMAGINARY NUMBERS APPLIED TO QUANTUM MECHANICS!11!1!!".

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