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Does quantum mechanics need imaginary numbers?
- Qem 5y agoNo. Geometric Algebra provides a nice alternative to them. See http://worrydream.com/refs/Gull%20-%20Imaginary%20Numbers%20are%20not%20Real.pdf http://worrydream.com/refs/Gull%20-%20Imaginary%20Numbers%20...
- jleyank 5y agoWhatever makes the formalism work (better). If they're not physical observables it doesn't matter what's involved. As long as the observables are real the formalism isn't critical.
- voldacar 5y ago>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
- CogitoCogito 5y ago> 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.
- jcranmer 5y agoTry 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).
- onethought 5y agoI 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.
- zardo 5y agoThen you can use a helpful little notational shorthand and write matrices of the form [[a,-b,],[b,a]] as a+bi.
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- marginalia_nu 5y agoYou 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)
- spyremeown 5y agoIt 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!!".
- injb 5y ago>> 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.
- voldacar 5y agoYeah the shitty terminology can be blamed on Descartes. I think Gauss wanted to rename them "lateral numbers" but I guess that never stuck
- dvsfish 5y agoWow, genius. This would have personally helped immensely
- octoberfranklin 5y agoTo 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/Surreal_number https://en.m.wikipedia.org/wiki/Transfinite_number https://en.m.wikipedia.org/wiki/Transfinite_number https://en.m.wikipedia.org/wiki/Hyperreal_number https://en.m.wikipedia.org/wiki/Hyperreal_number
- forum_ghost 5y agoThese are not scary enough. https://en.wikipedia.org/wiki/Schizophrenic_number https://en.wikipedia.org/wiki/Schizophrenic_number
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- Jack000 5y agoI 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.
- a9h74j 5y ago> 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?
- octoberfranklin 5y agoIt'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 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.
- WinterMount223 5y agoThose are complex numbers, just expressed differently. Complex numbers are not tied to a notation.
- ChrisLomont 5y agoTheir 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.
- kevinventullo 5y agoThe 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.
- ChrisLomont 5y agoLook, 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.
- aidenn0 5y agoGiven that imaginary numbers were invented in the 16th century, I think we can say that QM works just fine without imaginary numbers...
- rubidium 5y agoInvented or discovered is a debate in the philosophy of mathematics
- injb 5y agoGenerally true, but harder to make this case for imaginary numbers, because we definitely didn't need them when we adopted them in the 16th century or whenever it was. We adopted them because we had already adopted the (very imaginary) rule that - * - = +. Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. That's why I find it hard to accept that QM needs them, even though I lack the capacity to really understand the QM argument.
- drdeca 5y agouh, (-1) * (-1) = 1 holds in any ring. If you have multiplication distributing over addition, then the product of (-1) and (-1) is 1 . It isn't an arbitrary choice.
- not2b 5y agoHere's a physical example for those who have trouble intuitively grasping why the product of two negative numbers is positive. Let's say we have a water tank. We can fill it by opening a faucet on the top, or drain it by unplugging the bottom. For simplicity let's say that if we fill it we add 1 liter per minute, and if we drain it we lose 1 liter per minute (or equivalently, add -1 liter per minute). Q1: if we open the top faucet for 2 minutes, how much water do we add? 1 liter/min times 2 minutes: 2 liters. Q2: if we open the bottom faucet for 2 minutes, how much water do we add? -1 liter/minute times 2 minutes, -2 liters. What about the volume two minutes ago? That would be -2 minutes, right? So suppose the drain was open, how much more water did we have -2 minutes from now, or 2 minutes ago? -2 times -1: 2 liters more.
- waynecochran 5y agough... I really hate stupid comment about complex numbers > The square root of negative one doesn’t correspond to any physical quantity neither does pi
- mmarq 5y agoOr 19 for that matter, if one really thinks about it
- Koshkin 5y agoGod made the integers; everything else is man’s work. L. Kronecker
- voldacar 5y agoor the 99.999...% of real numbers which are uncomputable
- 8note 5y agoFor some definition of circle* If you're approximating, pi is incredibly useful for all kinds of physical things using circles
- waynecochran 5y agoi turns out to be incredibly useful as well ...
- waynecochran 5y agopi = i * ln(-1).
- mrjangles 5y agoEveryone in the comments repeating the basic facts about complex number they learned in high school needs to read the article. This is Physics Today. Every person who reads it or contributes is already aware of what complex numbers are.
- kevinventullo 5y agoI think the question itself doesn’t make sense. It’s like asking if quantum mechanics “needs” linear algebra, or if general relativity “needs” differential geometry. Like there’s probably some roundabout way to do the exact same thing but calling it something different (e.g. matrices or commutative algebra), but then this calls into question what exactly one means by “using complex numbers”. I guess if you thought there was a simplification to be had, that would be a good reason to search for a fundamentally different way of framing things, but in my experience as a mathematician, the whole point of fancy theoretical frameworks is to arrive at conceptual simplicity. You prove something very general and then you can take it as a conceptual axiom.
- WinterMount223 5y agoI’m not convinced. Complex numbers are a representation of a system. Yes, you can express it somehow else, but at its core it’s the same. As I understand it, they are saying that using complex numbers is the simplest (pun not intended) representation necessary and there’s no further simplification possible. That is meaningful because an ancient Greek would say that reals or even integers are sufficient to describe the world. They are not. Just like Euclides was not enough.
- kevinventullo 5y agoI guess I don’t find it that meaningful. The Greeks also didn’t have the differential geometry needed to describe relativity, but no one’s trying to remove that from physics. For the record, as I mentioned elsewhere, I think further simplifications will only come from higher levels of abstraction, not less abstraction.
- Kranar 5y agoThis is literally addressed verbatim by the article, and I quote: "One can always devise new mathematical constructs that behave in all the same ways as complex numbers even though they’re called something else." It's honestly as if people are just jumping to comment on the title of the article without actually reading it. The article is fundamentally about how trying to replace complex numbers by some real valued representation of them, for example using a vector of real numbers, or a matrix of real numbers, doesn't work due to the compounding of dimensions. The two examples they give where using pairs of real numbers fails is multiparticle states and a spin 1/2 qubit. At any rate, there's no point rehashing the article here, since it's all written in a fairly straight forward manner. It's just frustrating reading so many people write the same trivial comment dismissing the article when the article is written precisely to demonstrate why these simplistic dismissals fail.
- galaxyLogic 5y agoA complex number is an ordered pair of two real numbers. Right? What makes them be what they are is the rules for how to do arithmetic with them. Similarly vectors in the plane are ordered pairs of real numbers, and you call them vectors rather than imaginary numbers because the products of vectors are defined differently. I don't really see much ontological difference between vectors and complex numbers, except for the rules how to calculate with them. Due to historical reasons ones are called vectors and the others complex numbers. They could be called anything. What matters is the rules for calculating with them. When you program complex arithmetic on a computer it is easy to see that there is nothing "unreal" about them. Or is there?
- joe__f 5y agoIf you give vectors in the plane the same multiplication rule as complex numbers, then I think it's fair to say that they are ontologically equivalent. If you take your vectors and equip them with a different multiplication rule they will no longer form a field. Any other structure that your vectors may form under this multiplication rule is then ontologically different from the complex numbers. Wikipedia tells me that "Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra." The fact that the complex numbers can form the scalars of a vector space of all possible quantum states is an essential property for formulating quantum mechanics. I think your point that the complex numbers don't have anything unreal about them is probably fair. One key difference between the real and complex numbers is that the complex numbers have no canonical ordering, so you can't use them to measure distances between points in physical space. Maybe in this sense you could think of them as being less real than the reals or integers etc.
- lokedhs 5y agoI don't fully understand what you meant. Either I misunderstood it or there is a problem with your explanation. You can measure distance between complex numbers. But the distances between three numbers has a non-transitive relationship which is what prevents ordering. Was that what you meant, or is there some more fundamental relation between not having an order and the idea of distances?
- MichaelRazum 5y agoI think the question is the same if physics needs pi or sqrt(2) number's that kind off can't exist in the physical world.
- Koshkin 5y agoForget about quantum mechanics. The complex numbers play an important role in electrical engineering, even.
- fmvab 5y agoThis is literally addressed in the second paragraph: In electromagnetism and most other fields of physics, imaginary numbers are merely a mathematical convenience. All the relevant phenomena can still be described using nothing but real numbers.
- Koshkin 5y agoBut the “convenience” is important, and its very existence is not accidental.