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>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 compl

>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 ago
  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).
  - onethought 5y ago
    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.
    - jcranmer 5y ago
      > 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!"
      - onethought 5y ago
        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.
- kjeetgill 5y ago
  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?
zardo 5y ago
Then you can use a helpful little notational shorthand and write matrices of the form [[a,-b,],[b,a]] as a+bi.
- deleted 5y ago
  [deleted]
- marginalia_nu 5y ago
  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)
spyremeown 5y ago
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!!".
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 ago
  Yeah 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 ago
    Wow, genius. This would have personally helped immensely
- octoberfranklin 5y ago
  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/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 ago
    These are not scary enough. https://en.wikipedia.org/wiki/Schizophrenic_number https://en.wikipedia.org/wiki/Schizophrenic_number
  - 5y ago
deleted 5y ago
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Jack000 5y ago
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.
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 ago
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 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 ago
Those are complex numbers, just expressed differently. Complex numbers are not tied to a notation.
ChrisLomont 5y ago
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.
- kevinventullo 5y ago
  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.
  - ChrisLomont 5y ago
    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.

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 ago
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).
- onethought 5y ago
  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.
  - jcranmer 5y ago
    > 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!"
    - onethought 5y ago
      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.
kjeetgill 5y ago
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?

zardo 5y ago

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

deleted 5y ago
[deleted]
marginalia_nu 5y ago
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)

spyremeown 5y ago

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!!".

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 ago
Yeah 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 ago
  Wow, genius. This would have personally helped immensely
octoberfranklin 5y ago
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/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 ago
  These are not scary enough. https://en.wikipedia.org/wiki/Schizophrenic_number https://en.wikipedia.org/wiki/Schizophrenic_number
- 5y ago

deleted 5y ago

[deleted]

Jack000 5y ago

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.

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 ago

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 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 ago

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

ChrisLomont 5y ago

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.

kevinventullo 5y ago
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.
- ChrisLomont 5y ago
  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.