9 ms·

Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instea

Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter). But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.

deleted 5y ago
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midjji 5y ago
Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.
- kevin_thibedeau 5y ago
  'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.
  - gfody 5y ago
    you can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one
- dreamcompiler 5y ago
  The other side of that same coin is that we're describing wave functions of probability (with interference between probabilities) and complex numbers are exceedingly handy for wave functions, as any EE can attest.
  - phkahler 5y ago
    >> complex numbers are exceedingly handy for wave functions, as any EE can attest. Because they encode phase information. Also because they come about in the solution of differential equations. Physicists often talk about amplitudes, but I never hear them talk about phase. There was one paper that I can't find, complete with a diagram that suggested (to me) that phase was determining quite a bit.
    - kgwgk 5y ago
      A wave function describes - usually at least - the quantum state of an isolated quantum system. The phase has no physical meaning. The relative phase between wave functions could mean something... but not if the systems are isolated.
      - dreamcompiler 5y ago
        Phase doesn't really matter much in EE for an isolated sinusoid either. But when you compare the phase of a transmitted signal to a local reference, or you compare the phases of the sinusoids in an FFT to one another, or you're trying to synchronize the Texas electrical grid to the rest of the country, phase means a lot. Phases are almost always only useful in a relative sense.
- GoblinSlayer 5y ago
  Rotational symmetry a straightforward consequence of the wave equation.
- azalemeth 5y ago
  There are lots of equivalent representations of the same thing, and for non relativistic QM i is by far the simplest way of proceeding. Part of learning QM is learning that changing how you view the world without actually changing the world is a very powerful thing and sometimes more complex ways of thinking are actually easier "later on". A simple example is the equivalence of two complex parameters, alpha and beta, arranged in a 2-matrix and a real 3-matrix for the representation of rotation. Another example would be ladder operators for the simple harmonic oscillator -- arguably overcomplicated for the problem at hand, they form the basis of much of what follow (i.e. vacuum creation and annihilation operators). The whole point of the notational soup that one finds e.g. in an extended field theory is that it correctly generates a lot of these details "automatically". It's obtuse and makes doing simple things hard, but makes showing non-trivial relationships that are true in general very much easier than the alternative ;-).
gigatexal 5y ago
This is super clear. Thanks for clarifying.
asxd 5y ago
This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the paper, but this statement from the abstract confuses me: > Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. Maybe I'm interpreting "real numbers" and "need" differently than the authors, since in my head complex numbers are basically just a structure containing two real numbers and some modified behavior.
- robbedpeter 5y ago
  I think you still "need" the imaginary number, even if you're expressing it as a more granular/verbose set of fundamentals, and I always wonder if there's an order of operations issue, or some syntactic gotcha instead of anything real?
- itchyjunk 5y ago
  Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number can be thought of as representing the same thing), I think this way of thinking misses that we also have a field of complex numbers. Point being R^2 isn't a field but C is. If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).
  - asxd 5y ago
    "Field" is a term I've heard come up again and again since college, in engineering-adjacent math. Over the years, I've occasionally looked it up and tried to understand the importance, but I've never found any literature that made much sense to my admittedly short-sighted mind. I'm putting this bluntly (and sincerely), but this seems like a decent time to ask a question I should have asked long ago during college: What is a field and why do we care?
SideQuark 5y ago
Once you replace scalars with 2 vectors you aren't basing it on real numbers, you're basing it on operations on 2 vectors. Of course you can put real numbers at the base of nearly anything, but it the theory operates at a fundamental level on items that are more complex then real numbers, then it's not based on real numbers. You might as well argue it's based on surreal numbers or Dedekind cuts at that point. Somewhere Feynman has a quote that qm was the first theory that required complex numbers. Finding and understand that quote should explain this better.
mst_moonshine 5y ago
Agreed. So what you need is the 'complex structure' behind rather than just 'complex numbers'. Any form of representations (numbers, matrices, and so on) should correspond to a unique structure. The question why the complex structure emerges in quantum mechanics is more interesting.
- kgwgk 5y ago
  Skilling and Knuth have some interesting papers on the subject: The Symmetrical Foundation of Measure, Probability, and Quantum Theories https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800057 https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800... The ABC of Physics https://www.mdpi.com/2673-9984/3/1/9/htm https://www.mdpi.com/2673-9984/3/1/9/htm
- zarzavat 5y ago
  Complex numbers have two roles in mathematics. The first is as a number system based upon SO(2) the group of rotations in 2D, the second is as the algebraic closure of the reals. That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions). Physics uses complex numbers in the first sense. There's really nothing too special about SO(2), there's an SO(n) for all n. Whereas mathematics uses complex numbers in both senses. There is something rather special about complex numbers as the algebraic closure of the reals and it's what makes a lot of modern math tick.
  - amai 5y ago
    "That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions)." Can you elaborate on this? What is an algebraic closure of the reals in higher dimensions?
    - zarzavat 5y ago
      The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction. The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers. In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.
a9h74j 5y ago
The article reads to me as reductionistic in the sense that provided QM could use real numbers only, it would not matter if aerodynamics or whatever at a higher, emergent level required them -- almost suggesting the other levels are not part of physics. In the same sense I did not see reference to General Relativity in the article -- as another fundamental ground in physics besides QM. Never learned manifolds stuff for GR, so I don't know if complex numbers become naturally essential there.
resters 5y ago
You describe the one thing I don't like about complex numbers, that people often don't realize that the same things can be represented fully using other mathematical objects. Complex numbers are basically syntactic sugar for that more general type of object.
- Rerarom 5y ago
  So you would like them to not be representable?
  - resters 5y ago
    I think people often confuse the representation for the object itself, and thus create a limited mental model that must be undone later. Analogous to a world in which all cookie shaped objects are edible and delicious. Sure it would be nice to live in that world but it's not reality and a more rich (but less pleasant) representation makes reality more accessible. I'd say the same thing about any math that talks about "numbers" without defining which number system very explicitly, even at the Kindergarten level. The slop in these early abstractions is somewhat convenient but it erects serious mental barriers against more accurate abstractions. Look at the sloppy way that many programming languages handle lossy casts for an example of the pernicious nature of the idea that "number" should have a highly intuitive meaning. On the other hand, imagine all the nice notations/sugars we might invent that are highly intuitive and capture mathematical objects more elegantly than what we are using today. Representations are the interface between abstract concepts and brains evolved for eating, sex and lying.
edgyquant 5y ago
While you may be entirely right here I feel you’re taking liberties when using the word trivial

deleted 5y ago

[deleted]

midjji 5y ago

Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.

kevin_thibedeau 5y ago
'i' is a 90-degree rotation. It isn't hidden at all. Similarly, everywhere a complex exponential shows up, there's also some spinny/rotational thing happening.
- gfody 5y ago
  you can always imagine another dimension, with all the symmetry of every dimension so far plus some novel one
dreamcompiler 5y ago
The other side of that same coin is that we're describing wave functions of probability (with interference between probabilities) and complex numbers are exceedingly handy for wave functions, as any EE can attest.
- phkahler 5y ago
  >> complex numbers are exceedingly handy for wave functions, as any EE can attest. Because they encode phase information. Also because they come about in the solution of differential equations. Physicists often talk about amplitudes, but I never hear them talk about phase. There was one paper that I can't find, complete with a diagram that suggested (to me) that phase was determining quite a bit.
  - kgwgk 5y ago
    A wave function describes - usually at least - the quantum state of an isolated quantum system. The phase has no physical meaning. The relative phase between wave functions could mean something... but not if the systems are isolated.
    - dreamcompiler 5y ago
      Phase doesn't really matter much in EE for an isolated sinusoid either. But when you compare the phase of a transmitted signal to a local reference, or you compare the phases of the sinusoids in an FFT to one another, or you're trying to synchronize the Texas electrical grid to the rest of the country, phase means a lot. Phases are almost always only useful in a relative sense.
GoblinSlayer 5y ago
Rotational symmetry a straightforward consequence of the wave equation.
azalemeth 5y ago
There are lots of equivalent representations of the same thing, and for non relativistic QM i is by far the simplest way of proceeding. Part of learning QM is learning that changing how you view the world without actually changing the world is a very powerful thing and sometimes more complex ways of thinking are actually easier "later on". A simple example is the equivalence of two complex parameters, alpha and beta, arranged in a 2-matrix and a real 3-matrix for the representation of rotation. Another example would be ladder operators for the simple harmonic oscillator -- arguably overcomplicated for the problem at hand, they form the basis of much of what follow (i.e. vacuum creation and annihilation operators). The whole point of the notational soup that one finds e.g. in an extended field theory is that it correctly generates a lot of these details "automatically". It's obtuse and makes doing simple things hard, but makes showing non-trivial relationships that are true in general very much easier than the alternative ;-).

gigatexal 5y ago

This is super clear. Thanks for clarifying.

asxd 5y ago

This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the paper, but this statement from the abstract confuses me: > Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. Maybe I'm interpreting "real numbers" and "need" differently than the authors, since in my head complex numbers are basically just a structure containing two real numbers and some modified behavior.

robbedpeter 5y ago
I think you still "need" the imaginary number, even if you're expressing it as a more granular/verbose set of fundamentals, and I always wonder if there's an order of operations issue, or some syntactic gotcha instead of anything real?
itchyjunk 5y ago
Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number can be thought of as representing the same thing), I think this way of thinking misses that we also have a field of complex numbers. Point being R^2 isn't a field but C is. If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).
- asxd 5y ago
  "Field" is a term I've heard come up again and again since college, in engineering-adjacent math. Over the years, I've occasionally looked it up and tried to understand the importance, but I've never found any literature that made much sense to my admittedly short-sighted mind. I'm putting this bluntly (and sincerely), but this seems like a decent time to ask a question I should have asked long ago during college: What is a field and why do we care?

SideQuark 5y ago

Once you replace scalars with 2 vectors you aren't basing it on real numbers, you're basing it on operations on 2 vectors. Of course you can put real numbers at the base of nearly anything, but it the theory operates at a fundamental level on items that are more complex then real numbers, then it's not based on real numbers. You might as well argue it's based on surreal numbers or Dedekind cuts at that point. Somewhere Feynman has a quote that qm was the first theory that required complex numbers. Finding and understand that quote should explain this better.

mst_moonshine 5y ago

Agreed. So what you need is the 'complex structure' behind rather than just 'complex numbers'. Any form of representations (numbers, matrices, and so on) should correspond to a unique structure. The question why the complex structure emerges in quantum mechanics is more interesting.

kgwgk 5y ago
Skilling and Knuth have some interesting papers on the subject: The Symmetrical Foundation of Measure, Probability, and Quantum Theories https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800057 https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800... The ABC of Physics https://www.mdpi.com/2673-9984/3/1/9/htm https://www.mdpi.com/2673-9984/3/1/9/htm
zarzavat 5y ago
Complex numbers have two roles in mathematics. The first is as a number system based upon SO(2) the group of rotations in 2D, the second is as the algebraic closure of the reals. That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions). Physics uses complex numbers in the first sense. There's really nothing too special about SO(2), there's an SO(n) for all n. Whereas mathematics uses complex numbers in both senses. There is something rather special about complex numbers as the algebraic closure of the reals and it's what makes a lot of modern math tick.
- amai 5y ago
  "That these two are the same thing is somewhat of a fluke (it doesn't work in higher dimensions)." Can you elaborate on this? What is an algebraic closure of the reals in higher dimensions?
  - zarzavat 5y ago
    The complex numbers are the closure regardless of dimension. When I was writing that I was thinking of the Quaternions, which are the 4 dimensional analog of the complex numbers, in 2^N dimensions this is the Cayley Dickson construction. The fluke is this: Euclidean space of dimension N has N(N-1)/2 rotational dimensions. If you plug 2 into that you get 2x1/2 which is 1 dimension. i.e. the rotations in 2D space look like a circle. If you add an extra dimension (the radius) you get the polar form of complex numbers. In other dimensions this doesn't always work. In 3 dimensions we have 3x2/2 = 3 rotational dimensions, so we need a space with dimension 4 (the quaternions). In 4 dimensions we need a 6 dimensional rotation space. We just established that Cayley Dickson algebras only come in powers of 2, so it doesn't fit at all.

a9h74j 5y ago

The article reads to me as reductionistic in the sense that provided QM could use real numbers only, it would not matter if aerodynamics or whatever at a higher, emergent level required them -- almost suggesting the other levels are not part of physics. In the same sense I did not see reference to General Relativity in the article -- as another fundamental ground in physics besides QM. Never learned manifolds stuff for GR, so I don't know if complex numbers become naturally essential there.

resters 5y ago

You describe the one thing I don't like about complex numbers, that people often don't realize that the same things can be represented fully using other mathematical objects. Complex numbers are basically syntactic sugar for that more general type of object.

Rerarom 5y ago
So you would like them to not be representable?
- resters 5y ago
  I think people often confuse the representation for the object itself, and thus create a limited mental model that must be undone later. Analogous to a world in which all cookie shaped objects are edible and delicious. Sure it would be nice to live in that world but it's not reality and a more rich (but less pleasant) representation makes reality more accessible. I'd say the same thing about any math that talks about "numbers" without defining which number system very explicitly, even at the Kindergarten level. The slop in these early abstractions is somewhat convenient but it erects serious mental barriers against more accurate abstractions. Look at the sloppy way that many programming languages handle lossy casts for an example of the pernicious nature of the idea that "number" should have a highly intuitive meaning. On the other hand, imagine all the nice notations/sugars we might invent that are highly intuitive and capture mathematical objects more elegantly than what we are using today. Representations are the interface between abstract concepts and brains evolved for eating, sex and lying.

edgyquant 5y ago

While you may be entirely right here I feel you’re taking liberties when using the word trivial