Does quantum mechanics need imaginary numbers?
physicstoday.scitation.org
physicstoday.scitation.org
> The square root of negative one doesn’t correspond to any physical quantity
L. Kronecker
If you're approximating, pi is incredibly useful for all kinds of physical things using circles
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?
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.
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?
There are many possible orderings that you can define on the complex numbers. The fact that ordering them is not unique means that any such order is a choice rather than a property of the numbers themselves. You're thinking about measuring distances in the complex plane which is absolutely fine, and also has a canonical definition. I was referring to the fact that it's not intrinsically meaningful to say that 'London is (5000+250i)km from New York'
But this is crucially important. The complex numbers is an (algebraically closed) field. Vectors comprise, well, a vector (linear) space. Most useful vector spaces have the dimension greater than 2. (In fact, when the complex numbers are thought of - rarely - as vectors space, it is a 1-dimensional complex vector space.)
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.
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.
For the record, as I mentioned elsewhere, I think further simplifications will only come from higher levels of abstraction, not less abstraction.
"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.
My point about calling into question what one means by “using complex numbers” is that the article says it is “necessary to establish some ground rules that exclude real-valued quantum theories that restate the standard complex-valued theory by other names” but then handwaves the rules. Presumably you’re only allowed to tensor over R. But more importantly, it provides no motivation for why these ground rules wouldn’t be somewhat arbitrary. I guess I just don’t see the value of removing complex numbers from the formulation.
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
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.
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).
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.
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!"
> 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.
If you want to see some interesting applications of them, check out some explanations of the Riemann Hypothesis on YouTube.
Can you go a bit further? Magnitude or strength is not just the value (+5 or -5) itself?
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.
The jump to complex numbers suddenly introduces a whole bunch of things: letters AND numbers? i*i = -1? Thinking in 2 dimensions and rotations?
0 -1
1 0
"Matrices" sound booooorrriiiing next to "COMPLEX IMAGINARY NUMBERS APPLIED TO QUANTUM MECHANICS!11!1!!".
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.
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
Some are searching for signs of ET or God in irrational numbers: https://en.wikipedia.org/wiki/Schizophrenic_number
> 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.
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?
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.
These authors aren't complete idiots.
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.
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.
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, 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.
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).
If you have multiplication distributing over addition, then the product of (-1) and (-1) is 1 .
It isn't an arbitrary choice.
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.
If someone fails to count (+,-) and (-,+) separately, they're just counting wrong.
Complaining that this is asymmetric is like complaining that even/odd is asymmetric, on the basis that "even plus even is even, and odd plus odd is even, but only if one is even and the other is odd, is the result odd. This is unbalanced in favor of producing even numbers".
-1 * -1 = 1
-1 * x =
-1 * x + 0 =
-1 * x + -x + 1*x =
(-1 + 1)*x + -x =
-xThis is just an extension of ordinary addition and multiplication with natural numbers. Starting with “2 x 3 = 6” it is inevitable that you’d come up with “(-2) x (-3) = 6”. It’s not some kind of imaginary or weird rule. If we extend our numbers to include negative numbers, and we want to preserve as much behavior as we can from what we observed multiplying positive numbers, then this is the only sensible way of doing things.
When you start with something simple (like positive integers) and extend it, you keep some properties, gain some new properties, and lose some properties. For example, in the transition from rational to real numbers we gain the property that all Cauchy sequences converge. In the transition from real to complex we gain the property that the number of solutions to a nonzero polynomial is equal to the polynomial’s degree, but we lose the property that numbers are ordered.
There are some very deep reasons why complex numbers are a natural choice for doing things in functional analysis. It’s definitely a sweet spot… surprisingly, there’s a concept called “holomorphic functions” which is a very tight constraint on functions, yet simultaneously a right field of study, and it’s the foundation of QM. If you move down the ladder to real numbers, the concept of holomorphic functions does not exist. If you move up the ladder to quaternions or octonions, you lose some critical properties like commutativity.
> …we definitely didn't need them when we adopted them in the 16th century…
They were necessary for solving polynomial equations… even finding real solutions to polynomials with real coefficients.
To be clear (as you surely know but a reader might not), that's 'inevitable' in the sense of "a mathematical consequence", not 'inevitable' in the weaker human-events sense of "I can't imagine it winding up any other way". For one approach, 0 = (2 + (-2))3 = (2)(3) + (-2)(3) implies that (-2)(3) = -(2)(3); and then 0 = (-2)(3 + (-3)) = (-2)(3) = (-2)(-3) implies that (-2)(-3) = -(-2)(3) = (2)(3).
That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence.
If we start with the counting numbers and formulate multiplication as repeated addition, 2x3=6. We can arrive at this conclusion by constructing a model of natural numbers, or from an axiomatic approach. If we decide that we want to introduce negative numbers, we are going to end up with a different model or a different set of axioms to accommodate them.
The conclusion that (-2)x(-3)=6 is obvious only in the sense that our choice of axioms for negative numbers is obvious—it just seems like the right way to define negative numbers. For example, your proof relies on the distributive property of multiplication… something that we chose to preserve in our axioms. But we were not forced to preserve this axiom when defining negative numbers, hence it is begging the question. Just because an axiom is part of our formulation for natural numbers does not mean it is part of our formulation for positive and negative numbers.
For more information on this topic, the “transfer principle” article on Wikipedia is really interesting. It dives into other extensions of numbers, such as the hyperreals. This principle is the missing ingredient here.
> That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence.
I dare not argue with you what you meant, and apologise for assuming, but it very definitely is inevitable as a mathematical consequence. What I wrote was not a sketch or a hint, but a proof, from the ring axioms.
Hence "begging the question"... you're assuming the ring axioms hold for the negative numbers. But they do not hold for the natural numbers in the first place... so where did the ring axioms come from, and why must they hold?
Different people sat down over the last millennium or so and realised that the natural numbers could only express quantity and not direction. They looked around them at the physical reality that they found themselves in, and noticed that sometimes it was useful to count in the opposite direction (ie. decreasing). Eventually they realised that they could do this by adding in some new numbers that were ordered below 0, and which functioned as additive inverses to the natural numbers.
This process was constrained by the properties of reality; so the properties of the negative numbers are not arbitrary, they are fixed by properties of the world that different people noticed.
Eventually (around the end of the 19th century maybe?) people tried to formalise the properties of these numbers in axiomatic schema. They noticed that certain of these axioms could also describe other extended algebraic objects which are a bit like the integers, and called these objects rings. So that's where the ring axioms came from; by observing reality and then generalising
The natural numbers don’t satisfy the ring axioms. Perhaps this is the flaw in your reasoning.
Negative numbers need to model different aspects of physical reality; for example they need to describe where you end up relative to your starting point when you walk 5m in one direction, and then walk 6m back in the exact opposite direction. So the axioms for the negative numbers are not arbitrary. I think they are probably unique given certain physical requirements, but I haven't thought or read about this.
They don't "need" to, but it is very useful that they do.
Rephrasing the argument only helps if I misunderstand your argument, and I don’t.
You're free to choose your own whatever other axioms for your 'negative numbers' that you want. If they don't describe the process of counting downwards or walking in opposite directions like I described, then I won't call them negative numbers.
And yet,
> I'm describing the process of observing properties of nature and then finding a formalised description of those properties.
A "formalized description of those properties" is known as a set of axioms. That's just the name for it.
If you "found something in nature" which you call the "negative numbers," what you've done is constructed something, labeled it "negative numbers", and derived axioms from its properties. It does not really matter if you come up with the construction first and derive the axioms afterwards, or if you come up with the axioms first and later derive a construction that satisfies those axioms.
If you did not take upper division mathematics, you would likely not be familiar with this equivalence.
Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers.
> Aliens would also find negative numbers in this way, but they'd call them something different (presumably).
Yes, that's exactly what I was saying when I said that this definition was "inevitable". It's so obvious and makes so much sense that there's just no reasonable chance that we'd come up with something else.
But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property.
> You're free to choose your own whatever other axioms for your 'negative numbers' that you want.
If you think that the actual definition of "negative number" is somehow in dispute than you definitely misunderstood the argument.
Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers.
I think you're talking about something like a non-standard model of the Peano axioms? My understanding is that this can only put in new 'numbers' c such that n < c for all natural numbers n. So, sure I've chosen a specific model of these axioms, but the model agrees with all other models for every finite number.
Either way I don't really see why making a choice here detracts from my argument; numbers exist as a part of nature, and we can describe them with Peano axioms and a choice of model. I didn't assume any (specific mathematical) axioms in the first place, I assumed that reality exists and that anything that can be conceived of is a part of reality. I personally confirm for myself that this is true by observation, although you may not have done so for yourself.
But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property.
I don't really get what point you’re making here
The only "easy" thing is writing down an equation like (-2)x(-3)=6. That's not what we're doing, though. We're defining how multiplication works, and (-2)x(-3) is just an example.
As other sibling comments have said, what would your arithmetic look like that still worked, but with a different 'rule' (which is a rule just in the sense of a consequence of the axioms, not in the sense of an arbitrary choice)? If you wanted, for example, (3 + -2)(3 + -2) to equal (3)(3) + (3)(-2) + (-2)(3) + (-2)(-2), then you'd have a hard time getting that equality to hold.
Hmm.