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Quantum physics falls apart without imaginary numbers
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Re: Quantum physics falls apart without imaginary numbers
#102This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.
This is your regularly scheduled reminder that the author of these papers ( https://arxiv.org/abs/2101.10873 , https://arxiv.org/abs/2111.15128 ), which this article is based on know very well that complex numbers have real matrix representations. What they add which your comment discounts is the locality structure of quantum mechanics, i.e. what happens when you combine multiple quantum systems. Specifically if the…
Thanks, this is very helpful.
When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?
Re: Quantum physics falls apart without imaginary numbers
#103[deleted]
Re: Quantum physics falls apart without imaginary numbers
#104One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math." And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.
"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. " The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
Not really! Often you get a complex solution and both the real and imaginary components are valid.
Re: Quantum physics falls apart without imaginary numbers
#105Earlier quoted context omitted.
I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension. Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. G…
You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
You’re right about cross product vs outer product. (I do not know how I got that one wrong…)
Re: Quantum physics falls apart without imaginary numbers
#106I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all.
It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers.
Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved.
But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM.
But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math.
So I can't tell what any of this is supposed to mean at all?
Re: Quantum physics falls apart without imaginary numbers
#107Earlier quoted context omitted.
The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with…
Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them. In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary num…
We used them in the late 80s and early 90s for pretty fractals on home computers; but for SciFi world-building… it can be done, but I don't think you can really get into those worlds as a reader without already being somewhat familiar with the maths.
Reason I think this is having listened to Greg Egan's Dichronauts, in which spacetime is ++-- with all the counterintuitive hyperbolic rotations that this implies.
Other ways to get i into common use besides that, probably gives equally counterintuitive results if you don't already know the maths; for example, instead of a maximum debt you can take out based on your ability to repay, you get a region which, for certain interest and repayment rates, is the Mandelbrot set.
Re: Quantum physics falls apart without imaginary numbers
#108I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird. Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring…
> There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.
I think you couldn't have put it better. I'm just a physicist, though.
Re: Quantum physics falls apart without imaginary numbers
#109There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…
Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of fre…
I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?
Re: Quantum physics falls apart without imaginary numbers
#110I've always had problems with how complex numbers are taught.
The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything.
Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc.
I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved.
And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense.
Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.