Quantum physics is a description of reality, not reality itself.
Thank god - I wouldn't want to see the reality fall apart.
Quantum physics falls apart without imaginary numbers
141–150 of 166 posts
Re: Quantum physics falls apart without imaginary numbers
#142“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits…
You miss a key part of complex numbers if you think of them as just vectors: they are a field.
And not just a field, but the algebraic closure of real numbers.
Re: Quantum physics falls apart without imaginary numbers
#143There 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…
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…
> Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.
I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it.
So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.
Re: Quantum physics falls apart without imaginary numbers
#144Earlier quoted context omitted.
Given that Lifschitz wrote that before QED, he did not even begin to understand the modern understanding of QM and the electron. Nor did he see any inkling of QMs replacement, (T)QFTs. QM (and his quote, and your understanding) are nearly 100 years out of date. The entire quote is nonsense - QED (well after Landau wrote his text) shows that the opening sentence is as valid a Asimov book from the period with the wrong…
> all physical theories had to agree You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
> if it needs to be solved at all
You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it.
> You seem to have missed the whole point (which comes after "yet" at the end of the quote)
Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part.
Your initial claim was "A classical apparatus is part of the QM framework." It is not. QM can be completely defined (and was done so very early on) without any connection to classical, and it took (and is still taking) effort to show that classical things come from it.
As two examples, the Dirac-von Neumann axioms for QM (from which it all can be derived) are from the early 1930s, and have precisely zero mention of need for any classical physics. If you don't believe it, read them, or download von neumann's book and read it. There are subsequently axiomatic forms of QFTs, TQFTs, and so on, none needing anything more than pure math to define. There's a large collection of research over the past ~100 years with groups poking at different axiom sets or arguing if this or that set is complete, still ongoing (e.g., [2]), but AFAIK, there is no big group that claims QM is not based on axioms at this point. Or QFT or TQFTs (which Atiyah spent significant time axiomatizing before he died [3]). QM can be derived from TQFTs.
Care to show me which axiom in TQFTs is the classical apparatus? Say, as opposed to the zillion other math structures that use the same words to define things which coincidentally didn't match physics? (Unless you're Max Tegmark, for which all math is physics, a fringe view but a powerfully thought out one...)
It's nice when they correspond to nature, but there is zero need for nature to define them. They're pure math, and the agreement with nature has led to the entire "It from Bit" or "Unreasonable Effectiveness of Mathematics" views in physics.
As to some really important classical connections that took a long time to derive, Dyson's 1967 proof that matter is stable (which is an incredibly classical observation) under QM is a really neat result [1]. So the classical connections are not needed to state QM, and even historically the connections were interspersed over time, and most were found long after QM was axiomatized.
So, still claim the "yet" phrase is true? If so, how did D&vN make axioms from which all QM derives?
[1] https://fisherp.scripts.mit.edu/wordpress/wp-content/uploads...
[2] https://link.springer.com/article/10.1007/s10701-008-9230-4
[3] https://en.wikipedia.org/wiki/Topological_quantum_field_theo...
Re: Quantum physics falls apart without imaginary numbers
#145Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic closure, so that all polynomials have roots, we get the complex numbers.
Re: Quantum physics falls apart without imaginary numbers
#146Earlier quoted context omitted.
> all physical theories had to agree You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
> if it needs to be solved at all You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it. > You seem to have missed the whole point (which comes after "yet" at the end of the quote) Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part. Your initial claim was "A classical apparatus is part of th…
> They're pure math, and the agreement with nature
Sure, as far as the math is concerned, it appears that there is no need for "classical objects." But physics is not (and has never been) "pure math," which is why Landau and Lifshitz, in particular, keep insisting on the importance of understanding the "physical principles" behind the axioms, whatever they are, and the facts of the theory; and so the "agreement with nature" is all but expected; but in order to see that agreement we need to observe, and we can only make observations by looking at "classical objects"; then, to make a connection back to the theory we need to have a way of making the result of the observation directly available to the framework itself - its "physical content" and its math.
Re: Quantum physics falls apart without imaginary numbers
#147I had this thought years ago:
Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1
In other words, when you “break apart” i, you get 1 and -1
Re: Quantum physics falls apart without imaginary numbers
#148Earlier quoted context omitted.
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?
yes, specifically doing things without complex numbers breaks if you require composing systems to work like they do in normal quantum mechanics (composing with the tensor product).
If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
Re: Quantum physics falls apart without imaginary numbers
#149I'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'…
Back when I did olympiad math we would sometimes solve classical geometry problems with a "complex bash" --- you assign points complex coordinates and then do some algebra to show that the desired property must hold true.
Re: Quantum physics falls apart without imaginary numbers
#150I'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'…
I find this "spiral" concept showing up in other cases as well, and I agree that it's under-rated (or under-studied) as a way to understand the complex numbers. By thinking about periodic points on a complex spiral, you can extend many useful properties of polynomials to those with rational coefficients. We can begin to talk about the roots of equations like x^(3/2) + x^(1/2) = 0 and we find results that very naturally extend from roots of similar "standard" polynomials. We can see that x^(1/2) = 0 has half as many roots as x^1 = 0, and x^(1/4) = 0 has half again as many roots, even if they all "project" down to 1 -- their periodicity on the spiral is 4pi and 8pi instead of the standard 2pi. I feel like there's a lot going on there that hasn't been explored.