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Quantum physics falls apart without imaginary numbers

scientificamerican.com

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Re: Quantum physics falls apart without imaginary numbers

#61

Post by sfpotter »

rcme wrote: post #45
lanstin wrote: post #27

Earlier quoted context omitted.

Complex analysis is so much more regular than real analysis differentiability over a two dimensional quantity is so much strong than over a one dimensional quantity that you have much stronger results. Basically, if you know an analytic function in a neighborhood you know it over the entire plane. Plus you have functions like e ^ ( 1 / z ) which is pretty amazing around zero.

While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).

Following a proof step-by-step != understanding the proof

Re: Quantum physics falls apart without imaginary numbers

#63

Post by tgv »

dr_dshiv wrote: post #55
nathan_compton wrote: post #16

Earlier quoted context omitted.

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

Because they are separate but interacting with real numbers? I don’t understand.

Multiplying vectors differs from multiplying complex numbers.

Re: Quantum physics falls apart without imaginary numbers

#64

Post by tokai »

mjhay wrote: post #14
CottonMcKnight wrote: post #12

I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.

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…

>"real" numbers have never been controversial

Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.

Re: Quantum physics falls apart without imaginary numbers

#65

Post by eigenket »

whatshisface wrote: post #5

This 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 state of one system lives in A, and the state of another system lives in B then the state of a combined system lives in the tensor product A ⊗ B.

If you do the trick where you replace complex numbers by real matrices what you end up is having the state of the first system living in A = X ⊗ A', and the second is in B = X ⊗ B', where A' and B' are real and the X subsystem is the degree of freedom you're using to "fake" the complex numbers.

Then if you try to combine the two systems you end up with a state that lives in A ⊗ B = (X ⊗ A') ⊗ (X ⊗ B') but what you need in order to get the behavior you want is actually for the combined system to live in X ⊗ (A'⊗ B').

What they do is a bit more complicated because they show that all ways to fake complex numbers using real numbers breaks this way of combining systems, but this is the gist.

Re: Quantum physics falls apart without imaginary numbers

#66

Post by justin_ »

dr_dshiv wrote: post #55
nathan_compton wrote: post #16

Earlier quoted context omitted.

You miss a key part of complex numbers if you think of them as just vectors: they are a field.

Because they are separate but interacting with real numbers? I don’t understand.

He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0]

You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.

You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].

[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra

Re: Quantum physics falls apart without imaginary numbers

#67

Post by [deleted] »

dboreham wrote: post #9

Haven't read it, but it's obviously wrong. To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.

[deleted]

Re: Quantum physics falls apart without imaginary numbers

#68

Post by qwertox »

deviantbit wrote: post #24

Just remember that Rene Descartes coined the term Imaginary Numbers, and he also believed in ghosts. Stop calling them imaginary.

They are like negative numbers. You can have 1 apple, but you can not have -1 apples. You can owe +1 apple, and say that you therefore have -1 apples. So -1 does exist as a number, but it is a representation of something that is happening with a positive number. And the "i" is similar to this. Maybe calling it "perpendicular" would have been better suited, because imaginary is really confusing.

Re: Quantum physics falls apart without imaginary numbers

#69

Post by machina_ex_deus »

andrew_eu wrote: post #15

There 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…

If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity).

The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.

Re: Quantum physics falls apart without imaginary numbers

#70

Post by C-x_C-f »

nathan_compton wrote: post #17
BrandoElFollito wrote: post #8

Earlier quoted context omitted.

This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)). First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine. Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you…

There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.

> There really isn't anything weird or suspect about renormalization

This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.

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