Earlier quoted context omitted.
> x^n + y^n = z^n > There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n. I love this type of humour.
Care to explain? My maths are failing me.
Quantum theory based on real numbers can be experimentally falsified
91–100 of 130 posts
Re: Quantum theory based on real numbers can be experimentally falsified
#92Earlier quoted context omitted.
Care to explain? My maths are failing me.
https://en.wikipedia.org/wiki/Fermat's_Last_Theorem
Which is extra funny as the actual proof[1] is so involved, roughly ~130 pages as published and spanning multiple very complex mathematical subjects.
[1]: https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...
Re: Quantum theory based on real numbers can be experimentally falsified
#93Earlier quoted context omitted.
"Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that). It doesn't mean just any way of representing quantum mechanics with real numbers. Of course you can represent quantum mechanics with real numbers, for the reason you say; but for that very reason, that isn't what anyone means by "quantum theory based on real num…
I appreciate you're able to see the quandary. I think the crux of what you said is in your first sentence: > "Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that) I'm familiar with complex math as far as remedial DSP and electrical engineering goes, so this may be over my head. I'm not sure what a real amplitude is,…
I think the base idea here is something like this: if you want to describe, say, a sound-wave, you can use complex numbers to represent the wave, but you can also, in principle, use strictly real numbers to describe the behavior of each individual molecule of gas using Newton's equations of motion (assuming you can ignore quantum effects for your simulation). So, in classical mechanics, while complex numbers are a useful abstraction for waves, they are not fundamental.
The same is not true for QM: the wave part / the complex numbers are fundamental.
Also note, when people say "complex numbers" they refer to the algebraic field, which is the object composed of [a pair of reals, complex addition, complex multiplication]. In particular, complex multiplication is the key here, since it is what differentiates complex numbers from 2-vectors. To drive this point further, while 1-vectors, 2-vectors, 3-vectors, 4-vectors etc behave very similarly, only the reals and the complex numbers can form a field - there are no "numbers" in the same sense formed of 3-tuples or 4-tuples or other n>2-tuples of reals*.
This is why the discussion focuses on the complex numbers (meaning, again, not just a pair of reals, but also the particular +,-,*,/ operations for them that make them a field).
* to be fair, quaternions come pretty close - you can define a 4-tuple of reals + addition + multiplication that is almost a field, except that multiplication is anti-commutative instead of being commutative (p*q = -q*p).
Re: Quantum theory based on real numbers can be experimentally falsified
#94I'm surprised the article doesn't mention phases or the idea of projective Hilbert spaces. The QM formulation it gives is well known ambiguous: the condition =1 does not determine φ, because it's also satisfied by any other state ψ=exp(iλ)φ with λ real. So, the state of a physical system is really described by the ray (equivalent class) of all state vectors differing by a phase. I wonder if this has any consequence o…
You can work in projective space or with any representative in the full space, or keeping track of global phases, it really doesn't make any difference. But all physicists are aware of this afaik. As you say, it is well known, so no need to mention it.
Re: Quantum theory based on real numbers can be experimentally falsified
#95Earlier quoted context omitted.
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 thi…
"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: Wh…
To use C++ terminology, It's any type where operator+(), operator-(), operator*(), and operator/() are defined and behave as they do in rational numbers.
Re: Quantum theory based on real numbers can be experimentally falsified
#96Earlier quoted context omitted.
So not having yet read through OP I am not terribly surprised that this is true and I can kind of give a quick sketch in terms of a QM game that I want everyone to know, called Betrayal. The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativist…
I wish I understood enough QM to understand this comment! Any suggestions for where I might learn about GHZ states and the like?
If you want to understand math behind Quantum Mechanic and Quantum Computing I think this is a great resource: https://scottaaronson.blog/?p=4805
Re: Quantum theory based on real numbers can be experimentally falsified
#97Okay so for us laypeople - what does this mean? Quantum theory is wrong? :-)
It might mean that complex numbers (real plus imaginary part) are real in the physical sense.
Re: Quantum theory based on real numbers can be experimentally falsified
#98Earlier quoted context omitted.
> x^n + y^n = z^n > There's a cute little fact -- unfortunately I won't have time to prove it in class -- that the above equation has nontrivial integer solutions when n=1 or n=2, but not for any larger integers n. I love this type of humour.
Care to explain? My maths are failing me.
There is a good book on this, I forget the name of the book, was it biography of Andrew Wiles?
Re: Quantum theory based on real numbers can be experimentally falsified
#99Earlier quoted context omitted.
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.
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.
>> 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.
Re: Quantum theory based on real numbers can be experimentally falsified
#100Note 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…
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.
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.