The article is a bit sparse of technical details but am I misunderstanding what they're doing or are they describing a field that's isomorphic to C but described as a pair of real numbers? If so, I don't see how that meaningfully takes the imaginary numbers out of quantum mechanics any more than renaming imaginary numbers as extended numbers would.
Even then, all of chemistry DFT is based on the idea that the electron density contains the physical observable information and you and I both know that the overall phase of the wave function isn't physical except through interference. There is plenty of useful qm without C already out there!
This is referring to the fact that overall phase is not real (no observable difference) but relative phase has. The word “except” is not downplaying its importance, but to emphasize the fact that overall phase isn’t physical.
This might be the crux of the concern though -- "imaginary" numbers are terribly named and even mathematicians can conflate their concept with their name.
„the use of complex numbers helps to distinguish between quantities, that can be measured simultaneously and the one which can't. You would loose that feature, if you would formulate QM purely with real numbers.“
"our knowledge of quantities is necessarily accompanied by uncertainty. Consequently, physics requires a calculus of number pairs and not only scalars for quantity alone. Basic symmetries of shuffling and sequencing dictate that pairs obey ordinary component-wise addition, but they can have three different multiplication rules. We call those rules A, B and C. “A” shows that pairs behave as complex numbers, which is why quantum theory is complex."
"the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic"
I mean it's trivial to do quantum math without the imaginary units. Just rename the solutions to algebraic polynomials something else and continue.
There is nothing strange about i and claims contrary to that misunderstand what it even is. Partly terminology is to blame. I simply represents a 90° rotation of space. Really quite simple and easily measurable in our 3d world
No not at all. I is just something that behaves as if it is equivalent to negative one (that is, the additive inverse of the multiplicative identity) after combining it with itself in some way. We commonly call this multiplication. If such a thing comes with another operation called addition that behaves similarly to addition and multiplication (i.e. form a ring), then they will behave like i. Geometrically, multiplication by I can be seen as a 90deg rotation of a 2d vector. Complex numbers are simply 2-d coordinates (or rather, they are isomorphic to 2-d coordinates). Nothing special really. Easy to measure with a protractor and ruler.
In general there are many algebraic rings with an element that, when multiplied by itself, produces the additive inverse of the multiplicative identity.
In math, officially i is the "root" of x^2+1=0 or to be more precise, C is R[x]/x^2+1, i.e. you take all the polynomials in x and pretend that the polynomials A and B they are equivalent when A-B is a multiple of x^2+1.
There is also a construction with matices instead of polynomials.
And perhaps others. Each of them are useful in some cases.
X*X + 1 = 0 is a fundamental statement on an algebraic rings behavior with the additive and multiplicative identities and the additive and multiplicative group operations. Namely, it says that the ring contains an element that when multiplied by itself is equal to the additive inverse of the multiplicative identity . Plenty of rings have such an element. You can complete any ring with such an element and call it whatever you want. The use of the term imaginary for it is incredibly unfortunate. There's nothing strange or mystical about it. It's very real. In fact the rational complex numbers are more real than the non complex real numbers
Dummit and Foote is the classic abstract Algebra textbook to learn about how to precisely define these. Its treatment of ring theory is very well motivated and easy to grasp
In general, determining if two arbitrary reals are the same is impossible per the halting problem. People claim to measure 'real' numbers. This is a lie. People can only measure rational numbers. A real number is either a rational or the supremum of some arbitrary set of rationals (perhaps an infinite one). A set is described by whether or not a number is in it. To be able to determine what number is in your set you need to have some sort of decision procedure (a program). However, more real numbers exist than there are possible written programs. Thus, the full set of reals is inexpressible
On the other hand, it's very easy to see and measure rational complex numbers with a protractor.
The article is a bit sparse of technical details but am I misunderstanding what they're doing or are they describing a field that's isomorphic to C but described as a pair of real numbers? If so, I don't see how that meaningfully takes the imaginary numbers out of quantum mechanics any more than renaming imaginary numbers as extended numbers would.
Yup, swapping the complex numbers with matrices that encode the same transformations:
Here is the paper:https://arxiv.org/pdf/2504.02808
Even then, all of chemistry DFT is based on the idea that the electron density contains the physical observable information and you and I both know that the overall phase of the wave function isn't physical except through interference. There is plenty of useful qm without C already out there!
"except through the inference" is carrying a lot of weight there. That's pretty physical.
This is referring to the fact that overall phase is not real (no observable difference) but relative phase has. The word “except” is not downplaying its importance, but to emphasize the fact that overall phase isn’t physical.
This might be the crux of the concern though -- "imaginary" numbers are terribly named and even mathematicians can conflate their concept with their name.
„the use of complex numbers helps to distinguish between quantities, that can be measured simultaneously and the one which can't. You would loose that feature, if you would formulate QM purely with real numbers.“
https://physics.stackexchange.com/a/83219/1648
Related:
https://www.mdpi.com/2673-9984/3/1/9
"our knowledge of quantities is necessarily accompanied by uncertainty. Consequently, physics requires a calculus of number pairs and not only scalars for quantity alone. Basic symmetries of shuffling and sequencing dictate that pairs obey ordinary component-wise addition, but they can have three different multiplication rules. We call those rules A, B and C. “A” shows that pairs behave as complex numbers, which is why quantum theory is complex."
https://arxiv.org/abs/0907.0909
"the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic"
I mean it's trivial to do quantum math without the imaginary units. Just rename the solutions to algebraic polynomials something else and continue.
There is nothing strange about i and claims contrary to that misunderstand what it even is. Partly terminology is to blame. I simply represents a 90° rotation of space. Really quite simple and easily measurable in our 3d world
I or j don’t need to originate from Sqrt(-1) ?
No not at all. I is just something that behaves as if it is equivalent to negative one (that is, the additive inverse of the multiplicative identity) after combining it with itself in some way. We commonly call this multiplication. If such a thing comes with another operation called addition that behaves similarly to addition and multiplication (i.e. form a ring), then they will behave like i. Geometrically, multiplication by I can be seen as a 90deg rotation of a 2d vector. Complex numbers are simply 2-d coordinates (or rather, they are isomorphic to 2-d coordinates). Nothing special really. Easy to measure with a protractor and ruler.
In general there are many algebraic rings with an element that, when multiplied by itself, produces the additive inverse of the multiplicative identity.
In math, officially i is the "root" of x^2+1=0 or to be more precise, C is R[x]/x^2+1, i.e. you take all the polynomials in x and pretend that the polynomials A and B they are equivalent when A-B is a multiple of x^2+1.
There is also a construction with matices instead of polynomials.
And perhaps others. Each of them are useful in some cases.
X*X + 1 = 0 is a fundamental statement on an algebraic rings behavior with the additive and multiplicative identities and the additive and multiplicative group operations. Namely, it says that the ring contains an element that when multiplied by itself is equal to the additive inverse of the multiplicative identity . Plenty of rings have such an element. You can complete any ring with such an element and call it whatever you want. The use of the term imaginary for it is incredibly unfortunate. There's nothing strange or mystical about it. It's very real. In fact the rational complex numbers are more real than the non complex real numbers
> In fact the rational complex numbers are more real than the non complex real numbers
Fascinating. Can you say more about this or point me to where I may learn?
Dummit and Foote is the classic abstract Algebra textbook to learn about how to precisely define these. Its treatment of ring theory is very well motivated and easy to grasp
In general, determining if two arbitrary reals are the same is impossible per the halting problem. People claim to measure 'real' numbers. This is a lie. People can only measure rational numbers. A real number is either a rational or the supremum of some arbitrary set of rationals (perhaps an infinite one). A set is described by whether or not a number is in it. To be able to determine what number is in your set you need to have some sort of decision procedure (a program). However, more real numbers exist than there are possible written programs. Thus, the full set of reals is inexpressible
On the other hand, it's very easy to see and measure rational complex numbers with a protractor.
Easier to see without the square root:
What action when applied twice results in a sign change?"A 90 turn" is one answer. There are probably others.
“A -90 turn” would be another.