Comments by "Lepi Doptera" (@lepidoptera9337) on "Do Complex Numbers Exist?" video.

Complex numbers in quantum mechanics represent rotations. Of course we can represent rotations without complex numbers. Matrices work just fine. The paper is complete nonsense, anyway. We know where rotations and hence complex numbers come into play from first principles. The entire structure of quantum mechanics can be derived from a few trivial axioms with like half a dozen pages of high school level math and has been repeatedly by several authors over the years. The papers in which this was done are about as interesting as watching paint dry.
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1 is the loneliest number. ;-)
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@teezettsb Look up "covariant formulation of classical electrodynamics" on Wikipedia, for instance. That's the modern form that is in use by most theorists today if I am not mistake (I am not a theorists, just talking to some on occasion). In general most of the "interesting" things in theoretical physics are happening at the level of group representations, Lie-algebras and now even topology. There has been quite a bit of shift in physics from more or less "point and shoot" calculations to mathematically well founded formal structures.
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You don't need complex numbers for any of these things. Matrices do just fine. Complex numbers are just a convenience for the case of two dimensional rotations. Quaternions are a convenience for three dimensions. After that it's all matrix algebras anyway. There is no point in trying to push the hypercomplex algebras any further. They have their limited uses and after that the classic algebraic approach is more or less dead.
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Some numbers (the finite integers) can be implemented as cardinality of sets. Negative numbers can't. They can still be found in the differences between sets, which is a well defined concept (as a physical operation). You just described how it works. One set of apples has more members than the other. The one that has less can be assigned a differential cardinality relative to the other and that will be denoted by a negative number.
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The point of theoretical physics is to derive self-consistent representations of nature. Is this stuff theoretical physics? No, not really. It's more of a public admissions of these authors that they don't have a clue what quantum mechanics is. :-)
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It's the other way around. All of mathematics was originally an abstraction of physical observations. We didn't "invent" numbers. They dropped on us because classical objects can form collections that behave like sets. Set theory is literally an abstract extension of "fruits in baskets". One can say the same thing about the real and complex numbers, groups, Riemannian manifolds etc.. There is, of course, a "backreaction". Some deeper properties like the topological properties have non-trivial and not easily accessible consequences for physics, which wasn't understood until relatively lately (as in the last fifty years, roughly) and we are still working very hard on exploring these connections.
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@worldnotworld I didn't say that these things are not real. What I am saying is that all these ideas came from real things and real world problems. Mathematics, at the core, is physics that has been liberated of its experimental and observational basis. We can discuss whether things like the axiom of choice have any relevance to practical applications of set theory in science (I don't believe so), but these are, IMHO, fringe problems, even for most mathematicians. What will hit the mathematicians really hard in a short amount of time (less than a century) is the fact that physicists have already liberated themselves from the idea of objects with the advent of quantum theory. Sets are simply not what underlies nature. Implementations of sets in form of classical objects in classical baskets are merely an emergent property of nature. There has, as far as I can tell, not been any serious work in mathematics on the idea that maybe one should start math not with sets and formal logic but with a generalized version of quantum mechanics. Quantum math (which is NOT the same as the math of quantum mechanics but much, much more general) simply does not exist, yet, but eventually it will.
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So what happened? Did you hit your head and had a bleeding in your brain? Complex numbers are completely trivial, kid. :-)
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@ysf-d9i That all (non-trivial) polynomials of order n have a complete set of n zeros in the complex plane doesn't make sense to you? What??? Can I have whatever it is that you are smoking right now? ;-)
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@ysf-d9i What's up with the projection, kid? Are you really that lonely? Let me give you some attention then. :-)
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@pelasgeuspelasgeus4634 You can draw the imaginary axis in any direction on paper, just like you are drawing the real axis on paper. These things don't exist in nature. They are abstractions in your mind. You clearly don't understand how your mind works. ;-)
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@pelasgeuspelasgeus4634 You are asking about things that don't exist. The description of something is not a something. It's nothing. The distinction is kindergarten level categories. John may be your friend, but "John", his name, is not. If John, the kid, is not in the room, then you can't play with "John", the name of the kid who is not in the room. :-)
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@pelasgeuspelasgeus4634 I perfectly understood the first time around that your mental age is in the low single digit numbers. You keep asking childish questions. ;-)
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Yes, it would be. It's part of the complex numbers, too. Look up "Riemann sphere".
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@pelasgeuspelasgeus4634 There are no axes in physical space, either. Those are merely abstractions used to describe the motion of classical objects in spacetime.
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@pelasgeuspelasgeus4634 If I check my GPS I will see three numbers. If I look at a complex number I see a scale factor and a rotation. That you can't only tells me that you don't understand complex numbers. :-)
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@pelasgeuspelasgeus4634 No, it doesn't care about rotations. The devices that care about rotations are called "gyroscopes". Dude. Please. Pay. Attention. In. School. :-)
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@pelasgeuspelasgeus4634 I am saying that You. Need. To. Pay. More. Attention. In. School. :-)
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Why are you telling us that you weren't paying any attention in science class? ;-)
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Numbers are a human invention. Nature really doesn't care how we describe it.
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Complex numbers are one way to define rotations (and scaling) in two dimensions. That's all there is to them, even though a lot of interesting stuff follows from that.
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In physics it does. If you change the sign it means that you are changing the direction in which you are going in time (for many problems).
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No, but you can have one sheep less. ;-)
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@jasonthomas2908 Whether you had one sheep more or one sheep less could mean eating or starving in the past. ;-)
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@jasonthomas2908 I don't know what mathematical reality is supposed to be. Is it all the random thoughts in your mind? I don't care about those. I do care about the difference between one sheep less and one sheep more. It can mean survival. ;-)
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Complex numbers don't exist because numbers are abstracts. What does exist are rotations, which can be represented by complex numbers.
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Complex numbers aren't describing particles. They are describing rotations of representations of the states of ensembles.
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@idegteke Nothing is made of ensembles. An ensemble is the abstract notion that one can repeat the (exact) same experiment an infinite number of times. In practice that is completely impossible, but many physical systems (like a thermal light source, for instance) behave very similar to such an ensemble and can be approximated by one. Other systems (like a human body) can not be. The ensemble description is simply not a valid approach to every system, but it is valid for many important systems and that is why we are using it in quantum THEORY.
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@idegteke Physics is 100% dependent on observations. Just because you didn't take the time to learn enough about nature doesn't invalidate either nature or its scientific descriptions. At most it indicates that trying to get you a modicum of scientific education in school was a waste of taxpayer money. We should have used it on planting trees, instead. ;-)
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@idegteke Yes, you are a bullshitter. Why do I have to care? I don't. :-)
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Exactly. But mathematics has historically followed physics very closely. Set theory is the behavior of collections of classical objects and their properties. Euclidean geometry is the generalization of the physics of the tangent space etc.. Complex numbers are rotations (and scaling) in two dimensions. So, yes, mathematics borrows very heavily from physics. That's why it's useful for physicists to begin with.
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Maybe not, but it can still hit you in the face with a durian if you sleep under the wrong tree in the tropics. It's a miracle.
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Yes, but you don't want to collide with it. It's usually very painful and dangerous.
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The point is that they represent two dimensional rotations. Modern physics is all about rotations.
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Complex numbers are rotations in two dimensions, quaternions are rotations in three dimensions. In modern theory we reduce Maxwell to an equation using rank 2 4-tensors. That's much, much better. It also generalizes nicely to all other possible representations of the Poincare group.
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Why do you want to hear even more nonsense from Sabine? ;-)
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It's another nonsensical video by a YouTuber who is only interested in view count. ;-)
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See the post of Maciej Zwoliński above.
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Which form of multiplication exists in a vector space is highly dependent on the dimension of that vector space, though. In physics we are typically dealing with dimensions 1-4 and lately roughly up to 11.
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Quantization does not remove the need for real numbers. Energy is, for instance, not quantized. Neither is momentum. Only angular momentum and charges are and even they have non-trivial algebras that contain a lot of roots.
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