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This was my first thought on seeing the title. Complex numbers are just vectors with special behavior for some operations, right? I haven't read through the pa

• robbedpeter 5y ago
  I think you still "need" the imaginary number, even if you're expressing it as a more granular/verbose set of fundamentals, and I always wonder if there's an order of operations issue, or some syntactic gotcha instead of anything real?
• itchyjunk 5y ago
  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 this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).
  • asxd 5y ago
    "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: What is a field and why do we care?
    • thrtythreeforty 5y ago
      To use C++ terminology, It's any type where operator+(), operator-(), operator*(), and operator/() are defined and behave as they do in rational numbers.
• Vetch 5y ago
  > just vectors with special behavior for some operations > basically just a structure containing two real numbers and some modified behavior I think the issue might be that you're brushing away what is central to their utility and interestingness? Yes, you can take the view that it's a 2D euclidean vector space but it's not just. It's a 2D commutative algebra over the reals, an algebraically closed field and its algebraic properties and the addition of a notion of a rotation operation to our concept of number is what's of central importance.
• a9h74j 5y ago
  > Complex numbers are just vectors with special behavior for some operations, right? Decades since I took Complex Analysis, but: Not if you care about poles, zeros, residuals, cuts, conformal mappings, multiple layers of some sort overlaying the same point on the complex plane. It goes way beyond "declaring two variables" vs. "declaring a struct containing two variables." Not just a representation issue.
  • a9h74j 5y ago
    EDIT: To add a simple analogy: Is a parabola only x^2 -- just another polynominal -- or is a parabola a conic section first and foremost? polynominal in x == "flatland" view
• Sniffnoy 5y ago
  "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 numbers", because there's not much point in discussing trivial rephrasings like that!
  • prof-dr-ir 5y ago
    Then what is "quantum theory based on the real numbers"? I think your notion of "real amplitudes" cannot be the complete answer because the Schrodinger equation is linear, and complex linear algebra is just a special case of real linear algebra. I looked at the appendix of the paper and their claim seems to hinge on some form of non-decomposibility of a certain tensor product state? It is the paragraph below equation A2. More generally I got a bit frustrated reading the paper because the axioms of real quantum mechanics did not seem to be properly formulated.
  • asxd 5y ago
    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, since generally when I hear "complex number", my head thinks "compact way to represent a frequency, amplitude and phase", all of which are real. It seems like I was reeled in with familiar-sounding terminology that may in fact have a deeper meaning in this context.
    • simiones 5y ago
      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).
• crdrost 5y ago
  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 relativistically separated rooms (so they can’t communicate) they go, where they find a screen and buttons labeled 0 and 1. The screen displays a prompt, each teammate presses exactly one of the buttons once before time runs out, then the three numbers pressed get summed together into a number. 25% of the time we run a “control round,” everyone gets a prompt to make the sum of their numbers even, and they win if the sum is even. The easiest way is if everyone hits 0, 0+0+0 is even. But a team can also answer 0+1+1 or so and win. Otherwise we randomly choose one to be the traitor and send them the control prompt, to make the sum even. But we send the other two the prompt to make the sum odd! In this case the team will only win if their joint sum is odd. Long story short, classical players of this game have a success probability bounded from above by 75%. This is the Bell inequality. But quantum capable players can walk in with a GHZ state, |+++> + |–––>, which only collapses to even sums. If they have to do a control round they will all just measure this in the computational basis. The more interesting thing, where i really matters, comes during the traitor rounds. Here you want to perform the phase rotation gate in the Hadamard basis, |+> → |+>, |–> → i |–>, And any two of them can thereby switch the state to |+++> – |–––>, a state which only has odd configurations. Quantum players can win 100% of the time. Over multiple independent trials you should be able to observe the inequality violations even if quantum coherence were to limit your success probability to 90%. I suspect that the inequality here is something similar, quantum mechanics but you can only form real-coefficient superpositions, and therefore you cannot take the square root of a unitary transformation just by doing it for half the time, per Schrödinger.
  • selestify 5y ago
    I wish I understood enough QM to understand this comment! Any suggestions for where I might learn about GHZ states and the like?
    • 5y ago
• rstuart4133 5y ago
  I'm no an expert in complex numbers, or quantum theory or a heavy user of them. But I am a computer programmer. To a computer programmer, the fundamental difference between complex numbers is they can express infinite repetition succinctly. Now I try to write down what that means precisely, it's hard. An example of the effect is the polynomial for sin() is infinite, or you can express it as sin(x) = (e^(ix) - e^(-ix))/2i using complex numbers. To a computer programmer who makes his living from writing down formulas, the difference between having to write an infinite amount of code to express a concept or just use 20 characters above is profound. Different people find their profundity in different places, I guess - but this the sort of thing that drives us programmers to create entire new computer languages. That's not to say the primary observation I see being made here is wrong. That observation is that there complex numbers bring nothing really new to the table. You can do everything they do in other ways, say with matrices and a few extra rules. And indeed, mathematicians have come up with numerous other ways of expressing iteration, ∫ and Σ springing to mind. But nonetheless, the hyperoperations (counting, addition, multiplication, exponentiation, ...) are special. They are most heavily used mathematical operations, by a huge margin. They have one job - to capture the operation we programming nerds call iteration. But when restricted to real numbers, (I'm no mathematician, so this is conjecture), they can't capture self similarity. The result of function expressed as a finite number of hyperoperations operating on reals always flys off to infinity, or asymptotically approaches a constant, or is undefined over part of it's domain. When you add complex numbers you get another possibility: a possibly intricate pattern repeated infinitely. So yes, complex numbers are basically just a structure containing two real numbers and some modified behaviour, and these is nothing special about that. But there is something profound in the way they allow the planets favourite mathematical operations to finitely express infinitely more behaviours - with no more lexical overhead.