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Why are amplitudes complex numbers? (2018)
- kazinator 3y agoAmplitudes are complex numbers, because periodic signals exhibit phase. The modulus ("complex absolute value") gives us the amplitude, and the angle encodes phase. The way we usually encode phase is that it's frequency dependent. π or 180° of phase corresponds to a frequency-dependent amount of time. At any frequency, sine and cosine waves are at 90 degrees from each other, forming a quadrature. This 90 degrees has not only frequency and time interpretation, but a vector interpretation. If we chop the signal into samples to make a vector of numbers, the sine and cosine vectors will be at 90 degrees in the sense that their cross product is zero. I.e. actually perpendicular in the N-space they inhabit. Under Fourier analysis, we are projecting the signal onto these basis vectors: how much of the sin, and how much of the cos. We can combine the sine and cosine into a complex number thanks to Euler's formula e(ix) = cos(x) + i sin(x). By imagining the signal as being in polar coordinates, where its phase angle is the angle around the complex plane, and amplitude is the modulus, we simplify and condense the math. The two vectors at 90 degrees apart are combined into one vector of complex numbers for us to deal with. So "how much of this signal correlates with sin(x)" gives us the imaginary component, and "how much of this signal correlates with cos(x)" gives us the real component. We can just add these together to make a complex number. Its argument (angle) gives us the phase, and modulus the amplitude.
- xeonmc 3y agoBasically, either you have unsigned numbers (conjugated magnitude), with growth/shrink operations, or you have numbers with a polarity, with displacement operations . It is incomplete to have the notion of "negative numbers" without also including the imaginary parts.
- jojobas 3y agoIt's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
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- xeonmc 3y agoIf you have real numbers instead of integers, then it implies fractional applications of the arithmetic operators. Fractional negation is complex rotation, so if you have negative numbers and they are non-integers then it necessarily must also include complex numbers.
- jojobas 3y ago> implies fractional applications of the arithmetic operators Does it? Where does this implication come from?
- xeonmc 3y agoYou can’t generate a continuum without log/exp
- jojobas 3y agoSo log/exp or partial operators? And what do you mean by "generate a continuum"? Both log and exp are formulated without the need for complex numbers. Real numbers are a field for +-/ (although not a group for / because zero is weird).
- avalys 3y agoI would very much enjoy reading the pre-reading for this post.
- brookst 3y agoAnd the pre-pre, and probably pre-pre-pre. Much respect to those who can follow this.
- whatshisface 3y agoHere, try this: https://www.scottaaronson.com/democritus/ https://www.scottaaronson.com/democritus/
- lupire 3y agoA simpler preliminary question would be, why are complex numbers present in non quantum wave mechanics, and then how does this compare and contrast to quantum mechanics.
- thsksbd 3y agoWhy would this be weird? The classical waves have an amplitude and phase, which is all the QM probability functions really are.
- bdjsiqoocwk 3y ago> A simpler preliminary question would be, why are complex numbers present in non quantum wave mechanics I'm not sure there's anything deep here. Imaginary exponentials contain sines and cosines and so are the solution to a lot of differential equations, even in the complex solutions have no physical interpretations. See e.g. the case of electromagnetism.
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- ShamelessC 3y agoThe answer provided essentially requires you to know so much in advance that it would be unsurprising or even obvious to you. I'm not entirely sure who this writing style is for, if not simply Scott Aaronson's note-taking process...
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- sw1sh 3y agoComplex numbers are not a QM feature; they are just more convenient to use in computations. There are equivalent formulations, like Wigner's phase-space with quasi-probabilities and MIC-POVMs that don't have any complex numbers. The weird part is the negative probabilities, not imaginary numbers.
- whatshisface 3y agoThe equivalent formulations aren't the alternate theories that information theorists study when they say, "QM without complex numbers." They study very much non-equivalent theories, although they feel unnatural enough that it's hard to see what they're all about unless you closely study the technical details. To tell you the truth I don't have any picture of what restricting the matrices to real numbers means physically.
- MichaelZuo 3y agoWhat do matrices with complex numbers 'mean physically'?
- whatshisface 3y agoThe measurement operators are matrices that come about as a result of assigning real eigenvalues (these are your possible measurement outcomes) to orthonormal vectors (your arbitrary coordinate system). The results are hermitian, complex-valued matrices, because that's just what comes out if you try to engineer a matrix to have those eigenvalues and vectors. The rest follows from that. Trying to fit a real number constraint somewhere, other than the one that's already there (real measurement outcomes), to me seems like the step you would have to justify, not the absence of one.
- MichaelZuo 3y agoCan you elaborate on the physical meaning that follows? I just don't see how it links up with something tangible in the real world.
- tvst 3y agoIf "i" wasn't called "imaginary" I don't know if anyone would find it weird when it appeared in physics. In many ways i is as weird as negative numbers, irrational numbers, and transcendental numbers. But we're somehow ok with all of those. (By the way, I don't mean to imply Scott Aaronson finds complex numbers weird. He's just wondering why not other systems, and even mentions quaternions as an alternative — which could be called weird in their own right... So in a sense I'm attacking a straw man.)
- jojobas 3y agoNegative, irrational and transcendental numbers are all on the same number line. Getting out into a plane (and losing something as important as the order relation) is radically different from figuring out what other numbers are on a line.
- whatshisface 3y agoOrder relations don't appear in QM.
- jojobas 3y agoThey do, a lot, just not about wave-function amplitudes.
- whatshisface 3y agoThey don't appear in the mechanics, they appear in the things we say about the mechanics.
- bdjsiqoocwk 3y agoThis is in the context of QM, and that should be included in the title. "Why are amplitudes complex numbers" is meaningless, unless it's clear from the context that we're talking about QM amplitudes. I would suggest "Why are amplitudes [in quantum mechanics] complex numbers?"
- jl2718 3y agoThis seems to me as backward reasoning. Maxwell’s equations were descriptive of an observational phenomena, which was electromagnetic oscillation, and Schroedinger’s equations used amplitudes to unify them as an energy conservation law. The author asks “why not quaternions”, and gives the reasons, for which he concludes that complex numbers must be the only fabric of the universe. This is backward reasoning, and also narrow to the author’s specialty. Quaternions are just one of many choices for a 4th-degree extension field of the reals, where complex are the only choice of 2nd degree extension. Yang-Mills theory uses Lie algebras of higher degree to form an energy conservation law that unifies the observations of electromagnetism with other observations made at high energy, of which the complex numbers of electromagnetism are a subset. The article appears to suggest that QM phenomena exist because of the mathematical curiosities of the complex numbers applied to linear operators. Its not my place to say whether the universe “is” mathematics, but I do feel comfortable using math as an attempt to describe it, and there may be many choices, but our convention generally is to choose the description that eliminates possibilities that are rejected experimentally, and many such may be required, but our secondary preference is to choose the smallest set of descriptions that contain the observations.