Why are amplitudes complex numbers? (2018)(scottaaronson.blog)
scottaaronson.blog
Why are amplitudes complex numbers? (2018)
https://scottaaronson.blog/?p=4021
37 comments
This 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?"
The 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...
Amplitudes 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.
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.
I would very much enjoy reading the pre-reading for this post.
And the pre-pre, and probably pre-pre-pre. Much respect to those who can follow this.
Here, try this: https://www.scottaaronson.com/democritus/
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A 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.
> 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.
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.
Why would this be weird? The classical waves have an amplitude and phase, which is all the QM probability functions really are.
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Complex 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.
The 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.
These alternate theories that use quaternions and whatnot are just mathematical marvels that have nothing to do with physics, QM can be well explained with conventional probability theory already (even without generalized probability theories that people also study). Seems like negative conditional probabilities governing quantum processes can be understood as intrinsic Bayesian inference or Particle filter estimators. I wish there were more research in this direction ...
>QM can be well explained with conventional probability theory already
The subjective experience of a person performing QM experiments, sure, but not the actual universe, that's what Bell's theorem was about.
The subjective experience of a person performing QM experiments, sure, but not the actual universe, that's what Bell's theorem was about.
That's not what I'm saying. Bell or any CHSH-like experiment can be equivalently described using random variables and quasi-stochastic processes instead of quantum states, unitaries, and measurements. It would still involve non-local correlations and inequality violations, but without mentioning the Born rule, phases, and interference with imaginary numbers. It is just an equivalent mathematical framework.
This paper [1] doesn't stoop to providing an example, but isn't that just the thing where you can write 1 and i as 2x2 matrices? I don't think that's what Scott is talking about. Requiring that the elements of the density matrix be real (or allowing them to be quaternion) creates a non-equivalent theory.
[1] https://arxiv.org/pdf/1704.08525.pdf
[1] https://arxiv.org/pdf/1704.08525.pdf
Right, these are different questions indeed. Scott wonders what happens to amplitudes as they already appear in the theory but with numbers being no longer complex. But those lifted representations effectively change to a specific basis in higher dimensions (think qubit's 2x2 density matrix becoming a 4-dimensional distribution vector, with the same 3 real degrees of freedom) where everything is real and interpreted as probabilities.
Well, yes, but real matrices are also a subspace of complex matrices, you don't have to switch to a real valued representation of GL(n) to arrive at that.
What do matrices with complex numbers 'mean physically'?
The 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.
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.
Can you elaborate on the physical meaning that follows?
I just don't see how it links up with something tangible in the real world.
I just don't see how it links up with something tangible in the real world.
The complex numbers in the matrices appear as a consequence of trying to do something else. I don't think they have much physical meaning on their own, which is why I am surprised that people ask what it would mean if they had to all be real numbers.
The parent 'sw1sh' didn't ask that? Do you mean some sort of critique found elsewhere?
If "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.)
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.)
Negative, 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.
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.
Order relations don't appear in QM.
They do, a lot, just not about wave-function amplitudes.
They don't appear in the mechanics, they appear in the things we say about the mechanics.
Basically, 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.
It is incomplete to have the notion of "negative numbers" without also including the imaginary parts.
It's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
If 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.
> implies fractional applications of the arithmetic operators
Does it? Where does this implication come from?
Does it? Where does this implication come from?
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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.