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Why are amplitudes complex numbers? (2018)

scottaaronson.blog

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Re: Why are amplitudes complex numbers? (2018)

#11

Post by whatshisface »

sw1sh wrote: post #10

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.

Re: Why are amplitudes complex numbers? (2018)

#12

Post by MichaelZuo »

whatshisface wrote: post #11
sw1sh wrote: post #10

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 physica…

What do matrices with complex numbers 'mean physically'?

Re: Why are amplitudes complex numbers? (2018)

#13

Post by whatshisface »

MichaelZuo wrote: post #12
whatshisface wrote: post #11

Earlier quoted context omitted.

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 physica…

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.

Re: Why are amplitudes complex numbers? (2018)

#14

Post by sw1sh »

whatshisface wrote: post #11
sw1sh wrote: post #10

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 physica…

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 ...

Re: Why are amplitudes complex numbers? (2018)

#15

Post by whatshisface »

sw1sh wrote: post #14
whatshisface wrote: post #11

Earlier quoted context omitted.

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 physica…

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 estimator…

>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.

Re: Why are amplitudes complex numbers? (2018)

#16

Post by tvst »

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.)

Re: Why are amplitudes complex numbers? (2018)

#17

Post by jojobas »

tvst wrote: post #16

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 c…

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.

Re: Why are amplitudes complex numbers? (2018)

#18

Post by whatshisface »

jojobas wrote: post #17
tvst wrote: post #16

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 c…

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.

Order relations don't appear in QM.

Re: Why are amplitudes complex numbers? (2018)

#19

Post by sw1sh »

whatshisface wrote: post #15
sw1sh wrote: post #14

Earlier quoted context omitted.

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 estimator…

> 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.

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.

Re: Why are amplitudes complex numbers? (2018)

#20

Post by jojobas »

xeonmc wrote: post #2

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's only incomplete if you require particular operations, you're perfectly fine with +-*/ and real numbers.
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