Quantum theory based on real numbers can be experimentally falsified
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Re: Quantum theory based on real numbers can be experimentally falsified
#2Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc.
Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the time. Dirac figured he needed some 4×4 matrices in order to be able to take the square root of some operator at some point, which is how his equation predicted the existence of antimatter (because it implied the wavefunction had to be a vector with more components - some of the other components turned out to be antimatter).
But complex numbers happen to have the right properties that at least we can do without matrices and vectors for the lowest-level quantities in quantum mechanics: the state amplitudes or the values of wavefunctions of spinless, non-relativistic particles. This article is saying that you can't do away with these properties at the lowest level of quantum mechanics, whether or not you actually use complex numbers to represent them.
Re: Quantum theory based on real numbers can be experimentally falsified
#3Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
Re: Quantum theory based on real numbers can be experimentally falsified
#4Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
Re: Quantum theory based on real numbers can be experimentally falsified
#5Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
Unless we live in a computer simulation, of course, in which case we have to content ourselves with the IEEE floats.
Re: Quantum theory based on real numbers can be experimentally falsified
#6Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the…
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Re: Quantum theory based on real numbers can be experimentally falsified
#7Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
You can describe it to within measurement error without any irrational numbers. And with fewer decimal places than you'd probably imagine.
See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.
Re: Quantum theory based on real numbers can be experimentally falsified
#8Note that it is trivial to split the real and imaginary parts into two separate real-numbers and write quantum mechanics that way with only real numbers. Instead of i you get a 90 degree rotation matrix, instead of individual numbers you get a 2-element vector, etc. Lacking "numbers" with the right arithmetic properties for other things in quantum mechanics, we indeed use matrices and vectors for other stuff all the…
Sure is a complicated way to say that there are rotations somewhere in qm though. In particular once you realize that rotations are a good way to describe how a particular property is preserved under certain group operations, symmetry you might say.
Re: Quantum theory based on real numbers can be experimentally falsified
#9Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
...but Planck length!
Re: Quantum theory based on real numbers can be experimentally falsified
#10Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
Pi is the ratio of a mathematical circle to its diameter, but there are no physical circles which have that ratio as exactly Pi, and even if one existed, you'd never be able to distinguish it from one that was merely equal to Pi to the accuracy you are capable of measuring, because you'd need to measure with infinite precision, which you can't.