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> the chance of different configurations being realized are complex for quantum mechanics I don't think there is anything essential about complex numbers when

> the chance of different configurations being realized are complex for quantum mechanics I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. In the wiki article on the Heisenberg picture [1] of QM we read: > By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space. Then in the article on the Stone-von Neumann theorem [2] we read: > Informally stated, with certain technical assumptions, every representation of the Heisenberg group H2n + 1 is equivalent to the position operators and momentum operators on Rn And in the article on the Heisenberg group [3] we find out that this group consists of 3 by 3 upper triangular matrices with three independent elements that can be taken from any ring. This kind of high level lie group lie algebra approach is something I got really into in the past and continue to be fascinated by. I wish I could explain better why as a result I don't see complex numbers as essential at all in the foundations of quantum theory, but rather, just very convenient for doing calculations. I hope you find the linked wiki articles as interesting as I do. Cheers! [1] https://en.wikipedia.org/wiki/Heisenberg_picture https://en.wikipedia.org/wiki/Heisenberg_picture [2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#Representation_theory_formulation https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo... [3] https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group

jessriedel 8y ago
> I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. I'm well aware of this (as this is my profession). My other comment, which you already replied to, agrees that there exist representations of QM that do not feature complex numbers. That's besides the point I was making.
chriswarbo 8y ago
> Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. Slightly tangential, but whilst this is a perfectly valid way of thinking about things, to me it seems to be mixing up concepts (e.g. real numbers, complex numbers, matrices, etc.) with encodings-of-concepts-using-other-concepts (in software terms, it's conflating an interface with a particular implementation). In particular, I wouldn't say that a complex number is a pair of real numbers. I'd rather define complex numbers by their algebraic laws (e.g. i × i = -1), and note that some other concepts (like pairs of reals, certain matrices, clifford algebras, etc.) can satisfy these same laws. Similar to how fractions can be encoded as a pair of integers (where "(a, b)" encodes "a ÷ b"), integers can be encoded using a pair of naturals (where "(a, b)" encodes "a - b"), etc. yet we rarely claim that, say, -3 is the pair (0, 3). Likewise a vector/matrix/tensor isn't a 1/2/N dimensional arrangement of numbers. This is a total nit-pick, but notice that you're larger point is basically arguing the same thing w.r.t. quantum mechanics (that it doesn't rely on complex numbers, since other representations exist) ;)

jessriedel 8y ago

> I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. I'm well aware of this (as this is my profession). My other comment, which you already replied to, agrees that there exist representations of QM that do not feature complex numbers. That's besides the point I was making.

chriswarbo 8y ago

> Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. Slightly tangential, but whilst this is a perfectly valid way of thinking about things, to me it seems to be mixing up concepts (e.g. real numbers, complex numbers, matrices, etc.) with encodings-of-concepts-using-other-concepts (in software terms, it's conflating an interface with a particular implementation). In particular, I wouldn't say that a complex number is a pair of real numbers. I'd rather define complex numbers by their algebraic laws (e.g. i × i = -1), and note that some other concepts (like pairs of reals, certain matrices, clifford algebras, etc.) can satisfy these same laws. Similar to how fractions can be encoded as a pair of integers (where "(a, b)" encodes "a ÷ b"), integers can be encoded using a pair of naturals (where "(a, b)" encodes "a - b"), etc. yet we rarely claim that, say, -3 is the pair (0, 3). Likewise a vector/matrix/tensor isn't a 1/2/N dimensional arrangement of numbers. This is a total nit-pick, but notice that you're larger point is basically arguing the same thing w.r.t. quantum mechanics (that it doesn't rely on complex numbers, since other representations exist) ;)