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The question of the relationship of complex numbers and QM is extremely interesting imo, and a lot of the article is devoted to this question, not sure I totall

The question of the relationship of complex numbers and QM is extremely interesting imo, and a lot of the article is devoted to this question, not sure I totally agree the subject at hand is why is the universe the way it is. Additionally, if by real amplitudes he meant a completely different set of laws, I don't see how that is relevant to a discussion of why complex numbers play such an important role in QM. QM is really weird. One of its strangest properties is that although it contains classical mechanics as a limiting case, QM also required classical mechanics for its very formulation, see page 3 here [1] When most people encounter QM for the first time, they want to come up with some "intuitive sounding" principles from which they can derive everything else. That's natural. Most people just give up at that and work on other things, other people take it really really far. So far no one has been able to do it, but who knows right. In the article we can find this passage: > A major research goal in quantum foundations, since at least the early 2000s, has been to “derive” the formalism of QM purely from “intuitive-sounding, information-theoretic” postulates The entire approach of deriving QM from first principles without reference to classical mechanics may never work, not to mention that concept of "intuitive sounding" is highly subjective. Therefore, considering that there is no such set of axioms known at this time and the whole approach could be wrong, I find it objectionable to use this argument as a basis for explaining why amplitudes have to be complex. [1] http://power1.pc.uec.ac.jp/~toru/notes/LandauLifshitz-QuantumMechanics.pdf http://power1.pc.uec.ac.jp/~toru/notes/LandauLifshitz-Quantu...

jessriedel 8y ago
> not sure I totally agree the subject at hand is why is the universe the way it is. I'm confident that is what Aaronson is discussing here. > Additionally, if by real amplitudes he meant a completely different set of laws, I don't see how that is relevant to a discussion of why complex numbers play such an important role in QM. He's not having a discussion of why, assuming that the laws of QM are what they are, complex numbers feature in the most elegant representations. You brought that topic up.
- calhoun137 8y ago
  > In this post, I’d like to focus on a question that any “explanation” for QM at some point needs to address, in a non-question-begging way: why should amplitudes have been complex numbers? I will take your word for it that > The question in the post is "Why is the universe this way and not another way?" not "Why do we choose to use complex numbers in our description of the universe?" but why then does he so prominently raise this question at the beginning, answer it later, and title his post "why are amplitudes complex?". I must be really out of the loop. Did I really get the subject of this post so badly wrong? Sorry about that. Is this not a very confusing way to go about discussing why the universe is this way and not another?
  - jessriedel 8y ago
    "Why are amplitudes complex?" and "why should amplitudes have been complex numbers?" are not the same question as "Why do we choose to use complex numbers in our description of the universe?". Aaronson is using "amplitude" to refer to things out there in the external world. He explicitly refers to other ways the universe could have been: "quaternionic QM [i.e., QM with quaternion amplitudes] allows superluminal signaling". I don't know what to tell you except to read more carefully. It's philosophy of physics.

jessriedel 8y ago

> not sure I totally agree the subject at hand is why is the universe the way it is. I'm confident that is what Aaronson is discussing here. > Additionally, if by real amplitudes he meant a completely different set of laws, I don't see how that is relevant to a discussion of why complex numbers play such an important role in QM. He's not having a discussion of why, assuming that the laws of QM are what they are, complex numbers feature in the most elegant representations. You brought that topic up.

calhoun137 8y ago
> In this post, I’d like to focus on a question that any “explanation” for QM at some point needs to address, in a non-question-begging way: why should amplitudes have been complex numbers? I will take your word for it that > The question in the post is "Why is the universe this way and not another way?" not "Why do we choose to use complex numbers in our description of the universe?" but why then does he so prominently raise this question at the beginning, answer it later, and title his post "why are amplitudes complex?". I must be really out of the loop. Did I really get the subject of this post so badly wrong? Sorry about that. Is this not a very confusing way to go about discussing why the universe is this way and not another?
- jessriedel 8y ago
  "Why are amplitudes complex?" and "why should amplitudes have been complex numbers?" are not the same question as "Why do we choose to use complex numbers in our description of the universe?". Aaronson is using "amplitude" to refer to things out there in the external world. He explicitly refers to other ways the universe could have been: "quaternionic QM [i.e., QM with quaternion amplitudes] allows superluminal signaling". I don't know what to tell you except to read more carefully. It's philosophy of physics.