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I appreciate you're able to see the quandary. I think the crux of what you said is in your first sentence: > "Quantum theory based on real numbers" means a spe

by asxd 5y ago

I appreciate you're able to see the quandary. I think the crux of what you said is in your first sentence: > "Quantum theory based on real numbers" means a specific thing -- quantum mechanics with real amplitudes (and real anything-else-that-would-follow-from-that) I'm familiar with complex math as far as remedial DSP and electrical engineering goes, so this may be over my head. I'm not sure what a real amplitude is, since generally when I hear "complex number", my head thinks "compact way to represent a frequency, amplitude and phase", all of which are real. It seems like I was reeled in with familiar-sounding terminology that may in fact have a deeper meaning in this context.

simiones 5y ago
I think the base idea here is something like this: if you want to describe, say, a sound-wave, you can use complex numbers to represent the wave, but you can also, in principle, use strictly real numbers to describe the behavior of each individual molecule of gas using Newton's equations of motion (assuming you can ignore quantum effects for your simulation). So, in classical mechanics, while complex numbers are a useful abstraction for waves, they are not fundamental. The same is not true for QM: the wave part / the complex numbers are fundamental. Also note, when people say "complex numbers" they refer to the algebraic field, which is the object composed of [a pair of reals, complex addition, complex multiplication]. In particular, complex multiplication is the key here, since it is what differentiates complex numbers from 2-vectors. To drive this point further, while 1-vectors, 2-vectors, 3-vectors, 4-vectors etc behave very similarly, only the reals and the complex numbers can form a field - there are no "numbers" in the same sense formed of 3-tuples or 4-tuples or other n>2-tuples of reals*. This is why the discussion focuses on the complex numbers (meaning, again, not just a pair of reals, but also the particular +,-,*,/ operations for them that make them a field). * to be fair, quaternions come pretty close - you can define a 4-tuple of reals + addition + multiplication that is almost a field, except that multiplication is anti-commutative instead of being commutative (p*q = -q*p).
- asxd 5y ago
  Thanks so much for that analogy and explanation! It certainly made things a bit clearer. edit: I'm still a bit unclear on what it means to form a field. Texts around this topic seem pretty dense and encyclopedic. Is there a straightforward explanation of what an algebraic field is?
Sniffnoy 5y ago
"Amplitudes" here refers to the thing that QM uses instead of probabilities, not any other sort of amplitude.

by asxd 5y ago

simiones 5y ago

I think the base idea here is something like this: if you want to describe, say, a sound-wave, you can use complex numbers to represent the wave, but you can also, in principle, use strictly real numbers to describe the behavior of each individual molecule of gas using Newton's equations of motion (assuming you can ignore quantum effects for your simulation). So, in classical mechanics, while complex numbers are a useful abstraction for waves, they are not fundamental. The same is not true for QM: the wave part / the complex numbers are fundamental. Also note, when people say "complex numbers" they refer to the algebraic field, which is the object composed of [a pair of reals, complex addition, complex multiplication]. In particular, complex multiplication is the key here, since it is what differentiates complex numbers from 2-vectors. To drive this point further, while 1-vectors, 2-vectors, 3-vectors, 4-vectors etc behave very similarly, only the reals and the complex numbers can form a field - there are no "numbers" in the same sense formed of 3-tuples or 4-tuples or other n>2-tuples of reals*. This is why the discussion focuses on the complex numbers (meaning, again, not just a pair of reals, but also the particular +,-,*,/ operations for them that make them a field). * to be fair, quaternions come pretty close - you can define a 4-tuple of reals + addition + multiplication that is almost a field, except that multiplication is anti-commutative instead of being commutative (p*q = -q*p).

asxd 5y ago
Thanks so much for that analogy and explanation! It certainly made things a bit clearer. edit: I'm still a bit unclear on what it means to form a field. Texts around this topic seem pretty dense and encyclopedic. Is there a straightforward explanation of what an algebraic field is?

Sniffnoy 5y ago

"Amplitudes" here refers to the thing that QM uses instead of probabilities, not any other sort of amplitude.