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if you extend probabilities into the domain of complex numbers, you basically get quantum mechanics To me, the most fundamental question of quantum physics is

if you extend probabilities into the domain of complex numbers, you basically get quantum mechanics To me, the most fundamental question of quantum physics is this: Why not quaternions? As a mathematician, the distinguishing characteristic of C is that it is the algebraic closure of R; but I see no physical relevance to that.

deleted 12y ago
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tel 12y ago
Is there not a model of QM which uses quaternions probabilities? Is there a place where it breaks down?
- jaekwon 12y ago
  I remember reading things online where people claim that the natural world (probably including what's known in quantum mechanics) is better explained with quaternions.
  - judk 12y ago
    It's not broadly true. Quaternions are mainly useful for math on the surface of a sphere.
    - deleted 12y ago
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- lisper 12y ago
  Aaronson actually answers this question directly towards the end of the lecture. The short answer is that quaternions don't work for the same reason that real numbers don't work: the probabilities don't work out right for unitary transformations.
- dnautics 12y ago
  likely because the probabilities in QM need to be commutative (QM operators are not commutative, but the probabilities themselves are).
- stared 12y ago
  You can represent two-state systems (qubits) with quaternions. In general, quantum mechanics is about using complex vectors and unitary operations on them. As it is enough to implement quaternions, they are silently used in quantum mechanics.
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gus_massa 12y ago
(You must be careful because this probabilities use the normalization \sum(p_i)=1, but in quantum mechanics you must use \sum(|p_i|^2)=1. I think that it's an important difference, because it leads to interference and other interesting phenomena.) The idea is that all the quantum values are complex, and the measurable things are of the form f.f*=|x|^2, you have a symmetry. If you multiply all the universe by i (where i ~= \sqrt(-1) :) ) nothing changes. In general, you can multiply the universe by any complex z such that |z|=1. The global phase is irrelevant. This is the global U(1) symmetry of the universe. But this symmetry is local. You can choose the phase of each point of the universe independently. The problem is that to compensate the arbitrary picks you have to introduce a new field that compensate for the differences of the local election of the phases. This is just the electromagnetic field. (I'm hiding a technical detail here.) This is the quantum version of the old electromagnetic field of Maxwell. The particles of this field are the photons. I can't find an "easy" information source (for someone with a math degree), but you may start reading: http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathematics http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathema... The U(1) group has dimension 1, so you have only one "type" of photons. (This "type" is not related to the usual color of the light.) I think it's possible to extend this to the quaternions group of module 1, but it's a 3 dimensional group, so you would have ¿3? photons "types". If all the new "types" of photons are massless, this should be measurable in the experiments. (If the new "types" of photons are heavy, you need to get a bigger particle accelerator to see them, or a theoretical physicist to prove that this idea is actually impossible.) More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because. The other symmetries of the universe is the SU(3) "color" symmetry that is behind the strong force. It's the color of the quarks inside the nucleus it's not abelian so it's more complicated. The associated particles are the gluons, and there are 8 "types". The other symmetry is the SU(2) that is the cause of the weak force, and the particles are the W+, Z and W-. Actually the SU(2) symmetry is mixed with the U(1) symmetry. So the real U(1) group create the a version of the electromagnetic field that is related to the weak-hypercharge, and the photon and Z particles are a mix of the particles of the two fields. The mathematical idea is nice, because you have a copy of U(1) inside SU(2), so you can mix both U(1) groups and mix the particles. As an introduction, you can read http://en.wikipedia.org/wiki/Electroweak_interaction#Formulation http://en.wikipedia.org/wiki/Electroweak_interaction#Formula... but if you don't want to dive in the technical details you can think that the U(1) group is (almost) the electromagnetism an live more happily.

deleted 12y ago

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tel 12y ago

Is there not a model of QM which uses quaternions probabilities? Is there a place where it breaks down?

jaekwon 12y ago
I remember reading things online where people claim that the natural world (probably including what's known in quantum mechanics) is better explained with quaternions.
- judk 12y ago
  It's not broadly true. Quaternions are mainly useful for math on the surface of a sphere.
  - deleted 12y ago
    [deleted]
lisper 12y ago
Aaronson actually answers this question directly towards the end of the lecture. The short answer is that quaternions don't work for the same reason that real numbers don't work: the probabilities don't work out right for unitary transformations.
dnautics 12y ago
likely because the probabilities in QM need to be commutative (QM operators are not commutative, but the probabilities themselves are).
stared 12y ago
You can represent two-state systems (qubits) with quaternions. In general, quantum mechanics is about using complex vectors and unitary operations on them. As it is enough to implement quaternions, they are silently used in quantum mechanics.

deleted 12y ago

[deleted]

gus_massa 12y ago

(You must be careful because this probabilities use the normalization \sum(p_i)=1, but in quantum mechanics you must use \sum(|p_i|^2)=1. I think that it's an important difference, because it leads to interference and other interesting phenomena.) The idea is that all the quantum values are complex, and the measurable things are of the form f.f*=|x|^2, you have a symmetry. If you multiply all the universe by i (where i ~= \sqrt(-1) :) ) nothing changes. In general, you can multiply the universe by any complex z such that |z|=1. The global phase is irrelevant. This is the global U(1) symmetry of the universe. But this symmetry is local. You can choose the phase of each point of the universe independently. The problem is that to compensate the arbitrary picks you have to introduce a new field that compensate for the differences of the local election of the phases. This is just the electromagnetic field. (I'm hiding a technical detail here.) This is the quantum version of the old electromagnetic field of Maxwell. The particles of this field are the photons. I can't find an "easy" information source (for someone with a math degree), but you may start reading: http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathematics http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathema... The U(1) group has dimension 1, so you have only one "type" of photons. (This "type" is not related to the usual color of the light.) I think it's possible to extend this to the quaternions group of module 1, but it's a 3 dimensional group, so you would have ¿3? photons "types". If all the new "types" of photons are massless, this should be measurable in the experiments. (If the new "types" of photons are heavy, you need to get a bigger particle accelerator to see them, or a theoretical physicist to prove that this idea is actually impossible.) More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because. The other symmetries of the universe is the SU(3) "color" symmetry that is behind the strong force. It's the color of the quarks inside the nucleus it's not abelian so it's more complicated. The associated particles are the gluons, and there are 8 "types". The other symmetry is the SU(2) that is the cause of the weak force, and the particles are the W+, Z and W-. Actually the SU(2) symmetry is mixed with the U(1) symmetry. So the real U(1) group create the a version of the electromagnetic field that is related to the weak-hypercharge, and the photon and Z particles are a mix of the particles of the two fields. The mathematical idea is nice, because you have a copy of U(1) inside SU(2), so you can mix both U(1) groups and mix the particles. As an introduction, you can read http://en.wikipedia.org/wiki/Electroweak_interaction#Formulation http://en.wikipedia.org/wiki/Electroweak_interaction#Formula... but if you don't want to dive in the technical details you can think that the U(1) group is (almost) the electromagnetism an live more happily.