Why are amplitudes complex?
scottaaronson.com
scottaaronson.com
In quantum field theory, the lie algebra elements of certain lie groups (U(1), SU(2), and SU(3)) play a fundamental role. For example, the 8 quarks correspond to 8 lie algebra elements of SU(3).
A lie group is a continuous symmetry group (such as the rotation of a circle which leave it unchanged, as opposed to a finite group like the rotations of a square) and the lie algebra are the "generators" of lie groups in the same sense that finite groups can be generated, such as the symmetry group of rotations of a square can be generated by a single rotation of 90 degree's.
Here is the real kicker, the imaginary number i is actually equivalent in a very strong sense to the lie algebra element of O(1) (which is just the rotations of a circle) and can also be represented by the matrix:
0 -1
1 0
as can be proven by taking the powers of this matrix and the powers of i and simply checking everything.
Those are symmetries of the gauge fields, which apply just as well to classical fields as quantum ones.
https://physics.stackexchange.com/questions/352649/does-clas...
The fields describe possible physical configurations, which is the same for classical mechanics and quantum mechanics. In contrast, the numbers determining the chance of different configurations being realized are complex for quantum mechanics (amplitudes) but real and positive for classical Liouville mechanics (probabilities).
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
[2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo...
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.
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) ;)
A secondary complaint. None of the groups you mention play a central role in the machinery of quantum theory. They're important in some of the quantum mechanical models we use for particle physics. Arguably more important is the group R of translations in time. But even this isn't necessarily central. Quantum theory works just fine on curved spacetimes where time translation isn't a symmetry.
What's super interesting is that this fact is literally the reason why charge is conserved. This can be seen by expanding our horizons and instead of only looking at Ae^it, we instead replace i by its matrix representation, and then also add the matrix representation for the lie algebra elements of SU(2) i.e. the pauli matrices. Then for QCD use the 8 lie algebra elements of SU(3). This kind of progression seems very natural and intuitive to me, which is why I included this in the original comment.
I see math as very distinct from physics. Math is a language for understanding universal truths about abstract concepts, and as far as QM is concerned, I think what it says about nature is that there are a number of symmetries besides the ones found in classical physics. The mathematics of symmetry is all tied up with complex numbers, but it could all be done without it if you really wanted. I believe that by understanding the mathematics of lie groups and lie algebras we gain a much better understanding of the relationship between complex numbers and quantum amplitudes.
I even think the question "why are quantum amplitudes complex" potentially raises a number of red herring questions that wont help us understand nature better, when what we really should be focusing on is the role of the mathematics of symmetry in QM, not complex numbers.
(P.S. I am going from memory here and may have gotten some details wrong and apologize for being possibly misleading by accident, its hard to communicate this kind of thing but its fun to try)
You're mistaken that the existence of this U(1) is the reason that charge is conserved. The U(1) you're talking about is the group of units of the complex numbers. It acts trivially in all quantum mechanical calculations, essentially because quantum mechanical states are actually rays in a Hilbert space.
There is a different U(1) which is related to charge conservation. It appears in models where one can defined a charge, and acts via $e^{iq}$ on the subspace of states of charge $q$. Note that the action of this group on Hilbert space is non-trivial! It acts differently on different subspaces. In models where this group is a symmetry, by definition, time evolution doesn't alter the charge eigenstates, hence charge is conserved.
Um, what? Last I checked, there are 6 quarks, not 8.
Scott mentions in this article that his work on Quantum Computing and Quantum Information deals mostly with real quantum states, not so much complex.
However these are primarily discrete states. Ie, matrix operators and vector wavestates.
When working with position and momentum spaces, you need to deal with continuous wavefunctions, and the presence of complex numbers becomes more immediate. Eg, the momentum operator, as written in position space, has a factor of i, when showing that momentum is the generator of positional translations (p = -i hbar d/dx). This is also why the gradient term of Schrodinger’s equation is negative (kinetic energy is proportional to momentum squared, so the i^2 makes it negative).
Similarly complex numbers appear with the Quantum Harmonic Oscillator when rearranging the Hamiltonian suit the ladder operators (or creation/annihilation operators).
In signal processing we can take any periodic-in-time signal and pretend it is a function defined on a circle parametrized by angle measure.
Yes, but as always you can ask yourself: is this merely a coincidence (if there is such a thing as one) or is this reflecting some deeper relation between nature and mathematics?
After all, a "signal" is quite a general and abstract notion.
Complex numbers are a qubit the universe gives us for free.
I believe this part is what the answer is supposed to be:
> “local tomography”: i.e., the “principle” that composite systems must be uniquely determined by the statistics of local measurements. And while this principle might sound vague and unobjectionable, to those in the business, it’s obvious what it’s going to be used for the second it’s introduced. Namely, it’s going to be used to rule out quantum mechanics over the real numbers, which would otherwise be a model for the axioms, and thus to “explain” why amplitudes have to be complex
I will leave it to other to explain what this means in comments. For the record I don't think this explanation is correct, and strongly object. I don't think amplitudes "have" to be complex, it's just a mathematical convenience. If anything, the actual question we should be asking is: why are complex numbers so convenient.
You can of course formulate quantum mechanics in terms of strictly real numbers, e.g., the Wigner function. That's just a different representation of the same fundamental laws. (In that representation, it's easier to do some things and much harder to do other things.)
But that's not the alternative possible universe that Aaronson is considering. When he says "quantum mechanics with real amplitudes", he does not just mean a different representation of the same law; he means a totally different set of laws with dramatic experimental consequences (which of course aren't realized in the real world). 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?".
To see this, it may help to know that there is an operational (i.e., representation-independent) definition of the relative phase between states. For real-amplitude quantum mechanics, that phase has to take real values.
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-Quantu...
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.
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?
I don't know what to tell you except to read more carefully. It's philosophy of physics.
"But, while I remain less than 100% satisfied about “why the complex numbers? why not just the reals?,” there’s one conclusion that my recent circling-back to these questions has made me fully confident about. Namely: quantum mechanics over the quaternions is a flaming garbage fire, which would’ve been rejected at an extremely early stage of God and the angels’ deliberations about how to construct our universe."
There's a pecking order here.
I was at a tech party in Berkeley last fall. When various gamers (such as the founder of "Second Life") learned that I was a mathematician, they fawned over the octonions.
I'm sure there's something there (the octonions seriously screwed with my first research idea in college) but I can't say what it is.
BUT why is there a phase?