
Why are amplitudes complex? - beefman
https://www.scottaaronson.com/blog/?p=4021
======
calhoun137
The reason amplitudes are complex is because a fundamental role is played in
quantum theory by symmetry, and the imaginary number in an exponential (e^it)
is a very algebraically convenient way to capture an important symmetry.

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.

~~~
auntienomen
Ex particle theorist here. I think this answer is a bit misleading. Symmetries
in quantum mechanics do typically act via $e^{iHt}$, but this isn't why
amplitudes are complex. It's a consequence of the fact that symmetries
generally act via unitary transformations, but to even speak of unitary
transformations, one must already have amplitudes which are complex.

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.

~~~
calhoun137
Great points, let me try and explain better where the group O(1) appears in
basic QM. An amplitude in QM is only defined up to a global phase
transformation e^it. To be more clear, given some amplitude A and a real
number t, we will calculate the same probability for any measurement by using
the amplitude Ae^it instead. Well there is the group O(1) right there.

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)

~~~
auntienomen
You mean O(2), which is isomorphic to U(1). O(1) is a discrete group.

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.

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jeffwass
In a nutshell, a Hamiltonian matrix (ie, the energy “Operator” matrix) must be
Hermitian to have real eigenvalues. A Hermetian matrix means it’s equal to the
complex conjugate of its transpose.

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).

------
paulsutter
For a more intuitive explanation, this paper on the advantages of complex
wavelets switched on the lightbulb for me why signals are complex in the first
place:

[http://people.math.sc.edu/blanco/IMI/DTCWT0.pdf](http://people.math.sc.edu/blanco/IMI/DTCWT0.pdf)

~~~
whatshisface
I'm not sure how much of relationship there would be between the convenience
of complex numbers in signal analysis and the appearance of complex numbers in
the Schrodinger equation.

~~~
kgwgk
It can be seen as a matter of convenience in Schroedinger’s equation too.
Other representations of QM exist. For example, in the phase space
representation we have a real function of positions and momentums. In the wave
function representation we have a complex function of positions: the (square
of the modulus of the) amplitude is related to the position and the
(derivative of the) phase is related to the momentum. In the Scroedinger’s
formalism we can also go from the position representation to the momentum
representation via Fourier transform.

~~~
whatshisface
While true, the convenience in the Schrodinger equation is due to a symmetry
in nature which the algebra of complex numbers mirrors. The pattern of a
symmetry in nature corresponding to a convenient algebraic object used to
represent it happens in many other places including Pauli spin matrices. In
signal processing complex numbers are useful for more purely mathematical
reasons.

~~~
jacobolus
In most parts of physics there are literally things spinning.

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.

~~~
whatshisface
The complex precession in the Schrodinger equation doesn't involve angular
momentum.

~~~
jacobolus
Depends on who you ask.
[http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf](http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf)

------
IIAOPSW
As I am sure Scott knows, you can completely remove the complex amplitudes
from your equations at the cost of storing the phase on a single ancila qubit.

Complex numbers are a qubit the universe gives us for free.

------
tim333
A simpler possible explanation is that physical phenomena like the two slit
experiment look like waves interfering the the simplest way to represent that
stuff is with complex numbers. The fact that that's the simplest way to do the
math doesn't necessarily mean reality is fundamentally that way - you can use
the same math for interfering water waves and similar.

------
calhoun137
I imagine a lot of people are going to be checking the comments here to find
out what the answer to the question in the title is.

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.

~~~
jessriedel
> I don't think amplitudes "have" to be complex

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.

~~~
calhoun137
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-
Quantu...](http://power1.pc.uec.ac.jp/~toru/notes/LandauLifshitz-
QuantumMechanics.pdf)

~~~
jessriedel
> 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
> 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
"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.

------
empath75
Is it possible to to construct probability amplitudes out of quaternions, etc?

~~~
kgwgk
I’ve not read TFA, but it mentions quaternions multiple times...

~~~
auntienomen
It's worth reading.

"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."

------
notatcomputer68
Sure complex numbers are convenient for representing phase and amplitude.

BUT why is there a phase?

~~~
kgwgk
Because despite Schroedinger's equation being first-order in time, the
imaginary diffusion coefficient makes it also a wave equation.

------
kevin_thibedeau
The short answer is that the observable parts of the universe like to spin
around in circles and spirals.

