5 ms·

Why are amplitudes complex?

103 points by beefman 8y ago

paulsutter 8y ago
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
- aaaaaaaaaab 8y ago
  The most intuitive explanation is simply that in the complex domain sinusoids can be represented as exponentials, and a phase shift becomes a simple multiplication by a constant. Thus sinusoidal signals can be manipulated according to a fairly simple algebra.
- whatshisface 8y ago
  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 8y ago
    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 8y ago
      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 8y ago
        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 8y ago
        The complex precession in the Schrodinger equation doesn't involve angular momentum.
        jacobolus 8y ago
        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
      - feanaro 8y ago
        > In signal processing complex numbers are useful for more purely mathematical reasons. 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.
kevin_thibedeau 8y ago
The short answer is that the observable parts of the universe like to spin around in circles and spirals.
jeffwass 8y ago
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).
calhoun137 8y ago
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 8y ago
  > 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 8y ago
    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...
calhoun137 8y ago
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.
- jessriedel 8y ago
  > 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). 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-classical-electrodynamics-have-u1-symmetry-if-yes-how 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).
  - calhoun137 8y ago
    > the chance of different configurations being realized are complex for quantum mechanics 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 https://en.wikipedia.org/wiki/Heisenberg_picture [2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#Representation_theory_formulation https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo... [3] https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group
empath75 8y ago
Is it possible to to construct probability amplitudes out of quaternions, etc?
- kgwgk 8y ago
  I’ve not read TFA, but it mentions quaternions multiple times...
  - auntienomen 8y ago
    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."
- Syzygies 8y ago
  I was going to bring up the quaternions if no one had. When I was asked to bring in a fictional approach to the Riemann Hypothesis for the film "A Beautiful Mind", the quaternions were my first thought. 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.
- cryptonector 8y ago
  In the comments on TFA the author (and others) shows that the lack of commutativity in the quaternions leads to faster-than-light signalling.
IIAOPSW 8y ago
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.
notatcomputer68 8y ago
Sure complex numbers are convenient for representing phase and amplitude. BUT why is there a phase?
- kgwgk 8y ago
  Because despite Schroedinger's equation being first-order in time, the imaginary diffusion coefficient makes it also a wave equation.
tim333 8y ago
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.

103 points by beefman 8y ago

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

aaaaaaaaaab 8y ago
The most intuitive explanation is simply that in the complex domain sinusoids can be represented as exponentials, and a phase shift becomes a simple multiplication by a constant. Thus sinusoidal signals can be manipulated according to a fairly simple algebra.
whatshisface 8y ago
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 8y ago
  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 8y ago
    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 8y ago
      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 8y ago
        The complex precession in the Schrodinger equation doesn't involve angular momentum.
        jacobolus 8y ago
        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
    - feanaro 8y ago
      > In signal processing complex numbers are useful for more purely mathematical reasons. 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.

kevin_thibedeau 8y ago

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

jeffwass 8y ago

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).

calhoun137 8y ago

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 8y ago
> 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 8y ago
  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...

calhoun137 8y ago

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.

jessriedel 8y ago
> 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). 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-classical-electrodynamics-have-u1-symmetry-if-yes-how 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).
- calhoun137 8y ago
  > the chance of different configurations being realized are complex for quantum mechanics 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 https://en.wikipedia.org/wiki/Heisenberg_picture [2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#Representation_theory_formulation https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo... [3] https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group

empath75 8y ago

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

kgwgk 8y ago
I’ve not read TFA, but it mentions quaternions multiple times...
- auntienomen 8y ago
  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."
Syzygies 8y ago
I was going to bring up the quaternions if no one had. When I was asked to bring in a fictional approach to the Riemann Hypothesis for the film "A Beautiful Mind", the quaternions were my first thought. 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.
cryptonector 8y ago
In the comments on TFA the author (and others) shows that the lack of commutativity in the quaternions leads to faster-than-light signalling.

IIAOPSW 8y ago

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.

notatcomputer68 8y ago

Sure complex numbers are convenient for representing phase and amplitude. BUT why is there a phase?

kgwgk 8y ago
Because despite Schroedinger's equation being first-order in time, the imaginary diffusion coefficient makes it also a wave equation.

tim333 8y ago

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.