A Complex numbers in QM

akhmeteli

The only other assumption is that two level systems exist, i.e. qubits. That seems to be obviously true right?
This is not the only assumption. There is also the assumption of "factorizably HSD probabilistic theory". Look at the motivation of this assumption immediately after Proposition 1. Among other things, it looks like this motivation is only good for finite-dimensional systems.
Another thing. Is existence of two-level systems so obvious? I guess this depends on exact definitions, because all realistic systems are multilevel systems.

DarMM

Gold Member
Another thing. Is existence of two-level systems so obvious? I guess this depends on exact definitions, because all realistic systems are multilevel systems.
Yes but you only need a subsystem to behave as a qubit, which is obviously true experimentally, e.g. the polarization states of a photon. They factually are a qubit, I don't think this aspect of the assumptions is easy to deny.

This is not the only assumption. There is also the assumption of "factorizably HSD probabilistic theory". Look at the motivation of this assumption immediately after Proposition 1. Among other things, it looks like this motivation is only good for finite-dimensional systems.
That paper proves it only for finite-dimensional systems, the other paper deals with the general case. Even ignoring this, it would mean that you'd have to acknowledge the spin degrees of freedom or any other quantum number with a finite set of values as complex when considered on their own.

maline

There is also the case where local measurements and their joint statistics overdetermine the global state, which is what happens in quaternionic QM. This overdetermination means measurements on one system constrain the global state enough to be noticed in a spacelike separated system, allowing nonlocal communication.
Ok, I agree with this (with the quibble that No-Signalling is a stronger assumption than "relativity" meaning Lorentz covariance).

There the problem is due to how the global aspect of the state not learnable from local measurements interacts with Poincaré symmetry.
Can you elaborate on this? It appears not to be discussed in the Barnum-Wilce paper.

DarMM

Gold Member
Ok, I agree with this (with the quibble that No-Signalling is a stronger assumption than "relativity" meaning Lorentz covariance).
I'm never sure how to phrase this. Some people take Relativity to be more than simply "physics takes place in a Lorentzian spacetime", but to also include the lack of superluminal signalling. Do you have a reference for Relativity meaning solely the former?

This is a genuine question as I'd like to know standard phraseology on this, but I've never really seen anybody nail it down.

Can you elaborate on this? It appears not to be discussed in the Barnum-Wilce paper.
It's discussed in the Oppio & Moretti paper.

akhmeteli

Yes but you only need a subsystem to behave as a qubit, which is obviously true experimentally, e.g. the polarization states of a photon. They factually are a qubit, I don't think this aspect of the assumptions is easy to deny.
Again, it depends on the exact definitions. There is no realistic system that behaves exactly as a qubit.

DarMM said:
That paper proves it only for finite-dimensional systems, the other paper deals with the general case.
If "the other paper" is Moretti/Opio, again, it looks like that paper only deals with Wigner elementary relativistic systems - I guess there is no external field there.
DarMM said:
Even ignoring this, it would mean that you'd have to acknowledge the spin degrees of freedom or any other quantum number with a finite set of values as complex when considered on their own.
I don't have to consider the spin degrees of freedom on their own. I know that real numbers are enough for the Dirac equation, which defines an infinite-dimensional system with spin degrees of freedom. I also know that real numbers can be enough for multiple-particle systems. This is why I am not worried about Proposition 1 or the Moretti/Oppio paper.

A. Neumaier

There is no realistic system that behaves exactly as a qubit.
I know that real numbers are enough for the Dirac equation
If you define realistic so narrow as to exclude photon polarization as realizing a qubit then there is no realistic system that behaves exactly as the Dirac equation demands.

Already the electron of a realistic hydrogen atom exhibits radiative corrections to the Dirac equation, leading to an observable Lamb shift. In contrast, a deviation of photon polarization from the qubit model has never been observed.
As I said, I am not impressed.
Nobody here is trying to impress you. We simply state some facts that you prefer to ignore.

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DarMM

Gold Member
Again, it depends on the exact definitions. There is no realistic system that behaves exactly as a qubit.
Okay, well if the assumption that there are qubits is too much for you it is very difficult to proceed further. I don't fully understand why one would have more doubts about qubits, reproduced with that behavior in labs across the world and yet not have doubts about Hilbert spaces over the reals, not used in modelling experiments anywhere.

If "the other paper" is Moretti/Opio, again, it looks like that paper only deals with Wigner elementary relativistic systems - I guess there is no external field there.
The presence of external fields is irrelevant. Scattering states, even from external fields, are (Fock spaces over) Wigner reps. So either you're denying the basic scattering formalism of QFT or you think the Hilbert space "becomes complex" or something during scattering.

maline

I'm never sure how to phrase this. Some people take Relativity to be more than simply "physics takes place in a Lorentzian spacetime", but to also include the lack of superluminal signalling. Do you have a reference for Relativity meaning solely the former?
Oh, I also have no idea whether there is an accepted "standard usage" about this. I just mean that Lorentz symmetry and No Signalling are two very different concepts. I find myself much more comfortable accepting the first as an assumption, while I would prefer the second to be a derived result, since it is described in terms of a practical scenario rather than an ingredient in the fundamental mechanics.
In standard (not quaternionic) QFT, No-Signalling is basically a result of the locality of the Lagrangian. Of course the latter is itself hardly trivial. It is related to Lorentz-covariance of the S-matrix, but (I think) has not been shown to follow from it. Besides, the existence of an S-matrix itself requires a list of nontrivial assumptions. Perhaps locality of the Lagrangian can be derived from diffeomorphism invariance? Just speculating... Anyway, Lorentz symmetry by itself is almost certainly not enough.
I wonder if one can define a quaternionic QFT that will be fully Lorentz invariant yet have superluminal signalling? That would probably lead to circular causality... sounds like fun!
It's discussed in the Oppio & Moretti paper.
Oh? I understood that paper to be discussing a different claim - that even if the Hilbert space was real, the orthogonal operators of the symmetry group would define a complex structure (an "imaginary unit" operator J), and the physically relevant observables would respect that structure (commute with J), so that we would automatically end up with effectively complex QM. I like this idea very much, but I don't see a direct connection to the idea of local tomography vs. relativity.

DarMM

Gold Member
I wonder if one can define a quaternionic QFT that will be fully Lorentz invariant yet have superluminal signalling? That would probably lead to circular causality... sounds like fun!
You can, the details are in Adler's book "Quaternionic Quantum Mechanics and Quantum Fields". Although he has since abandoned this. His hope was that the superluminal signals died off faster than current experimental limits, but he never got it to work for detailed QFTs like the standard model.

I like this idea very much, but I don't see a direct connection to the idea of local tomography vs. relativity.
Quite right. Real Quantum Theory has global/nonlocal degrees of freedom from not obeying local tomography and separately getting it to carry Wigner representations causes it to become complex. However these seem separate as you say. There doesn't really seem to be anything about these global degrees of freedom that might cause signalling.

So it would be more accurate to say quaternions break local tomography in a manner that causes singalling. Where as real QM breaks local tomography in a way that isn't a problem, but separately violates Lorentz covariance unless you basically make it complex QM.

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maline

I've been looking at the Moretti/Oppio paper, and I'm afraid I am quite disappointed. Their entire analysis applies only to "elementary relativistic systems", meaning that an irrep of the Poincare group is being treated as the entire Hilbert space. But realistic Hilbert spaces are direct sums of many such irreps, and most physical observables are not block-diagonal but rather mix the different irreps. If we define J as the direct sum of the J's found for each irrep, we have no guarantee that these general observables will commute with J.
For instance, for a free real scalar field, some irreps are: the vacuum (for which we will need to invent an extra dimension as its "imaginary partner"); the space of one-particle states; the spaces of N-particle states where all the particles are in the same mode; the spaces of two-particle states that, when expressed at wavefunctions on momentum space, have support only for pairs of momenta with some fixed scalar product; etc. The Hamiltonian, and the rest of the Poincare generators, do of course act separately on each irrep, but the field operators emphatically do not.

Furthermore, the results here "follow from relativity" only in a rather weak sense. The J operator is well defined whenever we have time translation invariance. You just find the states that are periodic in time (what in complex QM are eigenstates of the Hamiltonian, but in real QM will be pairs of vectors that we hope to identify as real and imaginary parts of the eigenstate). Then J moves each such pair back by 1/4 of a period. J commutes with the linear and angular momenta because these commute with the Hamiltonian. The role played by relativity is just to add the boost generators to the list of things J commutes with! This gives the authors a more nontrivial group to work with, and so irreps that are much bigger. Commuting with the generators means it commutes with the von Neumann algebra the unitaries generate, which (in the complex version) is all the operators on the Hilbert space of the irrep.
So relativity forces all the operators on the space of one-particle states, say, to have enough in common that they share a single complex structure. Okay, that's nice, but it's fundamentally a statement about the irrep, not so much about the complex structure. There is no reason to expect the result to generalize to full realistic Hilbert spaces, or if it does, it will be for reasons unrelated to relativity.

akhmeteli

If you define realistic so narrow as to exclude photon polarization as realizing a qubit then there is no realistic system that behaves exactly as the Dirac equation demands.

Already a realistic hydrogen atom exhibits radiative corrections to the Dirac equation, leading to the Lamb shift.
I agree with that. And I have repeatedly said that I cannot be sure real numbers are sufficient for QM. That does not mean I have to accept any specific "proof" of them not being sufficient without properly examining such proof.

As for "photon polarization as realizing a qubit"... You see, qubit is a good and popular model. But my understanding is whatever reality qubit can model, the Dirac equation can model it better, so if real numbers are enough for the Dirac equation, we don't care if they are enough for qubit. Again, I agree that QED is better than the Dirac equation, but I don't know a proof that real numbers are not enough to model the reality that we now model with QED.
A. Neumaier said:
Nobody here is trying to impress you. We simply state some facts that you prefer to ignore.
Of course, my opinion does not matter. But when I told @DarMM that something did not impress me, I tried to analyze his/her "facts" and explain why they did not seem conclusive, so I don't think I ignored those "facts". As for you, I am not sure what "facts" that you stated in this thread I ignored. I was indeed reluctant to discuss your opinion that one real function instead of four complex functions does not provide conceptual simplicity, but that was an opinion, not a fact.

akhmeteli

Okay, well if the assumption that there are qubits is too much for you it is very difficult to proceed further. I don't fully understand why one would have more doubts about qubits, reproduced with that behavior in labs across the world and yet not have doubts about Hilbert spaces over the reals, not used in modelling experiments anywhere.
As I explained in my answer to A. Neumaier, it is my understanding that whatever reality qubit can model, the Dirac equation can model it better (if you disagree, please let me know). As I know that real numbers are sufficient to describe physics described by the Dirac equation, I don't care if real numbers are enough for qubit.

Note also that qubit has a finite-dimensional state space, and one cannot even have the standard commutator of coordinate and momentum in a finite-dimensional space state.

DarMM said:
The presence of external fields is irrelevant. Scattering states, even from external fields, are (Fock spaces over) Wigner reps. So either you're denying the basic scattering formalism of QFT or you think the Hilbert space "becomes complex" or something during scattering.
This is an excellent remark. However, the presence of external fields is indeed relevant. The approach to using one real function instead of the Dirac spinor function in the Dirac equation in electromagnetic field requires that some component of the electromagnetic field does not vanish identically, although it can be arbitrarily small. So this approach does not go through for the free Dirac equation. However, in practice, this is not a significant limitation, as if there is at least one charged particle in the Universe, the electromagnetic field does not vanish identically, so, strictly speaking, there are no scattering states, although they are great approximations.

akhmeteli

it is my understanding that whatever reality qubit can model, the Dirac equation can model it better
I need to take back this statement as too general, but it seems to be correct at least for modeling electron spin and some other two-state systems.

A. Neumaier

I need to take back this statement as too general, but it seems to be correct at least for modeling electron spin and some other two-state systems.
Your view of quantum mechanics (which gives the Dirac equation for a single particle an undue importance) is too narrow.

Photon polarization is a 2-state system not modeled by the Dirac equation.

A. Neumaier

I mentioned the approach to multiple particles that does not require complex numbers.
Try the relativistic helium atom (i.e., two Dirac electrons in a Coulomb potential, ignoring radiative corrections for simplicity) with your approach. If you succeed to reproduce the few lowest levels of helium to a few decimals of accuracy you have an application and can claim that you did something useful. If not, well, your work will be useless.

akhmeteli

Try the relativistic helium atom (i.e., two Dirac electrons in a Coulomb potential, ignoring radiative corrections for simplicity) with your approach. If you succeed to reproduce the few lowest levels of helium to a few decimals of accuracy you have an application and can claim that you did something useful. If not, well, your work will be useless.
I cannot do what you want for the helium atom. Your conclusion is that my work is useless? Fine. But I believed we were discussing a different issue: does QM require complex numbers? You seem to be sure that it does. And I am trying to explain why it is not obvious, why one can do with real numbers (not pairs of real numbers) much more than most people believe is possible. Maybe you are right, and complex numbers are a must for QM, but you cannot prove that (or can you?), and such a proof, if it exists, seems to need more sophisticated arguments than what you offered before (such as your argument about stationary states).

A. Neumaier

I cannot do what you want for the helium atom. [...] But I believed we were discussing a different issue: does QM require complex numbers?
The helium atom (and all of relativistic quantum chemistry) is part of quantum mechanics. If you cannot do it without complex numbers then, at least until someone proves the opposite, QM requires complex numbers.

That the equations for a few special problems can be reformulated without complex numbers doesn't change this fact.

akhmeteli

The helium atom (and all of relativistic quantum chemistry) is part of quantum mechanics. If you cannot do it without complex numbers then, at least until someone proves the opposite, QM requires complex numbers.

That the equations for a few special problems can be reformulated without complex numbers doesn't change this fact.
This thread started with the following quote: "To explain why complex numbers are necessary." So your answer seems to be: "because so far there is no theory of all quantum phenomena using real numbers only (not pairs of real numbers)." I readily agree that this is a reasonable position, but this is not an explanation and not a guarantee for the future. Seventy years ago there was no real-numbers-only theory of the phenomena described by the one-particle Klein-Gordon equation. Now there is.

A. Neumaier

Seventy years ago there was no real-numbers-only theory of the phenomena described by the one-particle Klein-Gordon equation. Now there is.
One can only discuss past and present. Thus let us continue this discussion after another seventy years.

stevendaryl

Staff Emeritus
This thread started with the following quote: "To explain why complex numbers are necessary." So your answer seems to be: "because so far there is no theory of all quantum phenomena using real numbers only (not pairs of real numbers)." I readily agree that this is a reasonable position, but this is not an explanation and not a guarantee for the future. Seventy years ago there was no real-numbers-only theory of the phenomena described by the one-particle Klein-Gordon equation. Now there is.
I don't understand exactly what is being debated. Imaginary numbers come into play from the beginning in quantum mechanics when one writes:

$H |\psi\rangle = i \hbar \frac{d}{dt} |\psi\rangle$

You can certainly perform tricks to try to eliminate the $i$, but what's the motivation for that?

maline

I don't understand exactly what is being debated. Imaginary numbers come into play from the beginning in quantum mechanics when one writes:

$H |\psi\rangle = i \hbar \frac{d}{dt} |\psi\rangle$

You can certainly perform tricks to try to eliminate the $i$, but what's the motivation for that?
The question is a philosophical one: why did Nature "choose" complex amplitudes over real ones? The theory is more elegant the more inevitable its premises are. To use Scott Aaronson's phrase, we would like QM to be "an island in theoryspace"; the only possible version of what it could be.
Also, if one likes interpretations that see the quantum vector as ontologically real, then it seems to be the only complex-valued "beable" in Nature. I personally am not completely comfortable with this. The real numbers are our conceptualization of "quantities", but the complex numbers seem to live purely as an abstract concept. I don't really know what it "means" for a value to actually "be" complex. This isn't a terrible strong objection, but it has always niggled me.
If a result like Moretti & Oppio's would hold for all the operators in realistic quantum systems, I would find that very satisfying: the complex structure need not be built into the "beable" amplitudes; it emerges on its own from the symmetry structure and in particular from unitary time evolution.

As for Schroedinger's equation, the modification for a real Hilbert space is very minor: instead of $\frac{d}{dt} |\psi\rangle =\frac{-i}{\hbar}H |\psi\rangle$ with $H$ Hermitian, we would write $\frac{d}{dt} |\psi\rangle =\frac{1}{\hbar}\tilde H |\psi\rangle$ with $\tilde H$ anti-Hermitian (i.e. antisymmetric). This general form works for the real, complex, and quaternionic cases. Stone's theorem tells us that it follows from unitarity of the evolution, which in turn follows from time translation symmetry and reversibility.

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A. Neumaier

it seems to be the only complex-valued "beable" in Nature.
The electromagnetic field in its Silberstein representation is another, classical complex-valued beable!

A. Neumaier

As for Schroedinger's equation, the modification for a real Hilbert space is very minor: instead of $\frac{d}{dt} |\psi\rangle =\frac{-i}{\hbar}H |\psi\rangle$ with $H$ Hermitian, we would write $\frac{d}{dt} |\psi\rangle =\frac{1}{\hbar}\tilde H |\psi\rangle$ with $\tilde H$ anti-Hermitian (i.e. antisymmetric).
But then $i$ appears in the formula for the $\tilde H$ of a harmonic oscillator. So your proposed recipe doesn't help at all.

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"Complex numbers in QM"

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