A Complex Numbers Not Necessary in QM: Explained

[vanhees71]

The intuitionistic reals behave mathematically very different from the reals taught in any analysis course.

In intuitionistic math, most concepts from ZFC ramify into several meaningful nonequivalent ones, depending on which intuitionistic version of the axioms one starts with (all of which would become equivalent if the axiom of choice were assumed in addition). Thus one has to be very careful to know which version of the reals one is talking about.

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

 

[A. Neumaier]

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

Well, it shows the extent to which things can be made fully constructive. Thus it gives insight into the structure of mathematical reasoning. A physicist doesn't need it, of course...

 

[stevendaryl]

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

I spent a good number of years studying intuitionistic and constructive mathematics. I think it's interesting, but I'm not convinced that anything worthwhile comes from it.

An interesting fact about intuitionistic mathematics is the isomorphism between intuitionistic proofs and computer programs. Intuitionistically, if you prove a statement of the form

$\forall x \exists y: \phi(x,y)$

you can extract a program (expressed as a lambda-calculus expression) that given any $x$ returns a $y$ satisfying $\phi(x,y)$.

Every proposition in constructive logic (I'm a little hazy about the exact distinction between constructive and intuitionistic) corresponds to a type, in the computer-science sense, and the proofs of those propositions correspond to mathematical objects of that type. So for example:

$A \wedge B$ corresponds to the set of ordered pairs $(a,b)$ where $a$ is a proof of $A$ and $b$ is a proof of $b$.
$A \rightarrow B$ corresponds to the set of functions which given a proof of $A$ returns a proof of $B$.
$A \vee B$ corresponds to the disjoint union of proofs of $A$ and proofs of $B$.
etc. (quantifiers correspond to product types and tagged unions).

I think it's all very interesting, and it gives some insight into logic and programming and the connection between them. But ultimately, I don't see the whole endeavor as being tremendously useful.

 

[stevendaryl]

The biggest difference between constructive and classical logic is the extent to which it is possible to prove that something exists without being able to give an example. You can't do that in constructive logic. So a proof that there exists a nonmeasurable set doesn't go through. However, you can recover most of classical mathematics by doing a "double negation". In almost all cases where statement $A$ is provable classically, $\neg \neg A$ is provable constructively. A double-negation is not equivalent to the original statement in constructive logic.

 

[A. Neumaier]

In almost all cases where statement A is provable classically, ¬¬A is provable constructively.

In all cases, this is provable in the intuitionistic setting.

This shows that there is no loss of quality in assuming classical logic. One can only gain, never inherit a contradiction that is not already there on the intuitionistic level.

This is the reason why intuitionistic logic is irrelevant in practice.

 

[DarMM]

You gave a complex reason :-)

The real reason is that amplitudes satisfy a simple differential equations, probabilities don't. Knowing all probabilities at a fixed time is not even enough to determine the future probabilities, since probabilities lack the phase information at each point in configuration space.

I've been thinking about this a bit more and I'm not sure it is correct. Quaternionic quantum mechanics for example can be given simple differential equations. Also the differential equations are a dynamical feature where as the presence of complex numbers is a Kinematical feature of quantum theory.

I think the reason for complex numbers is the fact of QM being a multiple sample space probability theory obeying local tomography. Conservation of probability in such a generalized probability model implies those simple evolution equations.

 

[A. Neumaier]

I think the reason for complex numbers is the fact of QM being a multiple sample space probability theory obeying local tomography. Conservation of probability in such a generalized probability model implies those simple evolution equations.

This cannot have been the reason in the early days of QM when there was no notion of multiple sample space.

The true reason is that the Schrödinger equation already contains $i$ (and needs it for unitary evolution with a Hermitian generator), and it worked exceedingly well, so nothing more complicated was superior (or even competitive).

Of course, one could think of the Schrödinger equation of a single particle with spin as being quaternionic, but this does not extend nicely to the multiparticle case. Quaternionic quantum mechanics, p-adic quantum mechanics, and other exotics were never found interesting - except by the few who tried it and reported their meager results.

 

[DarMM]

I don't disagree with what you wrote, but that seems to be more the reason why historically QM has complex numbers. I more answering "Why couldn't it have been another field?" or "What's wrong with using another field?"

Similar to somebody asking why General Relativity has no Torsion. One can of course answer that Einstein didn't use torsion and there was never any need for it. However you can also explain what might be wrong with such an approach.

I'm basically driving at why the other approaches didn't work and QM must be complex.

 

[Fra]

This cannot have been the reason in the early days of QM when there was no notion of multiple sample space.

The true reason is that the Schrödinger equation already contains $i$ (and needs it for unitary evolution with a Hermitian generator), and it worked exceedingly well, so nothing more complicated was superior (or even competitive).

The historical reason is for things is interesting on its own, but if we seek reasons involving a deeper understanding of QM i think the notion of "multiple sample spaces" has the potential to be a better framework for deeper understanding. Then it not argument that this insight was lacking almost a century ago.

I now understand you want to avoid completely the "statistical" or "probabilistic" interpretation of things in the thermal interpretation, and in that process i can see why one sees no value in "multiple P-spaces", but I see another way. We can attach the statistical foundation to subjective statistics of an agent, which is also a bit related to underdetermined systems and reasoning upon incomlpete information. Here the notion of multiple samples spaces which are related makes excellent sense IMO, as one can motivate them from the perspective of compressed sensing, for final encoding in memory records - from which probabilistic expectations of future follows.

So I would insist we ask ourselfs, WHY multiple sample spaces? When one is required to discard information, also due to limited storage, the question is WHICH information can be throw away? (or which information about its environment can an electron throw again) with minimal consequences? Here one may evolve a recoding into conjugate samples spaces or otherwise any alternative coding, and each transformations may have benefits from survival perspective dependin how the environment acts.

In this evolutionary picture, I see the "multiple sample space" mental picture as the one allowing for more natural insight.

/Fredrik

 

[akhmeteli]

I don't disagree with what you wrote, but that seems to be more the reason why historically QM has complex numbers. I more answering "Why couldn't it have been another field?" or "What's wrong with using another field?"

Similar to somebody asking why General Relativity has no Torsion. One can of course answer that Einstein didn't use torsion and there was never any need for it. However you can also explain what might be wrong with such an approach.

I basically driving at why the other approaches didn't work and QM must be complex.

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

 

[A. Neumaier]

it is not quite obvious that "QM must be complex"

Simplicity forces complex numbers. Of course, one can rewrite things equivalently in a more cumbersome conceptual setting, but what's that good for??

 

[akhmeteli]

it is not quite obvious that "QM must be complex"

Simplicity forces complex numbers.

Your statement may be too strong, as it seems to be equally applicable to classical electrodynamics, where complex numbers certainly make life much easier. But that would suggest that the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Of course, one can rewrite things equivalently in a more cumbersome conceptual setting, but what's that good for??

Let me emphasize again that this is not about replacing complex numbers with pairs of real numbers. For example, for the Schrödinger equation or the Klein-Gordon equation we have just one real wave function instead of one complex function after the gauge transformation. For the Dirac equation, we can have just one real function instead of four complex components of the spinor wave function, and so on. For the Klein-Gordon-Maxwell electrodynamics you can algebraically eliminate the matter wave function altogether, which may be in sync with your emphasis on fields, rather than on particles. So the above seems to be a tentative answer to your question: "what's that good for??" and suggests that real numbers can even provide some conceptual simplicity.

 

[A. Neumaier]

the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Yes, it is a matter of convenience. In classical electrodynamics one uses complex numbers often, but there are many situations where they don't give an advantage and are not used.

In quantum mechanics avoiding complex numbers almost never gives an advantage, so it is hardly ever used. But the real formulation in terms of real and imaginary part is sometimes useful for the numerical solution of the Schrödinger equation as one can then use symplectic integrators.

that real numbers can even provide some conceptual simplicity.

No, since one usually needs all components of the spinor wave function and its transformation properties, which are ugly and inconvenient in your formulation.

 

[DarMM]

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

This might work for a fragment of QM, but in general it will cause states to violate local tomography and thus relativity.

 

[akhmeteli]

the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Yes, it is a matter of convenience. In classical electrodynamics one uses complex numbers often, but there are many situations where they don't give an advantage and are not used.
In quantum mechanics avoiding complex numbers almost never gives an advantage, so it is hardly ever used.

So I conclude that complex numbers are actually not a "must" in QM.

But the real formulation in terms of real and imaginary part is sometimes useful for the numerical solution of the Schrödinger equation as one can then use symplectic integrators.

I cannot emphasize enough that what I describe has nothing to do with replacing complex numbers with pairs of real numbers.

that real numbers can even provide some conceptual simplicity.

No, since one usually needs all components of the spinor wave function and its transformation properties, which are ugly and inconvenient in your formulation.

So you don't think using one real function instead of four complex functions can provide conceptual simplicity. Let us agree to disagree about that.

 

[akhmeteli]

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

​

​

This might work for a fragment of QM

You could call the Dirac equation "a fragment of QM", but I would say it is quite a large part of QM.

, but in general it will cause states to violate local tomography and thus relativity.

Let me note that even if this is so, it suggests that one needs pretty sophisticated arguments to prove that complex numbers are a must for QM. Anyway, could you please give a reference to the proof?

 

[DarMM]

You could call the Dirac equation "a fragment of QM", but I would say it is quite a large part of QM.

The point is with the multiparticle case. Also the Dirac equation as a wavefunction equation is seriously limited. The Dirac equation conceived of as the equation for the wave function of a single relativistic spin-1/2 particle is indeed a small fragment. One runs into problems with existence of bound states, positivity of energy and many other features.

Let me note that even if this is so, it suggests that one needs pretty sophisticated arguments to prove that complex numbers are a must for QM. Anyway, could you please give a reference to the proof?

I gave two in post #58

 

[akhmeteli]

The point is with the multiparticle case.

I consider the case of multiple particles in my work http://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf (published in EPJC), section 4. The treatment there may be not comprehensive enough for your taste, but it is not obvious that it cannot be generalized.

Also the Dirac equation as a wavefunction equation is seriously limited. The Dirac equation conceived of as the equation for the wave function of a single relativistic spin-1/2 particle is indeed a small fragment. One runs into problems with existence of bound states, positivity of energy and many other features.

The Dirac equation does not include all QED, but "a small fragment" of QM? Come on:-) Let me also add that a similar result can be obtained for the Dirac equation in the Yang-Mills field (see the reference at https://www.physicsforums.com/threa...d-of-spinor-field-in-yang-mills-field.960244/)

I gave two in post #58

And I am not impressed...

The Moretti/Oppio work seems to be about Wigner elementary relativistic systems - I guess there is no external field there, so it is not quite realistic (see also my short discussion with Moretti at https://physics.stackexchange.com/q...le-to-the-electromagnetic-field/268971#268971 (after his answer)).

The work on "local tomography" that you quoted contains an awful lot of "fairly natural constraints", which constraints may be good for eliminating approaches using real numbers but it is not obvious why one has to accept them.

 

[DarMM]

The Dirac equation does not include all QED, but "a small fragment" of QM? Come on

I don't understand what this means. The Dirac equation appears in QED as an equation of motion for field operators, this is separate to its use as a single particle relativistic wave equation (just like the Klein-Gordon equation). As the latter it is severely limited and runs into problems. As the former it's not really related to the discussion as its complexity or reality is unrelated to the complexity/reality of the Hilbert space.

In fact I'm confused as to what you are talking about. Are you arguing for the field of real numbers in the equations of motion of the field operators or as the underlying field for the Hilbert space of states. These are separate issues.

The work on "local tomography" that you quoted contains an awful lot of "fairly natural constraints", which constraints may be good for eliminating approaches using real numbers but it is not obvious why one has to accept them.

Could you describe what you are talking about here?

 

[akhmeteli]

I don't understand what this means. The Dirac equation appears in QED as an equation of motion for field operators, this is separate to its use as a single particle relativistic wave equation (just like the Klein-Gordon equation). As the latter it is severally limited and runs into problems. As the former it's not really related to the discussion as its complexity or reality is unrelated to the complexity/reality of the Hilbert space.

I mean that, while the single-particle Dirac equation does not describe all the physics described by QED, it describes an awful lot of quantum phenomena. Yes, the Dirac equation has its share of problems, but I would say it is better than, e.g., the nonrelativistic Schrödinger equation or the Klein-Gordon equation.

In fact I'm confused as to what you are talking about. Are you arguing for the field of real numbers in the equations of motion of the field operators or as the underlying field for the Hilbert space of states. These are separate issues.

I consider the following question: are complex numbers a must for QM (remember that the title of the thread is "Complex numbers in QM")? To this end, I consider non-second-quantized single-particle equations (Schrödinger, Klein-Gordon, Dirac). Traditionally it was believed that they required complex wave functions. It is not well-known that, as Schrödinger showed, complex wave functions (or pairs of real wave functions) are not required for, say, the Klein-Gordon equation. It is not well-known that they are not required for the Dirac equation either. Are they required for multiple particles? I am not sure. So far I am mostly speaking about solutions of single-particle equations. I believed this was relevant to your question "I basically driving at why the other approaches didn't work and QM must be complex."

Could you describe what you are talking about here?

Say, Proposition 1 in https://arxiv.org/abs/1202.4513 has a lot of assumptions. As the authors try to further motivate them, it is not obvious one must accept them. Furthermore, they write in the abstract: "orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which..." So it looks like if one agrees with their assumptions, one gets a finite-dimensional quantum mechanics. I guess that does not even cover the quantum mechanics of the nonrelativistic Schrödinger equation.

 

[DarMM]

So far I am mostly speaking about solutions of single-particle equations

Say, Proposition 1 in https://arxiv.org/abs/1202.4513 has a lot of assumptions.

The point is that it's for multiple particle states where one gets problems. The problem is that one violates local tomography. And what is the main part of that paper's proposition 1? That the theory obeys local tomography.

In any of the axiom systems derived for QM thus far the complex structure comes from local tomography, i.e. one can have reals and quaternions only if for composite systems (e.g. multiple particles) one breaks local tomography. Which is what I've been saying.

Without local tomography one will violate relativity.

 

[akhmeteli]

The point is that it's for multiple particle states where one gets problems. The problem is that one violates local tomography. And what is the main part of that paper's proposition 1? That the theory obeys local tomography.

In any of the axiom systems derived for QM thus far the complex structure comes from local tomography, i.e. one can have reals and quaternions only if for composite systems (e.g. multiple particles) one breaks local tomography. Which is what I've been saying.

Without local tomography one will violate relativity.

As I said, I am not impressed. Even if I accept local tomography, Proposition 1 contains other assumptions, which are not obvious, to say the least, and the article seems to be about finite-dimensional QM.

And again, I mentioned the approach to multiple particles that does not require complex numbers.

 

[DarMM]

As I said, I am not impressed. Even if I accept local tomography, Proposition 1 contains other assumptions, which are not obvious, to say the least, and the article seems to be about finite-dimensional QM.

The only other assumption is that two level systems exist, i.e. qubits. That seems to be obviously true right?

 

[maline]

it will cause states to violate local tomography and thus relativity.

Local tomography is not required by relativity. Not having local tomography just means that there are not enough local measurements to fully learn the state from their joint statistics. Why would this be any fundamental problem?

 

[DarMM]

Local tomography is not required by relativity. Not having local tomography just means that there are not enough local measurements to fully learn the state from their joint statistics. Why would this be any fundamental problem?

This is not all local tomography means. 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.

Only the real case involves underdetermination as you have mentioned and there are genuine "global statistics" not recoverable from local ones. There the problem is due to how the global aspect of the state not learnable from local measurements interacts with Poincaré symmetry.

 

[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]

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]

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.

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.

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.

 

[DarMM]

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]

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.

 

[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.

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.

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).

 

[akhmeteli]

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.

Yes, and I took back the relevant statement earlier (https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-8#post-6145920). Nevertheless, the statement is correct for other systems, such as electron spin.

 

[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]

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.

 

[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.

 

[PeterDonis]

Moderator's note: A subthread about Haag's Theorem and the Poincare group has been moved to a new thread:

https://www.physicsforums.com/threads/haags-theorem-and-the-poincare-group.967937/