# A Complex numbers in QM

#### stevendaryl

Staff Emeritus
The question is a philosophical one: why did Nature "choose" complex amplitudes over real ones?
The ontological question of what the wave function means is independent, it seems to me, of whether or not it uses complex numbers. If you had a physical interpretation that said something like "the wave function is a length of some object" or "the wave function is a probability", then certainly complex values are unphysical. But it seems to me that worrying about whether it's complex prior to coming up with a physical meaning for it is putting the cart before the horse.

Certainly, you can always eliminate complex numbers by using matrices instead of numbers, or by using quaternions, or whatever. But I don't see that such a change is anything more than aesthetic.

On the other hand, Hestenes had a program, spacetime algebra, for replacing all occurrences of complex numbers in physics by geometric objects such as elements of a Clifford algebra. To a large extent, it can be done. I'm not sure, though, that it actually has helped in figuring out the ontology of the wave function. Hestene's interpretation of the Schrodinger equation, for example, interprets the $i$ in $H |\psi\rangle = i \frac{d}{dt} |\psi\rangle$ not as an imaginary number, but as a bivector representing the spin of the particle. That makes the Schrodinger equation into an approximation to the Pauli equation. So it's sort of interesting.

#### DarMM

Gold Member
Certainly, you can always eliminate complex numbers by using matrices instead of numbers, or by using quaternions, or whatever. But I don't see that such a change is anything more than aesthetic.
Well quaternionic quantum theory violates no-signalling. It's not entirely aesthetic. @maline is talking about a similar reason for the reals.

#### DarMM

Gold Member
Just an update @maline , I did a bit of a literature search. I can't see an a priori argument eliminating the reals from first principles unlike the quaternions, although it is under investigation if local tomography hints at something and if the global degrees of freedom cause some problems to develop.

There is a paper by Adán Cabello here: https://arxiv.org/abs/1801.06347
He reconstructs quantum theory as the most general probability theory for an agent who can perform idealized measurements with discrete outcomes. Any more general theory leads to inconsistencies (probabilities sum to over unity) for repeated copies of an experiment. Just for interest Cabello uses the fact that it's the most general probability theory to argue that this implies it says little about nature.
Related to this discussion his argument only shows the most general probability theory is either Real or Complex QM. He has some comments at the end about trying to find the reason for the Hilbert space being complex.

#### maline

The electromagnetic field in its Silberstein representation is another, classical complex-valued beable!
I don't think the Silberstein representation should be thought of as fundamental or ontological (even if classical EM would be true of Nature). It is not Lorentz-covariant, and it also does not nicely express the Lorentz force law, which is the main observable effect of the fields.
But then $i$ appears in the formula for the $\tilde H$ of a harmonic oscillator. So your propsed recipe doesn't help at all.
Indeed, once we start listing Hamiltonians for systems that actually turn up in the world, we quickly see that they look much more natural as operators on complex Hilbert space. That is part of what we wish to explain. However, note that if necessary, was can always do things in real Hilbert space, by having separate dimensions for (what in complex QM we call) the real and imaginary parts of each dimension coefficient. Then the imaginary number $i$ is replaced by the operator $J$ which exchanges them anti-symmetrically.
The ontological question of what the wave function means is independent, it seems to me, of whether or not it uses complex numbers. If you had a physical interpretation that said something like "the wave function is a length of some object" or "the wave function is a probability", then certainly complex values are unphysical. But it seems to me that worrying about whether it's complex prior to coming up with a physical meaning for it is putting the cart before the horse.
I am referring to interpretations like MWI or dBB, or even GRW, where the wavefunction is seen as a fundamental object, like a field. Indeed in may be the only fundamental object. So that answers what its "meaning" is, and we are free to ruminate on whether complex values seem appropriate.
I do magnanimously allow others to not share my niggling feelings and ruminations.
Certainly, you can always eliminate complex numbers by using matrices instead of numbers, or by using quaternions, or whatever. But I don't see that such a change is anything more than aesthetic.
There is one important difference between complex QM and the real QM generated by thinking of the real and imaginary parts as different dimensions: the question of which operators count as linear. Operations like complex conjugation, or selecting the real or imaginary part, are linear on $\mathbb R^2$ but not on $\mathbb C$, and similarly for larger Hilbert spaces. When we say that QM is complex rather than real, we are referring to the fact that all of the unitaries and observable operators that show up in QM are complex-linear, rather than merely linear in the corresponding real space. In other words, looking from the real-QM perspective, there is a pairing of dimensions into effective complex dimensions, that is maintained by all physical operators. One effect of this is that the state ends up only being fixed (by physical outcomes) up to a complex phase, or in real-QM terms, up to a 2-D rotation. On the face of it, this seems like good enough reason to conclude that the real-QM description is just unnatural and our world does use complex QM. This is something that could have been otherwise, and we are left to asking God how the choice was made. Perhaps He wants us to be able to do local tomography of states.

But if it turns out that this pairing is actually completely natural and even inevitable, for any real Hilbert space with the right symmetries, then that gives us a different perspective. That would mean there really are not two options that could have been, but only two self-sufficient ways of expressing the same theory. The complex version will be the neater and more convenient one, but God need not commit to it.

And yes, this would explain why actual Hamiltonians like the harmonic oscillator look so ugly in real QM: it is because we are expressing things in terms of an observable position operator, and that operator must comply with the emergent complex structure - meaning it cannot have nondegenerate real-QM eigenstates (I mean in the limiting, rigged-Hilbert-space sense in which the standard non-relativistic position operator does have eigenstates), but only degenerate and indistinguishable pairs of states.

#### akhmeteli

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?
Again, the key question of this thread is: "Why complex numbers are necessary for QM?" Everybody agrees that they are very convenient, but are they necessary? While your argument looks reasonable, it is not quite waterproof. To see that, let us consider, following Schrödinger, the case of the Klein-Gordon equation in electromagnetic field. While your formula is still valid for that case, one can do without complex numbers, as any solution of this equation for the wave function can be made real by a gauge transformation (the 4-potential of the electromagnetic field will also change as a result, but the electromagnetic field will not, so we will get a physically equivalent solution). Thus, if you choose a certain gauge (the unitary gauge, where the wave function is real), you don't need complex wave functions.

It is also possible (but more difficult) to show that one real function is also sufficient for the Dirac equation in the electromagnetic field or in the Yang-Mills field. Again, let me emphasize that this is not about replacing complex numbers with pairs of real numbers.

As for the motivation... As this thread shows, many people would like to know the answer to the question: "Why complex numbers are necessary for QM?" The above shows that, to answer this question, one needs more elaborate arguments than yours.

There may be some additional motivation. For example, if we consider Klein-Gordon-Maxwell electrodynamics (scalar electrodynamics), we can use the gauge transformation to make the matter wave function real (this will not change the physics) and then algebraically eliminate the matter field altogether. The resulting modified Maxwell equations will describe independent evolution of the electromagnetic field. This is unexpected and looks pretty neat.

#### Auto-Didact

As it was brought up earlier, complex numbers in QM seem to have a very different role in QM than they have for example in electrical engineering theory (e.g. phasors) where their usage is completely artificial. In contrast, the role they play in QM seems to be fundamental - at least fundamental to the theory if not to Nature herself.

Over the years, my understanding on this matter is from reading multiple sources, which all seem to argue that complex numbers arise in QM due to, or more accurately in order to, model non-commutativity. A good example is the fundamental role they play in defining the Pauli matrices and so define the spinors in Dirac theory; in fact, the Pauli matrices are literally not much more than generators of the Lie algebra of SO(3) multiplied by a factor of $i$.

#### A. Neumaier

complex numbers arise in QM due, or more accurately in order to model non-commutativity. A good example is the fundamental role they play in defining the Pauli matrices
But this cannot be a fundamental reason since general real matrices are also noncommutative.

#### Auto-Didact

But this cannot be a fundamental reason since general real matrices are also noncommutative.
Not just non-commutativity in general but the specific case giving rise to Heisenberg's uncertainty principle.

Personally, I'd argue that the existence of spin is what requires complex numbers.

#### akhmeteli

Not just non-commutativity in general but the specific case giving rise to Heisenberg's uncertainty principle.

Personally, I'd argue that the existence of spin is what requires complex numbers.
These statements seem plausible, but they are not correct. I quoted the relevant articles several times.

Schrödinger (Nature, v.169, 538 (1952)) showed that
Heisenberg's uncertainty principle does not require complex numbers, as the Klein-Gordon equation in electromagnetic field satisfies the principle, but its wave function can be made real by a gauge transformation.

I (Journ. Math. Phys., 52, 082303 (2011),
the existence of spin does not require complex numbers, as, for example, the Dirac equation in electromagnetic field, which certainly describes spin, is generally equivalent to an equation for just one real function.

#### Auto-Didact

Regardless if it can be rewritten in real form in a highly ad hoc manner, the very derivation of the Dirac equation certainly requires complex numbers, wouldn't you agree?

If we accept the necessity of complex numbers for the derivation of the Dirac equation, then the complex numbers mostly seem to arise to ensure the Lorentzian structure of spacetime directly reflected in the wave operator.

Admittedly, this feels like a weak argument, similar to the point that Woit is making here: https://www.math.columbia.edu/~woit/wordpress/?p=7773

#### akhmeteli

Regardless if it can be rewritten in real form in a highly ad hoc manner, the very derivation of the Dirac equation certainly requires complex numbers, wouldn't you agree?

First, we discuss if complex numbers are required, rather than just convenient, so it does not matter if the Dirac equation "can be rewritten in real form in a highly ad hoc manner",
it is important if it can be rewritten in real form in principle.

To agree or to disagree that "the very derivation of the Dirac equation certainly requires complex numbers", I would need to look at the original derivation of the Dirac equation, as I forgot the derivation long ago:-). However, I don't understand how this could be relevant: if the Dirac equation in electromagnetic field is equivalent to an equation for just one real function, that means that complex numbers are not required for the Dirac equation.

Another thing about the derivation. To derive his equation, Dirac required that the equation be of the first order. However, Feynman and Gell-Mann proved later (Phys. Rev. 109, 193 (1958)) that the Dirac equation can be written as a second-order equation, I showed that it is generally equivalent to a fourth-order equation, therefore, Dirac's derivation was based on a wrong assumption (of course, that does not change the value of Dirac's accomplishment). So why is the original derivation important for this thread?

Auto-Didact said:
If we accept the necessity of complex numbers for the derivation of the Dirac equation, then the complex numbers mostly seem to arise to ensure the Lorentzian structure of spacetime directly reflected in the wave operator.

Admittedly, this feels like a weak argument, similar to the point that Woit is making here: https://www.math.columbia.edu/~woit/wordpress/?p=7773
Again, we are not trying to decide in this thread if complex numbers are convenient for the Dirac equation (they are convenient for classic physics as well), we are trying to decide if they are necessary. My take is they are not.

#### Auto-Didact

I am explicitly speaking about necessity, not convenience, as I already clarified in the beginning of post #156.

The original derivation of Dirac is important because it explicitly shows the principles he is adhering to, namely respecting the Lorentzian structure of the wave operator; this brings with it loads of mathematical bagage which is either fundamental to Nature or not.

The point I am making about ad hoc rewriting is precisely that the equation can be rewritten as second order or fourth order and so removing its complex nature. This procedure screams to me to be the same kind of mathematical trick such as the Wick rotation, i.e. of the variety often useful for tractability reasons, but certainly not universally valid.

#### meopemuk

Quantum mechanics = quantum logic.
Quantum logic = orthomodular lattice.
Orthomodular lattices can be realized as sets of subspaces in a Hilbert space with real, complex, or quaternionic scalars. This is Piron's theorem. So, there are 3 legitimate versions of quantum mechanics with good logical structures. For example, you can possibly build an octonionic Hilbert space, but this will not be a good quantum mechanics due to violation of some axioms of logic.

There were some attempts to develop real or quaternionic versions of quantum mechanics but, as far as I know, they didn't bring any new physics.

Eugene.

#### akhmeteli

I am explicitly speaking about necessity, not convenience, as I already clarified in the beginning of post #156.

The original derivation of Dirac is important because it explicitly shows the principles he is adhering to, namely respecting the Lorentzian structure of the wave operator; this brings with it loads of mathematical bagage which is either fundamental to Nature or not.

I am not saying the original derivation is not important, I am saying it is not relevant to the question: are complex numbers necessary for the Dirac equation? The answer cannot depend on Dirac's beliefs and principles: complex numbers are either necessary for the Dirac equation or not.

As for "Lorentzian structure" and "Wick rotation", let me ask you: do you think complex numbers are necessary for special relativity?

Auto-Didact said:
The point I am making about ad hoc rewriting is precisely that the equation can be rewritten as second order or fourth order and so removing its complex nature. This procedure screams to me to be the same kind of mathematical trick such as the Wick rotation, i.e. of the variety often useful for tractability reasons, but certainly not universally valid.
I don't understand this. Rewriting the Dirac equation as a fourth order equation plus gauge transformation allows using just one real function instead of four complex functions. However dirty this trick may be, it proves that complex numbers are not necessary for the Dirac equation.

#### Auto-Didact

The answer cannot depend on Dirac's beliefs and principles: complex numbers are either necessary for the Dirac equation or not.

As for "Lorentzian structure" and "Wick rotation", let me ask you: do you think complex numbers are necessary for special relativity?
The answer to the former is very much an answer to the latter: the hyperbolic structure of Minkowski space has mathematical and experimental consequences, e.g. the unobservable nature of Lorentz contraction; this is a consequence of the hidden conformal structure in SR. If complex numbers aren't fundamental to SR, it means that this is purely a coincidence.

I know that the consensus of the field is that complex numbers aren't necessary for SR; I accept this, but believe this consensus might be premature and that the complex structure brings far more with it that cannot really be appreciated from a purely pragmatic viewpoint; I am not arguing for convenience here, but for deep mathematical consequences.

My point is the following: unless we specifically push the theory not merely to its limits but beyond, i.e. attempt fundamental analysis of the concepts using the deepest rigorous theories of pure mathematics that we have, we cannot know whether something is necessary or merely convenient; the major programmes of theoretical physics that do this clearly show that the complex structure is not merely convenience but a necessity.

As for QT, the case for the necessity of complex numbers seems to be much clearer than the case for complex numbers in SR. Orthodox QM is patently laden with complex numbers, complex vector spaces and holomorphic functions. There are tonnes of equations in QFT with explicit complex notions such as the gamma function.

The positive frequency condition of QFT - i.e. positive energy - can be deeply demonstrated to be dependent on the underlying complex structure. Moreover, the scale invariance of renormalization group theory is also deeply dependent on the notion of complex numbers.
I don't understand this. Rewriting the Dirac equation as a fourth order equation plus gauge transformation allows using just one real function instead of four complex functions. However dirty this trick may be, it proves that complex numbers are not necessary for the Dirac equation.
Does this fourth order characterization with the gauge transformation correctly carry over to QFT in curved spacetime?

#### akhmeteli

The answer to the former is very much an answer to the latter: the hyperbolic structure of Minkowski space has mathematical and experimental consequences, e.g. the unobservable nature of Lorentz contraction; this is a consequence of the hidden conformal structure in SR. If complex numbers aren't fundamental to SR, it means that this is purely a coincidence.

I know that the consensus of the field is that complex numbers aren't necessary for SR; I accept this, but believe this consensus might be premature and that the complex structure brings far more with it that cannot really be appreciated from a purely pragmatic viewpoint; I am not arguing for convenience here, but for deep mathematical consequences.

My point is the following: unless we specifically push the theory not merely to its limits but beyond, i.e. attempt fundamental analysis of the concepts using the deepest rigorous theories of pure mathematics that we have, we cannot know whether something is necessary or merely convenient; the major programmes of theoretical physics that do this clearly show that the complex structure is not merely convenience but a necessity.

Well, let us assume for a moment that you are right, and special relativity does require complex numbers. In that case it is not the uncertainty principle or spin that require complex numbers, as there is no uncertainty principle or spin in special relativity, so your statements in post 158 are still incorrect.

Auto-Didact said:
As for QT, the case for the necessity of complex numbers seems to be much clearer than the case for complex numbers in SR. Orthodox QM is patently laden with complex numbers, complex vector spaces and holomorphic functions. There are tonnes of equations in QFT with explicit complex notions such as the gamma function.

The positive frequency condition of QFT - i.e. positive energy - can be deeply demonstrated to be dependent on the underlying complex structure. Moreover, the scale invariance of renormalization group theory is also deeply dependent on the notion of complex numbers.
The standard Dirac equation is also "patently laden with complex numbers", but it turns out it does not require them. Again, I cannot be sure that QFT does not require complex numbers, but then one needs more sophisticated arguments to prove the necessity.
Auto-Didact said:
Does this fourth order characterization with the gauge transformation correctly carry over to QFT in curved spacetime?
I don't know, but this is irrelevant to whether it is the uncertainty principle and spin that require complex numbers.

#### PeterDonis

Mentor
the unobservable nature of Lorentz contraction
What are you talking about here? Why do you think Lorentz contraction is unobservable?

#### PeterDonis

Mentor
The positive frequency condition of QFT - i.e. positive energy - can be deeply demonstrated to be dependent on the underlying complex structure.
Please give a reference for this demonstration.

#### PeterDonis

Mentor
the hyperbolic structure of Minkowski space
the hidden conformal structure in SR
You appear to be saying that these characteristics somehow require (or at least strongly imply) a complex structure underlying SR. Why?

#### Auto-Didact

Well, let us assume for a moment that you are right, and special relativity does require complex numbers. In that case it is not the uncertainty principle or spin that require complex numbers, as there is no uncertainty principle or spin in special relativity, so your statements in post 158 are still incorrect.
That doesn't follow. The argument for complex numbers in SR is in principle completely seperate from the argument for complex numbers in QM.

More importantly, I see that you are focussing alot on #158; I would like to paraphrase Bohr by saying "Every sentence I utter must be understood not as an affirmation, but as a question."

This thread is about complex numbers in QM in general; I think therefore that the arguments I give in other posts in favor of the fundamental place of complex numbers in QT cannot be dismissed so lightly.
I don't know, but this is irrelevant to whether it is the uncertainty principle and spin that require complex numbers.
It is obvious to me that we have very opposing views of how we basically regard what a physical theory is; the difference between our views are what historically was called the applied mathematics view versus the pure mathematics view of physics.

The derivation of spin from first principles by Dirac is to me clearly a result of the mathematical existence of spinors, whether that is/was acknowledged or not by physicists at the time or even today. The existence of spin as a mathematical object can be demonstrated to be a consequence of the existence of spinors, with the gamma matrices operators which act on spinors. In this sense, gamma matrices are non-commuting elements of the Clifford algebra, giving spinors more degrees of freedom than scalar wavefunctions. This is what my post in #158 was alluding to.

In contrast, from what I understand from your point of view, it seems you would claim that spin is just a physical quantity following from any mathematical model capable of describing aspects of the physics, whether or not these descriptions can on the face of it immediately be shown (through trivial efforts/arguments) to be equivalent to some other purely mathematical model of spin we already have; indeed, such a pragmatic view is referred to as an applied mathematics view of physics.

#### Auto-Didact

What are you talking about here? Why do you think Lorentz contraction is unobservable?
Lorentz contraction as an actual contraction of length is not directly observable; I thought this was more widely known?
Please give a reference for this demonstration.
Positive and negative frequency solutions can be naturally split on the Riemann sphere, a distinctly complex analytic notion.
See https://doi.org/10.1098/rspa.1982.0165

In principle, this also has alot to do with creation and annihilation operators satisfying the rules of a Grassman algebra.
You appear to be saying that these characteristics somehow require (or at least strongly imply) a complex structure underlying SR. Why?
The underlying properties of SR in Minkowski space and curved spaces has key properties allowing advanced mathematical treatment with conformal manifolds and the theory of Riemann surfaces which expose more of physical theory; is this a coincidence?

#### PeterDonis

Mentor
Lorentz contraction as an actual contraction of length is not directly observable
That's not what the reference you gave (which is just Terrell's classic paper introducing what is now known as Penrose-Terrell rotation) says. It says something more limited: that if you confine yourself to observations made using light rays emitted by a moving object and arriving at your eye, or some equivalent detector corresponding to a single timelike worldline, then you will not observe the object to be contracted, but rotated. But this is by no means the only possible way to make measurements on an object that is moving relative to you; there are other methods that allow you to measure the object's length and show that length to be contracted.

#### Nugatory

Mentor
Lorentz contraction as an actual contraction of length is not directly observable; I thought this was more widely known?
That paper that you are citing says that the Lorentz contraction cannot be observed using one particular technique: forming an image of reflected light on a screen (photographic film, retina of the eye, ...) and interpreting the image as if all the incident light was reflected at the same time. Of course this assumption is only valid when the speed of the moving object is negligible compared with the speed of light; the point of this paper is that (if some other reasonable conditions also apply) the bogus assumption leads to a misinterpretation of the image that precisely hides length contraction in the image.

However “cannot be observed using one particular technique”is a very different and much weaker statement than “not directly observable”, even when the technique in question is naked eye observation.

In principle something like the pole-barn experiment would allow direct observation of length contraction; the impediment is the practical difficulties of manipulating a pole moving at relativistic velocities.

#### Auto-Didact

However “cannot be observed using one particular technique”is a very different and much weaker statement than “not directly observable”, even when the technique in question is naked eye observation.
Fully agreed.
In principle something like the pole-barn experiment would allow direct observation of length contraction; the impediment is the practical difficulties of manipulating a pole moving at relativistic velocities.
Any examples of direct observations of the current experimental state of the art? The only observational "example" I can think of from the top of my head is the perspective from arriving muons which have an extended lifetime due to time dilation.

#### Nugatory

Mentor
Any examples of direct observations of the current experimental state of the art? The only observational "example" I can think of from the top of my head is the perspective from arriving muons which have an extended lifetime due to time dilation.
The appearance of magnetic fields around a current-carrying wire can be explained as a length-contraction effect; see, for example, the first section of http://www.physics.umd.edu/courses/Phys606/spring_2011/purcell_simplified.pdf

"Complex numbers in QM"

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