A Complex Numbers Not Necessary in QM: Explained

[stevendaryl]

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]

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]

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),
http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf) showed that

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.​

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[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?

​

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?

​

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.

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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.

​

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]

the unobservable nature of Lorentz contraction

What are you talking about here? Why do you think Lorentz contraction is unobservable?

 

[PeterDonis]

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]

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 separate from the argument for complex numbers in QM.

More importantly, I see that you are focussing a lot 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?

https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1041Lorentz 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 a lot 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]

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]

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]

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

 

[akhmeteli]

That doesn't follow. The argument for complex numbers in SR is in principle completely separate from the argument for complex numbers in QM.

Then why were you telling me about "Lorentzian structure" and "Wick rotation" in the context of QM?

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More importantly, I see that you are focussing a lot 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."

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Sorry for taking your post seriously.

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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 read your post #158 and commented on it. As far as I remember, I did not read your earlier posts, so I could not have dismissed them, neither lightly nor heavily. I don't think I have to read the entire long thread to post anything.​

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.

OK, I got it. I have a bad habit to read texts exactly as they are written. You have explained that your post #158 means something different from what is written there. I give up.

 

[Auto-Didact]

Then why were you telling me about "Lorentzian structure" and "Wick rotation" in the context of QM?

Lorentzian structure: because I was talking about QFT, seeing that the discussion was focusing on the Dirac equation. There is an argument to make that QM is always a limiting case of QFT; if one argues this way, whether or not they believe in the argument, then the argument about the complex nature of SR is extremely relevant.

Wick rotation: as I said, convenience techniques, such as e.g. the Wick rotation, which make the mathematics more tractable tend to give results of limited validity. By analogy, I was making the case that your algebraic technique and gauge transformation, resulting in a real fourth order version of the Dirac equation, may have similar limitations. This is why I asked if the resulting equation carries over correctly to QFT in curved spacetime.

Sorry for taking your post seriously.

You misunderstand my intent for clarification; we are speaking about what is fundamental in physical theory, i.e. not just contemporary physics but what should or will likely be continued to be regarded as fundamental in future physics yet to be discovered. As I remarked before this requires a certain view of physics.

In fact, the usual attitude of certainty learned during training is almost never warranted in such discussions, as opposed to discussions about non-fundamental (textbook level) physics, where the level of certainty is rather high because it can almost always trivially be empirically (through experiment) or mathematically justified (e.g. even by a smart undergraduate student).

In other words, discussing foundations necessarily brings with it more uncertainty than in the rest of physics, exactly as Bohr remarked. The only guide theoretical physics has ever known in this intrinsically uncertain endeavor is to rely on pure mathematics not yet appreciated to be relevant to physics; this is often called 'relying on mathematical beauty' and is a concept deeply misunderstood by many physicists today (e.g. Hossenfelder).

Relying on mathematical beauty is truly an art, which cannot be reduced to brainless application of simple techniques and equation churning; it requires development of correct esthetic senses from pure mathematics and therefore requires a finer touch. Importantly, using this methodology in fundamental physics to generate hypotheses often results in highly mathematically elaborate models which stretch across all of physics making them extremely specific and therefore extremely open to falsification.

This is in stark contrast to more specialized topics in physics, which are better served by the applied mathematics view in conjunction with experimental reasoning. I am a strong advocate that thinking about foundational issues in the more pure mathematics manner, and therefore doing fundamental physics in such a way, is the most productive theoretical methodology available and the only tried and true one with precedent of success; the results of all the biggest names in physics attest to this.

 

[Jimster41]

The $\pi$ of course is definable. But to define any conrete number, you must use some language (English, mathematical language, or whatever). A definition can be viewed as a sentence in the chosen language. The set of all possible sentences is countable, hence the set of all possible definitions is countable.

Still reading this wonderful thread but how is the set of all possible sentences countable? Based on what constraint? Countable words in some vocabulary of some language, countable number of languages, countable number of phonemes? Countable number of shapes the tongue can make? I don't see how a typographic theory is any more immune to the problem of Reals shown by Cantor (the infinite regression of... the diabolical diagonal... the monstrous recursive... meta...) than a typographical number theory.

 

[PeterDonis]

how is the set of all possible sentences countable?

Because it's built from a finite, discrete set (the alphabet of the language--or even some general alphabet like the Unicode character set that can represent any language), and the number of possible ways you can form sets from a finite, discrete number of elements is countable.

In other words, the concept of a "sentence", at least as I think @Demystifier was using it, ignores the actual continuous variability of the process that produces sentences (people writing them down in longhand or pronouncing them vocally) and focuses on the discrete set of items that carry the intended meaning.

 

[Jimster41]

Because it's built from a finite, discrete set (the alphabet of the language--or even some general alphabet like the Unicode character set that can represent any language), and the number of possible ways you can form sets from a finite, discrete number of elements is countable.

In other words, the concept of a "sentence", at least as I think @Demystifier was using it, ignores the actual continuous variability of the process that produces sentences (people writing them down in longhand or pronouncing them vocally) and focuses on the discrete set of items that carry the intended meaning.

Thanks @PeterDonis I jumped in too enthusiastically perhaps. Had to rapid google. Just got to the punchline of Escher, Godel, Bach. My take is that is is a transcendental punchline. There are no closed alphabets I don’t think (modern coding languages are a perfect example of hybrid you mention) IOW at least for the moment I having deep skepticism that there are countably many formulas - languages are a real as the Reals... just running in their shadow.

 

[PeterDonis]

There are no closed alphabets

If you mean there aren't any finite alphabets, this is obviously wrong. The alphabet we are using to write these posts is finite. Even if you want an alphabet to represent all human languages instead of just one, it's still finite.

But even leaving that aside, alphabets are certainly countable. Alphabets aren't continuous. They're discrete. Letters don't continuously change from one to another. They're like integers, not real numbers.

 

[A. Neumaier]

at least for the moment I having deep skepticism that there are countably many formulas - languages are a real as the Reals... just running in their shadow.

An alphabet is by definition finite. It distinguishes a finite set of images as being the characters of a particular language. If you extend the alphabet you change the language.

 

[Jimster41]

My reaction was/is confused. It seems clear to me that the process of communication is served by both that immutability and the mutability of it. We try to not change the alphabet but it is invented. Utility motivates its constraint. But to the point of countability of the “Definable Reals” based on the enforced finitude of an invented formal system... that just feels like bootstrapping a solution. I keep thinking that you could at some point give a computer the problem of coming up with a mathematical formula for every Real. Unless you very carefully constrained it wouldn’t it dutifully go blur the line between language, alphabet and the Reals. You could say, “I can’t understand what you mean by this symbol or word - string thing”. But then it could have a friend who is also a computer that is has been working with and they might just shake their heads and with shared clarity of their co-invented system, “that’s the formula for the Real Number r of course”. So what does “Definable” really mean? Isn’t it just interpretation.

This is a great thread I need to go continue reading. After years of just being totally baffled so far my (very incomplete) cartoon is that they (complex numbers) are symbols that allow for compact and expressive descriptions of wavy things, and especially helical screwy things. Their invention has naturally shaped what has thereafter been expressed.

 

[stevendaryl]

My reaction was/is confused. It seems clear to me that the process of communication is served by both that immutability and the mutability of it. We try to not change the alphabet but it is invented. Utility motivates its constraint. But to the point of countability of the “Definable Reals” based on the enforced finitude of an invented formal system... that just feels like bootstrapping a solution. I keep thinking that you could at some point give a computer the problem of coming up with a mathematical formula for every Real. Unless you very carefully constrained it, Wouldn’t it dutifully go blur the line between real language, alphabet and the Reals. You could say, “I can’t understand what you mean by this symbol or word - string thing”. But then it could have a friend, who is also a computer, that is has been working with and they might just shake their heads and with shared clarity of their co-invented system, “that’s the formula for the Real Number r of course”. So what does “Definable” really mean? Isn’t it just interpretation.

This is a great thread I need to go continue reading. After years of just being totally baffled so far my (very incomplete) cartoon is that they are symbols that allow for compact and expressive descriptions of wavy things, and especially helical screwy things. Their invention has naturally shaped what has thereafter been expressed.

I'm not sure exactly what the question is. With the usual notion of "alphabet", there are finitely many different characters in the alphabet. The extreme case is binary, where there are only two characters, 0 and 1. If the alphabet is finite, and the length of any particular formula is finite, then that means there are only countably many formulas.

Since computers use binary, they are intrinsically limited to countable languages. So whatever notion you have of a language, if it can be communicated through the internet in a forum such as this one, it's expressible using a countable language.

You could imagine that maybe in person, people could communicate using continuously many different symbols. If you idealize a character as a zero-thickness curve, then there are theoretically uncountably many different characters possible. However, actual communication requires error-correction. What I mean by that is that if I'm writing something on a chalkboard with a piece of chalk, I might make the letter "S". You can recognize that I have written the letter "S". But when you get down to the details, no two people draw an "S" in exactly the same way. Even with one person, no two instances of drawings of an "S" are exactly like. So for you to be able to reliably recognize that I have drawn an "S", you have to be forgiving of small variations. So the letter "S" is not a single curve. If you imagine the mathematical space of all possible curves, the letter "S" would not be a point in that space, but a region, containing all possible variations of "S" that would still be recognized as the same letter.

If you have a bound on the size of a letter (you're not going to consider a letter that doesn't fit on a sheet of paper, for example) and you have a bound on the resolution (how different must two letters be and still be considered the same letter) then there are only finitely many different letters possible.

 

[Jimster41]

I was thinking the extreme case is more the alphabet of all defined characters.

Not sure I get the 0,1 argument. We are made of atoms but I don't think you'd argue they are the building blocks of our formal system or that the limited number of atom types makes the manifestations of physical reality finite. Also, all those 0's and 1's are made of atoms (and photons I guess) - same stuff as us. What's the difference?

I'm not disagreeing with your third paragraph. Kind of what I was trying to say. After thinking about my initial knew jerk reaction to the proposal that a map from the Reals onto some definitely countable formal system can help tame the Reals my observation was just that the infinite mutability of your zero thickness curve and our formal agreement on what is an "S" in the space of all shapes of that curve (learned by the way) are both important. I don't see a natural law that says where to draw a line between those two poles, between infinitely mutable symbols and a single forever fixed perfectly detailed alphabet. Though you could argue that because we naturally do draw that line there is a natural law that says a line must be drawn.

Your last paragraph outlines some potentially useful constraints to character building but aren't they sort of arbitrary. I mean who uses paper? And what if we moved in 100,000 years to a fully animated sign-language/emojii like symbol space (traded by our virtual avatars to and from our visual cortex's via our implanted vision augmentation systems of course) - what is the resolution of an animation?

Probably you already know BTW there is a whole category of Art that plays with characters - fonts and symbols. They are being pushed to the limit all the time. Some of it is pretty amazing in the way it plays with our methods of making and breaking agreements on recognizably useful symbols. I've seen some in the past that really blew my mind - when I was putting of off my engineering studies to wander the art library at my school. Just now I googled "Art of symbols and alphabets" /images. It definitely gives me pause to then say - oh, yeah, that's a countable set.

 

[Auto-Didact]

I was thinking the extreme case is more the alphabet of all defined characters.

Not sure I get the 0,1 argument. We are made of atoms but I don't think you'd argue they are the building blocks of our formal system or that the limited number of atom types makes the manifestations of physical reality finite. Also, all those 0's and 1's are made of atoms (and photons I guess) - same stuff as us. What's the difference?

I'm not disagreeing with your third paragraph. Kind of what I was trying to say. After thinking about my initial knew jerk reaction to the proposal that a map from the Reals onto some definitely countable formal system can help tame the Reals my observation was just that the infinite mutability of your zero thickness curve and our formal agreement on what is an "S" in the space of all shapes of that curve (learned by the way) are both important. I don't see a natural law that says where to draw a line between those two poles, between infinitely mutable symbols and a single forever fixed perfectly detailed alphabet. Though you could argue that because we naturally do draw that line there is a natural law that says a line must be drawn.

Your last paragraph outlines some potentially useful constraints to character building but aren't they sort of arbitrary. I mean who uses paper? And what if we moved in 100,000 years to a fully animated sign-language/emojii like symbol space (traded by our virtual avatars to and from our visual cortex's via our implanted vision augmentation systems of course) - what is the resolution of an animation?

Probably you already know BTW there is a whole category of Art that plays with characters - fonts and symbols. They are being pushed to the limit all the time. Some of it is pretty amazing in the way it plays with our methods of making and breaking agreements on recognizably useful symbols. I've seen some in the past that really blew my mind - when I was putting of off my engineering studies to wander the art library at my school. Just now I googled "Art of symbols and alphabets" /images. It definitely gives me pause to then say - oh, yeah, that's a countable set.

You are mixing up two distinct semiotic ideas: the sign which can be (dis)continuously varied, with the symbol which is discrete.

As others have already stated each alphabet is finite, ultimately because each letter is a sign which represents a discrete object: a symbol. Changing the signs - i.e. the representation of the symbols - a little bit, doesn't change the symbol itself.

There need not be a bijective relationship between representations and objects and generally speaking, there isn't. The meaning of the sign in the form of concatenated signs i.e. as words is of course strongly context dependent.

 

[A. Neumaier]

So what does “Definable” really mean?

In the present context it has a precise meaning. Pick a formal language in which you have a specification of what the concept of a real number means. Say HOL light. In this language you can create certain formulas that define particular real numbers. The collection of all these numbers is the set of definable reals in this language. It is countable.

Note that like all mathematical notions, this notion depends on the formal specification and hence the language used, in the same way as the notion of a set is dependent on its precise specification.

 

[Jimster41]

In the present context it has a precise meaning. Pick a formal language in which you have a specification of what the concept of a real number means. Say HOL light. In this language you can create certain formulas that define particular real numbers. The collection of all these numbers is the set of definable reals in this language. It is countable.

Note that like all mathematical notions, this notion depends on the formal specification and hence the language used, in the same way as the notion of a set is dependent on its precise specification.

HOL and ML are two things I had not heard of. Pretty interesting.

 

[*now*]

I missed much adding to the btsm bibliography thread and one of the papers I missed adding there seems relevant here too,

https://arxiv.org/abs/1902.03026

Natural discrete differential calculus in physics

We sharpen a recent observation by Tim Maudlin: differential calculus is a natural language for physics only if additional structure, like the definition of a Hodge dual or a metric, is given; but the discrete version of this calculus provides this additional structure for free.

 

[*now*]

Given that link, an earlier paper might also be worth mentioning here too,
https://arxiv.org/abs/1508.00001
Michelangelo's Stone: an Argument against Platonism in Mathematics

If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent from us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra.

 

[bhobba]

I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra.

That is philosophy which we do not discuss here. There are all sorts of views about eg conventionalism. On this forum we simply note that the fundamental science as far as we can tell today is written in the language of math. Why it's like that we, by forum rules, do not go into.

As far as the original question goes I have stayed out of it mostly, but mathematically QM is a generalized probability model. These have what are called pure states. In ordinary probability theory pure states are the outcomes of what probabilities are assigned to. There is no way to continuously go from one pure state to another, so if we want to model some situation and be able to do that to use the powerful methods of calculus you need to go to complex numbers. See:
https://www.scottaaronson.com/democritus/lec9.html
https://arxiv.org/abs/quant-ph/0101012
It is strange that nature is accommodating like that. Gell-Mann thinks its part of the self similarity we see as we delve deeper and deeper into different layers of nature:


Thanks
Bill

 

[SSequence]

There was some discussion regarding definability in the first few pages of this topic. I have posted a question about it (in-case someone might be interested).

 

[A. Neumaier]

If there is a "platonic world" M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

 

[microsansfil]

It is strange that nature is accommodating like that. Gell-Mann thinks its part of the self similarity we see as we delve deeper and deeper into different layers of nature:

Why would the world we perceive through our consciences be the world as it is in itself ? And so why would "aliens" with totally different sensory experiences than ours (color, sound, touch,...) build the same theoretical models as ours ?

/Patrick

 

[Auto-Didact]

Why would the world we perceive through our consciences be the world as it is in itself ? And so why would "aliens" with totally different sensory experiences than ours (color, sound, touch,...) build the same theoretical models as ours ?

/Patrick

Even with different sensors, most or at least some of the objects that they are detecting will have the same underlying characteristics and dynamics; this is what is captured by our mathematical models.

There are very strong arguments based on mathematical theorems, based on the unity of mathematics, as well as based on metrological constraints, that these models will be therefore quite similar, possibly approximate subsets of each other, or even different approximations from different angles to the same underlying mathematical structure.

Any key mathematical differences at a superficial stage of theorisation will be based on the different idealizations of the different assumed key mathematical properties which underly the differently constructed/evolved methods of mathematics.

 

[microsansfil]

Even with different sensors, most or at least some of the objects that they are detecting will have the same underlying characteristics and dynamics; this is what is captured by our mathematical models.

I'm not talking about different sensors, but different first-person experiences. First-person experiences by which we perceive the world before any act of objectification, by inter-subjectivity between us, as human beings.

For example about color.

http://www.ler.esalq.usp.br/aulas/lce1302/visao_aves.pdf : It is true, as many youngsters learn in school, that objects absorb some wavelengths of light and reflect the rest and that the colors we perceive “in” objects relate to the wavelengths of the reflected light. But color is not actually a property of light or of objects that reflect light. It is a sensation that arises within the brain.

you can explain to Tommy Edison, who has been blind since birth, the theory behind the operation of a color sensor, He still won't be what a color is, i.e living the experience of the feeling that colour is.

in addition, it should be noted that Chromesthesia or sound-to-color synesthesia is a type of synesthesia in which heard sounds stimuli automatically and involuntarily evoke an experience of color. And therefore not to confuse the cause with the effect. We, as humans being, have no choice but to start from the effects (first-person experiences) and seek to do "reverse engineering".

/Patrick

 

[Auto-Didact]

I'm not talking about different sensors, but different first-person experiences. First-person experiences by which we perceive the world before any act of objectification, by inter-subjectivity between us, as human beings.

That is not a question about physics but a question about physiology, specifically the distinction of sensation and perception. It is a widely studied phenomenon, with much more known than just thought experiments like Mary's room, e.g. colour blind people already satisfy all the criteria.

Even worse, there are people in Africa who can easily distinguish far more shades of green than the rest of the world can due to having grown up around many trees, even causing them to have names for those different greens. We are all colour blind to their many greens.

This hardly invalidates any physics, since in terms of physiology it can be explained more or less; the difficult part is to accurately reduce this purely to biophysics and natural selection, but there is little doubt this can be done apart from an explanation for consciousness itself.

 

[microsansfil]

This hardly invalidates any physics

There is no question in my remarks of seeking to invalidate physics I use every day indirectly. Just to become aware that consciousness is the starting point of any inquiry.

I don't remember who wrote this: Lived experience is where we start from and where all must link back to, like a guiding thread.

Human Being can question his beliefs, but not the one in which he believes deeply.

/Patrick

 

[Jimster41]

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

I think he was pretty much just quoting Rovelli... in the paper... so. Cute as the Gell-Mann Ted Talk is Rovelli seems to disagree pretty directly, giving examples of how basic geometric axioms we take to be universal aren't that obviously so.

Rovelli is a bit of a whacko tho.

 

[*now*]

Your argument is not cogent.
You can ask the same question about the physical world, and find that it is far too large but most interesting, because the value is in the selection, not in the totality!

Although Bhoba said this sort of discussion isn’t wanted, just there could be some confusion. The words I quoted are the abstract of the paper I linked in order to provide a wider impression than just linking the previous paper. So, the argument belongs with that second paper, not with me. Also, given that possible misunderstanding, I’m not sure if the comment concerns more of the sentence and argument of the second paper linked, which I think argues that an independent world of mathematical truths may be reduced to something “trivial, or contradictory”, and I think giving an alternative example.

I think he was pretty much just quoting Rovelli... in the paper... so. Cute as the Gell-Mann Ted Talk is Rovelli seems to disagree pretty directly, giving examples of how basic geometric axioms we take to be universal aren't that obviously so...

I’m not sure about disagreement, Jimster, there seems to be at least some agreement between the essay and the clip like solidity of mathematical theorems and a similar lack of requirement for humans/mind, and there might not be further disagreement either. Although the notions involved vary somewhat, Gell-Mann’s definition of fundamental law - a law unifying fundamental particles and forces, “is not a theory of everything”. It would apply to some but wouldn’t apply to the many chance outcomes that occur. So, it continues, assuming the law exists it only applies to some part excluding much that provides a huge amount of information. As notions of universality here seem to be with respect to some part and not all, perhaps there isn’t disagreement with the paper.