I've read that complex numbers are a convenience . . . — Art48

True enough, but the value of complex analysis lies far more in the concept of holomorphic (or analytic) functions, where differentiation is over areas and not simply points in the complex plane. For example, when studying work one hopes for conservative force fields in which the amount of work required in going from one point to another point is not a function of the path taken. This is essentially a holomorphic field. However, it's usually a topic in the reals involving gradients, so is of little importance here. Just casting about I'm afraid.

I have read that Roger Penrose asserts a stronger position in which complex numbers and functions are somehow fundamental in reality.

But, I'm not a physicist. However, in going to 2D Euclidean space you lose multiplication of "points" as well as division.

I dabble in complex analysis all the time, but @apokrisis knows much more about the physics of this topic than I do.

All I know is this:

Where i = $\sqrt{-1}$ and all of the following takes place in the complex plane

1 Start with 1
2. 1 × i = i (90 degree anticlockwise rotation)
3. 1 × i × i = -1 (another 90 degrees anticlockwise rotation for a total of 180 degrees, same direction)
4. 1 × i × i × i = -i (another 90 degrees anticlockwise rotation for a total of 270 degrees, same direction)
5. 1 × i × i × i × i = 1 (another 90 degrees anticlockwise rotation for a total of 360 degrees, same direction. We're back to square 1)

Clyclical transformation (?), much like sine and cosine.

Question: are complex numbers required for QM, or could Schrodinger's equation be reformulated exclusively in terms of sin and cos? — Art48

Recent experiments seem to show that complex numbers are required for QM.

Two independent studies demonstrate that a formulation of quantum mechanics involving complex rather than real numbers is necessary to reproduce experimental results.
...
[Renou and colleagues] considered two theories that are both based on the postulates of quantum mechanics, but one uses a complex Hilbert space, as in the traditional formulation, while the other uses a real space. They then devised Bell-like experiments that could prove the inadequacy of the real theory. In their theorized experiments, two independent sources distribute entangled qubits in a quantum network configuration, while causally independent measurements on the nodes can reveal quantum correlations that do not admit any real quantum representation.

Chen and colleagues and Li and colleagues now provide the experimental demonstration of Renou and co-workers’ proposal in two different physical platforms.
...
Despite the difficulties inherent in each implementation, both experiments deliver compelling results. Impressively, they beat the score of real theory by many standard deviations (by 43 sigma and 4.5 sigma for Chen’s and Li’s experiments, respectively), providing convincing proof that complex numbers are needed to describe the experiments. — Quantum Mechanics Must Be Complex - Alessio Avella, INRIM (January 24, 2022)

Thanks for the reference. At the end, the author adds:

We must be careful, however, in assessing the implications of these results. One might be tempted to conclude that complex numbers are indispensable to describe the physical reality of the Universe. However, this conclusion is true only if we accept the standard framework of quantum mechanics, which is based on several postulates. As Renou and his co-workers point out, these results would not be applicable to alternative formulations of quantum mechanics, such as Bohmian mechanics, which are based on different postulates. Therefore, these results could stimulate attempts to go beyond the standard formalism of quantum mechanics, which, despite great successes in predicting experimental results, is often considered inadequate from an interpretative point of view

The Avella paper is a great resource. Thanks everyone.

Well, maybe this is not a Philosophy forum after all ...

At the end, the author adds:

"We must be careful, however, in assessing the implications of these results. One might be tempted to conclude that complex numbers are indispensable to describe the physical reality of the Universe. However, this conclusion is true only if we accept the standard framework of quantum mechanics, which is based on several postulates. As Renou and his co-workers point out, these results would not be applicable to alternative formulations of quantum mechanics, such as Bohmian mechanics, which are based on different postulates. Therefore, these results could stimulate attempts to go beyond the standard formalism of quantum mechanics, which, despite great successes in predicting experimental results, is often considered inadequate from an interpretative point of view" — jgill

Yes, though those alternative formulations are non-local (i.e., require faster-than-light communication for measurements) and therefore don't integrate nicely with relativity. What these experiments demonstrate is that a local-relativistic universe must be based on complex-valued amplitudes.

The Avella paper is a great resource. Thanks everyone. — Art48

:up:

Well, maybe this is not a Philosophy forum after all ... — Alkis Piskas

It's philosophy of physics - specifically the interpretation of quantum mechanics, the nature of space and time, and the relation of mathematics to the universe.

Philosophy of physics is the study of the fundamental, philosophical questions underlying modern physics, the study of matter and energy and how they interact. The main questions concern the nature of space and time, atoms and atomism. Also included are the predictions of cosmology, the interpretation of quantum mechanics, the foundations of statistical mechanics, causality, determinism, and the nature of physical laws.[80] Classically, several of these questions were studied as part of metaphysics (for example, those about causality, determinism, and space and time). — Philosophy of physics - Wikipedia

What these experiments demonstrate is that a local-relativistic universe must be based on complex-valued amplitudes — Andrew M

Feynman's path integrals involve ${{e}^{i\theta}}$ heavily and I suspect that the simple evaluation of products of terms like these: ${{e}^{it}}\cdot {{e}^{is}}={{e}^{i(t+s)}}$ compared with their counterparts in sines and cosines plays a huge role in application of theory.

In his "QED" he explains that the "arrows" he describes (vectors):

Arrows on a plane can be "added" by putting the head of one arrow on the tail of another, or "multiplied" by successive turns and shrinks.

Complex numbers and their properties facilitate this.

Again, this is an almost trivial argument for convenience rather than necessity.

(If you are a real quantum physicist, speak up and clarify this issue :chin: )

A Bit More Readable Version . . .

It's a much more complicated subject than I had imagined.

A Bit More Readable Version . . .

It's a much more complicated subject than I had imagined. — jgill

Yes, me too.

Complex numbers and their properties facilitate this.

Again, this is an almost trivial argument for convenience rather than necessity. — jgill

Yes, so Chen and Li's experiments are an argument for necessity. The gist is that an entanglement swapping protocol is followed such that two observers (Alice and Charlie) end up with one each of a pair of entangled qubits. The subsequent measurements on those qubits (as confirmed by Chen and Li's experiments) match the statistical predictions of standard quantum mechanics.

However the entanglement isn't fully swapped under real-valued QM, and so it makes different predictions to standard QM (and fails to be confirmed by experiment). As the Physics Today article says:

Bob then makes a joint measurement on his two qubits, with four possible outcomes. In complex-valued quantum mechanics, that measurement halves the number of dimensions of the system and cuts the number of entangled pairs from two to one. That is, it transfers the entanglement to qubits A and C. But in a real-valued formulation, Bob’s four-outcome measurement doesn’t cut the dimensionality by enough to fully swap the entanglement—he’d need an eight-outcome measurement to do that—so qubits A and C don’t end up fully entangled. — Does quantum mechanics need imaginary numbers? - Physics Today

— Does quantum mechanics need imaginary numbers? - Physics Today https://physicstoday.scitation.org/doi/10.1063/PT.3.4955 — Andrew M

Final paragraph of the article is:
Neither Pan’s nor Fan’s group has yet closed the loopholes in their experiments. Technically, therefore, the jury is still out on whether real or complex numbers are the better descriptors of the quantum world. Still, it seems likely that future students of quantum mechanics will have no choice but to continue to grapple with the mathematics of imaginary numbers.

Seems like it's still an open question (with evidence leaning towards that complex numbers are necessary).

Update: A YouTube clip that addresses the question.
Do Complex Numbers Exist? (Sabine Hossenfelder)
https://www.youtube.com/watch?v=ALc8CBYOfkw&t=615s

Final paragraph of the article is:
Neither Pan’s nor Fan’s group has yet closed the loopholes in their experiments. — Art48

The loopholes have been closed in a more recent experiment. The remaining aim is to close all the loopholes simultaneously.

By ensuring events in the experiment happened quickly and far enough apart, the researchers say in the paper that, given reasonable assumptions, they have closed three loopholes: locality, independent source, and measurement independence.
...
The new experiment left open the detection loophole, which was closed in Lu and his colleague’s previous experiment. Lu says the team hopes to develop new techniques to enable an experiment that simultaneously closes the loopholes. — New Experiment Suggests Imaginary Numbers Must Be Part of Real Quantum Physics - APS

It's philosophy of physics - specifically the interpretation of quantum mechanics, the nature of space and time, and the relation of mathematics to the universe. — Andrew M

OK. There's also a "Philosophy of kitchen", a "Philosophy of animals", ... in short, a "Philosophy of Everything".

Since you referred to Wikipedia, here is what it says about "Philosophy":

"Philosophy is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language."
(https://en.wikipedia.org/wiki/Philosophy)

You can also find in there the branches of philosophy: Aesthetics, Ethics, Epistemology, Metaphysics, Logic, Mind and language, Philosophy of science, Political philosophy, Philosophy of religion, Metaphilosophy.

Funny, I've seen the argument that even real numbers may need to be excluded from QM.

The first paper I recall with this argument was Paul Davies', "Universe From Bit." The argument runs like this:

1. Real numbers with infinite decimals require infinite bits to encode.
2. Our universe appears to be made up of a finite number of bits.
3.If the universe "computes itself," as many have suggested, then beam splitter experiments with a very large number of photons would have to encode numbers so small (with so many digits) that doing so would appear to require more information then exists in the entire universe.
4. If the "universe computes the universe," (Landauer) then the reals aren't going to be real. And with very large experiments, you might even be able to test this proposition.
5. The general acceptance of inviolable Platonic "laws of physics" that are invariant and do not depend on physical reality is the result of the religious inclinations of early pioneers in physics. That is, the modern idea of physical laws, now embraced by many atheists and now the "conservative" position in physics, is the result of the religious instincts of individuals such as Newton.

Notably, theories of quantum gravity also tend to look to a finitist view of the universe.

However, and I may be totally over my head here, it seems like the objections to the reality of the Reals had to do with their infinite/infinitesimal nature. So, would it be possible to have a physics that requires complex numbers, i, but not the Reals?

The issues discussed in this thread primarily concern the necessity of complex valued integers and rationals in relation to entangled quantum states, their interactions and the Born rule.

The issues you raise concerning the existence, usefulness and intelligibility of the continuum of reals as part of the foundations of QM is valid albeit tangential to that discussion. Furthermore, the issues you raise are avoided in quantum computer science that is grounded in alternative mathematical foundations for QM that are constructive, computable and usually finite, such as Categorical Quantum Mechanics that is the underlying foundation for the ZX calculus. Those theories retain the essential underlying logical properties of complex Hilbert Spaces that are necessary for formalising quantum computing applications, including the conjugate transpose operator and unitary and self-adjoint operators, but without retaining the continuum of reals and the non-constructive propositions of complex Hilbert spaces.

So, would it be possible to have a physics that requires complex numbers, i, but not the Reals? — Count Timothy von Icarus

IMO, no. For example, 5i x 6i = -30. You lose multiplication and division.

Furthermore, the issues you raise are avoided in quantum computer science that is grounded in alternative mathematical foundations for QM that are constructive, computable and usually finite, such as Categorical Quantum Mechanics that is the underlying foundation for the ZX calculus. — sime

I continue to learn things in my old age . . . https://en.wikipedia.org/wiki/ZX-calculus

But I have nothing to do with category theory in general. Too lofty and fluffy for me. :roll: