Saturday, March 06, 2021

Do Complex Numbers Exist?

[This is a transcript of the video embedded below.]

When the world seems particularly crazy, I like looking into niche-controversies. A case where the nerds argue passionately over something that no one knew was controversial in the first place. In this video, I want to pick up one of these super-niche nerd fights: Are complex numbers necessary to describe the world as we observe it? Do they exist? Or are they just a mathematical convenience? That’s what we’ll talk about today.

So the recent controversy broke out when a paper appeared on the preprint server with the title “Quantum physics needs complex numbers”. The paper contains a proof for the claim in the title, in response to an earlier claim that one can do without the complex numbers.

What happened next is that the computer scientist Scott Aaronson wrote a blogpost in which he called the paper “striking”. But the responses were, well, not very enthusiastic. They ranged from “why fuss about it” to “bullshit” to “it’s missing the point.”

We’ll look at the paper in a moment, but first I will briefly summarize what we’re even talking about, so that no one’s left behind.

The Math of Complex Numbers

You probably remember from school that complex numbers are what you need to solve equations like x squared equals minus 1. You can’t solve that equation with the real numbers that we are used to. Real numbers are numbers that can have infinitely many digits after the decimal point, like square root of 2 and π, but they also include integers and fractions and so on. You can’t solve this equation with real numbers because they’ll always square to a positive number. If you want to solve equations like this, you therefore introduce a new number, usually denoted “i” with the property that it squares to -1.

Interestingly enough, just giving a name to the solution of this one equation and adding it to the set of real numbers turns out to be sufficient to make all algebraic equations solvable. Doesn’t matter how long or how complicated the equation, you can always write all their solutions as a+ib, where a and b are real numbers. 

Fun fact: This doesn’t work for numbers that have infinitely many digits before the point. Yes, that’s a thing, they’re called p-adic numbers. Maybe we’ll talk about this some other time.

Complex numbers are now all numbers of the type a plus I time b, where a and b are real numbers. “a” is called the “real” part, and “b” the “imaginary” part of the complex number. Complex numbers are frequently drawn in a plane, called the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. i itself is by convention in the upper half of the complex plane. But this looks the same as if you draw a map on a grid and name each point with two real numbers. Doesn’t this mean that the complex numbers are just a two-dimensional real vector space?

No, they’re not. And that’s because complex numbers multiply by a particular rule that you can work out by taking into account that the square of i is minus 1. Two complex numbers can be added like they were vectors, but the multiplication law makes them different. Complex number are, to use the mathematical term, a “field”, like the real numbers. They have a rule both for addition AND for multiplication. They are not just like that two-dimensional grid.

The Physics of Complex Numbers

We use complex numbers in physics all the time because they’re extremely useful. There useful for many reasons, but the major reason is this. If you take any real number, let’s call it α, multiply it with I, and put it into an exponential function, you get exp(Iα). In the complex plane, this number, exp(Iα), always lies on a circle of radius one around zero. And if you increase α, you’ll go around that circle. Now, if you look only at the real or only at the imaginary part of that circular motion, you’ll get an oscillation. And indeed, this exponential function is a sum of a cosine and I times a sine function.

Here’s the thing. If you multiply two of these complex exponentials say, one with α and one with β, you can just add the exponents. But if you multiply two cosines or a sine with a cosine… that’s a mess. You don’t want to do that. That’s why, in physics, we do the calculation with the complex numbers, and then, at the very end, we take either the real or the imaginary part. Especially when we describe electromagnetic radiation, we have to deal with a lot of oscillations, and complex numbers come in very handy.

But we don’t have to use them. In most cases we could do the calculation with only real numbers. It’s just cumbersome. With the exception of quantum mechanics, to which we’ll get in a moment, the complex numbers are not necessary.

And, as I have explained in an earlier video, it’s only if a mathematical structure is actually necessary to describe observations that we can say they “exist” in a scientifically meaningful way. For the complex numbers in non-quantum physics that’s not the case. They’re not necessary.

So, as long as you ignore quantum mechanics, you can think of complex numbers as a mathematical tool, and you have no reason to think they physically exist. Let’s then talk about quantum mechanics.

Complex Numbers in Quantum Mechanics

In quantum mechanics, we work with wave-function, usually denoted Ψ, which are complex valued, and the equation that tells us what the wave-function does is the Schrödinger equation. It looks like this. You’ll see immediately, there’s an “i” in this equation, which is why the wave-function has to be complex valued.

However, you can of course take the wave-function and this equation apart into a real and an imaginary part. Indeed, one often does that, if one solves the equation numerically. And I remind you, that both the real and the imaginary part of a complex number are real numbers. Now, if we calculate a prediction for a measurement outcome in quantum mechanics, then that measurement outcome will also always be a real number. So, it looks like you can get rid of the complex numbers in quantum mechanics, by splitting the equation into a real and imaginary part, and that’ll never make a difference for the result of the calculation.

This finally brings us to the paper I mentioned in the beginning. What I just said about decomposing the Schrödinger equation is of course correct, but that’s not what they looked at in the paper, that would be rather lame.

Instead they ask what happens with the wave-function if you have a system that is composed of several parts, in the simplest case that would be several particles. In normal quantum mechanics, each of these particles has a wave-function that’s complex-valued, and from these we construct a wave-function for all the particles together, which is also complex-valued. Just what this wave-function looks like depends on which particle is entangled with which. If two particles are entangled, this means their properties are correlated, and we know experimentally that this entanglement-correlation is stronger than what you can do without quantum theory.

The question which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.

I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!

Well, kind of. Here is the fineprint: This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.

Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.

Or, of course, it might be that that the experiment will not agree with the predictions of quantum mechanics, which would be the most exciting of all possible outcomes. Either way, I am sure that this is a topic we will hear about again.

128 comments:

  1. In the YouTube comments for this video, people are discussing and debating whether numbers are real, and if so, what makes them real. To me, it's like something existing in a dimension that consists of the consciousness of humanity that we project out into the world via language and symbols.
    I'd like to see what other commenters think.

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  3. Complex numbers don't exist. For that matter, natural numbers don't exist. They are merely useful fictions.

    I say this because I am a mathematical fictionalist. However, there are also many mathematical platonists who would disagree with me.

    Honestly, it doesn't make any important difference. Platonists and fictionalists do their mathematics in pretty much the same way. Their philosophical differences don't actually affect the mathematics.

    And then there's the Quine - Putnam indispensibility thesis, which argues for platonism to explain why mathematics works so well in physics. However, I happen to think that fictionalism makes more sense of the role of mathematics in physics.

    So it is really much ado about nothing. Go with whatever makes most sense to you.

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  4. This is a very interesting topic, and I am looking forward for the peer-reviewed release.

    I would like to add one issue of classical physics which is often neglected and which could not exist without imaginary numbers: The spacetime metric of general relativity - according to MTW page 305:

    delta s squared = minus delta tau squared

    A positive square equals a negative square (!). Many authors are supposing that the spacetime interval is a square - and they are not extracting the root - in order to avoid a result which would mean that a real term equals an imaginary term. But whatever is the sign convention of the different authors (since Minkowski), it is impossible to formulate the spacetime metric without negative squares.

    The current theories of quantum gravity are wondering why curved spacetime is not compatible with quantum mechanics, but perhaps it is rather our current concept of spacetime (implying the grotesque equation "real = imaginary") which needs to be reviewed, and perhaps this could be an interesting topic to talk about.

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    1. The "delta s squared = minus delta tau squared" sign has to do with the metric signature. With signature [-,+,+,+] this happens. but not with [+,-,-,-].

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    2. Lawrence, negative squares are required whatever is the author you are considering. Minkowski ("Raum und Zeit") followed the signature [+,-,-,-], and he had to distinguish between timelike and spacelike because his timelike metric became imaginary for spacelike intervals. By consequence, you find in section 3 the strange equation:

      Minus F = [spacelike metric] = k squared

      This principle of a twofold metric was adopted by MTW, although they used the opposite signature.

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    3. I suppose somehow, I am not seeing the mystery you point out. The [+,-,-,-] signature leads to the line element

      ds^2 = g_{00}c^2dt^2 - dΣ_3^2

      where g_{00} is the time-time metric element and dΣ_3^2 is the spatial metric interval. We then have ds^2 = c^2dτ^2 = c^2g_{00}dt^2 for stationary clocks in this metric and the infinitesimal spatial distance at any instance of coordinate time, eg dt = 0, as dD^2 = dΣ_3^2. One has to remember the spacetime line element subtracts the spatial line element. In the case of the signature [-,+,+,+] we now have the line element

      ds^2 = -g_{00}c^2dt^2 + dΣ_3^2

      We then have ds^2 =- c^2dτ^2 = -c^2g_{00}dt^2 for stationary clocks in this metric and the infinitesimal spatial distance at any instance of coordinate time, eg dt = 0, as dD^2 = dΣ_3^2. In this case the spacetime line element subtracts the coordinate time part.

      There is nothing really deep here.

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  5. Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place.

    Again, please let's not 'forget' the other camp - those of us who think there's nothing wrong with QM but are very much interested in [the reasons for*] its particular mathematical structure.

    * Including, for example, Hardy's 5th "reasonable axiom".

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    1. There is of nothing wrong with quantum mechanics. There is something odd though with how quantum mechanics and macroscopic physics of statistical mechanics and classical mechanics work together.

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    2. From the "QM is just probabilistic mechanics" perspective there's not really any oddness even there. (At worst there's the "small measurement problem" and "small" is the operative word.)

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    3. @ Paul Hayes: It is more than just a small problem. QM is really a dynamics of amplitudes A and probabilities are P = A^*A or |A}^2. Classical probability is also an L^1 measure theory, while QM is L^2.

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    4. Lawrence, it really isn't more than just a small problem for the class of interpretation I'm talking about (see e.g. Jeff Bub's "The Measurement Problem from the Perspective of an Information-Theoretic Interpretation of Quantum Mechanics").

      Classical - Kolmogorovian - probability is a subset of (C*-)algebraic "quantum" probability. The classical theory is just what you get when the algebra of observables / random variables is commutative, and in this formalism the classical probability state representation space is L^2 too.

      This is what I've been alluding to when saying e.g. that "QM is just probabilistic mechanics", not the naive and archaic approach to QM which only knows about the Kolmogorovian model of probability (and from which perspective such statements will seem mysterious).

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  6. Well, there is no technical problem to work with complex numbers. And I do not really understand why these numbers should be the origin of controversal debates. I would also note that the historical starting point justifying the creation of i is related to a procedure looking solutions for polynomials of degree three. See the history around the Tartaglia Cardan method... No relation with the quantum theory at this time !

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  7. Isn't the assumption of "independent sources" the most obvious weakness in the proof, as it is with many other no-go theorems in QM? Everything originated in the Big Bang, so all matter has a common origin and so there is no such thing as an independent source. This seemingly leads right back to superdeterminism.

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    1. Sandro Magi,
      I wonder about this, actually took "independent sources" as a breath of fresh air and certainly something critical to get clear about. Reading about decoherence, it seems that it pivots on the condition of there being a definite phase relationship between subsystems. In my preferred universe, composite entities would manifest their own unique phase pattern through their internal dynamics. One gets some sense of that here:

      Decoherence-Free Subspaces and Subsystems – https://arxiv.org/abs/quant-ph/0301032

      If one must make effort to engineer decoherence-free subspaces and subsystems in order to do quantum computing, it follows that decoherence between subsystems in the normative condition.
      Trust me to perhaps have it all wrong.

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    2. Sandro Magi,

      Correction: “…it follows that decoherence between subsystems is the normative condition.”

      And, I failed to quote the paper, “Decoherence is the phenomenon of non-unitary dynamics that arises as a consequence of coupling between a system and its environment.”

      I read this to mean that, in interactions between subsystems lacking phase coherence the universe actually has to figure out what happens next, determinism is thereby punctuated, maybe with just a comma, but punctuated. With similar caveat.

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  8. The question as to what is meant by the actual existence of numbers is a difficult one.

    We can just as well ask what is meant by the existence of manifolds, of vector spaces etc, etc. The list goes on and on.

    One may as well say what do we mean by mathematical knowledge tout court.

    I would suggest it comes down to what we mean by the existence of 'abstract' things. These aren't limited to just numbers. How about words or thoughts?

    Can you point to the word 'and'?

    Or to the thought 'green'?

    Nominalism suggests that we merely think of the number three, as the class of all things that are three in number. In other words, it is an equivalence class of all three things in the physical and mental world. Personally, I'm not very taken by this.

    This leaves the ontological attitude, where numbers actually exist. This relates to Plato's notion of the realm of forms and which is only reachable by the intellect. It's important to realise that he doesn't limit himself to numbers or mathematical forms as ideas. More over arching forms would be the idea or form of Justice, that of the Good and of the One.

    Plato even describes the creation of the world as an intermingling of nous (intellect) and ananke (neccessity). Today, the opposite move is very much in evidence where the realm of neccessity is declared to be all that there is, with even human minds reduced to computing machines.

    Probably the last major Platonist in the Western world was Hegel with his emanationist philosophy predicated on the notion of Aufhebun, the enfolding of a motion and an opposite motion in a higher synthesis. It's pretty much symbolised by the famous picture of ying-yang that encapsulates a similar notion in Taoist philosophy, that of a white and black teardrop changing into each other.

    Niels Bohr, when he was knighted in 1947, incorporated this Ying-Yang symbol in his coat-of-arms to symbolise wave-particle duality. His motto was: Contraria Sunt Complementa, or Contraries Are Complementary.

    Interestingly, Hegel synthesised the notion of the One and the not One in his philosophy. Being and non-Being, he declared, were simply opposite sides of the same coin.

    What has physics found in its analytic enquiry into the being of matter? Matter which for us is signified by the quality of mass? It's ontological quiddity, it's thisness and isness disappears. The Higgs shows that matter is not an 'is' or a 'thos' but part of a relational event.

    This, by the way, is not a million miles away from the Buddhist notion of Pratitsayamuda, or dependent origination. The notion that ontological existence is relational - and hence arises.

    Personally, I'm curious as to whether Hegel was influenced by Eastern philosophy when he integregated 'nothingness' into the 'substance' of Western philosophy. After all, Liebniz was known to be a Sinophile.

    It would be interesting to find out why and how Buddhist philosophy was led to this. No doubt, that will come in time. It's not easy, after all, to grapple with philosophy in a different idiom, different language and different world view. It would mean taking Buddhist philosophy seriously and not some kind of interesting or not interesting medieval or feudal hangover, as the non-commutative geometer, Mathilde Marcolli thought in her Heart in the Machine.




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    1. "Can you point to the word 'and'?"

      and ←


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    2. See, that's why I asked what people think. It's fascinating to me what people come up with to describe concepts, and what meaning and even spirituality they are imbued with.

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    3. Interesting reflection. Although, I haven't read much of Plato, it was peculiar that what he called idea in Republic is not something that is usually meant by idea (i.e. perception-grounded reduction to shed 'rough' details), but what is usually covered by another word - abstraction. Not as in a generalized concept, but as in an invariant relation.

      Which also hints at another intuition to approach numbers - as relation between objects (scalars). Then number represents some specific relation, where relation has a priority over description and measure. Not just as a "basket of 3-ness" as a concept, i.e. description is set and detached objects are aggregated according to it.

      In that way numbers and other mathematical descriptions can be seen as abstractions, or the most compressed and accurate representations of relations discovered (through observation or contemplation) in nature. Hence, in a sense... the most 'real' of all representations, transferring the essence.

      Korzybski attempted to integrate mathematical thinking as the basis for sane relations and the development of proper language which may help to overcome our inner inconsistencies (developed a method akin to cognitive therapy, successfully tried on PTSD people, no proper trials). He postulated that any knowledge can be approached only through relations and processes, which will form the structure of that knowledge. As no object may be seen on its own but only in relations, that will help, as he also postulated, to readjust our cognitive model to more sane thinking. He called it the consciousness of abstracting.

      And since you've mentioned Pratityasamutpada it should look familiar as it's indeed similar. The difference is that Korzybski's attempted to integrate scientific thinking with everyday knowledge in order to form the theory of sanity. And Pratityasamutpada is a sort of empirical model invented to structure life of an already selected group of people, who has chosen a life of asceticism and contemplation, and so developed as to provide them with suitable methods/objects for contemplation.

      It's interesting that you've mentioned Buddhism from a philosophical perspective (not religious, i.e. ritualistic), which is rarely seen as such. There were few people who approached it so (properly including the context, from the perspective of external observer reflecting another thinking and its own thinking as another context). I think the idea of nothingness (shunya), among other things, was an invention of early ascetics to relativize Brahmanic absolutes (self, eternity, constancy) in order to de-program many recluses indoctrinated in this knowledge. As many symbols (impermanence, dissonance, non-self) were created as guides from the perspective of contemplation (dhyana) and, most likely, from the state of deep contemplation (ASC from the perspective of neurophysiology).

      It's also interesting that you've mentioned such opposites as Being and Non-Being, Ying and Yang, etc. in that context. As that is exactly one of the instructions for contemplation in early suttas (one of basic texts on instructions to ascetics) - to start contemplating the opposites in order to... neutralize both of them, so that eventually conceptual thinking releases its grasp and the duality is pacified (basically, slowing or shutting down running processes in thinking), aka the Middle Way. It looks like primarily it meant to be a practice manual, not a philosophical text. Yet, it does not deny symbols' depth.

      It looks like some objects in philosophy compress that neutralizing capacity which is beneficial to maintenance of our brains (to release stuff), when contemplated long enough. And some good philosophers have discovered the best indications which are some sort of invariant. I haven't read Hegel, but it seems that some insights may well be discovered independently.

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    4. @Steven Evans:

      Steven Evans <-

      Is that pointing to Steven Evans or to the word that references Steven Evans?

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    5. @Vadim:

      I hadn't heard of Korzybski, so I looked him up on Wikipedia. I can see why you say he integrates scientific thinking in real life but, for me, personally speaking, he also limits by excluding much else, like religion and spirituality. I'd say, again from my own point of view, that this appears similar to the school of analytic philosophy which also exluded a great deal by their operational idea of what constitutes meaning. It was all part of the positivist philosophy that was all the rage at that time.

      The phrase that I'm thinking about from Hegel is what he begins with his Science of Logic where he equates non-Being and Being:

      >Nothing, pure nothing. It is simply equality with itself,absence of all determination and content, undifferentiatedness in itself. Insofar as thinking or intuiting can be mentioned here, it counts as a distinction as to whether something or nothing is intuited or thought. To intuit nothing or think nothing has, therefore, a meaning. Both are distinguished and therefore nothing is (exists), in our intuiting or thinking. Or rather it is empty intuition and thought itself, and the same empty intuition or thought as pure being.

      >Nothing is, therefore, the same determination, or rather absence of determination, and thus altogether the same as pure being.

      It's difficult language, but then again he's trying to intuit what cannot really be thought. Language cracks and bends under such pressure. Nevertheless, he's given a good shot at it.

      Given what he says, it's not surprising to see parallels between Hegels thought here and the philosophical Buddhist position of sunyata (nothingness). There nothingness, it must be stressed, is not nothingness per se. It is actually something as Hegel points out. It's also, I think, has parallels in Aristotles notion of potentiality as the underlying substrate of reality, or the apeiron (the boundless). The apeiron, the unlimited, the boundless, has played a minor role in Western philosophy but a major one in Buddhist. It would be certainly interesting to discover whether Hegel was influenced by Buddhist philosophy. It wouldn't be suprising if that was the case. On the other hand, it could as well be independently discovered, after all it has, as I'vecalreafy poubted out, played a minor role in Western philosophy. But it is worrh noting that he does mention Buddhism in his lectures on the philosophy of religion andcso he is acquainted with it. It's taken from this that he see's Buddism as 'nihilistic' because they think on 'nothing'. But this appears to be because of a very literal minded reading of nothing. The nothingness of Buddhism is not the very literal nothing of Parmenides who pointed out it cannot be thought. Like Hegel points out, it's merely the obverse of ontological substance, that is being. Given this, I can't see the characterisation of Buddhism as being nihilistic as being particularly correct. It's rather like taking to heart the modern physical conception of matter as being mostly energy or force rather than matter and then thinking that reality does not exist - when it most assuredly does.

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    6. Mozibur5:39 AM, March 08, 2021

      "Steven Evans <-
      Is that pointing to Steven Evans or to the word that references Steven Evans?"

      It's pointing to a member of the equivalence class of my name, which consists of 2 words. "And" is a member of the equivalence class of the word "and". Words have meanings or functions.
      Whether mathematical concepts are "real" comes down to whether the mathematical concept is essential in describing the particular aspect of physical observation in question. The point of the blog post is that complex numbers may be essential to describing a certain aspect of reality viewed at a certain level of precision. If this is the case, there would be no way of describing that aspect of reality at that level of precision without using the complex numbers or a mathematically isomorphic structure - so the complex numbers.

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    7. "It's difficult language, but then again he's trying to intuit what cannot really be thought."
      Nah, it's an alright indication (as far as SVO languages can go). A Buddhist philosopher may've only nitpicked around his attempts to 'ontologize' undifferentiatedness. The line of argument on example of determination would be to indicate that the absence of determination is not really an absence of anything of substance, but an indication of a cessation of arising, which is just used as an antidote to those who are under the spell (trance) of determination. Hence, their ontological status (so to speak) is not on the same footing. Like a scaffolding or a raft metaphor, "You've used the raft to get through? Good. Now, break it! Why do you need to carry it anymore."
      So it's basically an instruction for a beginning of contemplation for ascetics. Master-meditator would've probably told Hegel, "Good, now you've understood the instruction. Go and meditate for 10 years!" :-)

      "nothingness, it must be stressed, is not nothingness per se"
      Yeah, the same difficulty of indication. Some ascetic traditions approached directly (Upanishads, some Buddhists, Jains), some from negation (Buddhists). It's interesting to ponder about their approaches from the perspective of functional completeness in logic, then Upanishadic tradition (neti,neti) can be considered as NAND operator, and Buddhist tradition - especially, Nagarjuna's Mulamadhyamakakarika - as NOR operator.

      "'nihilistic' because they think on 'nothing'"
      Yeah, typical (for value-based approaches, e.g. objectification, Western materialism, etc.) misconception from inability to reflect one's own context. It's interesting that some modern and even bright intellectuals don't get it. But nihilism as 'nothing matters', 'whatever', and other permanent fatalistic tendencies is possible only when some "grand meaning" (usually, emotional and nostalgic content which can be reduced to withdrawal from oxytocin and dopamine-reinforced responses) is previously pre-set and is thwarted. So in Buddhist terms it's exactly grasping (upadana).

      "limits by excluding much else, like religion and spirituality"
      Not sure what you mean by those words (e.g. religion can be seen differently, as a superstition, as social engineering, as a way to organize life around ritual, etc.) but he doesn't specifically target them. His main concern was to restructure thinking by means of reflection on language we use in everyday life and to get rid of metaphysical content, in order to release inconsistencies between phenomenal world and our model of it, while developing a habit (a physiological inhibition) to remember the difference at all times, "The map is not the territory."

      And if some area (and its terms) would prove itself beneficial we may include it properly. As when thinking is clarified it does not need beliefs and superstitions (hence, what is commonly called religion and ideology become redundant, not excluded, but "grown out from"). That does not mean necessarily positivistic as some abstractions may be useful (empirical or otherwise), though not measurable. So it's more methodological approach to the reflection of thinking itself, than specifically to its content (ideological, filtering). But still not as strong as in Buddhist tradition (where in "right view" is ironically included abandoning religion).

      Here is a serious question. Is Schrödinger's cat a rebirth of Nansen's one or just a rebranding? :-)

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    8. @Steven Evans @Mozibur
      What evidence is there to confirm that the entity 'Steven Evans' is not a mere bot? ;)

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    9. Well, I don't imbue the complex numbers with spirituality, so guilty as charged.

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    10. Delightfully informative review of the philosophy and physics here. Thanks.

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    11. “It's rather like taking to heart the modern physical conception of matter as being mostly energy or force rather than matter and then thinking that reality does not exist - when it most assuredly does. “

      There does seem to be a preference for some universal wavefunction description over that of a universal particle-function description, if you will pardon the expression. Particles do function. The distinction at depth created by physical boundaries is an essential ingredient to how the universe functions. Prosaically, consider two possible games. One is chase-the-ball where you throw the ball and chase it. Or, there is wall-ball where you throw the ball against a wall and catch in on the rebound.

      The universe plays wall-ball. There is a distinction at depth between enduring, composite physical entities. They are coherent unto themselves. Their internal dynamics are a manifold adaption of their parts to a unique whole and their phase coherence with their environ is not global.

      Hope this settles the matter.

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    12. @Vadim:

      When I said "nothingness is not nothingness per se", I was harking back to Parmenides and what he had to say about it, which was that it couldn't be thought because it couldn't be actually referred to. This isn't the nothing that Hegel refers to and which I said was comparable to the sunyata of Buddhism.

      It's got nothing to do with ascetism.

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    13. @Steven Evans:

      I'm not putting spirituality into the complex numbers, but simply pointing out that the ontology of abstract thongs, such as numbers and words is a difficult one. Plato, resolved this as step in his intellective dialectic towards the One. If you're not keen about Platonist philosophy, thats fine with me. Plenty of people have been.

      Even for nominalists this is difficult. There is, after all, no set of all two things of which you can take an equivalence class of ...

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    14. Mozibur10:17 PM, March 11, 2021

      I was responding to C. Thompson's bot remark.

      I don't think the ontology is difficult. The complex numbers are precisely defined. They are useful for describing aspects of the physical world. In the application described in the blog post, it turns out they may even be essential, relative to the real numbers at least.
      There are some questions not worth pondering. We gather them together in a dustbin marked "philosophy". And there are some things which don't exist e.g. humans don't have a "spirit" i.e. we're all bots ;)

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    15. @Steven Evans:

      Ontology is difficult, even the nominalist case leads us to such difficult questions as the set of all sets and the like.

      Given that people like Newton and Einstein thought philosophy as important, I take your dismissal of philosophy as similar to your dismissal of super-symmetry, and which has nothing to do with philosophy, and every thing to do with physics, as being based upon nothing other ignorance.

      If you want to think of yourself as a bot - go ahead ... no skin of my nose.

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    16. "When I said "nothingness is not nothingness per se", I was harking back to Parmenides and what he had to say about it"
      I wasn't objecting (or agreeing for that matter), just reflecting from the perspective of Buddhist knowledge. As far as I can remember, I enjoyed Parmenides (it was some unfinished text). And he approached the indication to 'unspeakable' level directly, i.e. from pure being (same as Brahmanic and Upanishadic texts).

      "nothing to do with ascetism"
      I wanted to stress the proper context of those traditions. As both Upanishadic texts, which grew out of reflecting Brahmanic knowledge directly, and Buddhist texts, which reflected Brahmanic knowledge from negation, were strictly ascetic traditions (shramana). So I think it's important to keep that in mind when considering them, as they cannot be considered as just intellectual systems.

      All those traditions were directed to a particular end, not to establish a proper cosmology or any other. Namely, they were designed to end attachment and suffering for an aspirant (i.e. already a selected person, who *consciously* and himself wants to end suffering, and who 'has left home'). And I think (but, who am I, right?) that this context cannot be neglected when considering those traditions.

      Why do I think it's of importance from our perspective? As those traditions (especially ascetic, not the initial religious) were designed with some practice in mind, which stresses the importance of primary work with attention, and selects particular methods/objects for that. And that, as far as I know, is a very different approach from that of Western philosophy. As it [the ascetic approach] is practice-oriented.

      And, surprisingly, most of what is interesting in that knowledge comes exactly from those ascetic practice-oriented traditions. Not the initial religious (i.e. ritualistic and mythological) contexts. But those ascetic traditions in a way started from reflecting and processing religious traditions (some developed in parallel), i.e. initiating proper philosophic thinking. And distilling thinking and thought out of them [religious contexts]. In that regard, I don't think there is Eastern philosophy outside of ascetic (i.e. practice-oriented) tradition. Hence, different approach to epistemology is guaranteed (to say the least).

      What does it mean from the perspective of modern knowledge ('why' I think it's important)? We can indirectly check (fMRI studies, neurophysiology, etc.) if there are any reasonable correlations with such practices. And whether or not they are beneficial, and all those insights correlate with something tangible. Not just some fancy of a wacky mind. The most interesting part is that they do correlate with real changes in behavior, and they do change our cognitive models.

      That's what I thought is important to keep in mind, and attempted to highlight in a light tone. I have no problems with any philosophy or any philosophers (nor do I have any reasonable knowledge of those, only that some were curious). It was written in the spirit of consideration.

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    17. Mozibur5:29 AM, March 12, 2021

      Physics reduces all observed phenomena to the quantum and GR, up to the precision of measurement and excluding a few open questions. This all-encompassing, mind-bogglingly successful ontological reduction is all you need to know on the topic of "ontology".
      Remind me. Why are Newton and Einstein famous? For discovering general empirical facts about nature or philosophy?
      The set of all sets is a contradictory idea which is why ZFC is based on the concept of the iterative set. There is no evidence for supersymmetry, so it's not known to be physical. These are just basic facts. The "nominalist case" is the kind of nonsense philosophers spend centuries discussing to no end, and belongs in the aforementioned dustbin.

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  9. I think this paper would have made Dr. Isaac Asimov very happy. In his column in "The Magazine of Science-Fiction and Fantasy", circa the 1960's, he wrote an essay about complex numbers which started with an anecdote from some (non-science) course he took in college. The professor had listed a number of professions on the board in two columns, something like "Realists" vs. "Fantasists" Those weren't the exact words, which I forget, but mathematicians were listed in the latter category. When asked why, the professor said something like, "They have a concept called imaginary numbers, which don't actually exist, but they believe in them anyway."

    Of course Dr. Asimov objected to that strongly, but at that time had no concrete examples and used the debating trick of showing that the professor did not understand the mathematics of numbers and therefore had no basis for his remark. He had since thought of better arguments for his essay, but would be glad to have the additional ammunition.

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  10. That is why my mother said that a well-made jacket does not show the seams!

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  11. The question of whether or not complex numbers exist is interesting to me since I don’t even consider non complex numbers to “exist”. Well actually I do consider non complex numbers to exist as a tool of human language, just as terms in the English language do, though not beyond such as for a constructed mental realm.

    So then do I consider complex numbers to exist in the weak sense of language? Well of course I do! And when we can combine them so that they cancel out we may also construct coherent mathematical statements. Perhaps this could be helpful regarding the bizarreness of quantum mechanics. The point would be that if you’re left with a complex number then even here you shouldn’t have delivered a coherent statement.

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  12. Hi Sabine, very interesting blog post. By now you know in which camp you can find me.
    Many greetings Stefan

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  13. The role of complex numbers is a part of the general unitary and Hermitian structure of quantum mechanics. The evolution of a wave ψ(t) = U(t)ψ(0) requires U^† = U^{-1} and U(t) = e^{-iHt} with i∂_tψ(t) = Hψ(t).

    I honestly do not think anything is wrong with QM. I would say it is more classical mechanics that is wrong. Quantum mechanics is an L^2 theory with probability p = 〈ψ_i|ψ_i〉, which curiously is the same with gravitation. Convex structures with measure p and q are dual with 1/p + 1/q = 1, and this raises the hypothesis that QM and GR are dual in this sense. Standard probability is L^1 or straight forwards linear probabilities. This means the dual system is q → ∞ or no integration measure. This would correspond to deterministic systems of classical mechanics or Turing machines. In some way we have I think a difficulty in getting an understanding of how classical mechanics emerges from QM.

    In the Zurek setting classical states are einselected states stable against quantum noise. The interface between QM and CM is where I think most of these problems lie. I would tend to say that for einselected states the complex phase of quantum amplitudes are negligibly small and we just ignore them.

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    1. "I honestly do not think anything is wrong with QM. I would say it is more classical mechanics that is wrong."
      From a layman's perspective there is something odd with time in the equation in comparison to classical mechanics with bumping particles. It does not 'feel' right. Namely, we've got probabilistic distribution of some kind (micro) smeared over some space (yeah, just a model and all that, not the point). Yet, we're slicing time as in classical macro case (derived from 'bumps'). And that just doesn't feel right (yeah, not very good argument).

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    2. Your first paragraph about unitary structure does not prove that complex numbers are necessary, rather than just very convenient, for quantum mechanics. Schrödinger noted in his article (Nature (London) 169, 538 (1952)) that a scalar wave function can be made real by a gauge transformation and commented: "That the wave function of (3) can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation."

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    3. From the perspective of QM classical mechanics is obviously wrong. From the perspective of classical physics QM is obviously wrong.

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  14. First, a small typo: "There useful..." should perhaps be "They are useful..."

    The whole issue of where some mathematical concepts are real seems to be a bit misplaced. Is a concept that exists in our minds real? Well mathematics is something that exists in our minds. We construct mathematical theories in our minds in terms of which we try to understand the physical world outside our minds. That does not mean that the mathematical concepts in our minds are physically real in world outside our minds. Even the difference between classical physics and quantum physics is something that only exists in the minds of people and not in the physical world outside our minds.

    So, the whole issue which complex numbers boils down to whether it is possible to model the physical world (or some aspects of it) without complex numbers. Perhaps seeing it like this makes it a little less interesting.

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  16. Complex numbers are routinely employed in an area that is usually considered part of classical physics: the design of AC circuits. But they are used there for exactly the same reason as in quantum physics. It's only that the "quanta" there are so tiny and numerous that nobody cares about their discreteness and exact numbers. Fundamentally, everything is quantum physics.

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    1. The same with EM fields and waves. The Maxwell equations permit complex valued solutions. The calculations are done in complex variables because things are simpler, and at the end of it all the physical solution is the real part. I complete this below in a stand alone post.

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  17. It seems to me that if "something" can express the regularity and correlations that are seen in reality, in that sense it can be said that it is "something" it is real too, it is like the image in the mirror, the difference would be that everything that is in the mirror exists in reality; But not every mathematical "trick" has a real equivalent or we don't always see the "trick"!

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    1. I think of these mathematical things as existing the same way a piece of furniture does, made with a use in mind. The user needs to know what the object/number is meant for before it's more than a random, abstract object. I think that's related to what you're saying?

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  18. Exceptional topology of non-Hermitian systems

    Rev. Mod. Phys. 93, 015005

    DOI: https://doi.org/10.1103/RevModPhys.93.015005

    © 2021 American Physical Society

    https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.015005

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  19. Amazing post :) Some technical nitpicks:

    "You can’t solve this equation with real numbers because they’ll always square to a positive number."

    "A non-negative number" would be more precise. Also, some of the imaginary units are capitalized, though I can see some ways to justify that...

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    1. Another technical language nitpick: "yin/yang," not "Ying-Yang." NB: to be more etymologically precise, one should at very least use the pinyin designations, which my keyboard doesn't accomodate ...

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    2. Another technical language nitpick: 'yin/yang,' not 'Ying-Yang.'

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  20. what is role of complex numbers in space and time ?

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  21. Map 1 to {{1,0},{0,1}} and i to {{0,-1},{1,0}}. It's an isomorphism. Tadah! No imaginary numbers needed. As for numbers existing, can anyone show me negative three apples?

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    1. Your question reminded me of an account of a mathematical dream involving 'negative spoons'. Search for 'World Dream Bank: Spoon Challenge' to find it.

      Warning: The main site has plenty of Not Safe For Work Content

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    2. As for numbers existing, can anyone show me negative three apples?

      Lend me three apples and I'll show you.

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  22. Does Hilbert space exists? Do linear operators exist? Do Grassmann numbers exist?

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    1. Well, it depends on what you mean by "exist". But if you think that space and time exist, then Hilbert spaces (at least some of them) do also exist. If you don't like my definition of "exist", that's fine, but please be consistent.

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  23. I would start with a simpler question. Physics is impossible without pi. Does this mean that pi does exist? And if yes, what does "exist" in this context means?

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    1. "Physics is impossible without pi."

      No one's ever noticed a difference between truncating pi after, say, the 100th digit and pi. So, physics is totally possible without pi.

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    2. Well, the question then is just transformed: whether the truncated pi exists.

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  24. Werner wrote in a post on how complex numbers are used in AC circuits. The same with EM fields and waves. The Maxwell equations permit complex valued solutions. The calculations are done in complex variables because things are simpler, and at the end of it all the physical solution is the real part.

    With quantum mechanics the Hermitian structure of operators and observables is such that things are more subtle. A quantum wave is intrinsically complex valued, which fueled the Bohr insistence on its epistemic nature, which makes it different from an EM wave.

    In quantum mechanics the electric field with a momentum p = ħk is E ~ E_0(exp(-ikx) a + exp(ikx) a^†). The lowering and raising operators act on Hilbert space as a|n〉 = √n|n-1〉 and a^†|n〉 = √(n+1)|n+1〉, and the magnetic field is similar. This means electric field is not as a matrix diagonal, but rather off diagonal. It is then not a Hermitian matrix and not a proper observable. This in principle is the case no matter how large n is. However, classically we think of the electric field as an observable, and a wire with a length (say acting as an antenna) gives a current with volts, and its length defines a volts/m or units of electric field.

    There is then a big disconnect between macroscopic physics and microscopic or quantum physics. Usually, we think of QM as on the small, though that is not always the case, and for the EM field we usually think of a classical EM wave as having many photons. The conundrum is that we are loath to throw away either quantum or classical physics. The macroscopic world of classical mechanics and standard thermodynamics is not something we are entirely willing to throw out, though its foundations seem different from QM. It appears macroscopic physics, in particular thermodynamics, has a subjective foundation. This may be argued from a Bayesian perspective.

    This in part is where the dichotomy between real valued and complex valued physics lies. We might think of our experience of the world is where the imaginary part of the substratum is “collapsed away,” which takes us into the matter of decoherence and measurement of QM. We might from a QM perspective say that constructive and destructive interference of quantum waves pushes these quantum phases into a physics “dungeon,” but we cannot argue consistently that they completely disappear.

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    1. Lawrence Crowell wrote: "There is then a big disconnect between macroscopic physics and microscopic or quantum physics."

      The still mysterious "quantum-classical transition"!
      Isn't there just one world for which we seek a unified description? At which frequency do we need to switch from the classical to the quantum description?

      I think the distinction between classical and quantum phenomena is without foundation. Plasma instabilities, for example, can be described in both classical and quantum language. Classically, one talks of the growth rate of a plasma wave, quantum mechanically of stimulated emission. There's a paper by E.G. Harris, "Classical Plasma Phenomena from a Quantum Mechanical Viewpoint", which I read many years ago. Perhaps you'll find it as interesting as I did.

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    2. I have not studied quantum plasmas. I suspect it is topological, maybe cohomology of categories. L^2 systems, such as metric space or the modulus square of QM, somehow map into L^1 of probability theory or L^∞ of classical mechanics. Convex hulls with 1/p + 1/q = 1 for L^p dual to L^q measure theories imply GR and QM have a duality. These somehow get mapped to L^1 and L^∞. There really is not even a mathematics on this.

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  25. Hi All,

    Complex numbers have the same reality as any of the other categories of numbers. All are just structures required to solve equations. Say you start with just whole counting numbers. 1,2,3,... Addition of any two gives you another whole number. But, the inverse of addition is subtraction. One has to add the number zero and negative numbers to be able to solve all subtraction possibilities. This set is now all the integers. Now try multiplication, which is repeated addition. Multiply any two integers and the result is an integer. The inverse of multiplication is division. A new set, the fractions, is required to be able to solve all divisions, except for division be zero. This larger set is now the rational numbers. Now go to exponentiation, which is repeated multiplication. It has two types of inverses, logs and roots. One of the new sets of numbers required are the irrationals, for example trying to take the root of 2. Rationals and irrationals together are the real numbers, which make up all the points on a 1D number line. The other set is the imaginaries, for example trying to take the root of -1. The real numbers and imaginary numbers together make up the complex numbers, which make up the 2D complex plane. The complex numbers allow all sorts of great solutions to various math problems, including their use in standard quantum mechanics. Now, if you throw black holes into the mix, you now have a solution to division by zero...

    In anyone is interested, I wrote an online complex-capable scientific calculator in Javascript years ago, so the source code is viewable by anyone with a webbrowser. You can find it by googling John's scirealm calculator. I also have some thoughts on the relation of special relativity to quantum mechanics, which very much involves the use of imaginary and complex numbers. Google SRQM.

    Cheers,
    John Wilson

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  26. Dr Hossenfelder,
    This might not be a sensible question, but:
    I've wondered for a while how mathematicians and phycisists come up with concepts and models that can be described and utilised but might not have true equivalents that can be made. Things like spaces, manifolds, shapes in extra dimensions (e.g. tesseracts) and objects like the Klein bottle and Mobius strip, etc.
    What cognition and intuition leads from numbers and symbols to visualising these objects? Have you created anything like these, and if you don't mind, how did that process happen, what is it like?

    Can anyone else explain anything about this?

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    1. Much of the intuition comes from geometry, which comes from the real world. (I'm a mathematician.)

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  27. The following is an argument for the existence of the natural numbers. The existence of photons in discrete energy levels calls for the independent existence of the natural numbers. This immediately implies the existence of rationals. Is there an argument for continuity of space, unless gravity is viewed as a continuity theory and as ultimately valid? This would call for the existence of real numbers.

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    1. Many experiments have been done looking for discontinuities in space, but none have been found.

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    2. https://arxiv.org/abs/0801.0287v3 (Data Tables for Lorentz and CPT Violation)

      "...coefficient for Lorentz violation with a Lorentz-violating operator. The Lorentz-violating physics associated with any operator is therefore controlled by the corresponding coefficient, and so any experimental signal for Lorentz violation can be expressed in terms of one or more of these coefficients."

      "To date, there is no compelling experimental evidence supporting Lorentz violation. A few measurements suggest nonzero coefficients at weak confidence levels."

      I think it is just as fair to say that there is compelling evidence that the granularity of space is extremely small, but no compelling evidence that it does not exist.

      In any case, there is a practical limit to how finely we can calculate or measure anything. From that point of view and in accordance with Dr. Hossenfelder's criterion, real numbers do not exist. However the concept of them does. (As patterns of neuron activation in our brains.)

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  28. It would be best to not redefine what "physically exist" means. Numbers, being concepts, don't physically exist. Particles physically exist. The wave function does not physically exist.

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    1. I don't think you've thought this through. Try to figure out what's the difference between a wave-function and a particle and maybe you'll see the problem with your statement.

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    2. I still think that it becomes very dangerous when we are going to identify a descriptive tool (math) with what it is trying to describe (nature). Which s-theorists already became trapped in and may be lost in math forever.

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    3. Particles are part of the ontology. They are the things that exist in three-space. The wave function is part of the physical law that determines how particles behave. The wave function is a function on the many-dimensional configuration space (it is a function of all the particle positions). Chairs exist in three-space. Chairs are made of particles.

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    4. Sabine wrote: "Try to figure out what's the difference between a wave-function and a particle and maybe you'll see the problem with your statement."

      There is a problem only if you think that a particle "is" a wave-function. But why should anybody assume "particle" and "wave-function" to be synonymous?

      Freeman Dyson has explained it well:
      "A physical object can collapse when it bumps into an obstacle. But a wave-function cannot be a physical object. A wave-function is a description of a probability, and a probability is a statement of ignorance. Ignorance is not a physical object, and neither is a wave-function. When new knowledge displaces ignorance, the wave-function does not collapse; it merely becomes irrelevant."

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    5. As Nima Arkani-Hamed said in a recent interview, “In physics very often we slip into a mistake of reifying a formalism”.

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    6. David, you really don't want to sink in the swamp of the wave-particle-duality. But is it worth of trying?...

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    7. Eusa, it's not really a swamp and no-one needs to sink in it. It's well known that the ontic status of the wavefunction is interpretation-dependent and people just need to acknowledge that. Personally, I've little sympathy for the more intelligent species of "psiontology" (and none at all for "werewaves") but from the perspective of a superdeterministic interpretation or theory it certainly would be reasonable to consider the wavefunction "physically existent".

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    8. People should get out of the swamp. John Bell wrote, "Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave?" I don't know why this isn't clear to some people. Maybe those people can explain why they don't find this to be clear. It is important when discussing a physical theory to be explicit as to what the ontology is, i.e., what things in the theory are physical objects that exist in three-dimensional space. You need those objects to make chairs, etc. Clearly (yes, clearly) the wave function is not a physical object because its domain is not three-dimensional space (plus time). If there are N particles, then the wave function's domain is R^{3N+1}.

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    9. David Marcus wrote: ¨It is important when discussing a physical theory to be explicit as to what the ontology is, i.e., what things in the theory are physical objects that exist in three-dimensional space.¨

      Exactly. But already the term ¨object¨ carries too many misleading connotations. For most people it implies existence continuous in time; we assume that chairs, billiard balls, or the moon are there all the time. But for electrons this is an unwarranted extrapolation to the shortest time scales. There is no evidence that the ¨world lines¨ of electrons are continuous. On the contrary, QED suggests that at the zeptosecond scale pair creation occurs and the trajectory of an ¨individual¨ electron loses all meaning. And although photons are assumed to carry polarization information from the source to the detectors in the Aspect et al. experiments, this leads into a ¨swamp¨ teeming with quantum objects having ¨undefined¨ properties and engaging in superluminal communication.

      I think the only way to stay out of this morass is to stick to a statistical description of absorption and emission events, as Einstein considered already in 1916. ¨Electrons¨ and ¨photons¨ are just names we give to special patterns of events in spacetime. QFT is a statistical theory describing the correlations of events (points) in spacetime. I´m convinced that all of QED follows naturally from the mathematics of point processes; alas my grasp of that kind of mathematics is just too feeble. Perhaps you could offer a helping hand? ;-)

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    10. Another way to categorize which 'things' exist is to ask what are we able to do not just say. We cannot create or destroy material objects. We can only modify them but only to the preexisting possibilities (cut & try). But we can create immaterial things e.g. math. We can and do modify them, even destroy (disuse). However they are 'existing' objects and very useful in our ongoing quest to modify our environment, both material and social. The salient thing is how helpful are these abstract things for improving our existence.
      Debating reality of abstract ideas is an interesting hobby if that's what you like to do.

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    11. According to your definition, space and time don't exist, and neither does the universe, because we can't create or destroy it. Frankly I don't think you've thought this through.

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    12. Somehow my point got misunderstood. Material objects such as the universe or space certainly exist. I have reread what I had written to see where the confusion arises but cannot find it. Maybe you can point that out? Summary: both material and abstract objects exist but our interaction with them is different and that is the salient issue IMHO.
      Very possible that I have not thought this issue through but who has. For my practical purposes I treat the 2 categories differently. I cannot dismiss material objects from existence but can ignore/dismiss ideas that are irrelevant to my existence.
      Thanks for replying.

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    13. Morris,

      I don't know what you mean by space being a material object, or what distinction you want to draw between the mathematical description of space and space. We do not know there is any. I am not saying there is none, I am saying we do not know one. The same goes for particles or literally anything. When we say something "exists" we mean that it is necessary to describe our observations. That's exactly the sense in which space-time exists, and gravitational waves exist, and quarks exist and so on. If you want to propose any other notion, fine with me, but please make sure it's consistent and meaningful.

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    14. Thanks for replying. I see that I am contributing to confusion here but that is not my intent. So there is nothing in your above answer that I disagree with. I included space in my category of material objects b/c I thought that it was your opinion and I did not wish to contradict. In fact I don't know, as you said. For practical reasons a distinction between space and space-time is not relevant to my existence.

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  29. True that maths is just a descriptive tool. A language, to be precise. But unlike ordinary language, it is unambiguous, it is meant to be clear, undeceptive.
    Now, the essential difference btw complex and real numbers should tell us something useful about the difference btw classical and quantum world. And what is the difference? Mathematically, complex numbers are a non-ordered field. You cannot just put them out there in a row.
    Why? What does this tell us?
    What is the fine structure of reality that defies being ordered inline? Any hints?

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    1. This comment has been removed by the author.

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    2. No, I mean that you can sort all points in spacetime with a few simple tricks, like a metric for distance from origin plus time for same-distance points. I guess you could take even the SR interval plus time as a basis for an ordering -haven't checked this. And still, QM sems to reject this. Think about quantum jumps, function collapses, emtangled 'spooky actions', etc. QM seems to tell us 'Thou shall not order spacetime - not under all circumstances at least'.
      And this 'she' tells us by imposing complex nums.

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  30. Its impossible to think of nothing because once you think of it, it is no longer nothing. It has become something. You can however kludge the problem and think of an imperfect qualified nothing. That's life in the big city. The junction of philosophy, science, and the imperfections of language

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  31. I don't know whether complex numbers exist or not. But I do know that there are people who get a complex from being treated like a number.

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  32. If things need a physical substrate of some kind to exist, then I can propose one or at least suggest it: the physical settings and arrangement and connections of a group of neurons which produce the understanding of the concept in a brain. (it may not be a unique arrangement from person to person, but as long as the concept is useful in a practical way to many people, that is strong evidence that equivalent arrangements exist.)

    It seems to be the function of most brain neurons to produce such arrangements in order to solve problems. For example, if there is a waterhole surrounded by dense shrubbery and you are very thirsty, but you have seen seven lions enter the shrubbery and only four come out, then the concept of minus three lions will be useful, and your neurons can see (detect) its occurence.

    Years ago I saw an experiment on a nature show which indicates wild monkeys also have this concept.

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  33. In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann, after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.


    The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics.


    The complex number system on which this conjecture is based was thought to have no meaning or application to any physical property in reality. But this feeling has turned out to be wrong.


    In 1999, it was suggested by David Hilbert and George Pólya that in the nontrivial zeros form a set of real and discrete numbers in the Riemann zeta function are just like the eigenvalues of another function called a differential operator, which is widely used in physics.


    This special newly discovered operator has close ties with quantum physics. The special operator in quantum physics is not Parity / Time (PT) symmetric in the complex number domain. If it can be shown that the PT symmetry is broken for the imaginary part of the operator, then it would follow that the eigenvalues are all real numbers, which would finally constitute the long-awaited proof of the Riemann hypothesis.


    PT-symmetric quantum mechanics is an extension of conventional quantum mechanics into the complex domain. (PT symmetry is not in conflict with conventional quantum theory but is merely a complex generalization of it.) PT-symmetric quantum mechanics was originally considered to be an interesting mathematical discovery but with little or no hope of practical application, but beginning in 2007 it became a hot area of experimental physics.

    One of the applications involving PT symmetry is whispering gallery waves.


    Because of their complex number based quantum behavior, whispering gallery waves (WGW) have mysterious properties that are seen in nanoplasmonics. In whispering gallery waves, the complex number system relates to the index of refraction of the light contained in WGW type of the optical cavity. When two WGWs are near each other, there is a one way flow of energy between them and a translation of frequencies associated with that transfer. This energy extraction process is unleashed by PT symmetry breaking and the decay that this symmetry breaking produces.

    The WGW is the structure that gives form to the condensate of the Surface Plasmon Polariton (SPP). This is called a petal condensate.

    Physicists have observed spontaneous symmetry breaking in an optical microcavity, they have demonstrated experimentally the emergence of spontaneous symmetry breaking in an ultrahigh-Q whispering-gallery microresonator. The Optical whispering gallery (WGW) microcavity is the structural form that the Surface Plasmon Polariton assumed in Nanoplasmonics. . These whispering gallery modes are analogous to the acoustic resonances in the whispering gallery in St. Paul Cathedral in London.

    A critical clue to the role of symmetry breaking in Nanoplasmonics is the observation that the application of an electrostatic field catalyzes spontaneous symmetry breaking in the WGW via the Kerr effect.

    When this electrostatic field is applied, the WGW produces symmetry breaking which induces a energy transfer between the right and left handed polariton spin currents which convert a dipole spin condensate into a monopole spin condensate.

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    1. Axil, Hilbert and Polya were not alive in 1999!

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    2. Correction as follows: In 1999, Michael Berry and Jonathan Keating formed the Berry-Keating conjecture investigating the Hilbert-Pólya conjecture.

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  34. Hobbes is on it....

    https://www.gocomics.com/calvinandhobbes/1988/01/06/

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  35. Complex numbers could be describing a reality such as in a dynamic system inside a small space and temporal interval where time and space are closed dimensions, that is, time does not go from past to future and the result is always the same with respect to real time; but any dynamic process expressed in real numbers always goes from past to future.

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  36. I think that this question and questions about the nature of numbers in general is like asking what music is, there's no one simple, straight answer that everyone can agree with.

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    Replies
    1. The axioms that define number systems like the complex numbers are precise and objective. So, in fact, there is one simple answer that everyone can agree on.
      The topic of the blog post is whether complex numbers are essential for describing the quantum. Again, this is a well-defined question for physicists. It's just a question of probing nature deeply enough.

      Delete
    2. Steven Evans,
      Thanks for answering, I see my analogy wasn't adequate. I got distracted by pondering 'how are different sorts of numbers real', although it's apparent how they function to describe real and theoretical phenomena.
      I'm slowly learning how mathematics are used in physics.

      Delete
  37. “Either embrace complex numbers or accept that nature is even more non-local than quantum mechanics.”

    Imagine a ground state atom floating in a hard vacuum in an exceptionally dark vessel floating in deep space. Its localized Schrödinger wave function consists of a spectrum of momentum waves that share one phase angle inside the vessel.

    Since every momentum component rotates at a different rate in phase space, this localized wave is inherently unstable. It diffuses over time, making the location of the atom less and less definite. The expansion is far more subtle than just “not knowing” where the atom is since the resulting wave can do strange things such as interfere with itself.

    The slowly expanding wave function captures everything that is experimentally meaningful about the atom’s current location. So what happens when it encounters the walls of the container?

    In the simplest case, it reflects back to form a standing wave. However, in a universe with absolute conservation of linear momentum, this transformation cannot happen in isolation. It requires an exchange of momentum. The roughly spherical inward momentum transfer compresses the atom ever so slightly to form a standing wave. The outward momentum sphere heats or expands the container’s walls in response to the “pressure” created by the one-atom gas inside.

    Notably, since the inward compressive and outward expansive momentum spheres form a conserved momentum pair, the compressed atomic wave function becomes entangled with the slightly expanded container. The inward momentum component remains simple and transforms the atomic wave function according to Schrödinger’s equation. The outward momentum sphere typically becomes heat, which is another way of saying it is shredded by the continual exchange of phonons in the container’s thermal matter. The result is an astronomically low-amplitude momentum wave function component of astronomically high complexity. Yet, it remains quite real no matter how diluted it grows compared to its entangled standing wave partner. Recall, for example, that even a single photon with far lower wave amplitudes still refracts according to Schrödinger’s equation as it passes through a galaxy-spanning Einstein lens.

    If the container is in touch with the rest of the universe, the amplitudes of the original outbound momentum sphere originating from the atom spread at up to the speed of light across the rest of the local universe, growing more complicated each time they encounter thermal matter. If you multiply this expansion and shredding operation across many atomic wave functions, the more exact names for it are entropy growth or the arrow of time.

    The amusingly human-centric question of whether any of our primate brain cognitive number concepts, including real numbers themselves, are “real” in a physics sense is a delightful issue for some other time. Intuitionists such as Gisin have in the past made interesting points on this very topic.

    However, with regards to the more straightforward question of whether “nature is even more non-local than quantum mechanics,” the oddly simple answer is of course it is. The more common name for this pathologically extreme version of non-locality, one that owes its very existence to the presence of incomprehensibly complex entanglements, is classical physics. It is where sufficiently large numbers of quantum waves have become so diluted, pureed, and statistically constrained that the only thing left for them is to provide a very convincing approximation of smooth differential behavior while focusing primarily on the absolute conservation of all the underlying quantum numbers (Rehteon’s Theorem).

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  38. When we say something "exists" we mean that it is necessary to describe our observations.

    Sabine,
    Would you be willing to also say the above if "is real" were substituted for "exists"?

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    Replies
    1. I normally use these phrases (is real/does exist) to mean the same thing. I have avoided the phrase "is real" just because the topic of this thread might make some people think I am actually referring to real numbers.

      Delete
  39. One more question. Mathematicians told me that there are two ways to represent complex numbers by structures made from real numbers (by 2 × 2 matrices and by Clifford_algebra). The final expressions will be much more complicated but they will be made from real numbers and will be equivalent to structures with complex numbers. Does this statement concerns the current discussion or not?

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    Replies
    1. Evgenii, first: That’s some remarkable work you’ve done on order reduction in modeling, TEG for medical implants more recently, and long-term promotion of Lyapunov scale-up. MIT Russ Tedrake et al.’s delightful and excellent robotic control systems work comes to mind on that last point. Power harvesting for medical implants is a great topic where any innovation helps. The power problem is even worse for prosthetics. For years, Dean Kamen has had the most amazing prosthetic arms and interfaces to them, but powering them at useful levels for sufficiently long periods has been challenging. (I need to check back to see how that area is doing; I haven’t looked in a while.)

      Regarding the use of matrices to represent complex numbers, I think it’s safe to say that quantum mechanics has had a powerful tendency ever since Born to treat matrices as more fundamental than other representations. Born bragged about the opacity of his matrices, asserting that they reflected the incomprehensibility of quantum mechanics.

      That’s always bothered me due to the simple phenomenon of gimbal lock. When controlling 3D rotations with software, gimbal lock exemplifies the risks of casually assuming that infinitely precise numbers are “real” and therefore have no associated resource costs. Any given matrix representation of a rotation system introduces a capricious non-physical axis of precision asymmetry for which the level of precision required approaches infinity as the desired rotation axis approaches the asymmetry axis. The computational simplicity and reliability of switching to quaternions vividly demonstrate that, at least for 3D rotations, matrices introduce noise and complexity that have nothing to do with physics and everything to do with lousy modeling choices.

      So here’s a scary thought: What if a substantial chunk of modern mathematical physics is nothing more than navigating the human-generated mathematical noise introduced by decades of seemingly innocuous decisions such as whether to use matrices or quaternions to represent 3D rotation? To take a hint provided by the symmetries of quantum mechanics, what if precision symmetry and computational symmetry are just as fundamental to classical physics as they are to the computational modeling of physics?

      For just such reasons, I suspect work such as yours on reducing model complexity is far more relevant to theoretical physics than most physicists might imagine.

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    2. Thank you, Terry. Yet, for model reduction you should thank mathematicians, I have just applied their algorithms to engineering problems.

      Delete
    3. What do you mean "represent complex numbers by structures made from real numbers"?

      Delete
    4. The key term in this respect is isomorphism. If two mathematical structures are isomorphic, they are mathematically equivalent. One can use one or another without a difference in underlying mathematics.

      For example, matrices 2x2 made from real a and b as below will be mathematically equivalent to complex numbers a + ib.

      a b
      -b a

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  40. For instance, just look at the concept of imaginary time.

    Is it applicable to think that in statistical theory you need a "scanner" to probe simultaneity planes for path integral bundle compilation achieving most accurate probabilitiy amplitudes?

    Still, i wonder if behind the Planckian uncertainty there does exist a dot productive plane of energy in particle with imaginary-like ordering not only statistically but rhythmically spinning in global beat as the origin of the material existence... We could find an analogue in black hole studies.

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  41. An Introduction to Geometric Algebra with some Preliminary Thoughts on the Geometric Meaning of Quantum Mechanics

    DOI
    https://doi.org/10.1088/1742-6596/538/1/012010

    https://iopscience.iop.org/article/10.1088/1742-6596/538/1/012010

    A unified mathematical language for physics and engineering in the 21st century

    https://doi.org/10.1098/rsta.2000.0517

    https://royalsocietypublishing.org/doi/10.1098/rsta.2000.0517

    GAME2020 3. Professor Anthony Lasenby. A new language for physics.

    https://youtu.be/x7eLEtmq6PY

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  42. I believe and it is only my opinion that our conscience has a strong classical bias, it does not help us to directly judge the quantum world, it can only be done using mathematics; so there are no guarantees about what is real and what is disposable in the equations using our intuition, the only option is experiment and mathematics, and develop an intuition about the mathematics implicit in the phenomenon. In that sense, mathematics is more real than our judgment about that reality. I don't think quantum physics can be explained creating a fundamental reality with little bits of classical concepts, that is a trap of our consciousness.

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  43. Luis, I like your point, but I have to ask: What if our mathematics also have a strong classical bias?

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    Replies
    1. Hi Terry, yes, I think they have them, I think there is no choice but to trial and error and how much fantasy and mathematical intuition and not to trust classical concepts including time and space. Thanks .

      Delete
  44. Physics undergraduate proposes solution to quantum field theory problem

    More information: Jiani Fei et al. Nevanlinna Analytical Continuation, Physical Review Letters (2021).

    DOI: 10.1103/PhysRevLett.126.056402

    In examining this problem, Fei realized that to accurately convert quantum mechanic theories from imaginary to real numbers, physicists needed a class of functions that are causal. This means that when you trigger the system you're examining, a response in the function only happens after you've set off the trigger. Fei realized that the Nevanlinna functions—named after Finnish mathematician Rolf Nevanlinna's Nevanlinna theory, which was devised in 1925—guarantees that everything is always causal.

    https://phys.org/news/2021-03-physics-undergraduate-solution-quantum-field.html

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  45. A complex number is equivalent to a pair of real numbers. Any equation involving complex numbers can be converted to pairs of equations involving pairs of real numbers. I'm an electrical engineer by training. Complex numbers are are at the heart of signal processing and communication theory. Filter design is based on the analysis of poles and zeros on the complex plane; the amplitude and phase of a sinusoid is most cleanly expressed by a complex number. The math works out much better if we use complex numbers, but if someone forced us to, we could reformulate with uglier systems of equations that keep amplitude and phase separate. But all we would achieve by that is to represent complex numbers in polar coordinates: x = r cos theta, y = r sin theta.

    So I'm not convinced that complex numbers are more "real" because of an alleged requirement that they be used in quantum theory. Just as in digital communication theory you can always come up with a formulation in terms of reals. It's just that if you do so you give up the power of complex analysis and have to recreate it from scratch.

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    Replies
    1. @ Joe,
      If you have time, I have a little off topic curiosity here. Do you know of any studies of micro-voltage variation in the power grid. Could not find anything. Wonder what the signal looks like.
      Thanks

      Delete
    2. Don, I'm not that kind of electrical engineer and know little about power distribution. These days I work in electronic design automation.

      Delete
    3. You can also reduce problems in gravitation to 10 coupled non-linear PDEs, but the content of the theory is in the idea that matter induces changes in the metric structure of spacetime, so the metric tensor as an algebraic object in a Riemann space becomes a physical thing, as real as the dielectric constant in paraffin. Just counting equations and solving specific problems is not enough - the algebraic structures are always important in fundamental physics, and the "i" in quantum mechanics can be traced back to the geometry of spacetime, which IMO makes it as real as the metric tensor in GR.

      -drl

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  46. >But we don't have to use them. In most cases we could do the calculation with real numbers. It's just more cumbersome.

    Since nature doesn't appear to allow arbitrary precision, we could just as well dispense with the reals and calculate only with rationals. That would be even more cumbersome as well. Imagine trying to work with calculus with only the rationals. And what about pi? Or the square root of two? Where should we truncate these numbers?

    We could say this is prima facie evidence that nature likes real numbers as opposed to just the rationals ... ;-).

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    Replies
    1. All calculations anyone has ever done, by pencil and paper, abacus, slide rule (I could never get more than three significant figures from one), calculator, or computer have used only rational numbers; and the same for all physical measurements.

      The reason for the utility of calculus is that it is the limit of finite-difference equations as the minimum increment goes to zero, and therefore works well for all systems where such increments are very small, without having to worry about the actual electrons or molecules or alloy-grains.

      The fact is, finite-difference equations work just well as differential equations (and give the same form of results), without hiding the fact that minimum increments can exist. (I had to do a finite-difference calculation of a structural system for a course in Numerical Analysis; it gave the same answer as the corresponding differential equation, with no more effort involved.)

      You know all of this, of course, and for you it seems to produce a mystical reaction? (Or maybe that was just a joke.) For me it just seems like straight-forward mathematical logic, necessarily true in any deterministic universe. One can always strike mysticism by digging deeply enough, but for me it is at another level.

      Thanks for your knowledgeable comments in general, though.

      Delete
    2. All calculations anyone has ever done, by pencil and paper, abacus, slide rule (I could never get more than three significant figures from one), calculator, or computer have used only rational numbers;

      Not sure this is true. e^πi = -1. That calculation only works if e and pi are real.

      Delete
    3. I amend my statement from "All calculations" to "All arithmetic calculations". I agree that "calculations" is a broader term than that, although in an abstract way.

      Delete
  47. Hi Sabine, such things are fun, but the classical theory of diffraction is intractable without the essential appearance of the complex plane. No QM, just waves. You could argue that you are dealing with the Schroedinger equation for photons, but that would be retrograde history.

    -drl

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  48. Also - Dirac eqn goes to Pauli eqn in the non-rel limit, which goes to Schroedinger eqn for the spinless limit, so you can trace the "i" in Schroedinger all the way back to spacetime geometry as encoded in the Dirac Clifford algebra, making it very real in the direct sense.

    -drl

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  49. It's unfortunate these numbers were called "imaginary", people now a days take it so literally.

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  50. There is a 3rd direct sense in which complex numbers show up. You can think of Minkowski space as a projective space made metric by introducing an ideal domain in the Cayley-Klein sense (fundamental quadric). The ideal domain can be identified with the light cone. Same thing for Euclidean geometry, where the ideal domain can be identified with the points at infinity (celestial sphere). Such projective-metric geometries are characterized by a global parameter. For Minkowski, it is 1 - the speed of light. For Euclid, it is "i". This to me is the true metaphysics of "i". :)

    -drl

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  51. Dear Sabine,

    I wonder if this is like asking if possibilities (or impossibilities) exist ...

    The possibility that the particle will exist at 'x'
    and (logical and- as in the multiplication of 2 numbers)
    the impossibility that it will exist anywhere else,
    in the infinitesimally near future,
    equals
    the probability of the fact that the particle will have existed at 'x',
    in the infinitesimally close past.

    (the Born rule, interpreted using the language of nonstandard analysis, in a dialect where each monad of time involves a nonstandard future as well as a nonstandard past)

    ReplyDelete

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