Now on to negative probabilities :P

April 28, 2021 3:11 AM

Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists! - "A Polish-Chinese-Canadian research team has proven that the imaginary part of quantum mechanics can be observed in action in the real world."

Using a device with lasers and crystals, the game master binds two photons into one of two quantum states, absolutely requiring the use of complex numbers to distinguish between them. Then, one photon is sent to Alice and the other to Bob. Each of them measures their photon and then communicates with the other to establish any existing correlations.

also btw...

• Bacteria know how to exploit quantum mechanics, study finds - "A group of scientists at the University of Chicago discovered that certain bacteria deliberately use the rules of quantum mechanics to save their photosynthetic equipment from damage by oxygen."[*]
• New Research Reveals That Quantum Physics Causes Mutations in Our DNA - "The research team found that atoms of hydrogen, which are very light, provide the bonds that hold the two strands of the DNA's double helix together and can, under certain conditions, behave like spread-out waves that can exist in multiple locations at once, thanks to proton tunneling."
• Do quantum effects play a role in consciousness? - "The role of biophotons in the brain is a growing area of research in neurobiology – and where there are photons there might be quantum mechanics."

posted by kliuless (30 comments total)

Cool!

But what are these "negative probabilities" of which you speak? I thought quantum logic was bad enough.
posted by Alex404 at 4:37 AM on April 28, 2021 [1 favorite]

I'm curious to understand how this is significant compared to other regular uses of complex math to represent the world. For instance, power systems in electrical engineering rely on complex variables as a representation tool (the specifics I'm a little rusty on, having taken that course more than 20 years ago). In general, complex number abstractions have substantial representative power for a variety of real-world phenomena. Is this different?
posted by simra at 6:55 AM on April 28, 2021 [7 favorites]

certain bacteria deliberately use

One might argue that bacteria do not "deliberately" do anything at all.
posted by aramaic at 7:07 AM on April 28, 2021 [6 favorites]

Alex404: negative probabilities are taken to be an indication your theory is wrong. See under bad ghosts.
posted by edd at 7:48 AM on April 28, 2021

I mean it's not always that bad as long as you don't see them. Much like the imaginary bits. But often it's... well it's quantum mechanics and it's weird.
posted by edd at 7:50 AM on April 28, 2021

One might argue that bacteria do not "deliberately" do anything at all.

evidence to the contrary
posted by lalochezia at 7:51 AM on April 28, 2021 [2 favorites]

I don't know if it was the journalist not entirely understanding the story themself, or dumbing it down too much in an attempt to explain it, but I couldn't make any sense out of this. As far as I know, the fact that the quantum wavefunction is an essentially complex-valued entity had been understood for the better part of a century. And complex numbers are no less (or more) "real" than the so-called real numbers: both are abstractions that describe regular patterns in reality. There are compelling theories about particle physics built around number systems with even more structure, like quaternions or even octonions. Interesting research but I just don't understand how it was described here.
posted by biogeo at 8:56 AM on April 28, 2021 [3 favorites]

But what are these "negative probabilities" of which you speak?


I didn't make this post or that title, but I'm thinking it's a joke. (Note the ":P" at the end.) The mathematical definition of a probability function is such that probabilities are constrained to the [0,1] interval of the real number line. There's definitely no negative probabilities, no matter how imaginative the physicists might be.
posted by mikeand1 at 9:17 AM on April 28, 2021

Ugh, I'd have to read the original paper to see what they actually found because the article that summarises it I'm not sure tells me! I've always been assured that you don't have to use complex numbers for quantum mechanics problems, you can use the sum over histories approach instead - the only problem with this method is that, generally, it suuuucks to do calculations this way and that complex numbers provide a far easier shortcut to doing such problems. If the finding is that you do need complex numbers for (certain) quantum physics problems, and that is a real phenomenom and not just, a trick of clever maths, what then is "imaginary time" - which crops up in a number of quantum physics problems.
posted by BigCalm at 9:29 AM on April 28, 2021 [2 favorites]

can be observed in action in the real world.

so we're just smuggling in that the word is "real"?
posted by thelonius at 10:11 AM on April 28, 2021

Complex numbers are just as real as "real numbers"....

Too much is made of the "square root of negative numbers" business. It's much better to think of complex numbers as two-dimensional numbers. Instead of "positive", "negative" and (ugh!) "imaginary" numbers, Gauss wanted to name them "direct", "inverse", and "lateral".

(Speaking for myself, I don't even believe that "real" numbers exist; they're just a convenient abstraction to help us do science and engineering. The universe we live in is discrete. There is no continuum.)
posted by phliar at 10:22 AM on April 28, 2021 [4 favorites]

To quote Kronecker, Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers, all else is the work of Man.)
posted by phliar at 10:26 AM on April 28, 2021 [2 favorites]

It's much better to think of complex numbers as two-dimensional numbers.

My high school math teachers were pretty good but I always felt they gave short shrift to complex numbers. I don't think I even learned that multiplying by i is just rotating a vector in the complex plane. So you have i, pointing straight up, and you multiply it by i, and that rotates it 90 degrees and hey, -1. I'm sure there is much more to understand, but this would have been helpful to be told right at the top.
posted by thelonius at 10:32 AM on April 28, 2021 [3 favorites]

Gauss wanted to name them "direct", "inverse", and "lateral".

Ooh, I like that. Unfortunately "inverse" has come to be more strongly associated with the multiplicative, rather than additive, inverse. We should call negative numbers "adverse". And since the imaginary unit corresponds to a leftward rotation from the "direct" semiaxis, we could call those "sinister numbers", with the whole axis being "lateral".

Now we need fun names for the unit values and semiaxes of the quaternions!
posted by biogeo at 1:47 PM on April 28, 2021 [3 favorites]

...so now explain this to me like I'm not math-inclined and frankly don't quite get the whole... quantum thing (I mean I saw the Bond movie but I don't think there's much of a connection, really.)

Is there a 'simple' precis somewhere?
posted by From Bklyn at 1:50 PM on April 28, 2021

I think Kronecker didn't give humans enough credit. Or maybe too much. Anyway I don't believe the integers are any less an invention than any other mathematical construct. Or any more.
posted by biogeo at 1:51 PM on April 28, 2021

> I don't even believe that "real" numbers exist; they're just a convenient abstraction

> Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.

when is the last time you've observed an integer in the wild? as i sit here drinking my morning coffee i can see four birds circling out my window, but i don't see the integer four anywhere. yet the abstract concept of the integers one two three four and the addition operation are useful tools to help me count birds. it gives me a way to perform counting independently of the type of natural thing i am counting.
posted by are-coral-made at 2:54 PM on April 28, 2021

Don’t worry, we all know there’s no such thing as 2.
posted by Huffy Puffy at 4:19 PM on April 28, 2021 [2 favorites]

here are links to preprints of the actual papers:

the experimental paper ("Operational Resource Theory of Imaginarity"): https://arxiv.org/abs/2007.14847

corresponding theory paper ("Resource theory of imaginarity: quantification and state conversion"): https://arxiv.org/pdf/2103.01805.pdf

earlier theory paper ("Quantifying the Imaginarity of Quantum Mechanics") https://arxiv.org/pdf/1801.05123.pdf

survey paper ("Quantum Resource Theories") : https://arxiv.org/pdf/1806.06107.pdf


From what I can gather, "resource theories" are in the economic sense: what if some quantum states or operations or properties were much more precious (difficult to manufacture) than others? e.g. suppose it is very very difficult or expensive to manufacture quantum states that have the property of being entangled? if some states / operations are more expensive or impossible to do, but other states / operations are very cheap or "free" to do, what kinds of information processing can you do? so this seems to be some kind of abstract theory that may help you think about how to do useful and efficient computations with quantum stuff in the real world.

Then it seems like there was a theoretical idea of how to define the property of a quantum state containing a complex/imaginary part as a resource theory -- e.g. what if quantum states that had an imaginary part were much more expensive or impossible to manufacture? purely as a mathematical puzzle, you could see if you could define that in some kind of self-consistent way.

the "operational" paper demonstrates that the concepts of "real" versus "imaginary" quantum states, and operations (e.g. measuring properties of quantum states in an experiment) that only require "resources" of real states vs imaginary states as inputs, can be made concrete in the world, experimentally, using linear optics -- it seems like they set up an experimental game where they are trying to distinguish certain "real" quantum states from other "real" quantum states by measuring things, and if they are only allowed to measure in ways that consume "real" quantum resources, then their ability to distinguish between different kinds of real states is a lot worse than if they are allowed to use "imaginary" quantum state resources when measuring things.

this is not at all my field so i don't have a handle on the significance of this result.

an uncharitable interpretation could be: there's some abstract mathematical game about treating "imaginarity" of quantum states as an economic resource, which might not be at all useful in practice, but this demonstrates there is at least one way to realise that abstract theory as a real world experiment with beam splitters and stuff you can poke, not merely equations. a more charitable interpretation could be: if you are trying to invent a different theory to quantum mechanics that eradicates the use of imaginary or complex numbers, then your new theory is now going to need to be consistent with this experimental setup and experimental results -- so perhaps this result kills off potential theories.
posted by are-coral-made at 4:20 PM on April 28, 2021 [4 favorites]

From the article: There is nothing in the physical world that can be directly related to the number i.

That's easily disproven. Suppose you have a sheet cake. If I remove a square with sides, say, 5cm, it's a square of cake with area 25 square cm. But look at the remaining cake. You would say that it's lacking 25 square cm, i.e. it now has -25 sq. cm., which I must have consumed. How big is a square of area -25 sq. cm.? Why, each side is 5i. The apparent hole in your cake is entirely imaginary. QED.
posted by Joe in Australia at 5:07 PM on April 28, 2021 [9 favorites]

There are only 10,004 things.

The Tao gives birth to the One.
The One gives birth to the Two.
The Two give birth to the Three.
The Three give birth
to the ten thousand things.
posted by zengargoyle at 5:07 PM on April 28, 2021 [3 favorites]

> when is the last time you've observed an integer in the wild?

I may not see the integer "2", but I can see twoness -- it's a property, the thing in common between "two beers" and "two birds". The continuum, though, exists only in our minds, since all reality is discrete. We like to pretend that a distance, ilke "from here to there", is infinitely divisible, but when you look closely it's just an integer number of planck lengths.

Speaking of beers, I need one. Excuse me while I step away....
posted by phliar at 5:12 PM on April 28, 2021

A constructivist would say that real numbers don't exist as such, processes for generating more and more decimal places exist. And there are only the same number of those processes as there are integers, rather than the bigger infinity of reals.

(A heavy majority of CS theorists think this way, I don't know about math departments.)
posted by away for regrooving at 8:38 PM on April 28, 2021

zengargoyle, that's a cool quote, what's its source? There was a school of thought in Ancient Greek philosophy, I think the Pythagoreans, which held that numbers, and reality itself, were the result of interactions between principles they called Monad, a principle of unity, and Dyad, a principle of multiplicity. Monad creates Dyad, and the two together create the remaining numbers. Seems very similar to the idea in that quote. The Pythagoreans, or maybe later Platonists, identified Monad with divine perfection, and Dyad with imperfect matter, which is probably quite different from any Taoist thinking on the subject, though.
posted by biogeo at 11:06 PM on April 28, 2021 [1 favorite]

I may not see the integer "2", but I can see twoness -- it's a property, the thing in common between "two beers" and "two birds".

Are you sure there is anything in common between "two beers" and "two birds", though, outside the workings of your own mind? Even "one beer" is pretty clearly hard to define; if you take a sip from it, does it stop being "one beer"? How much of it do you have to drink before it's no longer "one beer"? If you pour "two beers" into a larger glass, do they become "one beer"? If you order "two beers" and the server brings you a single pitcher from which you can fill exactly two glasses, did the server bring you "two beers"? The "oneness" of the beer is something your mind imposes upon it by virtue of an understanding of how to treat it as a unit (because it's in a glass, because it has the appropriate volume to be a serving, etc.). Neither the oneness nor the twoness of the beers are intrinsic properties of the beers: they are properties of how you treat the beers.

From a certain standpoint, the same thing is true of the birds. Each bird is constantly exchanging matter with its surroundings, its gut is filled with microbiota that are an integral part of its functioning as a healthy organism and yet not truly "part" of the bird, it may lose feathers. The unity of the bird as an object is to a certain extent a matter of perspective: on sufficiently microscopic or macroscopic scales, the concept of "bird" as a unit becomes nonsensical. And furthermore, suppose you spoke a language that lacks a word for "bird," instead only having more specific words like "sparrow" and "duck". If you have "two birds," might you instead have "one sparrow and one duck"? You might say "well I can still say I have two things," but now you have to explain why the sparrow and the duck are "things" that you count, and the cells that constitute them are not.

From another standpoint, of course, it is obvious that a bird has a certain intrinsic unity, a "thing-ness" that makes it a natural subject of enumeration. A bird is a natural unity in the way that an arbitrary system like, say, a collection of a few ants, a computer, and the Sun are not. Philosophical challenges aside, if you say "two birds" no one is going to have trouble understanding why each bird is a unit and together they are two. And indeed, the "natural numbers" are truly natural in a certain sense, in that we can demonstrate that even nonhuman animals are sensitive to the numerosity of a collection, and being able to judge the approximate cardinality of a collection of objects has clear survival value in the natural world.

Any argument that the natural numbers are "real" needs to address the first set of counter-arguments, and any argument that they are "invented" needs to address the second. At the moment, I'm not satisfied that either position adequately addresses its challenges, but I also find both positions to be compelling on their own merits.
posted by biogeo at 11:36 PM on April 28, 2021 [2 favorites]

I'm not a physicist, but I am very interested in quantum theory. Here's my understanding of the relationship between complex values in the quantum wave equation and the probability of finding a particle at a particular location. Taking the example of an electron in a hydrogen atom, at any location around the nucleus the quantum wave equation of the electron can be calculated as a complex number. The complex value can be converted to the probability of the electron being at that location using the probability amplitude. Max Born proposed this in 1926 and won the Nobel Prize for it in 1954. (Olivia Newton John is Born's granddaughter, BTW.)
Proving the imaginary part of a complex number in quantum theory has a physical meaning is a big deal. It shows using complex numbers is not just a convenient mathematical trick and imaginary values mean something in the real world.
posted by Metacircular at 3:20 AM on April 29, 2021

We like to pretend that a distance, ilke "from here to there", is infinitely divisible, but when you look closely it's just an integer number of planck lengths.

Sort of. It's true that below the Planck length things are different because you have to take account of gravity, since relativity allows for arbitrary Lorentz spacetime transformations (changes in reference frame) and we can get arbitrarily close to the speed of light with increasing energy. This universe-wide Lorentz symmetry breaks down below 1.616 x 10^−35 m, the scale at which metric fluctuations become increasingly extreme (aka quantum foam).

However it acts like a minimum length because trying to explore the Planck length requires particles with so much energy in so little volume that they collapse into black holes. So your accelerator would just be firing over-energetic particle after over-energetic particle at your target where they would collapse and merge into a bigger singularity, making your problem worse with each shot instead of creating particle jets like usual.

Also such an accelerator would have to be a light-year in diameter with current technology; that hampers the effort somewhat...
posted by likethemagician at 8:05 AM on April 29, 2021 [3 favorites]

Speaking for myself, I don't even believe that "real" numbers exist; they're just a convenient abstraction to help us do science and engineering.

Do you not think that the ratio of the circumference of a circle to its diameter exists?
posted by thelonius at 1:27 PM on April 29, 2021

What does it mean to say that a ratio "exists"? Do circles exist? Can you show me one? Not an object that is circular, mind you, but a circle.
posted by biogeo at 1:35 PM on April 29, 2021 [2 favorites]

I'm kind of with Biogeo on this: we think we see things and interactions between them, but what we're really do is modelling abstractions. The example of a circle and its diameter is a great example: there are no circles and no diameters, only our idea of a circle and its implied relationship to its diameter. It's possible that at some fundamental level our reality is actually precise and mechanistic and that any small portion of it could in theory be perfectly modelled by the gross probabilistic chemical reactions in our brain. Or maybe not! Maybe we're holographic ripples on the surface of an event horizon and no one part can be modelled without the whole; maybe we're all just fugitive thoughts in the Mind of God and there is no underlying reality. But we have all this beautiful maths and it works so well at large scales: it's hard to avoid the feeling that it actually means something more than a statistical and contingent truth brought about by the Law of Large Numbers.
posted by Joe in Australia at 3:33 PM on April 29, 2021 [2 favorites]