Quantum physics falls apart without imaginary numbers(scientificamerican.com)
scientificamerican.com
Quantum physics falls apart without imaginary numbers
https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/
165 comments
https://archive.md/L1SeH
There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane.
The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.
And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.
[0] https://en.m.wikipedia.org/wiki/Complex_number#History
The intuition I developed with them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale.
And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox.
[0] https://en.m.wikipedia.org/wiki/Complex_number#History
Whether a physical theory can be consistent without them is an interesting question
I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers.
If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix:
1 0
0 1
Instead of i, use a matrix that corresponds to a 90 degree rotation:
0 -1
1 0
This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them.
I do like complex numbers though. They are just a bit more concise and convenient than using matrices.
I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers.
If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix:
1 0
0 1
Instead of i, use a matrix that corresponds to a 90 degree rotation:
0 -1
1 0
This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them.
I do like complex numbers though. They are just a bit more concise and convenient than using matrices.
I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension.
Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.
EDIT: the difference between complex numbers and 2D vectors (as I understand it) is just that you can’t really multiply 2 2D vectors together to get another 2D vector (you need to multiply a scalar times the vector to get a vector), but you CAN multiply two complex numbers together to get another complex number.
To do multiplication with 2 vectors you need to introduce new things like the “inner product” (dot product) to produce another vector (as opposed to “outer product” which produces yet ANOTHER kind of thing, a matrix) whereas with complex numbers you can use regular multiplication without introducing new types of multiplication.
Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.
EDIT: the difference between complex numbers and 2D vectors (as I understand it) is just that you can’t really multiply 2 2D vectors together to get another 2D vector (you need to multiply a scalar times the vector to get a vector), but you CAN multiply two complex numbers together to get another complex number.
To do multiplication with 2 vectors you need to introduce new things like the “inner product” (dot product) to produce another vector (as opposed to “outer product” which produces yet ANOTHER kind of thing, a matrix) whereas with complex numbers you can use regular multiplication without introducing new types of multiplication.
> Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.
I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it.
So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.
I always liked that one. Gimbal lock comes in because to be of any practical use you gotta have a reference axis from which you start to rotate, which necessarily creates poles with a singularity. Since you gotta have the axis there's no way around it.
So, how to solve that? Easy: stuff the axis outside of 3d space. It doesn't even matter what the axis is, it's just there to stash the singularity away and you can rotate every which way continuously.
You've gone up a degree for each example.
The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product.
I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product.
I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
You’re right about cross product vs outer product. (I do not know how I got that one wrong…)
Or, as someone I know (Gnomon on h2g2) remarked, they're just as real as real numbers, which is to say, not real at all.
These[0] two[1] videos are probably the best overview of imaginary numbers I've ever come across. The whole channel is a treasure trove of the history behind math, notation, and conventions used in physics.
[0]: https://www.youtube.com/watch?v=CdwxpSInhvU
[1]:https://www.youtube.com/watch?v=M12CJIuX8D4
[0]: https://www.youtube.com/watch?v=CdwxpSInhvU
[1]:https://www.youtube.com/watch?v=M12CJIuX8D4
Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of freedom, like idk call it "spin" or something I guess. Oh shit.
Put another way, complex numbers are a qubit the universe gives you for free.
Put another way, complex numbers are a qubit the universe gives you for free.
I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?
Its simple. Start with your circuit and add one ancila qubit. Then in your set of basis gates (universal for quantum computation), replace every phase shifting operation with a controlled X rotation targeted on that ancila.
For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers.
If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers.
Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers.
If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers.
Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
I am super confused. Why can't I take this purely classical circuit and run it on a classical computer? Somewhere, there should be some blowup into exponential time?
The state space still grows like 2^n in the number of qubits. Again, all this mapping does is rename some variables from "imaginary number" to "degree of freedom on an imaginary system". But, the computer doesn't care what you call your vars.
I often wonder about how much mystery and skepticism would surround them if they were simply called "complete numbers" instead. Of course that's not a great name either, but it's vaguely motivated them being algebraically closed and most importantly it's a neutral, or even slightly positive name.
For me it’s more intuitive to think of numbers as being either unsigned (magnitude-only) or signed (has magnitude and polarity). It never really made sense to me to accept the concept of negative numbers without also accepting imaginaries.
Maybe my understanding is too limited, but when I had them explained as "rotational" numbers they seemed to reduce to a simple logic shortcut to get signs changing correctly around our arbitrary axes-based coordinate system.
No less useful, but kind of mundane.
Are there other things they fundamentally do, or is everything else rooted in that property? (Or have I just misunderstood them?)
No less useful, but kind of mundane.
Are there other things they fundamentally do, or is everything else rooted in that property? (Or have I just misunderstood them?)
I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe.
In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position.
For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability.
Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus
In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position.
For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability.
Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus
Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics.
That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics.
I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them.
Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes.
A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics.
I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them.
Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes.
A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
The main insight of complex numbers is the fact that it relates exponentiation (partial multiplication) to rotation. Take that away then you might as well just use matrix algebra instead.
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
Thank you, a great angle and explanation I will remember.
So I guess the fractionality is used by quantum mechanics as a simple convenience that allows probabilities to stay close to related attributes.
So I guess the fractionality is used by quantum mechanics as a simple convenience that allows probabilities to stay close to related attributes.
Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...
If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity).
The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.
The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.
There is a really good visual and intuitive explanation of complex numbers, and the sub-title is "a tale of numbers that like to turn":
https://acko.net/blog/how-to-fold-a-julia-fractal/
https://acko.net/blog/how-to-fold-a-julia-fractal/
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imaginary velocity can fit!
My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...
My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...
Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong.
You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.
You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.
> Classical mechanics are fundamentally wrong.
All physical theories are “fundamentally wrong.”
> You can’t
A classical apparatus is part of the QM framework.
Ergo: the commenter doesn’t know what they are talking about.
All physical theories are “fundamentally wrong.”
> You can’t
A classical apparatus is part of the QM framework.
Ergo: the commenter doesn’t know what they are talking about.
I'll make sure to let my PhD advisor know he was wrong about me since that's clearly under your jurisdiction.
It is not known if all physical theories are fundamentally wrong.
> A classical apparatus is part of the QM framework
This sounds like nonsense. Care to elaborate, preferably with a link to a good source?
> A classical apparatus is part of the QM framework
This sounds like nonsense. Care to elaborate, preferably with a link to a good source?
I am not an expert, but, from listening to physicists and reading popular works, I thought they generally agreed that physics was radically incomplete.
For instance, the two most powerful physical theories, Quantum Mechanics and General Relativity, contradict each other. Quantum mechanics has no explanation for what the collapse of the wave function means (it's really QM + Collapse) and it cannot account for gravity. General relativity, by contrast, assumes continuous space (which is incompatible with quantization) which leads it to predict point singularities (which is incompatible with the uncertainty principle and the Planck limit for physical distance.)
As I said, I am ignorant, but someone more knowledgeable could expand on this.
For instance, the two most powerful physical theories, Quantum Mechanics and General Relativity, contradict each other. Quantum mechanics has no explanation for what the collapse of the wave function means (it's really QM + Collapse) and it cannot account for gravity. General relativity, by contrast, assumes continuous space (which is incompatible with quantization) which leads it to predict point singularities (which is incompatible with the uncertainty principle and the Planck limit for physical distance.)
As I said, I am ignorant, but someone more knowledgeable could expand on this.
From Quantum Mechanics by Landau and Lifshitz:
The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a "classical object", the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterize it quantitatively...
We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed.
Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation.
The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy. If an electron interacts with such a "classical object", the state of the latter is, generally speaking, altered. The nature and magnitude of this change depend on the state of the electron, and therefore may serve to characterize it quantitatively...
We have defined "apparatus" as a physical object which is governed, with sufficient accuracy, by classical mechanics. Such, for instance, is a body of large enough mass. However, it must not be supposed that apparatus is necessarily macroscopic. Under certain conditions, the part of apparatus may also be taken by an object which is microscopic, since the idea of "with sufficient accuracy" depends on the actual problem proposed.
Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation.
Given that Lifschitz wrote that before QED, he did not even begin to understand the modern understanding of QM and the electron. Nor did he see any inkling of QMs replacement, (T)QFTs. QM (and his quote, and your understanding) are nearly 100 years out of date.
The entire quote is nonsense - QED (well after Landau wrote his text) shows that the opening sentence is as valid a Asimov book from the period with the wrong number of moons for various planets.
Classical QM was much more classical than modern QM, which has removed a lot of the weasel words used in the above ("large enough mass," "sufficient degree of accuracy," etc. - none of which were defined in Landau's time and all of which have been greatly extended beyond anything he could see).
A trivially simple example is asking why gold is yellow instead of silver like nearby metals. It's a very obvious property, on any mass and sufficient degree of accuracy, but has no classical explanation (since it's due to the interplay of QM and relativity.) There are tons of things like this where your claim (and Landau's handwaving) fail, so no, QM does not approximate classical here, since classical is wrong and QM is right, even at macro scales.
QM also doesn't occupy a very unusual space - all physical theories had to agree with previous knowledge under overlapping domains. Relativity did. Maxwell did. Thermo did. Stat mech did. And ALL of those (there are plenty more) were before QM. And most of those have had more improvements since then, also agreeing with previous theory on overlapping domains, e.g., QFTs have replaced QM for all modern physics and agree on some things, but go vastly beyond what was possible with QM.
QM is not special here.
The entire quote is nonsense - QED (well after Landau wrote his text) shows that the opening sentence is as valid a Asimov book from the period with the wrong number of moons for various planets.
Classical QM was much more classical than modern QM, which has removed a lot of the weasel words used in the above ("large enough mass," "sufficient degree of accuracy," etc. - none of which were defined in Landau's time and all of which have been greatly extended beyond anything he could see).
A trivially simple example is asking why gold is yellow instead of silver like nearby metals. It's a very obvious property, on any mass and sufficient degree of accuracy, but has no classical explanation (since it's due to the interplay of QM and relativity.) There are tons of things like this where your claim (and Landau's handwaving) fail, so no, QM does not approximate classical here, since classical is wrong and QM is right, even at macro scales.
QM also doesn't occupy a very unusual space - all physical theories had to agree with previous knowledge under overlapping domains. Relativity did. Maxwell did. Thermo did. Stat mech did. And ALL of those (there are plenty more) were before QM. And most of those have had more improvements since then, also agreeing with previous theory on overlapping domains, e.g., QFTs have replaced QM for all modern physics and agree on some things, but go vastly beyond what was possible with QM.
QM is not special here.
> all physical theories had to agree
You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
You seem to have missed the whole point (which comes after "yet" at the end of the quote). It has nothing to do with approximation. Also, QFT has not brought anything new in terms of solving the measurement problem (if it needs to be solved at all), so the point stands (just as it did "100 years" ago).
> if it needs to be solved at all
You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it.
> You seem to have missed the whole point (which comes after "yet" at the end of the quote)
Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part.
Your initial claim was "A classical apparatus is part of the QM framework." It is not. QM can be completely defined (and was done so very early on) without any connection to classical, and it took (and is still taking) effort to show that classical things come from it.
As two examples, the Dirac-von Neumann axioms for QM (from which it all can be derived) are from the early 1930s, and have precisely zero mention of need for any classical physics. If you don't believe it, read them, or download von neumann's book and read it. There are subsequently axiomatic forms of QFTs, TQFTs, and so on, none needing anything more than pure math to define. There's a large collection of research over the past ~100 years with groups poking at different axiom sets or arguing if this or that set is complete, still ongoing (e.g., [2]), but AFAIK, there is no big group that claims QM is not based on axioms at this point. Or QFT or TQFTs (which Atiyah spent significant time axiomatizing before he died [3]). QM can be derived from TQFTs.
Care to show me which axiom in TQFTs is the classical apparatus? Say, as opposed to the zillion other math structures that use the same words to define things which coincidentally didn't match physics? (Unless you're Max Tegmark, for which all math is physics, a fringe view but a powerfully thought out one...)
It's nice when they correspond to nature, but there is zero need for nature to define them. They're pure math, and the agreement with nature has led to the entire "It from Bit" or "Unreasonable Effectiveness of Mathematics" views in physics.
As to some really important classical connections that took a long time to derive, Dyson's 1967 proof that matter is stable (which is an incredibly classical observation) under QM is a really neat result [1]. So the classical connections are not needed to state QM, and even historically the connections were interspersed over time, and most were found long after QM was axiomatized.
So, still claim the "yet" phrase is true? If so, how did D&vN make axioms from which all QM derives?
[1] https://fisherp.scripts.mit.edu/wordpress/wp-content/uploads...
[2] https://link.springer.com/article/10.1007/s10701-008-9230-4
[3] https://en.wikipedia.org/wiki/Topological_quantum_field_theo...
You're right - it's likely a made up issue due to human psych, not anything due to physics, thus it's weird you get hung up on it.
> You seem to have missed the whole point (which comes after "yet" at the end of the quote)
Ok, so you agree the first part of the quote is sufficiently incorrect? Let's invalidate the next part.
Your initial claim was "A classical apparatus is part of the QM framework." It is not. QM can be completely defined (and was done so very early on) without any connection to classical, and it took (and is still taking) effort to show that classical things come from it.
As two examples, the Dirac-von Neumann axioms for QM (from which it all can be derived) are from the early 1930s, and have precisely zero mention of need for any classical physics. If you don't believe it, read them, or download von neumann's book and read it. There are subsequently axiomatic forms of QFTs, TQFTs, and so on, none needing anything more than pure math to define. There's a large collection of research over the past ~100 years with groups poking at different axiom sets or arguing if this or that set is complete, still ongoing (e.g., [2]), but AFAIK, there is no big group that claims QM is not based on axioms at this point. Or QFT or TQFTs (which Atiyah spent significant time axiomatizing before he died [3]). QM can be derived from TQFTs.
Care to show me which axiom in TQFTs is the classical apparatus? Say, as opposed to the zillion other math structures that use the same words to define things which coincidentally didn't match physics? (Unless you're Max Tegmark, for which all math is physics, a fringe view but a powerfully thought out one...)
It's nice when they correspond to nature, but there is zero need for nature to define them. They're pure math, and the agreement with nature has led to the entire "It from Bit" or "Unreasonable Effectiveness of Mathematics" views in physics.
As to some really important classical connections that took a long time to derive, Dyson's 1967 proof that matter is stable (which is an incredibly classical observation) under QM is a really neat result [1]. So the classical connections are not needed to state QM, and even historically the connections were interspersed over time, and most were found long after QM was axiomatized.
So, still claim the "yet" phrase is true? If so, how did D&vN make axioms from which all QM derives?
[1] https://fisherp.scripts.mit.edu/wordpress/wp-content/uploads...
[2] https://link.springer.com/article/10.1007/s10701-008-9230-4
[3] https://en.wikipedia.org/wiki/Topological_quantum_field_theo...
> They're pure math, and the agreement with nature
Sure, as far as the math is concerned, it appears that there is no need for "classical objects." But physics is not (and has never been) "pure math," which is why Landau and Lifshitz, in particular, keep insisting on the importance of understanding the "physical principles" behind the axioms, whatever they are, and the facts of the theory; and so the "agreement with nature" is all but expected; but in order to see that agreement we need to observe, and we can only make observations by looking at "classical objects"; then, to make a connection back to the theory we need to have a way of making the result of the observation directly available to the framework itself - its "physical content" and its math.
Sure, as far as the math is concerned, it appears that there is no need for "classical objects." But physics is not (and has never been) "pure math," which is why Landau and Lifshitz, in particular, keep insisting on the importance of understanding the "physical principles" behind the axioms, whatever they are, and the facts of the theory; and so the "agreement with nature" is all but expected; but in order to see that agreement we need to observe, and we can only make observations by looking at "classical objects"; then, to make a connection back to the theory we need to have a way of making the result of the observation directly available to the framework itself - its "physical content" and its math.
Your claim was "A classical apparatus is part of the QM framework."
I just pointed out it is not, and gave you axioms that define QM, and asked where the classical part is. You ignored, as expected.
>But physics is not (and has never been) "pure math,
This is not clear. I pointed out three pretty solid places that seem to disagree. In fact, a huge chunk of advances over the past 150 years came from where? Pure math. Electron oribital structure - forced by spherical harmonics (we didn't measure the orbits, then write equations - we looked for harmonics, the math game them all, then we experimented and found them all). Spin, forced by the math (relativity + qm had no solutions - to get them forced the representations to have a spin component, forced by pure math, then we found them). Positrons and anti-particles - forced by the math (the equation for electrons had two roots, pure math, at first people ignored the second root, then they decided to look, and found anti-matter). Relativity - forced by the math (the underlying Riemannian geometry is extremely forced - and the "pure math" of it led to a century of predictions, all from the pure math, that later became observed). Noether's theorem underlying all conservation principles - forced by the math (and it's an astounding, very general priciple, devoid of any physical content, but once applied, out pops all sorts of conservation laws, some we had not even observed, all forced by pure math). Quarks, Higgs Boson, nearly all of modern physics - forced by the math, not the other way around.
This list is incredibly extensive, far beyond what is here.
This is why physicists reached the "Unreasonable effectiveness of mathematics" and "It from Bit." More and more of physics is being recast as information theory and computation, since it is becoming more clear that a significant amount of what we call physics and nature is forced by math. Some, like Tegmark as I pointed out, go so far as to claim all math is physics and we just don't see some yet, because our inputs, timeframe, and objervations of multi-verse things are so small. So to claim physics is not math (another of your claims that is too hand wavy and short-sighted to believe) is not demonstrable - it's your belief, one that is slowly but surely giving way to realizing how intertwined they are.
The idea that we simply observe nature than add more terms to model it is long gone. What to look for as pointed to by math, not observation, has led to so many physical discoveries that it is why physicists (and mathematicians, and philosophers) deeply question why is math directing nature, not nature directing math.
Since I provided axioms for QM without your requirement of a "classical apparatus," and you cannot show where in those axioms your "classical apparatus" resides, I guess that part is done now too.
I just pointed out it is not, and gave you axioms that define QM, and asked where the classical part is. You ignored, as expected.
>But physics is not (and has never been) "pure math,
This is not clear. I pointed out three pretty solid places that seem to disagree. In fact, a huge chunk of advances over the past 150 years came from where? Pure math. Electron oribital structure - forced by spherical harmonics (we didn't measure the orbits, then write equations - we looked for harmonics, the math game them all, then we experimented and found them all). Spin, forced by the math (relativity + qm had no solutions - to get them forced the representations to have a spin component, forced by pure math, then we found them). Positrons and anti-particles - forced by the math (the equation for electrons had two roots, pure math, at first people ignored the second root, then they decided to look, and found anti-matter). Relativity - forced by the math (the underlying Riemannian geometry is extremely forced - and the "pure math" of it led to a century of predictions, all from the pure math, that later became observed). Noether's theorem underlying all conservation principles - forced by the math (and it's an astounding, very general priciple, devoid of any physical content, but once applied, out pops all sorts of conservation laws, some we had not even observed, all forced by pure math). Quarks, Higgs Boson, nearly all of modern physics - forced by the math, not the other way around.
This list is incredibly extensive, far beyond what is here.
This is why physicists reached the "Unreasonable effectiveness of mathematics" and "It from Bit." More and more of physics is being recast as information theory and computation, since it is becoming more clear that a significant amount of what we call physics and nature is forced by math. Some, like Tegmark as I pointed out, go so far as to claim all math is physics and we just don't see some yet, because our inputs, timeframe, and objervations of multi-verse things are so small. So to claim physics is not math (another of your claims that is too hand wavy and short-sighted to believe) is not demonstrable - it's your belief, one that is slowly but surely giving way to realizing how intertwined they are.
The idea that we simply observe nature than add more terms to model it is long gone. What to look for as pointed to by math, not observation, has led to so many physical discoveries that it is why physicists (and mathematicians, and philosophers) deeply question why is math directing nature, not nature directing math.
Since I provided axioms for QM without your requirement of a "classical apparatus," and you cannot show where in those axioms your "classical apparatus" resides, I guess that part is done now too.
> This is not clear.
Of course it is. In physics, experiment (observation) remains the sole criterion of truth, no matter what math may be saying. (And I will continue to stress that observations, experiments, and measurements - all are inevitably "classical" by nature, and so in order to draw connections from those back to the underlying physical theory, the latter must include the notion of a classical measurement device.) Also, math is not physics because a mathematical theory includes a lot of scaffolding - things that do not correspond to anything real, and it's a physicist's job to tell the difference.
Of course it is. In physics, experiment (observation) remains the sole criterion of truth, no matter what math may be saying. (And I will continue to stress that observations, experiments, and measurements - all are inevitably "classical" by nature, and so in order to draw connections from those back to the underlying physical theory, the latter must include the notion of a classical measurement device.) Also, math is not physics because a mathematical theory includes a lot of scaffolding - things that do not correspond to anything real, and it's a physicist's job to tell the difference.
Incredible analogy! The closer is a keeper:
> The mechanisms of Newton’s laws are first order terms of a proper QM solution.
> The mechanisms of Newton’s laws are first order terms of a proper QM solution.
Except that this is only true when Newton's laws can be reasonably applied at all. In many (most?) quantum situations Newton's laws are completely nonsensical (while, conversely, in most classical situations QM is plain useless).
Momentum is mv, mv^2 is two times the kinetic energy. (Energy is real in quantum systems as well as classical ones.)
This makes no sense whatsoever.
Classical momentum is mv, not mv^2. Also, momentum is a vector and generally can't be negative.
Quantum mechanical velocity is complicated as well.
Quantum mechanical velocity is complicated as well.
I thought quantum tunneling was due to uncertainty, allowing the position of a particle to resolve on the other side of a barrier sometimes because it is undefined before the tunneling.
Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?
Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?
That what Carl bender talks about[0]
That quantized nature is due to are ability to only measure only real values, which can be a sparse subset(Null set) placed on different sheets of the complex function, like qunatized energy levels of the electron in atoms.
He also talk about research showing that exactly what happens when setting an experiment such that there is an interference which I briefly looked at a while back.
[0] https://www.youtube.com/watch?v=_Sm7SNlNUOI&list=PLOFVFbzrQ4...
[0] https://www.youtube.com/watch?v=_Sm7SNlNUOI&list=PLOFVFbzrQ4...
Thank you for sharing this concept. I find it very illuminating.
I prefer "subtle" over "imaginary"
Can complex numbers be seen as just a tuple of length 2, containing two real numbers?
I guess people felt the same when negative numbers were introduced.
I've always had problems with how complex numbers are taught.
The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything.
Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc.
I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved.
And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense.
Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.
The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that's used for geometry or 2D coordinates or anything.
Then you progress into all the complex e^ functions where you basically forget about a plane, and it's just a convenient shorthand for math that would otherwise involve a bunch of trigonometric functions. But again, this never provides any actual intuition... it's just convenience. You could still write everything as sin() and cos() etc.
I finally felt like I understood complex numbers when I asked myself, how do you create a continuous solution to the function y(x) = (-1)^x? Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...]. But you can work out the math such that it necessarily produces a spiral through them. And it's different from the e^ equations because there's no pi involved.
And so rather than thinking of complex numbers as a "plane", it's much better to think of them simply as the inherently spiral motion required to continuously join alternating values. And when I look at how complex numbers are actually used in things with physical correlates -- e.g. signal processing -- the spiral intuition always continues to make sense.
Of course, at the end of the day, it's all the same math. But the idea of a continuous spiral path between otherwise discontinuous real numbers has wound up clicking for me in a way that the metaphors of a 2D plane, or of more generalized arbitrary "rotation", never has. That complex numbers are about spiral oscillation, not about planes.
I find this "spiral" concept showing up in other cases as well, and I agree that it's under-rated (or under-studied) as a way to understand the complex numbers. By thinking about periodic points on a complex spiral, you can extend many useful properties of polynomials to those with rational coefficients. We can begin to talk about the roots of equations like x^(3/2) + x^(1/2) = 0 and we find results that very naturally extend from roots of similar "standard" polynomials. We can see that x^(1/2) = 0 has half as many roots as x^1 = 0, and x^(1/4) = 0 has half again as many roots, even if they all "project" down to 1 -- their periodicity on the spiral is 4pi and 8pi instead of the standard 2pi. I feel like there's a lot going on there that hasn't been explored.
You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical? As in:
https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral
And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.
And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.
Ah, well that's fair! In any case though "spiral" is a pretty general term; I'm really just trying to get a sense of the shape the op was referring to, assumedly not something like an Archimedean spiral. Thank you for that idenfication.
> Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...].
y = cos(pi * (x-1)). No complex numbers needed.
Yes, you can make a 3-D spiral / helix by mapping from R to C, but (a) that's only when your domain is R; ignoring when the domain is C.
The spiral doesn't help you with a simple task like solving this: 0 = x^2 +1
A helix is a time-based view of drawing a circle, in the plane.
Your favorite use of complex numbers isn't the only use of complex numbers.
https://brilliant.org/wiki/complex-numbers-in-geometry/
https://web.evanchen.cc/handouts/cmplx/en-cmplx.pdf
y = cos(pi * (x-1)). No complex numbers needed.
Yes, you can make a 3-D spiral / helix by mapping from R to C, but (a) that's only when your domain is R; ignoring when the domain is C.
The spiral doesn't help you with a simple task like solving this: 0 = x^2 +1
A helix is a time-based view of drawing a circle, in the plane.
Your favorite use of complex numbers isn't the only use of complex numbers.
https://brilliant.org/wiki/complex-numbers-in-geometry/
https://web.evanchen.cc/handouts/cmplx/en-cmplx.pdf
Are you familiar with Visual Complex Analysis?
Your spiral motion intuition is really cool and complements the amplitwist model that book introduced me to nicely.
On a random tangent, one of the reasons why I love Common Lisp is that it has functions like CIS[1] in its library. That got me realizing you can literally view SIN and COS as CAR and CDR for what's essentially a cons of non-Complex numbers. I don't claim there's any benefit to doing so, but it's not wrong.
[1] http://clhs.lisp.se/Body/f_cis.htm
Your spiral motion intuition is really cool and complements the amplitwist model that book introduced me to nicely.
On a random tangent, one of the reasons why I love Common Lisp is that it has functions like CIS[1] in its library. That got me realizing you can literally view SIN and COS as CAR and CDR for what's essentially a cons of non-Complex numbers. I don't claim there's any benefit to doing so, but it's not wrong.
[1] http://clhs.lisp.se/Body/f_cis.htm
If you should be interested, 3Blue1Brown has a video primer on complex numbers that includes describing multiplications by i as rotations on the plane, showing why a plane is useful for its representation. I always enjoy videos from that channel. https://www.youtube.com/watch?v=5PcpBw5Hbwo
Back when I did olympiad math we would sometimes solve classical geometry problems with a "complex bash" --- you assign points complex coordinates and then do some algebra to show that the desired property must hold true.
i is a number defined to have i² = -1. Everything else follows from that. The complex plane arises naturally as a useful representation of complex numbers, it is not the starting point.
I do not understand at all that stuff about "spiral path between otherwise discontinuous real numbers", but I suspect you may be making things more complicated than they need.
I do not understand at all that stuff about "spiral path between otherwise discontinuous real numbers", but I suspect you may be making things more complicated than they need.
I didn't have time to read all of it, but it seems to focus a bit much on the imaginary numbers aspect of complex numbers, and how they're all spooky and weird.
Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.
Then again, I'm just a programmer.
[1]: https://www.nature.com/articles/s41586-021-04160-4
[2]: https://physics.stackexchange.com/questions/691623/how-does-...
Reading this abstract[1] and this[2] StackOverflow discussion about the topic, it seems the point is rather more along the lines that complex numbers aren't just plain two-dimensional vectors of real numbers. There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.
Then again, I'm just a programmer.
[1]: https://www.nature.com/articles/s41586-021-04160-4
[2]: https://physics.stackexchange.com/questions/691623/how-does-...
> There's an extra constraint involved by requiring that i^2 = -1, which could be written other ways, that ties the two elements of the tuple together. It seems quantum physics requires this constraint in order to describe reality.
I think you couldn't have put it better. I'm just a physicist, though.
I think you couldn't have put it better. I'm just a physicist, though.
This Scott Aaronson lecture I really liked is relevant. It's like a "why quantum mechanics probably had to do the weird probability amplitudes (which can be negative and complex) instead of just normal probabilities even without experimental results" lecture:
https://www.scottaaronson.com/democritus/lec9.html
I like SA's blog; and, based on that I bought this book (Quantum Computing Since the Time of Democritus). It's expensive and bad. Really mind-numbingly awful. I can't tell if his writing has improved dramatically since he wrote the book, or what. The entire book is done in this tongue-in-cheek pseudo-first-person, chatty, pseudo-Socratic dialogue style. That sort of stuff is fine for, say, a couple of tightly-written pages. But ... not for hundreds of pages. It's a pity, since the information in the book is good.
“Complex numbers” are rather poorly named. They are more naturally understood as simply a vector which has a magnitude, can be rotated and scaled. As geometric objects they are much more intuitive. The subject geometry algebra takes a great approach of generalizing this idea and augmenting basic linear algebra to unify complex numbers and beyond (quaternions, ect) with geometric objects and operations. This also fits in nicely with group theory, which organizes all kinds of objects which also have the same properties as numbers.
You miss a key part of complex numbers if you think of them as just vectors: they are a field.
Because they are separate but interacting with real numbers? I don’t understand.
Multiplying vectors differs from multiplying complex numbers.
He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0]
You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.
You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].
[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra
You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.
You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].
[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra
Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up.
I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).
And not just a field, but the algebraic closure of real numbers.
[deleted]
A vector field?
It's a different concept [0], regrettably the word "field" is vastly overworked in math (and physics)
[0] https://en.wikipedia.org/wiki/Field_(mathematics)
[0] https://en.wikipedia.org/wiki/Field_(mathematics)
[deleted]
IMO there is nothing “natural” in interpreting complex numbers as a vector. The fact you get a thing out of them that looks a lot like a vector is one of the stupefying ‘mysteries’ of math which make the discipline so cool.
Complex numbers afaik began as an attempt to solve polynomial equations. They begin from the agreement to invent a number i whose square is -1 so you can solve equations having sqrt(-1) in them.
The jump from sqrt(-1) to plane rotations is to my feeble mind one of the most flabbergastingly unintuitive things in ‘basic’ maths. “A rotation you say? Who ordered that!?”
Complex numbers afaik began as an attempt to solve polynomial equations. They begin from the agreement to invent a number i whose square is -1 so you can solve equations having sqrt(-1) in them.
The jump from sqrt(-1) to plane rotations is to my feeble mind one of the most flabbergastingly unintuitive things in ‘basic’ maths. “A rotation you say? Who ordered that!?”
Complex analysis is so much more regular than real analysis differentiability over a two dimensional quantity is so much strong than over a one dimensional quantity that you have much stronger results. Basically, if you know an analytic function in a neighborhood you know it over the entire plane. Plus you have functions like e ^ ( 1 / z ) which is pretty amazing around zero.
While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).
Following a proof step-by-step != understanding the proof
I've read the whole article twice now and, at the end of the day, it doesn't seem to actually explain anything at all.
It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers.
Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved.
But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM.
But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math.
So I can't tell what any of this is supposed to mean at all?
It explains how standard (complex) quantum theory comes up with the right answers, but how you can also just rewrite the equations as a less-elegant "real" quantum theory that involes no complex numbers, that also comes up with the right answers.
Which makes perfect sense, of course, because all of the rules of complex math are written in terms of real numbers at the end of the day. When CPU's are calculating math operations, it's not like any complex/imaginary bits or bytes are involved.
But then the article describes an experimental setup where the results are somehow only consistent with standard/complex QM, and are inconsistent with real QM.
But it totally neglects to say how or why. The entire premise behind real QM is that the complex stuff can just be rewritten as real. But this article seems to provide zero explanation, analogy, or intuition whatsoever as to why there's a case where this would ever not be possible. At no point does the article define what it even means to not be reducible to real math.
So I can't tell what any of this is supposed to mean at all?
> So I can't tell what any of this is supposed to mean at all?
Quantum mechanics is defined by three axioms:
1. States are unit vectors. Vectors whose 2-norm is 1.
2. Operations preserve the 2-norm. They are described by unitary matrices.
3. Systems are combined using the tensor product. If system A has state u and system B has state v, then the combined system (A, B) has state u⊗v. If you apply operation x to system A and operation y to system B, the operation on the combined system is x⊗y.
What this paper proved is that, if you use these axioms but limit yourself to unit vectors and unitary matrices with real entries, you cannot explain some experiments. This is surprising because a complex number can be thought of as just a pair of real numbers. For example, in numpy, you can turn any complex ndarray into a real ndarray by making an ndarray with one additional index of length 2, like this:
The problem is axiom (3), where systems are combined using the tensor product. The issue is that complex_to_real(A ⊗ B) has one more index than A⊗B, but complex_to_real(A)⊗complex_to_real(B) has two additional indices because you gained one real-vs-imaginary index from A and also one from B. The tensor product doesn't understand that these indices should be merged, instead of concatenated. Using complex numbers tweaks the definition of the tensor product so that it does merge these indices. The experiment is basically a way of checking that you contracted those extra indices instead of keeping them both.
So really this result is not about complex numbers vs real numbers. It's about how the states of quantum systems are combined. If you want to use real numbers, you can't use the normal tensor product; you have to use a modified one that understands every system has a special real-vs-imaginary index and that when combining systems you must contract these indices together. Complex numbers just happen to have a tensor product that packages this contract-one-index functionality nicely.
Quantum mechanics is defined by three axioms:
1. States are unit vectors. Vectors whose 2-norm is 1.
2. Operations preserve the 2-norm. They are described by unitary matrices.
3. Systems are combined using the tensor product. If system A has state u and system B has state v, then the combined system (A, B) has state u⊗v. If you apply operation x to system A and operation y to system B, the operation on the combined system is x⊗y.
What this paper proved is that, if you use these axioms but limit yourself to unit vectors and unitary matrices with real entries, you cannot explain some experiments. This is surprising because a complex number can be thought of as just a pair of real numbers. For example, in numpy, you can turn any complex ndarray into a real ndarray by making an ndarray with one additional index of length 2, like this:
def complex_to_real(complex_ndarray):
real_ndarray = np.zeroes(shape=(*complex_ndarray.shape, 2), dtype=np.float64)
real_ndarray[..., 0] = np.real(complex_ndarray)
real_ndarray[..., 1] = np.imag(complex_ndarray)
return real_ndarray
The problem is axiom (3), where systems are combined using the tensor product. The issue is that complex_to_real(A ⊗ B) has one more index than A⊗B, but complex_to_real(A)⊗complex_to_real(B) has two additional indices because you gained one real-vs-imaginary index from A and also one from B. The tensor product doesn't understand that these indices should be merged, instead of concatenated. Using complex numbers tweaks the definition of the tensor product so that it does merge these indices. The experiment is basically a way of checking that you contracted those extra indices instead of keeping them both.
So really this result is not about complex numbers vs real numbers. It's about how the states of quantum systems are combined. If you want to use real numbers, you can't use the normal tensor product; you have to use a modified one that understands every system has a special real-vs-imaginary index and that when combining systems you must contract these indices together. Complex numbers just happen to have a tensor product that packages this contract-one-index functionality nicely.
In philosophy of science and mathematics there is actually a famous argument by Hartry Field to the effect that Physics doesn't need to posit the existence of any numbers at all, not even natural numbers:
https://academic.oup.com/book/26363
As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories.
Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers.
Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.
https://academic.oup.com/book/26363
As far as I know (I haven't read the book) he argues for this by logically reconstructing part of Newtonian Mechanics without any numbers (without assuming the Peano axioms) and suggesting that in principle similar things could be done for other theories.
Now it would be very surprising if Quantum mechanics would even require the existence of imaginary numbers, since these seem to be much less basic than natural or real numbers.
Anyway, I agree that popular articles haven't made it clear what the relevant physicists mean with their thesis. And as a non-physicist it is hard to read the original source.
You definitely can do quantum mechanics fine without complex numbers. A good article would explain why it becomes much simpler/elegant with them, but apparently this isn't one. Clickbait title.
One of the first things we were taught in physics was "don't think that imaginary or complex numbers have physical significance. just do the math."
And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.
And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out.
This is one of the complicated steps in physics (I have a PhD in physics (and forgot everything since)).
First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine.
Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard.
Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason.
Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools.
The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you.
But then I forgot everything because I left academia and my memories may be faulty.
First you have some math that goes along discovering physics. You split vectors, multiply mass by something and it's fine.
Then you have math that helps you with physics. Simple differentials equations that uncover while laws of nature (cooling down speed for instance). This is the golden time for many because you're at this sweet spot where it is exciting but not too hard.
Then comes the travel in desert of abstract things you have no idea about and winner why someone hates you by shoving Abel groups down your throat for no reason.
Finally comes that sight of relief when you can binding do some maths to end up with a real life solution without too much thinking because you have solid tools.
The last part is a bit morally complicated because you have the feeling that you are cheating. Renormalization, I am looking at you.
But then I forgot everything because I left academia and my memories may be faulty.
There really isn't anything weird or suspect about renormalization (except the name, perhaps). Read A. Zee's book on Quantum Field Theory.
> There really isn't anything weird or suspect about renormalization
This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
That's always been my take too.
Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed of light" gives the correct answer even though the theory implied by this method is naive and wrong.
Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed of light" gives the correct answer even though the theory implied by this method is naive and wrong.
Read "Quantum Field Theory in a Nutshell."
You are certainly right, like I said it was a long time ago.
But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))
But doing splits and advanced acrobatics to get rid of infinities always felt like a hack (Feynman felt the same do at least I am not alone :))
"shut up and calculate", also known as the Feynman approach to quantum mechanics.
It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.
It's not imprecise. It reproduces experimental results from theory, so it's in fact the most precise approach in existence.
> don't think that imaginary or complex numbers have physical significance.
Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.
Yeah I'd say that's the most common approach but I think it's misguided. Complex numbers aren't any less physical than any other number. It just turns out that for historical reasons, it makes sense to define observable quantities using self-adjoint operators (which have real eigenvalues, and the latter are used to measure things like energy). But that doesn't mean the rest is not physical. Just because we can't take a picture of an object in the dark, it doesn't mean the object isn't there when the lights are off.
I'd say that complex numbers are the only ones that have physical significance. They are what's actually happens in the real world until we disturb it with experiment.
"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. "
The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
Not really! Often you get a complex solution and both the real and imaginary components are valid.
A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.
>A complex solution can be valid but you never measure a complex number.
I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
I see where you are coming from, and I'm asking this as a genuine question rather than to argue, but what's stopping me from measuring the length and the mass of an object and saying the "length-mass" of it is length + i(mass)? I suppose it isn't useful since complex numbers are not ordered, but aren't "numbers" arbitrary? In measure theory, measures are defined as outputting positive real numbers and +infinity because those happen to align with our intuition about how measures work, but as far as I know, maths(and physics here I guess) does not care about the representation of my quantity which I'm measuring, but it only cares about it's properties.
Well, for one thing, for such quantity to make physical sense, both the real part and the imaginary part should be of the same dimension, e.g. "length." Also, the result of a measurement is supposed to come from (be an eigenvalue of) an observable - an operator, and, on the one hand, I think I'd have a hard time conjuring one up; on the other hand, the eigenvalues are "supposed to be" real anyway! So, no, that doesn't work.
Nothing stops you from representing the value that way, but when you go to a meter or lay a yardstick against something, you are measuring a real number (or, at the very least, a number which has no complex character to it). "This many ticks on a ruler" or "this many clicks on a clock."
Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)
I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.
The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with everyday intuition.
> Of course the "real" numbers have never been controversial
Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
In 21st century hindsight, being annoyed by irrational numbers seems a bit odd to me. I mean this very much as an opinion. I actually partially understand where they were coming from; it makes a bit more sense than the 21st century perspective would indicate, but still, obviously, not something we'd agree with today.
Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree.
Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves.
Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree.
Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves.
Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
Plus Pythagoreans had a much more religious attitude towards number than any present-day mathematician. In fact I think it's almost misleading to remember them chiefly for their mathematical contributions (many of which are disputed anyway) while they were first and foremost mystic philosophers.
> that first number that isn't something that can be a number of meters or apples is a shock.
That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.
That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.
Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them.
In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary numbers to understand? Are they just fundamentally outside ordinary experience, or do we not see them because we have workarounds that hide them?
It would be interesting to read a SciFi novel where regular people commonly encounter imaginary numbers and other similarly far from everyday life stuff like calculus.
In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary numbers to understand? Are they just fundamentally outside ordinary experience, or do we not see them because we have workarounds that hide them?
It would be interesting to read a SciFi novel where regular people commonly encounter imaginary numbers and other similarly far from everyday life stuff like calculus.
Man, "outside of science" is doing a lot of work here. We also don't see quantum mechanics "outside of science," why would they also not be fundamental? The fact of the matter is that there are A LOT of things we can't explain without complex numbers or quantum mechanics, so people believe that they're fundamental. That's not weird.
Yeah, it definitely is. I'm not denying that they're fundamental, I have no reason to doubt that, I just think it's odd that they are so deeply embedded into everything, simple enough to kind of understand the basic idea of, and yet totally separate from everyday experience.
Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people.
Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people.
Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
real things heating up and down (i.e not an idealized particle but a thing with volume), the motion of a pendulum, electricity flowing through basically anything (same as heating, when you consider actual volume), these are a few of the things I have worked with where I literally have no option but to use imaginary numbers to model them.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts.
Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
We used them in the late 80s and early 90s for pretty fractals on home computers; but for SciFi world-building… it can be done, but I don't think you can really get into those worlds as a reader without already being somewhat familiar with the maths.
Reason I think this is having listened to Greg Egan's Dichronauts, in which spacetime is ++-- with all the counterintuitive hyperbolic rotations that this implies.
Other ways to get i into common use besides that, probably gives equally counterintuitive results if you don't already know the maths; for example, instead of a maximum debt you can take out based on your ability to repay, you get a region which, for certain interest and repayment rates, is the Mandelbrot set.
Reason I think this is having listened to Greg Egan's Dichronauts, in which spacetime is ++-- with all the counterintuitive hyperbolic rotations that this implies.
Other ways to get i into common use besides that, probably gives equally counterintuitive results if you don't already know the maths; for example, instead of a maximum debt you can take out based on your ability to repay, you get a region which, for certain interest and repayment rates, is the Mandelbrot set.
Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.
>"real" numbers have never been controversial
Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.
Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.
I have the same attitude towards the use of the term "significant" in statistics, where it has a technical meaning that doesn't correspond to the common language use of the term to mean "important" or "large".
Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.
Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.
`i` has a geometric meaning! It's explained by Geometric Algebra: https://en.wikipedia.org/wiki/Geometric_algebra
In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space.
David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgo...
In three dimensional Geometric Algebra, `i` is defined as the product of three orthogonal unit vectors, `abc`. You have `abc = bca = cab = i` and `acb = cba = bac = -i`. So `i` defines a chirality on the space.
David Hesternes has a paper relating that to QM: https://web.archive.org/web/20120406093531/http://www.montgo...
Just remember that Rene Descartes coined the term Imaginary Numbers, and he also believed in ghosts.
Stop calling them imaginary.
Stop calling them imaginary.
They are like negative numbers. You can have 1 apple, but you can not have -1 apples. You can owe +1 apple, and say that you therefore have -1 apples. So -1 does exist as a number, but it is a representation of something that is happening with a positive number. And the "i" is similar to this. Maybe calling it "perpendicular" would have been better suited, because imaginary is really confusing.
My favorite property of "imaginary numbers":
i^i = 0.20787957635... (Spoiler alert: it's real!)
No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are.
[1]: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
[2]: https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...
i^i = 0.20787957635... (Spoiler alert: it's real!)
No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are.
[1]: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
[2]: https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...
Physics and computer science share the feature that it seems easier to start by explaining linear motion (linear programs) but all the really interesting stuff is circular (loops). Complex numbers are the simplest representation for describing and combining rotations in a consistent way. (Other representations like "r theta" are not as simple.)
Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.)
See Clifford Algebra for generalization of the complex numbers
Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.)
See Clifford Algebra for generalization of the complex numbers
> Later, complex numbers, which are the sum of a real and an imaginary number, gained wide acceptance by mathematicians because of their usefulness for solving complicated mathematical problems. They aren't part of the equations of any fundamental theory of physics, however—except for quantum mechanics.
I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.
I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.
Could you be more specific, because I was definitely under the impression that the original quote was correct.
Where are imaginary numbers required for the undamped mass-spring system? Because a lot of "complex" equations are using complex e^ simply as an alternative to trigonometric functions (for aesthetics or convenience), where there's nothing inherently imaginary whatsoever. The same as much of signal processing.
I'm less familiar with using complex numbers in linear algebra, but I know that when I studied it in college we never touched them, so I don't understand how we'd lose most of linear algebra?
But I think the point the article is making is that, except for QM, there are no physical instantiations of complex/imaginary values. Rational numbers physically "exist" as a fraction of a distance between two points; real numbers "exist" as actual geometric proportions, and negative numbers "exist" as an opposite direction. But complex/imaginary numbers are just intermediary tools for solving equations (or conveniences to replace trigonometric functions), they don't correspond to anything physical (except, it seems, in QM).
Where are imaginary numbers required for the undamped mass-spring system? Because a lot of "complex" equations are using complex e^ simply as an alternative to trigonometric functions (for aesthetics or convenience), where there's nothing inherently imaginary whatsoever. The same as much of signal processing.
I'm less familiar with using complex numbers in linear algebra, but I know that when I studied it in college we never touched them, so I don't understand how we'd lose most of linear algebra?
But I think the point the article is making is that, except for QM, there are no physical instantiations of complex/imaginary values. Rational numbers physically "exist" as a fraction of a distance between two points; real numbers "exist" as actual geometric proportions, and negative numbers "exist" as an opposite direction. But complex/imaginary numbers are just intermediary tools for solving equations (or conveniences to replace trigonometric functions), they don't correspond to anything physical (except, it seems, in QM).
This video by Sabine Hossenfelder is probably more informative on this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw (it discusses this paper).
Sabine's theories look like a New Age bullshiter parroting "quantumagical" nonsense to sell books, sorry.
Well, physics (including classical mechanics) already uses irrational reals, which are pretty spooky themselves. Imaginary numbers don't seem so much worse.
Well, multiple disciplins including and/or associated with electrical engineering and electronics would fall apart without imaginary numbers.
Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic closure, so that all polynomials have roots, we get the complex numbers.
There is a video depicting the story of a search for a cubic equation solution and invention of complex numbers. Here it is described in a few sentences but really it was a novel, with secrets passed from dying masters to apprentices, duels (mathematical), broken oaths and suchlike.
https://www.youtube.com/watch?v=cUzklzVXJwo
https://www.youtube.com/watch?v=cUzklzVXJwo
This reminds me of this paper even though it's probably only tangentially related:
https://ieeexplore.ieee.org/abstract/document/6875117
On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.
https://ieeexplore.ieee.org/abstract/document/6875117
On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.
I had this thought years ago:
Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1
In other words, when you “break apart” i, you get 1 and -1
Real number 1 is 1 OR -1 Imaginary number 1 is 1 AND -1
In other words, when you “break apart” i, you get 1 and -1
This is your regularly scheduled reminder that complex numbers have (real) matrix representations, and what matters in any model is the properties it has not its identity as an object.
This is your regularly scheduled reminder that the author of these papers (https://arxiv.org/abs/2101.10873, https://arxiv.org/abs/2111.15128), which this article is based on know very well that complex numbers have real matrix representations.
What they add which your comment discounts is the locality structure of quantum mechanics, i.e. what happens when you combine multiple quantum systems. Specifically if the state of one system lives in A, and the state of another system lives in B then the state of a combined system lives in the tensor product A ⊗ B.
If you do the trick where you replace complex numbers by real matrices what you end up is having the state of the first system living in A = X ⊗ A', and the second is in B = X ⊗ B', where A' and B' are real and the X subsystem is the degree of freedom you're using to "fake" the complex numbers.
Then if you try to combine the two systems you end up with a state that lives in A ⊗ B = (X ⊗ A') ⊗ (X ⊗ B') but what you need in order to get the behavior you want is actually for the combined system to live in X ⊗ (A'⊗ B').
What they do is a bit more complicated because they show that all ways to fake complex numbers using real numbers breaks this way of combining systems, but this is the gist.
What they add which your comment discounts is the locality structure of quantum mechanics, i.e. what happens when you combine multiple quantum systems. Specifically if the state of one system lives in A, and the state of another system lives in B then the state of a combined system lives in the tensor product A ⊗ B.
If you do the trick where you replace complex numbers by real matrices what you end up is having the state of the first system living in A = X ⊗ A', and the second is in B = X ⊗ B', where A' and B' are real and the X subsystem is the degree of freedom you're using to "fake" the complex numbers.
Then if you try to combine the two systems you end up with a state that lives in A ⊗ B = (X ⊗ A') ⊗ (X ⊗ B') but what you need in order to get the behavior you want is actually for the combined system to live in X ⊗ (A'⊗ B').
What they do is a bit more complicated because they show that all ways to fake complex numbers using real numbers breaks this way of combining systems, but this is the gist.
Thanks, this is very helpful.
When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?
When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?
yes, specifically doing things without complex numbers breaks if you require composing systems to work like they do in normal quantum mechanics (composing with the tensor product).
If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
Help a lapsed mathematician out here... the complex numbers can be represented as skew-symmetric 2x2 real matrices, right? And this is an isomorphism? So if the 2x2 real skew-symmetric matrices are "the same" as the complex numbers, how do we end up having this issue that comes up in quantum physics where you can only use the actual complex numbers and not the representation?
Any explanation I can give in a comment here will be pretty flawed, if you want more detail than my previous comment I think your best bet is to read the papers at the arxiv links I posted above.
Did someone claim otherwise?
If the title doesn't count then we need to question what it even means for a quantum theory to 'use' imaginary numbers.
Because if real skew symmetric matrices count, then a harmonic oscillator also inescapably uses imaginary numbers.
Because if real skew symmetric matrices count, then a harmonic oscillator also inescapably uses imaginary numbers.
"Quantum physics falls apart without imaginary numbers."
>Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
>Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.
[deleted]
Something I've always wondered is: what is the weakest algebra that could be used to model physics?
E.g.: Are nationals sufficient? Integers? Finite integers?
E.g.: Are nationals sufficient? Integers? Finite integers?
Sure, but like what they say about Lisp, you are then bound to reinvent the complex numbers, poorly.
But you can't really disagree with that. It's wrong to say that physics "needs" any one thing in particular when you can construct it from other things, and use them instead.
This is not about experimentally falsifying real quantum theory, but nice anyway:
Why are amplitudes complex?
https://scottaaronson.blog/?p=4021
Why are amplitudes complex?
https://scottaaronson.blog/?p=4021
The utility behind complex numbers (for physicists at least) is really that they are a model for certain algebraic and geometric properties that are together very useful.
Congratulations, BTW it also falls apart without the even numbers, same thing for the odd ones.
As a random aside, if anyone wants a fun read, check out An Imaginary Tale: The Story of √-1
I get a redirect loop when I try to view this page. I'll try the archive.md link.
You should know that physicists don't talk all day long about quantum mechanics.
Websites seem not to work without them as well.
The page isn’t redirecting properly
An error occurred during a connection to www.scientificamerican.com.
The page isn’t redirecting properly
An error occurred during a connection to www.scientificamerican.com.
This problem can sometimes be caused by disabling or refusing to accept cookies.That it says “imagination” at first
it also falls apart without imaginary superpositions
[deleted]
Haven't read it, but it's obviously wrong.
To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.
To expand: any time you read "...magical complex numbers" just mentally replace "complex" with "negative" and then examine how odd the original text now reads. There's nothing fundamentally different between the concept of negative numbers, and complex numbers.
[deleted]
"Quantum Physics Falls Apart Without Negative Numbers" sounds a bit obvious, but reasonable IMO.