[HN Gopher] Quantum physics falls apart without imaginary numbers
___________________________________________________________________
Quantum physics falls apart without imaginary numbers
Author : wrycoder
Score : 90 points
Date : 2023-04-24 14:41 UTC (1 days ago)
(HTM) web link (www.scientificamerican.com)
(TXT) w3m dump (www.scientificamerican.com)
| neonate wrote:
| https://archive.md/L1SeH
| throwaway81523 wrote:
| I get a redirect loop when I try to view this page. I'll try the
| archive.md link.
| magicalhippo wrote:
| 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-...
| icapybara wrote:
| > 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.
| crazygringo wrote:
| 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.
| zuminator wrote:
| 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...
| dbelford wrote:
| 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.
| zuminator wrote:
| 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.
| gowld wrote:
| > 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
| djsavvy wrote:
| 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.
| feoren wrote:
| 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.
| knome wrote:
| 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
| xchip wrote:
| Congratulations, BTW it also falls apart without the even
| numbers, same thing for the odd ones.
| m3kw9 wrote:
| That it says "imagination" at first
| crazygringo wrote:
| 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?
| jessriedel wrote:
| 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.
| Strilanc wrote:
| > 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:
| 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
|
| These real ndarrays can represent all states and operations
| that the complex ndarrays could. So how could they possibly
| fail to explain any experiment?
|
| 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.
| cubefox wrote:
| 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.
| dimitrios1 wrote:
| As a random aside, if anyone wants a fun read, check out An
| Imaginary Tale: The Story of [?]-1
| retrocryptid wrote:
| 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.
| C-x_C-f wrote:
| > 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.
| scotty79 wrote:
| 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.
| bmacho wrote:
| 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.)
| wrycoder wrote:
| But, how would one handle positive and negative charge?
| amelius wrote:
| You just do the bookkeeping, taking into account that if
| charges are opposite then they attract, etc.
| [deleted]
| sdfghswe wrote:
| "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.
| BrandoElFollito wrote:
| 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.
| nathan_compton wrote:
| There really isn't anything weird or suspect about
| renormalization (except the name, perhaps). Read A. Zee's
| book on Quantum Field Theory.
| BrandoElFollito wrote:
| 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 :))
| C-x_C-f wrote:
| > 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.
| thatnerd wrote:
| 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.
| nathan_compton wrote:
| Read "Quantum Field Theory in a Nutshell."
| nathan_compton wrote:
| "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.
| computerfriend wrote:
| Not really! Often you get a complex solution and both the
| real and imaginary components are valid.
| nathan_compton wrote:
| 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.
| wadd1e wrote:
| >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.
| nathan_compton wrote:
| 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."
| Koshkin wrote:
| 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.
| 0xBABAD00C wrote:
| 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.
| xchip wrote:
| You should know that physicists don't talk all day long about
| quantum mechanics.
| deviantbit wrote:
| Just remember that Rene Descartes coined the term Imaginary
| Numbers, and he also believed in ghosts.
|
| Stop calling them imaginary.
| qwertox wrote:
| 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.
| javajosh wrote:
| 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
| CottonMcKnight wrote:
| 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.
| mjhay wrote:
| 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.
| tokai wrote:
| >"real" numbers have never been controversial
|
| Some controversy does exist. NJ Wildberger is most famous for
| not believing in the real numbers.
| mtlmtlmtlmtl wrote:
| 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.
| kgwgk wrote:
| > 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.
| jerf wrote:
| 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.
| kgwgk wrote:
| > 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.
| C-x_C-f wrote:
| 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.
| eternityforest wrote:
| 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.
| p_j_w wrote:
| 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.
| ben_w wrote:
| 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.
| RugnirViking wrote:
| 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.
| pcrh wrote:
| 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.
| ordu wrote:
| 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
| tehsauce wrote:
| "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.
| nathan_compton wrote:
| You miss a key part of complex numbers if you think of them as
| just vectors: they are a field.
| [deleted]
| 0xBABAD00C wrote:
| And not _just_ a field, but the algebraic closure of real
| numbers.
| dr_dshiv wrote:
| Because they are separate but interacting with real numbers?
| I don't understand.
| tgv wrote:
| Multiplying vectors differs from multiplying complex
| numbers.
| justin_ wrote:
| 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
| nathan_compton wrote:
| 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).
| adammarples wrote:
| .
| mort96 wrote:
| A vector field?
| C-x_C-f wrote:
| 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)
| [deleted]
| fsloth wrote:
| 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!?"
| lanstin wrote:
| 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.
| rcme wrote:
| 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).
| sfpotter wrote:
| Following a proof step-by-step != understanding the proof
| reeboo wrote:
| So does algebra.
| contravariant wrote:
| And ordinary differential equations.
| hackandthink wrote:
| This is not about experimentally falsifying real quantum theory,
| but nice anyway:
|
| Why are amplitudes complex?
|
| https://scottaaronson.blog/?p=4021
| whatshisface wrote:
| 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.
| sebzim4500 wrote:
| Did someone claim otherwise?
| contravariant wrote:
| 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.
| whatshisface wrote:
| "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?"_
| sebzim4500 wrote:
| By that logic, it doesn't even need real numbers. Just do
| everything with cauchy sequences of rationals.
| jiggawatts wrote:
| 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?
| Koshkin wrote:
| Sure, but like what they say about Lisp, you are then
| bound to reinvent the complex numbers, poorly.
| [deleted]
| whatshisface wrote:
| 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.
| eigenket wrote:
| 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.
| civilized wrote:
| Thanks, this is very helpful.
|
| When I took undergrad quantum physics, we saw that the
| Schrodinger 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?
| eigenket wrote:
| 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.
| civilized wrote:
| 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?
| itvision wrote:
| 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. This problem can
| sometimes be caused by disabling or refusing to accept cookies.
| patrick451 wrote:
| > 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.
| crazygringo wrote:
| 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).
| esalman wrote:
| Well, multiple disciplins including and/or associated with
| electrical engineering and electronics would fall apart without
| imaginary numbers.
| hsnewman wrote:
| Quantum physics is a description of reality, not reality itself.
| Koshkin wrote:
| Thank god - I wouldn't want to see the reality fall apart.
| sleepyams wrote:
| 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.
| stametseater wrote:
| Well, physics (including classical mechanics) already uses
| irrational reals, which are pretty spooky themselves. Imaginary
| numbers don't seem so much worse.
| 2overengineered wrote:
| [dead]
| rprenger wrote:
| 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
| thechao wrote:
| 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_.
| odette4 wrote:
| [dead]
| ubj wrote:
| 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...
| bookofjoe wrote:
| it also falls apart without imaginary superpositions
| andrew_eu wrote:
| 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
| fullstackchris wrote:
| 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...
| lanza wrote:
| 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.
| TechnicolorByte wrote:
| Incredible analogy! The closer is a keeper:
|
| > The mechanisms of Newton's laws are first order terms of
| a proper QM solution.
| Koshkin wrote:
| 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).
| Koshkin wrote:
| > _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.
| SideQuark wrote:
| 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?
| Koshkin wrote:
| 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.
| SideQuark wrote:
| 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.
| Koshkin wrote:
| > _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).
| SideQuark wrote:
| > 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_fie
| ld_theo...
| Koshkin wrote:
| > _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.
| Zuider wrote:
| 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.
| lanza wrote:
| I'll make sure to let my PhD advisor know he was wrong
| about me since that's clearly under your jurisdiction.
| mr_mitm wrote:
| 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.
| whatshisface wrote:
| Momentum is mv, mv^2 is two times the kinetic energy. (Energy
| is real in quantum systems as well as classical ones.)
| tanseydavid wrote:
| Thank you for sharing this concept. I find it very
| illuminating.
| user070223 wrote:
| 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=PLOFVFbz
| rQ4...
| digging wrote:
| 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?
| xeonmc wrote:
| 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.
| detrites wrote:
| 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?)
| bottom999mottob wrote:
| 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
| detrites wrote:
| 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.
| xeonmc wrote:
| 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'etre of complex numbers is the derivative
| theorem of Fourier transforms. In fact, perhaps imaginary
| numbers should really be renamed "impetus numbers"
| dist-epoch wrote:
| 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/
| MengerSponge wrote:
| I prefer "subtle" over "imaginary"
| lacker wrote:
| _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.
| albrewer wrote:
| 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
| TRiG_Ireland wrote:
| 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.
| Robotbeat wrote:
| 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.
| tomrod wrote:
| Eh, RE: your edit, "multiplication" itself has several
| different definitions in a vector space. For example, a dot
| product, outer product, Hadamard product, etc.
| Robotbeat wrote:
| Exactly. You have to introduce these new definitions, which
| isn't required for complex numbers.
| lloeki wrote:
| > 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.
| computerfriend wrote:
| 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.
| Robotbeat wrote:
| You're right about cross product vs outer product. (I do
| not know how I got that one wrong...)
| machina_ex_deus wrote:
| 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.
| dxuh wrote:
| 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.
| amelius wrote:
| I guess people felt the same when negative numbers were
| introduced.
| IIAOPSW wrote:
| 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.
| bollu wrote:
| 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?
| IIAOPSW wrote:
| 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.
| bollu wrote:
| 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?
| reeboo wrote:
| 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...
| user8501 wrote:
| 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
| dboreham wrote:
| 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.
| sebzim4500 wrote:
| "Quantum Physics Falls Apart Without Negative Numbers" sounds a
| bit obvious, but reasonable IMO.
| [deleted]
| [deleted]
| justinpombrio wrote:
| `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...
| ars wrote:
| This video by Sabine Hossenfelder is probably more informative on
| this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw (it
| discusses this paper).
| anthk wrote:
| Sabine's theories look like a New Age bullshiter parroting
| "quantumagical" nonsense to sell books, sorry.
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