Are imaginary numbers required to describe reality?

sjastro · Jan 16, 2022

Complex numbers z = x + yi where x,y are real numbers and i is the imaginary number i = √-1 play a major role in the mathematics behind quantum mechanics.
Generally complex numbers simplify the mathematics but in the case of entangled systems, complex numbers may be a necessity indicating a physical reality behind them.

Sabine Hossenfelder gives an introductory description of complex numbers and brief overview of the topic.

A more detailed description is given here.
Imaginary numbers could be needed to describe reality, new studies find

SkyWriting · Jan 16, 2022

sjastro said:
Complex numbers z = x + yi where x,y are real numbers and i is the imaginary number i = √-1 play a major role in the mathematics behind quantum mechanics.
Generally complex numbers simplify the mathematics but in the case of entangled systems, complex numbers may be a necessity indicating a physical reality behind them.

Sabine Hossenfelder gives an introductory description of complex numbers and brief overview of the topic.

A more detailed description is given here.
Imaginary numbers could be needed to describe reality, new studies find

Sabine Hossenfelder is striking!

sjastro · Jan 16, 2022

SkyWriting said:
Sabine Hossenfelder is striking!

Yeah she has got a mean left hook.

This is one of her more poignant videos.

HantsUK · Jan 17, 2022

sjastro said:
Complex numbers z = x + yi where x,y are real numbers and i is the imaginary number i = √-1 play a major role in the mathematics behind quantum mechanics.
Generally complex numbers simplify the mathematics but in the case of entangled systems, complex numbers may be a necessity indicating a physical reality behind them.

Sabine Hossenfelder gives an introductory description of complex numbers and brief overview of the topic.

A more detailed description is given here.
Imaginary numbers could be needed to describe reality, new studies find

I first discovered complex numbers when I was about 13 (what does that say about me?) but it wasn't until A-Levels (6th form, 16+) that it was covered in school. And not until Uni that I discovered their practical use in electrical circuit theory. In electrical/electronic engineering we use j instead of i, as i is used for current.

An inductor has an imaginary resistance of j2.pi.f.L and a capacitor has a negative imaginary resistance of -j/(2.pi.f.C). It makes what would otherwise be difficult maths into simple arithmetic using complex numbers.

I have always found Euler's identity intriguing, which combines i (or j), e, and pi.
Euler's identity - Wikipedia

The retort to anyone who claims what they do is complex, is to say, "yes, mostly imaginary" but that's lost on most people.

sjastro · Jan 17, 2022

HantsUK said:
I first discovered complex numbers when I was about 13 (what does that say about me?) but it wasn't until A-Levels (6th form, 16+) that it was covered in school. And not until Uni that I discovered their practical use in electrical circuit theory. In electrical/electronic engineering we use j instead of i, as i is used for current.

An inductor has an imaginary resistance of j2.pi.f.L and a capacitor has a negative imaginary resistance of -j/(2.pi.f.C). It makes what would otherwise be difficult maths into simple arithmetic using complex numbers.

I have always found Euler's identity intriguing, which combines i (or j), e, and pi.
Euler's identity - Wikipedia

The retort to anyone who claims what they do is complex, is to say, "yes, mostly imaginary" but that's lost on most people.

Euler's identity used to be my avatar in various forums until I took on a more mysterious identity.

In a physicsworld poll it ranked along with Maxwell's equations as the most beautiful.

FrumiousBandersnatch · Jan 18, 2022

sjastro said:
Euler's identity used to be my avatar in various forums until I took on a more mysterious identity.
In a physicsworld poll it ranked along with Maxwell's equations as the most beautiful.

Ramanujan's equation for 1/pi may be ugly, but is still astonishing...

sjastro · Jan 18, 2022

FrumiousBandersnatch said:
Ramanujan's equation for 1/pi may be ugly, but is still astonishing...

So is this only uglier.
The Lagrangian for the standard model of particle physics.

Screen_Shot_2016-08-03_at_3.20.12_pm.png

durangodawood · Jan 19, 2022

sjastro said:
Euler's identity used to be my avatar in various forums until I took on a more mysterious identity.
In a physicsworld poll it ranked along with Maxwell's equations as the most beautiful.

I contest the quote that says you dont need to be a mathematician to appreciate Eulers thing. I dont even know what it means to say X to the power of an imaginary number.

sjastro · Jan 19, 2022

durangodawood said:
I contest the quote that says you dont need to be a mathematician to appreciate Eulers thing. I dont even know what it means to say X to the power of an imaginary number.

I suppose the appeal is to non mathematicians who happen to be scientists as illustrated by the Feynman and Einstein quotes.

Incidentally one doesn't have to make the case of using quantum mechanics to show complex numbers apply to the real world.
From complex numbers z mathematicians developed complex functions f(z).
You are familiar with the elementary functions such as the exponential function exp(x), trigonometric functions such as sin(x) and cos(x) and the logarithmic function log(x) where x is a real number.
There are also the complex elementary functions exp(z), sin(z), cos(z) and log(z) which forms the broader subject of complex analysis.
Complex analysis which initially started off in the esoteric world of pure mathematics is now used in other fields of physics apart from quantum mechanics such as fluid mechanics and thermodynamics.

durangodawood · Jan 19, 2022

sjastro said:
I suppose the appeal is to non mathematicians who happen to be scientists as illustrated by the Feynman and Einstein quotes.

Incidentally one doesn't have to make the case of using quantum mechanics to show complex numbers apply to the real world.
From complex numbers z mathematicians developed complex functions f(z).
You are familiar with the elementary functions such as the exponential function exp(x), trigonometric functions such as sin(x) and cos(x) and the logarithmic function log(x) where x is a real number.
There are also the complex elementary functions exp(z), sin(z), cos(z) and log(z) which forms the broader subject of complex analysis.
Complex analysis which initially started off in the esoteric world of pure mathematics is now used in other fields of physics apart from quantum mechanics such as fluid mechanics and thermodynamics.

Ah right. I was being sloppy considering physicists as "mathematicians". Compared to me they are. But thats not saying much I guess.

SkyWriting · Jan 19, 2022

durangodawood said:
I contest the quote that says you dont need to be a mathematician to appreciate Eulers thing. I dont even know what it means to say X to the power of an imaginary number.

You get to make up your own meaning.

sjastro · Jan 19, 2022

durangodawood said:
Ah right. I was being sloppy considering physicists as "mathematicians". Compared to me they are. But thats not saying much I guess.

According to pure mathematicians they are the only mathematicians around.
Historically they have looked down at the application of mathematics by physicists and applied mathematicians.
One of the greatest mathematicians in history David Hilbert stated "Physics is too hard for physicists" implying the maths was beyond them.
Hilbert co-authored a book which was the 1920s version of "Mathematics for Dummies" except it was directed towards educating physicists.

Perhaps this quote from a pure mathematician explains why there is some animosity directed to those that apply mathematics.

Here’s an analogy. Let’s say you compose music, maybe symphonies or classic jazz. You are among a small group of successful composers. By success, you mean that orchestras or jazz groups perform your music. There are appreciative crowds. There is a good income, but mostly you are thrilled and deeply satisfied by the joy you have brought and the recognition you have attained among your peers.

Then somebody comes forward and says “I reworked some of your stuff and sold it to Coca-cola for a new jingle. They gave me a lot of money.”

SkyWriting · Jan 19, 2022

sjastro said:
According to pure mathematicians they are the only mathematicians around.Historically they have looked down at the application of mathematics by physicists and applied mathematicians.One of the greatest mathematicians in history David Hilbert stated "Physics is too hard for physicists" implying the maths was beyond them.Hilbert co-authored a book which was the 1920s version of "Mathematics for Dummies" except it was directed towards educating physicists.Perhaps this quote from a pure mathematician explains why there is some animosity directed to those that apply mathematics.

Here that is illustrated:

HILBERT

durangodawood · Jan 20, 2022

sjastro said:
According to pure mathematicians they are the only mathematicians around.
Historically they have looked down at the application of mathematics by physicists and applied mathematicians.
One of the greatest mathematicians in history David Hilbert stated "Physics is too hard for physicists" implying the maths was beyond them.
Hilbert co-authored a book which was the 1920s version of "Mathematics for Dummies" except it was directed towards educating physicists.

Perhaps this quote from a pure mathematician explains why there is some animosity directed to those that apply mathematics.

Sounds like academic department rivalry. Maybe the chemists should resent the art school that uses their lovely acrylic paints.

sjastro · Jan 20, 2022

durangodawood said:
Sounds like academic department rivalry. Maybe the chemists should resent the art school that uses their lovely acrylic paints.

It goes far deeper than that.
Pure mathematicians are mostly Platonists who believe mathematics is discovered rather than invented and is not part of physical reality.
They criticize the use of abstract mathematics in the sciences to describe physical reality.

G.H. Hardy who collaborated with Srinivasa Ramanujan to produce ugly mathematics (post #5) reflected this general attitude.

Hardy said:
It is quite common, for example, for an astronomer or a physicist to claim that he has found a ‘mathematical proof’ that the physical universe must behave in a particular way. All such claim, if interpreted literally, are strictly nonsense. It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics.

He gives applied mathematicians a serve.

Hardy said:
One rather curious conclusion emerges, that pure mathematics is one the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot.

But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

durangodawood · Jan 20, 2022

sjastro said:
It goes far deeper than that.
Pure mathematicians are mostly Platonists who believe mathematics is discovered rather than invented and is not part of physical reality.
They criticize the use of abstract mathematics in the sciences to describe physical reality.

G.H. Hardy who collaborated with Srinivasa Ramanujan to produce ugly mathematics (post #5) reflected this general attitude.

He gives applied mathematicians a serve.

Hmm. Not sure about this:

It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics.

I dont think there are two (or more) worlds. In our world, mathematics seems to overlap with physics somewhat.

(And I hope your position isnt in some ways a little pathetic.)

sjastro · Jan 20, 2022

durangodawood said:
Hmm. Not sure about this:

I dont think there are two (or more) worlds. In our world, mathematics seems to overlap with physics somewhat.

Indeed it does in fact there is an interdisciplinary field mathematical physics which tries to preserve mathematical rigor in physics.
Mathematical physics can only take you so far; matrix mechanics is a mathematically rigorous method in quantum mechanics to determine the energy levels of the hydrogen atom.
For helium and all other elements which have more than one electron it is not as straightforward.
Electrons repel each other and can shield the outer most electron from the positive nucleus.
These variables cause a complex perturbation in the energy levels and the resulting mathematics is an approximation or a model which is the antithesis to the pure mathematician.

(And I hope your position isnt in some ways a little pathetic.)

Having done three years of pure mathematics as an undergraduate would hopefully make me less pathetic than your run of the mill applied mathematician.

FrumiousBandersnatch · Jan 21, 2022

durangodawood said:
Hmm. Not sure about this:

It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics.

Click to expand...

Perhaps the issue is that we cannot be certain about real-world events because they involve extremely complex objects and occur in an open system. It's Hume's problem of induction; we can infer that the sun will come up tomorrow or that there will be an eclipse, from past experience - but we can't be absolutely certain, because they are not simple point masses, and some unexpected, unpredicted event could occur (the sun could go nova, a massive rogue asteroid could mess things up, etc).

durangodawood · Jan 21, 2022

sjastro said:
Indeed it does in fact there is an interdisciplinary field mathematical physics which tries to preserve mathematical rigor in physics.
Mathematical physics can only take you so far; matrix mechanics is a mathematically rigorous method in quantum mechanics to determine the energy levels of the hydrogen atom.
For helium and all other elements which have more than one electron it is not as straightforward.
Electrons repel each other and can shield the outer most electron from the positive nucleus.
These variables cause a complex perturbation in the energy levels and the resulting mathematics is an approximation or a model which is the antithesis to the pure mathematician.....

Sounds like the complex perturbation is just "too hard" for mathematics, but not impossible in principle.

But then there's some truly random events in QM which render definite mathematical descriptions of reality impossible in principle, right? The best that can be achieved are probabilities.

durangodawood · Jan 21, 2022

FrumiousBandersnatch said:
Perhaps the issue is that we cannot be certain about real-world events because they involve extremely complex objects and occur in an open system. It's Hume's problem of induction; we can infer that the sun will come up tomorrow or that there will be an eclipse, from past experience - but we can't be absolutely certain, because they are not simple point masses, and some unexpected, unpredicted event could occur (the sun could go nova, a massive rogue asteroid could mess things up, etc).

Makes sense. Mathematical models examine certain key relationships. But to be perfectly accurate you need to model all the relationships, in other words.... build a 2nd reality.

Are imaginary numbers required to describe reality?

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