Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

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socrat44
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Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by socrat44 »

Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!
By Faculty of Physics University of Warsaw on Apr 27, 2021
-----
“In physics, complex numbers were considered to be purely mathematical in nature.
It is true that although they play a basic role in quantum mechanics equations,
they were treated simply as a tool, something to facilitate calculations for physicists.
Now, we have theoretically and experimentally proved that there are quantum states
that can only be distinguished when the calculations are performed with the
indispensable participation of complex numbers,” explains Dr. Streltsov.

https://scitechdaily.com/physicists-pro ... KGkIFzeblE
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Terrapin Station
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by Terrapin Station »

socrat44 wrote: Wed Apr 28, 2021 11:30 pm Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!
By Faculty of Physics University of Warsaw on Apr 27, 2021
-----
“In physics, complex numbers were considered to be purely mathematical in nature.
It is true that although they play a basic role in quantum mechanics equations,
they were treated simply as a tool, something to facilitate calculations for physicists.
Now, we have theoretically and experimentally proved that there are quantum states
that can only be distinguished when the calculations are performed with the
indispensable participation of complex numbers,” explains Dr. Streltsov.

https://scitechdaily.com/physicists-pro ... KGkIFzeblE
========
That would simply show that complex numbers are a necessary tool in order to account for some observed phenomena under our current mathematical paradigm. (Well, and at least until some clever person comes up with a workaround under the same paradigm.)

It in no way shows that any mathematical objects (or abstract objects in general) are real (in the objective of extramental sense).
socrat44
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by socrat44 »

Terrapin Station wrote: Thu Apr 29, 2021 12:16 am
socrat44 wrote: Wed Apr 28, 2021 11:30 pm Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!
By Faculty of Physics University of Warsaw on Apr 27, 2021
-----
“In physics, complex numbers were considered to be purely mathematical in nature.
It is true that although they play a basic role in quantum mechanics equations,
they were treated simply as a tool, something to facilitate calculations for physicists.
Now, we have theoretically and experimentally proved that there are quantum states
that can only be distinguished when the calculations are performed with the
indispensable participation of complex numbers,” explains Dr. Streltsov.

https://scitechdaily.com/physicists-pro ... KGkIFzeblE
========
That would simply show that complex numbers are a necessary tool in order to account for some observed phenomena under our current mathematical paradigm. (Well, and at least until some clever person comes up with a workaround under the same paradigm.)

It in no way shows that any mathematical objects (or abstract objects in general) are real (in the objective of extramental sense).
'' . . . complex numbers are an integral, indelible part of quantum mechanics . . . ''
/ Dr. Streltsov./
" The mathematics of QM is straightforward, but making
the connection between the mathematics and an intuitive picture
of the physical world is very hard"
/ Claude N. Cohen-Tannoudji . Nobel Prize in Physics 1997 /

Do Complex Numbers Exist?
175,946 views•6 Mar 2021
https://www.youtube.com/watch?v=ALc8CBYOfkw
=======.
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Terrapin Station
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by Terrapin Station »

socrat44 wrote: Thu Apr 29, 2021 11:51 am '' . . . complex numbers are an integral, indelible part of quantum mechanics . . . ''
/ Dr. Streltsov./
Sure, in the sense of most of it amounting to "things we do with mathematics."
" The mathematics of QM is straightforward, but making
the connection between the mathematics and an intuitive picture
of the physical world is very hard"
And sure, I agree with that as well, because mathematical objects, functions, etc. simply isn't what the external world is like, so trying to translate something that's essential mathematical, especially when we're dealing with advanced/very abstract mathematical ideas, into something that isn't mathematical is difficult.
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by RCSaunders »

socrat44 wrote: Thu Apr 29, 2021 11:51 am Do Complex Numbers Exist?
Nope! Not even plain old counting numbers exist materially or independently of human minds. Numbers are nothing but a method of identifying some characteristics of material reality: multiplicity (counting) and relative physical characteristics like dimension and time (measurement).

Attributing actual metaphysical (ontological) existence to numbers is superstition.
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

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Imaginary numbers are a fine and wonderful refuge
of the divine spirit, almost an amphibian between
being and non-being.
/ Gottfried Leibniz /
#
One might think this means that imaginary numbers are just a mathematical
game having nothing to do with the real world. From the viewpoint of
positivist philosophy, however, one cannot determine what is real.
All one can do is find which mathematical models describe the universe
we live in. It turns out that a mathematical model involving imaginary time
predicts not only effects we have already observed but also effects we have
not been able to measure yet nevertheless believe in for other reasons.
So what is real and what is imaginary?
Is the distinction just in our minds?
/ Stephen Hawking /
#
Is imaginary science real?
If this is true, then it is possible that we live in metaphysical reality
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wtf
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by wtf »

RCSaunders wrote: Fri Apr 30, 2021 12:53 am
Nope! Not even plain old counting numbers exist materially or independently of human minds. Numbers are nothing but a method of identifying some characteristics of material reality: multiplicity (counting) and relative physical characteristics like dimension and time (measurement).

Attributing actual metaphysical (ontological) existence to numbers is superstition.
I agree with this. I didn't watch the vid, but it seems like yet another example of "physicists doing bad math."

First, as @RCSaunders says, even a familiar positive integer like 5 has no claim to physical existence. We can of course instantiate the abstract concept of 5 as 5 apples, 5 planets, or Five Easy Pieces. But the number 5, all by itself, has no instantiation in the world. It's a mathematical abstraction. Depending on one's philosophy you can say 5 lives in the higher world of Platonic forms, along with the Flying Spaghetti Monster and the Baby Jesus. Or if you deny superstition, 5 is a product of the human mind's quality of abstraction.

But it gets worse. What of rational numbers like 1/2? In the physical world we can never cut a pie exactly in half. I assume people know that. All physical measurement is approximate. So no non-integer rational number can ever be instantiated in the real world. If the number 5 can at least be instantiated as 5 apples, the number 1/2 can never be instantiated.

How about the irrationals? Pi is not instantiated in the world. It only exists as an abstract mathematical object.

I used "exists" in the preceding sentence to mean mathematical existence. Numbers like 5, 1/2, and pi don't have physical existence; but they do have mathematical existence, meaning that a consensus of mathematicians regards it as a mathematical object. This notion is of course historically contingent. Negative numbers were denied till they were accepted, irrationals caused a scandal among the Pythagoreans, and so forth.

How about the noncomputable real numbers? A number is computable if it can be arbitrarily approximated by a computer program. Turing elucidated this notion in his famous 1936 paper where he defined what it means for something to be a computation.

Now there are only countably many computer programs; but uncountably many real numbers. So almost all real numbers can NOT be approximated by a program. (Substitute "Turing machine" for program if you want to be more precise). What kind of numbers are they? They are essentially random. They have no pattern whatsoever. Their digits cannot be predicted or characterized. Yet without them, the real number line would be full of holes and the Intermediate Value theorem would be false. You could drive the graph of a continuous function across the x-axis without ever intersecting the axis!

Even some mathematicians and computer scientists reject the mathematical existence of such numbers. These are the constructivists, who hold that an object only has mathematical existence if we have a recipe or algorithm for producing it. Clearly noncomputable numbers fail this standard. But in standard (non-constructive) math, we allow such non-algorithmic objects to claim mathematical existence.

And now we come to the complex numbers, as exemplified by the humble imaginary unit i. The number i is characterized by its defining property that i^2 = -1. You'll note that there are actually two complex numbers that satisfy that property, since (-i)^2 = -1 as well. It turns out that if you pick one and all it i, and call the other one -i, nothing changes. (Technical buzzphrase: Complex conjugation is an automorphism of the complex numbers). For this reason it's always better to call i, "a square root of -1" and not "THE" square roon of -1, as is so commonly (and wrongly) done.

Finally, we come to the point of this shaggy math story. The number i has a very simple and obvious instantiation in the real world. Let me explain.

In the real numbers, we can regard a number like 5 as an operator that stretches a length. If we take the line segment between 0 and 1 and stretch it by a factor of 5, we get the segment from 0 to 5. So a number can be viewed as a linear scaling factor. Likewise if I have a directed arrow from 0 to 1 and I multiply it by -1, the result is a directed arrow from 0 to -1. I've flipped the orientation of the vector.

So numbers are geometric operators.

Now, say we live in the plane. I'm facing east. I make a quarter turn left to face north. I make another quarter turn left to face west, and finally one more turn brings me back to my starting orientation facing east.

Suppose I denote a quarter left turn by the symbol "i". It doesn't mean anything, it's just a gadget I use to keep track of my quarter turns. If I turn from east to north, that's i. If I do it again, that's i x i, i times i, or i^2. And what does i^2 do? It is the identical operator to multiplying by -1. It's the "flip in the reverse direction" operator. Voila. That's it. It's that simple. i^2 = -1.

Now i^3 makes you point south, and i^4 brings you back to where you started, facing east.

This is exactly what the imaginary unit i is. I mention in passing that the terminology "imaginary" is just awful. It's been responsible for so much confusion. The number i has a stronger claim to mathematical existence than most other numbers! The number i is indeed instantiated in the world every time you turn left at an intersection, or stand up, face east, then turn to the north. You multiplied your position vector by i. Do it twice, and the result is exactly as if you'd multiplied your initial position vector by -1. i^2 = -1. Done.

And now we come to this article full of gee-whizitude and I can't for the life of me figure out why physicists are going on like this. Just as the real numbers are operators that perform stretching and flipping, the complex numbers are operators that perform stretching, flipping, and rotating. It's truly that simple.

It is the case that the complex numbers were first discovered algebraically, as a byproduct of solving polynomial equations. But their true nature is geometric. They're simply rotation and scaling operators in the plane. You see this in the polar representation of a complex number re^(it) where r is the radius, or distance from the origin; and t is the angle the line segment to the complex number makes with the positive x-axis.

This is the meaning of Euler's famous formula e^(it) = cos t + i sin t. A complex number is just a direction on the plane, along with a scaling factor. This is the meaning of the famous special case posted by @socrat44, e^(-i pi) + 1 = 0. Let's unpack it. We start at the point (1,0) in the plane. We rotate it through an angle of pi radians, bringing it to the point (-1,0). We then add one, which shifts it to the right one unit, bringing us to the point (0,0), which is the same as the complex number 0. This is the geometric meaning of e^(-i pi) + 1 = 0. Start at (1,0). Rotate yourself about the origin by pi radians. Take one step to the right. You are now standing at the origin. That's it.

Now we can't measure arbitrary angles with perfect precision; so by that criterion, most complex numbers only have pure mathematical existence and can't be physically instantiate.

But the number i can. It represents a quarter turn counterclockwise in the plane. Next time you're out driving and you make a left turn, remember that you just multiplied your position vector by i and all will be clear. The number i is physically instantiated in the real world and we routinely experience it every day of our lives. Just turn left.

Moral of the story: Don't listen to physicists when they're doing bad math.
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by RCSaunders »

wtf wrote: Sun May 02, 2021 8:43 pm ...

First, as @RCSaunders says, even a familiar positive integer like 5 has no claim to physical existence. We can of course instantiate the abstract concept of 5 as 5 apples, 5 planets, or Five Easy Pieces. But the number 5, all by itself, has no instantiation in the world. It's a mathematical abstraction. Depending on one's philosophy you can say 5 lives in the higher world of Platonic forms, along with the Flying Spaghetti Monster and the Baby Jesus. Or if you deny superstition, 5 is a product of the human mind's quality of abstraction.

...
An elegant exposition. I mean the whole piece (which was too long to quote.) Thanks!
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by Eodnhoj7 »

RCSaunders wrote: Fri Apr 30, 2021 12:53 am
socrat44 wrote: Thu Apr 29, 2021 11:51 am Do Complex Numbers Exist?
Nope! Not even plain old counting numbers exist materially or independently of human minds. Numbers are nothing but a method of identifying some characteristics of material reality: multiplicity (counting) and relative physical characteristics like dimension and time (measurement).

Attributing actual metaphysical (ontological) existence to numbers is superstition.
Numbers are the quantification of forms, they are inseperable from forms.

The most basic form is the loop given all phenomena, as traceable, end at the same point they begin.

These forms are the grounding of reason as reason is the manipulation (change) of forms. The change of one form to another form is reason. Reason and form are inseperable thus necessitating all phenomenon as having a rational (conscious) base behind them.

The forms behind human awareness direct human awareness as human awareness is grounded in form. Form acts as the point of change from one form to another, or rather one change to another change.

As modes of change where a form is first unobserved then observed, which is what a change in observation is: first something is unobserved then observed, numbers as forms (through the universal loop which exists through all forms) exist independent of human awareness.
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by RCSaunders »

Eodnhoj7 wrote: Tue May 18, 2021 7:33 pm
RCSaunders wrote: Fri Apr 30, 2021 12:53 am
socrat44 wrote: Thu Apr 29, 2021 11:51 am Do Complex Numbers Exist?
Nope! Not even plain old counting numbers exist materially or independently of human minds. Numbers are nothing but a method of identifying some characteristics of material reality: multiplicity (counting) and relative physical characteristics like dimension and time (measurement).

Attributing actual metaphysical (ontological) existence to numbers is superstition.
Numbers are the quantification of forms, they are inseperable from forms.
...
That means just as much as saying numbers are pickles. It means nothing at all.

[There is something wrong with that boy!]
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by Eodnhoj7 »

RCSaunders wrote: Tue May 18, 2021 9:46 pm
Eodnhoj7 wrote: Tue May 18, 2021 7:33 pm
RCSaunders wrote: Fri Apr 30, 2021 12:53 am
Nope! Not even plain old counting numbers exist materially or independently of human minds. Numbers are nothing but a method of identifying some characteristics of material reality: multiplicity (counting) and relative physical characteristics like dimension and time (measurement).

Attributing actual metaphysical (ontological) existence to numbers is superstition.
Numbers are the quantification of forms, they are inseperable from forms.
...
That means just as much as saying numbers are pickles. It means nothing at all.

[There is something wrong with that boy!]
Quantify something without attaching it to a form....you can't.
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Re: Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

Post by attofishpi »

wtf wrote:In the physical world we can never cut a pie exactly in half. I assume people know that.
I soon worked this out, my sister would always get the bigger half.
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