What is Fundamental is the area of the imaginary unit by Jouko Harri Tiainen

Hello Jouko,

I really love those essay which emphasizes mathematics. As soon as I saw your topic containing Imaginary units, I hurried to read your essay. A well-written essay. I had some questions to ask but already found the answer on above post.

My essay is also related to mathematics:Is Mathematics Fundamental?

Kind Regards

Ajay Pokharel

Dear Jouko Harri Tiainen, The imaginary unit is relevant for physics; however, I think it describes the rotation of space well. Multiplication by an imaginary unit gives a rotation of 900, and multiplication by a square of an imaginary unit rotates the space by 1800, and so on. The wave function describes the rotation of space as a function of the momentum and energy of the particle.The physical space, which according to Descartes is matter, serves as the foundation for the birth of life. Look at my essay, FQXi Fundamental in New Cartesian Physics by Dizhechko Boris Semyonovich Where I showed how radically the physics can change if it follows this principle. Evaluate and leave your comment there. I highly value your essay; however, I'll give you a rating as the bearer of Descartes' idea. Do not allow New Cartesian Physics go away into nothingness, which wants to be the theory of everything OO.

Sincerely, Dizhechko Boris Semyonovich.

Hi Jouko,

Wow, I really had to keep concentration to read your essay. I like your way of "logical thinking" and valued it in order to bring you up in the contest. For me, however, the real conclusions were in your appendix.

Some remarks:

"continuous observation" When talking about continuous as a process of time, we are always becoming aware of the results of our observations later as they happen, we are always observing the PAST. The continuity we are conscious of may be an illusion (emergent phenomenon). Does this influence your thoughts?

A little remark on "quantum theory of immortality". If you are taking more then the normal age a cat survives (for instance 20 years) then the cat will die of age. This theory, in my opinion, hasn't the right name. It IS all right when we could arrange an infinity of measurements in one moment (whatever length that may have, but take the Planck time)...But...as time is an emergent phenomenon and EACH measurement is taking place in this illusion, we can go on and on....because it is NOT THE SAME emergent cat...

"let there be a superposition of all heads that is pure, allow one impure state for the UWF Universal Wave Function itself, which can act via entanglement as the wave equation for the pure superposition.". This superposition in my model is just ONE of the Reality Loops available as a probability.

In my contribution, I propose a new model of emergent reality that has not the problems of the MWI, Schrödingers Cat, the double slit experiment, paradoxes of Xeno etc, the collapse of the wave function, etc. I wonder what are your thoughts about my essay "Foundational Quantum Reality Loops", so I hope that you can spare some of your time-area to read, comment and maybe rate it.

Best regards and good luck

Wilhelmus de Wilde

Dear Jouko Harri Tiainen, The imaginary unit is relevant for physics; It is very interesting - the area of the imaginary unit, however, I think it describes the rotation of space well. Multiplication by an imaginary unit gives a rotation of в€Џ/2, and multiplication by a square of an imaginary unit rotates the space by в€Џ, and so on. The wave function describes the rotation of space as a function of the momentum and energy of the particle.The physical space, which according to Descartes is matter, serves as the foundation for the birth of life. Look at my essay, FQXi Fundamental in New Cartesian Physics by Dizhechko Boris Semyonovich Where I showed how radically the physics can change if it follows this principle. Evaluate and leave your comment there

Sincerely, Dizhechko Boris Semyonovich.

Dear Jouko,

Thanks for leaving a comment on my blog. I looked at your paper and let me get the bad news out of the way:

If you are working on these ideas for yourself, then more power to you. But I have the impression that you mean your work to be considered and eventually accepted by others, in particular physicists and mathematician. I regret to tell you that in its present form, your work has pretty much no chance to be taken seriously by any physicists or mathematicians.

The problem is not necessarily with your overarching idea, that we should consider the reals in terms of areas. After all, both length and area are measures and so perhaps there is a valid novel interpretation of imaginary units there somewhere.

No, the problem is with how you go about trying to demonstrate it:

You take a large number of unjustified steps, make many illegitimate moves and mistakes so that your conclusions simply don't follow. There are far too many for me to point out, so I will limit myself to a couple.

You say that you want to make the speed of light the imaginary unit. It is true that one can work with special relativity in a Euclidean metric by substituting [math]ict[/math] for [math]ct[/math] but that does not mean that $c$ itself can be set equal to the imaginary unit. If we could, then we should write $E=m i^2=E=-m$, which means that rest energy is negative. That is simply incorrect. Physicists do use $c=1$ but that is just notation, in that notation it is $E=m$ which is consistent with what we know.

2. You introduce brackets into general relativity without clearly saying what they mean (you call the bra the "constant of closure" for GR, which makes me wonder whether you know the difference between the concept of a field in mathematics and the concept of a field in physics), only to later claim that they somehow represent probabilities. While brackets may be used in semi-classical quantum gravity, they are utterly foreign to general relativity. If you are going to introduce them, you have to, for example, show how it does not lead to any contradictions in the theory.

There is much, much more but it does not add much for me to go on.

My suggestion is as follows:

Like I said, your basic idea may have merit, but how you go about trying to show it, mixing math and physics in all sort of illegitimate ways will not get your work accepted. In my view, you should

1. Separate out the physics entirely until you have the math worked out.

To work out the math you need a stronger math background. I would suggest either taking courses or self-study (but then you have to do the problems still) of the following subjects:

a. Introductory set theory. The second half of such a course covers how sets code numbers, starting with the natural numbers all the way to complex. Learning this will broaden your horizon and take the focus off the numbers themselves to the relationships that define different number sets.

b. Introductory abstract algebra: In such a course, you will learn the difference between a group, a ring, a module and a field. Again, this will provide you with a perspective in which it is the relationships between the numbers that provides the deepest sort of understanding

c. Analysis: Really, what you might need is measure theory, but that is a grad-level subject, and analysis is sort of the the simpler version of it. In this course you will learn how to give proofs, mathematical arguments which do not suffer a lot of the problems your arguments currently suffer from.

2. Only after you have worked the math out, start looking for applications in physics

I should note that I have the impression that while you understand many isolated physics concepts, I am not sure you have an understanding that integrates them. As the saying goes, the amateur sees 1000 facts where the expert sees one fact structure. So keep learning physics, too, but don't try to apply your math idea until you have worked it out as a purely math concept.

I have provided you with tough but honest criticism because this is how I see our role as participants in search of deeper understanding. The search of truth requires criticism whenever warranted, not pats on the back when they are not warranted. I wish you good luck in your endeavor.

All the best,

Armin

Dear Jouko Harri Tiainen,

I admire Armin's work very very much, but I don't think I agree with all of his statements, perhaps because I ignored your use of bra and ket, and also your treatment of entanglement. I pretty much ignore everyone's treatment of entanglement, for reasons I have already published, but as it is a common belief today, I do not generally downgrade essays for expressing this belief, or even a novel way of trying to make sense of it. Armin makes some good points, such as E=-m if i=c. Perhaps i~c would be more appropriate? You do use +i and -i so one might get E=m...

Nevertheless, my perspective here is that you are simply letting the speed of light take on a unit value and similarly Planck's constant take on unit value and you are trying to make sense of the imaginary i in key physics equations.

Why is that i there?

I have concluded, with many others, that geometric algebra is the most powerful tool available for physicists today. In geometric algebra the function of i is that of a duality operator, which transforms the element it is operating on into its dual. That is how I'm interpreting your work. As I say below, your essay (for me) requires more study, but I do not dismiss it out of hand. Perhaps because physicists are so comfortable with complex analysis and so used to using the imaginary i in Minkowski geometry and Schrödinger's equation they see no need to think further. For pure geometry this is probably reasonable, but physicists tend to treat the i in quantum mechanics as somewhat mystical. Again, I want to spend more time thinking about this, and I will do so in the framework of the geometric algebra duality operator.

By equating i to the speed of light (i=c) you suggest that the speed of light is a "constant of motion" if "the laws of physics (or the equations) are the same in all inertial reference frames."

If one believes as Einstein, that "space does not exist absent of field" and that the gravitational field fills space, then the Galilean invariance of the Maxwell-Hertz equations implies only one time dimension, and this is consistent with constant speed of light in a local gravity frame. Coordinates fixed in the gravity frame see constant c. But for other objects moving in the frame with velocity v, the constant local c appears as c+v from the perspective of elapsed time. This preserves the geometry of the Minkowski differential, without implying different time frames.

You then postulate that the mathematical definition of +i and -i can be associated with GR (c=i) and QM (h=i). That is truly fascinating, and may relate to the energy-time conjugation I develop in my essay. My own interpretation of the relativity of a self-interacting field (such as gravity) leads to unidirectional time. I will try to see how to understand this in terms of your postulate. The Minkowski geometry does not imply multiple time dimensions. It is compatible with 'same time' Lorentz formulations in one inertial frame.

You interpret h=i in Schrödinger's equation to satisfy 'Planck's quanta is constant' and "all time is equal for all observers", compatible with time as universal simultaneity. As I mention above, my own interpretation of the 'imaginary' i is as represented in geometric algebra, i.e., i is the duality operator that transforms one element of geometric algebra into its dual.

I think this part of your essay is potentially very deep and requires thought. I plan to give it more thought and will score it accordingly. Congratulations.

Best regards,

Edwin Eugene Klingman

Thank You for your in-depth comments and for the long long list of the many courses I should take to overcome the conceptual hurdles in my essay and my complete and obvious lack of knowledge in all fields of maths, science and philosphy. I have to thank you for pointing out a few of the many assumptions and "leaps of logic and ad hoc bounds of thought" in the essay. I will consider "these conceptual faults again" after I have done many many years of learning and integrating the ideas in your recommended courses -- in basic maths and elementary science -- and then I will try to start again with these ideas so as to form a more cogent flow of ideas for others to follow. Cheers Thanks ... Harri.

The only point I have to make is that Minkowski was the one who "made c(m)=i(s)" here is a direct quote --- Minkowski's Paper - Minkowsky, Hermann, German paper Raum und Zeit (1909), Jahresberichte der Deutschen Mathematiker-Vereinigung, 75-88. In the 1920 English translation...We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula 3x108(metre)=в€љ-1(second)... www.en.wikisource.org/wiki/Space_and_Time

And since he can do that I thought why not do 6.63Г--10-34(Joules)=в€љ-1(second) which leads to the idea that space-time and energy-time conjugation come from the same source the indistinguishable "imaginary unit" which then goes to the idea .

Basically the essay is all about what is the "imaginary unit" all about and why it is in all of the basic equations of physics. The basic idea is if you want a dual maths then use an area we get duals easily that way.

Thank you for bringing to my attention to Minkowski's paper. The link you provided does not work, but his original paper in German is available here:

https://www.math.nyu.edu/~tschinke/papers/yuri/14minkowski/raum-und-zeit.pdf

the passage you quoted is on page 86, starting with the 2nd paragraph. I will translate it (German is my native language):

"One can from the outset choose the relationship between the unit length and unit time such that the natural speed limit is c=1. If one introduces $sqrt{-1}times t=s instead of $t$, then the quadratic differential expression

[math]dtau^2=-dx^2-dy^2-dz^2-ds^2[/math]

is completely symmetric in $x,y,z,s$, and this symmetry transfers to any law which does not contradict the world postulate [he used this term on p 82 apparently for a mathematical version of the principle of relativity, and which he uses as a justification for considering space and time on an equal footing]. One can then clothe the essence of the postulate mathematically in the pregnant mystical formula

[math]3times 10^5 km=sqrt{-1}s [/math]"

I understand now why you might you thought that one can substitute the imaginary unit for $c$. However: the metric in the equation above is Euclidean, as all the terms on the right have the same sign. If this is not easy to see, suppose we symbolize the spacetime interval by $r$ (normally we would use $s$, but Minkowski is already using it in this formula), then we can define

[math]dtau^2=-dr^2[/math]

then all the minus signs turn into plus signs if we set the right sides of the two equations equal to each other:

[math]dr^2=dx^2+dy^2+dz^2+ds^2[/math]

The point is, in his operation he did not actually consider $c=i$ but rather $s=it$, which turns the geometry into Euclidean four-space and his "mystical formula" does hold in that geometry. But if you take your geometry to be Euclidean fourspace instead of Minkowski spacetime, then you have to also modify all other special relativistic equations accordingly. For example, the momentum four-vector, which in Minkowski spacetime is

[math](E/c, mathbf{p})[/math]

under the Euclidean metric becomes

[math](mathbf{p}, iE/c)[/math]

so that the Minkowski norm of the momentum four-vector, which is

[math]frac{E^2}{c^2}-p^2=m_0^2c^2[/math].

under the Euclidean metric becomes

[math]-left ( frac{-E^2}{c^2} right )-p^2=m_0^2c^2[/math]

Notice that in order to make this come out right, the right hand side under the Euclidean metric has to have the opposite sign as the Energy, whereas under the Minkowski metric it has the same sign. That implies:

[math]m_0^2c^2_{Minkowski}=-m_0^2c^2_{Euclidean}{[/math]

and since in both cases we consider the rest mass $m_0$ to not change sign under either metric, it has to be the case that

[math]c^2_{Minkowski}=-c^2_{Euclidean}[/math]

From which it follows that

[math]c_{Euclidean}=ic[/math]

I used the unusual notation to drive home that Minkowski's "mystical formula" is just the result of considering special relativity in a Euclidean metric.

I am really sorry about this misunderstanding, this is an extremely subtle issue which is not easy to catch unless one has already thought about it for a fair bit of time. Euclidean spacetime is not popular these days, but one textbook that uses it is Lawden, Tensor calculus, Relativity and Cosmology.

Regarding your other remarks: The courses I recommended are not elementary. If I had thought that you are not up to the task, I would have either refrained from giving any suggestions or suggested courses like elementary algebra or arithmetic. The courses I recommended are those which tend to turn the student into a mathematician, so I was not as condescending as you seem to think. But be that as it may: your main idea might have merit, but your methods to show it need to improve. And I went through all this not to put you down, but to help you.

Armin

So the compiler on this website is crap. I will rewrite the formulas that did not come out right:

(E/c, p) is the Minkowski four-vector, (p, iE/c) is the Euclidean fourvector.

the next formula is

E^2/c^2-p^2=m^2c^2

the one after is

-(-E^2/c^2)-p^2=m^2c^2

then the formulas come out right.

One cannot set c=i. "c" is a physical quantity. "i" is a mathematical quantity. Physical quantities are given by the product of a value and an unit. So expressions as c=i are meaningless.

Yes, I saw Minkowski writting (or the people that traduced his work from German) expressions such as "3·105 km = sqrt(-1) Sec", but such expressions are meaningless.

What we can do is reparametrize time as (t --> it) and use a natural system of units just to get a more symmetric expression for the element of line

ds2 = - (dt2 + dx2 + dy2 + dz2)

Alternatively we could just reparametrize the speed of light as (c --> ic) and obtain again a symmetric expression by imposing a natural system of units.

But nothing of this modifies the physics. Physics does not vary by a change in units or by modifying the labels we use to represent things.

The above changes turn the Minkowskian element of line into a symmetric Euclidean form, but other parts of the formalism are antisymmetrized. For example the equation of continuity is broken because the rate term (d/dx0) transforms into an imaginary quantity when we do (t --> it) or (c --> ic).

The equations x2 + 1 = 0 and x2 + 12 = i2 02 are identical. The solution to the second is identical to the solution to the first.

Similar remarks about setting i=h. This is a meaningless expression. Even if we ignore that, when you write the Schrödinger equation as (ihbar ðPsi/ðt = H Psi) and next claim h=i, we can check that replacing h by i in the equation breaks the imaginary term and the equation no longer describes quantum phenomena.

The set of equations in the Hypothenuse box are also incorrect. One can add and subtract the equations and obtain invalid results. One can also just set b=0 and obtain the invalid result i=0.

"Conclusion Winger's question on the nature of physics and mathematics can be addressed rigorously using the fundamental concept -- a number is an area. What is fundamental? A number is an area not a length." A number is neither an area nor a length, a number is a mathematical quantity. Areas and lengths are represented by physical quantities.