By means of the derivatives of a 2×2 complex matrix, this post proposes that fermions and bosons would be the same topological spaces super symmetrically transformed through time, being fermions the +1/2 or -1/2 partial complex conjugate derivative of bosons and vice versa. Ordinary and complex conjugate equations of all variables could not operate independently of each other, but should be combined to avoid the deletion of half of the system on the description of the atomic nucleus.
A 2×2 complex matrix «A» whose 4 elements are + complex vectors, can be considered as a complex equation of second degree. The operations of transposition, complex conjugation, or inversion, performed by rotating the Z coordinates 90 degrees provide the derivatives of the cyclical complex function.
A first 90 degrees rotation of A gives us its first partial complex conjugate derivative on «Acc.»; it’s partial because although the whole coordinate rotate, only half of the vectors change their sign, becoming -. The complex conjugation effect is equivalent to a transposition in this context, but identifying the vectors with letters, we see the actual motion of the system:
Rotating Acc. 90 degrees, we get on «-A» the first partial derivative of Acc., changing their sign half of the vectors. As the whole system became negative in «-A», it can be thought that -A is the integer derivative of A, being a second-degree derivative. But Acc. is a partial derivative of degree 1, which implies that the first degree of derivation has not been yet completed. So, -A is not the second-degree derivative of A, it’s the first-degree complex derivative of A, where its imaginary part is the – ½ part of Acc and its real part is the -½ part of -A.
This distinction is fundamental because in this context A cannot be directly integrated by -A, bypassing the partial complex conjugate derivative that is Acc, because the degree 1 has not been completely derivate yet. Removing Acc. from the complex differential equation is going to imply the deletion of the description of half of the complex system, the part that is its mirror symmetric antisymmetric counterpart.
That will occur if a complex partial differential equation is used to describe complex systems where several rotational permutations are needed.
In this sense, continuing the rotations, we will see that:
Acc is the ½ i derivative of A
-A is the ½ r derivative of Acc
-Acc is the ½ i derivative of –A
A is the ½ r derivative of -Acc.
So, we could say that ½ Acc(i) + -½ -A(r) is the first-degree complex derivative of the complex function, and that –½ -Acc(i) + ½ A(r) is the second-degree complex derivative of the complex function.
It also could be thought that the function A could be integrated with the –½ part of the partial complex conjugate derivatives Acc and -Acc, and that -A could be integrated by the +½ part of both partial complex conjugate derivatives.
But what is fundamental is to be aware that removing from the differential equation the complex conjugate derivatives, or the derivative of any variable, the given description will be only of ½ of the complex system.
Each partial derivative will depend on the transformation of the previous partial derivative, and the integer derivatives will depend on the previous partial derivatives; all the variables, the 2×2 matrix elements, are in this way interconnected and cannot be bypassed without affecting the entire description of the whole complex system.
If we represent the sequential evolution of the system in a linear way, it would be something like this:
You will recognize on the above figure the classic representation of an electromagnetic wave, but here the «electric field» (on the Y coordinate) and the «magnetic field» (on the X coordinate) are not different spaces that are perpendicular nor conjugate, they are the same space that evolves through time, experiencing an orthogonal upward displacement of energy and mass on A, then a horizontal right handed displacement on A complex conjugate, then an orthogonal downward displacement on -A, and then a horizontal left handed displacement on -A complex conjugate, while completing a 270 degrees rotation.
Why do the classical EM fields experience such displacements? Are we sure the electromagnetic wave is formed by two interacting waves or is it a same wave evolving through time? Does the electromagnetic wave rotate? Very interesting.
A different attempt to represent simultaneously all the related complex functions would be this diagram:
Some points where the functions intersect would be equal zero. In this sense the differential equation could be thought as a Riemann Z function where the zero points, where the vectors or numbers are not divisible would indicate a prime number.
The functions would be holomorphic except in some points near a «critical line», specially at one pole of the differential equation where they would be meromorphic, non-complex differentiable.
At this respect it could be interesting to realize that the –½ A complex conjugate derivative of 1 order and the -½ -A complex conjugate derivative of 2 order will form an integer complex conjugate derivative with respect to A. But that possibility will not work for –A (which is one of the poles of the differential equation) because the +½ -A complex conjugate derivative is of a higher order than –A. So, there’s not option of forming an integer + complex conjugate derivative with Acc and –Acc for –A.
On the other hand, the Schrodinger equation is a partial differential equation of second degree, which implies it does not combine all its derivatives; or it uses the derivatives of a part of the variables.
Being of second order, it’s thought that it offers a separate complex conjugate alternative equation:
Fermions and bosons, have been developed by quantum mechanics as two different kinds of unrelated spaces or particles. The complex Schrodinger function probabilistically describes the mirror symmetric bosonic particles with integer spin not ruled by the Pauli Exclusion principle, while its alternative complex conjugate function probabilistically describes the mirror antisymmetric fermionic particles with 1/2 spin, ruled by the Pauli Exclusion principle.
But the 2×2 matrix derivatives shows that the complex conjugate equation is not an independent equation, it’s a part of the whole complex equation, and removing one of its derivatives or one of its variables, the picture will be incomplete.
The complex matrix derivations show that combining all the partial derivatives on a complex differential function, bosons and fermions would be the same topological spaces being transformed through time when synchronising and desynchronising their phases of variation while the nucleus rotates.
The periodical transformations of bosons in to fermions and vice versa would occur by means of each partial complex conjugate derivative of the previous variation.
Each permutation causes a change on ½ of the rotational system.
In this sense, the nuclear spaces will act as fermions after being –½ transformed by the complex conjugate partial derivative of A, or +½ transformed by the complex conjugate partial derivative of -A. That explains why fermions have ½ spin.
Those same spaces will act as fermions when the successive ½ derivatives complete a whole derivative, of first order in the case of –A or of second order in the case of A.
A periodic alternation between imaginary and real partial derivatives will occur.
If we mapped on a same space simultaneously the different vector status and quantum particles given by the functions that represent different successive moments of time, we see that the whole system is symmetric. Being a symmetry reached through time, it can be said the nucleus is supersymmetric.
Supersymmetric particles, that should link the separate types of fermionic and bosonic particles, have been predicted by mainstream models and looked for particle accelerators. But considering the forgotten mirror symmetric part of the nucleus no new supersymmetric particles are needed.
Looking at the diagram, bosons appear on the imaginary points, and fermions on the real ones. They are imaginary or real for us because we are looking the system from a real point of view given by the non-rotated XY coordinates. But considering A is placed on a real coordinate, then bosons will be on the real points will fermions will be on the complex conjugate or imaginary points. It will depend on our referential frame.
It also can be seen that to superpose all of the successive status of the atomic nucleus simultaneously it’s necessary to add two X coordinates displaced to a projected + or – imaginary point as X+i and a X-i coordinates. The same will be necessary with Y-I and Y+i .
X+i and Y-i will represent an expansion of the real space, while X-i and Y+i will represent a spacial contraction. That is a consequence of the up down and left right movements of the nucleus while rotating.
Keeping fixed XY coordinates, and creating a complex system by rotating the Z coordinates that introduce the imaginary part, implies an expansion or a contraction of the represented space.
The next diagram shows in a separate way the derivatives of each variable, the 4 elements of the 2×2 complex matrix, and how they would be arranged by pairs forming 16 groups of symmetry related to each subatomic space or particle.
I think the -1/2 partial complex conjugate derivatives of A (Acc and -A), and the +1/2 partial complex antiderivatives of -A (-Acc and A) can also be expressed in terms of Fourier transforms.
In that context, the -1/2 complex conjugate derivative Acc would be the Fourier transforms of A.
And +1/2 complex conjugate derivative -Acc would be the inverse Fourier transform. To get the inverse Fourier it would be necessary to perform a triple Fourier transform, a triple partial complex conjugate derivative of A (-1/2Acc + -1/2-A + 1/2-A).
A discrete time Fourier transform produces, from uniformly spaced samples, a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
https://en.wikipedia.org/wiki/Fourier_series
«A Fourier series is a way of writing a periodic function or signal as a sum of functions of different frequencies»:
On the other hand, this article is part of a dual atomic model where the atom is thought as a dual system of two intersecting spaces vibrating with same or opposite phases with a shared nucleus of 2 orthogonal and 2 transversal subspaces vibrating with same or opposite phases that synchronise and desynchronise periodically.
The vectors would represent the forces of pressure o decompression caused by the intersecting spaces while expanding or contracting.
In a QCD context they could be identify as ¨quarks¨.
The strong interaction of the system will be given by the inner orbital motion of the contracting subspaces, the weak interaction by the inner orbital motion of the expanding subspaces, and the EM interaction by the forces of pressure caused by the electron-positron subspace when moving left to right and vice versa.
From the rotatory nucleus it seems the nuclear spaces would synchronise or desynchronise their phases of vibration by means of the transformation caused by the rotation itself.
The A matrix represents a transversal space determined by ac, and its mirror symmetric space determined by bd, and two orthogonal spaces determined bb ab and cd respectively.
The ac and bd spaces have mirror symmetry at the same moment, their phases are synchronised. The picture given by A would be the state where the highest level of expansion of ac and bd, being the + limit of the function that represents the variation of the spaces given by their contraction or expansion.
The next quantum state given at a future moment at –A will represent the – limit of the function, when both transversal spaces, represented now by -db and -ca, reach their higher level of contraction.
But if the fermionic transformation is intercalate between the two bosonic moments, between each expansion at A or contraction at –A, by means of the complex conjugate partial derivatives that are Acc and –Acc, two additional limits must be added to the function: Acc will represent the moment of where the left transversal space reaches its highest level of expansion and the right transversal space its highest degree of contraction, while –Acc will represent the contrary case where the left transversal space reaches its highest level of contraction and the right transversal space reaches its highest degree of contraction.
A very wonderful book to get a historical understanding about how the classical notion of space radically determined the development of the two main models, the Heisenberg and Born’s purely abstract matrix and later the more visual Schrodinger’s wave function, the disputes that emerged between them, and the prevalence of the limiting probabilistic interpretation, is «Quantum Dialogue. The making of a revolution» by «Mara Beller».
A crucial moment that made Heissenberg and his Gottingen group was when they realized that it was the classical notion of spatial continuity given by the classical concept of wave what caused the main problems on the initial atomic developments, it was necessary to break with that classical continuity to be able to describe the discontinues quantum atomic phenomena. And they found the way to do it through the symbolic abstraction of matrices.
The rotational atom I propose on this post explains discontinuity by means of the rotation of the atomic nucleus that causes the synchronization and desynchronization of the phases of vibration of the nuclear spaces; and it lets relate the apparent abstraction of a complex matrix (the more visual vectorial matrix could be related to «jacobian» matrices, I think) the description of the differential equation and the representation of its wave functions, with a visually physical model of intersecting spaces – or quantum fields – where space and time have an essential role in the description of the atom as a «supersymmetric» topological structure.
I added a link to the Google preview page:
It’s very interesting to see that although the Gottingen group gave up altogether to find a geometrical representation of the atom, they still had their intuitions about what should the geometry of the atom be. So, the book mentions for example that for Born «virtual oscillators were the real primary thing» and the «interaction of electrons inside the atom consists of a mutual influence exerted by virtual resonators on each other»; Heisenberg assumed that «something in the atom must vibrate with the right frequency»; or Pauli described the atom as «a collection of harmonic partial vibrations associated with transitions between different stationary states, and not as constellation of particles tied kinematically to the occupation of certain stationary states».
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You also could find interesting this article by Carlo Rovelly about his «Relational interpretation» of quantum mechanics and the misleading notions of the Schrodinger equation: https://arxiv.org/pdf/2109.09170.pdf
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An interesting article, although quite technical, appeared recently at Nature showing the necessity of complex numbers in quantum mechanics. It was experimentally proved: Quantum theory based on real numbers can be experimentally falsified
Related to the same idea about how theories that use real numbers instead of complex numbers to describe quantum mechanics have been falsified, you can read this recent interview to Jian-Wei Pan: Jian-Wei Pan: ‘The next quantum breakthrough will happen in five years’
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Have a great week
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Pdf version here:
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