Quaternionic foundations of Quantum Mechanics and spin 1/2 visualization

Abstract: This talk explores the intriguing link between quaternion quantum mechanics and the model of the elastic continuum proposed by Augustin Cauchy, highlighting the potential of this interdisciplinary connection to advance our understanding of quantum physics. We elucidate the derivation of non- and relativistic wave equations from the Cauchy theory and Hamilton quaternion algebra, demonstrating how the quaternion representation of the elastic continuum provides a comprehensive picture of quantum mechanics. This innovative interpretation addresses the ontological challenges often associated with quantum theory, offering a means to generalize and test the theory. Transitioning from the theoretical to the practical, we delve into a unique visualization of spinors using quaternions. We elucidate how the quaternionic representation enables a comprehension of the Dirac equation and the concept of spin 1/2, pivotal to the Planck-Kleinert model and the Pauli Exclusion Principle. We present an interactive simulation of a periodically twisting 3D grid, visualizing the spin 1/2 behavior of fermions in an elastic solid.

By visualizing Dirac spinors using two orthogonal quaternions, we illuminate the complex rotation behavior of spin 1/2 particles, drawing analogies with the famous Dirac belt trick. We define two quaternion functions that represent rotations as functions of time and distance, respectively. The visualization of the resultant spinor function as a series of concentric shells offers a illuminating representation of quantum spin, enhancing our understanding of its behavior in both time and space.

Keywords and phrases: quaternion

Computer scienceMathematicsPhysics

Audience: researchers in the topic

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Hypercomplex Seminar 2023

Series comments: Zoom LINK:

Join Zoom Meeting

zoom.us/j/96430665805?pwd=UnR3N0paMUkvazhNZHZPTVFxWDFLdz09

Meeting ID: 964 3066 5805 Passcode: 658094

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Hypercomplex Seminar is organized by Lodzkie Towarzystwo Naukowe and by following people

1. Dariusz Partyka (Catholic University of Lublin )

2. Mariusz Zubert (Lodz University of Technology)

3. Malgorzata Kempczyk (Catholic University of Lublin)

4. Andrzej Michalski (Catholic University of Lublin )

5. Krzysztof Pomorski (Technical University of Cracow + Quantum Hardware Systems company [ www.quantumhardwaresystems.com/ ] )

Conference is free for Everybody and Everybody is invited. Please contact kdvpomorski@gmail.com or HyperComplexSeminar@gmail.com for further details. Conference webpage is www.hypercomplexseminar.com/ and YouTube channel is : www.youtube.com/@hypercomplexseminar .

Schedule is given in Warsaw Time

Day I: Invited Lectures [9th July 2023, Sunday] (17:45-21:00):

Day1.Opening (17:45-18:00) [9th July 2023, Sunday]: Dariusz Partyka and Mariusz Zubert: Welcome Talk.

Notation: A-Mathematics , B-Physics , C-Computer Science , D-Electronics ,

Day1.A: Sergiy Plaksa [9th July 2023, Sunday] (18:00-19:00):

Monogenic functions in infinite-dimensional vector spaces with a commutative multiplication and harmonic vectors

Day1.B: Marek Danielewski [9th July 2023, Sunday] (19:00-20:00):

Quaternion Quantum Mechanics II: Unraveling the Mysteries of Gravity and the Dirac Equation within the Planck-Kleinert Crystal

Day1.(C-D): Chantal Roth [9th July 2023, Sunday] (20:00-21:00):

Quaternionic foundations of Quantum Mechanics and spin 1/2 visualization

Day II: Hypercomplex Analysis [10th July 2023, Monday] (10:00-18:00)

Session A1 (Hypercomplex structures)

10:00-10:50: Lino F. Reséndis, Qp spaces for hyperholomorphic functions in the unit ball of R4

11:00-11:25: Serhii Gryshchuk, Representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra

11:30-12:20: Massimo Vaccaro, The full set of invariants characterizing the Sp(n)-orbits in the real Grassmannians GR(k, 4n)

Session A2 (Complex and real analysis of one variable),

13:00-13:25: Olga Rovenska, Approximation of classes of Poisson integrals by Fejér means

13:30-13:55: Ruslan Salimov, On the exponential asymptotic of solutions of the nonlinear Cauchy-Riemann-Beltrami type equation

Day III: Mathematics for physics [11th July 2023, Tuesday] (10:00-18:00)

Session A3 (Holomorphic functions of one variable),

10:00-10:25: Iryna Denega, Evolutionary type inequalities for products of inner radii

10:30-10:55: Maciej Parol, The Koebe radius for certain class of polynomials

11:00-11:25: Szymon Ignaciuk, Physical interpretation of the membership to Kaplan classes of certain functions

Session A4 (Complex and real analysis of several variables),

12:00-12:25: Anna Kimaczynska, The grad div operator

12:30-12:55: Valery Volchkov, Vitaly Volchkov, Continuous mean periodic extension of functions from a segment

Session A5 (Poster session),

13:00-13:30: Jacek Dziok, On multivalent prestarlike functions

13:00-13:30: Natalia Zorii, On the role of the point at infinity in Deny's principle of positivity of mass for Riesz potentials

Day IV: Physics [12th July 2023, Wednesday] (10:00-18:00)

Session B1 (Physics: Ontology of Quantum Mechanics),

10:00-10:50: Jaroslaw Duda, Maximal Entropy Random Walk to understand Born rule and Bell violation

11:00-11:50: Manfried Faber, Running coupling from a classical soliton model

12:30-12:55: Lukasz Stepien, On some exact solutions of certain nonlinear partial differential equations

13:00-13:50 Jaroslaw Duda, Topological charges with electromagnetic + gravitomagnetic interactions

Session A6 & B (Advanced mathematical methods in physics),

14:30-15:20: Osamo Suzuki, Binary and ternary structures in evolution theory (VI) An entropic description of the evolution of the universe

15:30-15:55: Malgorzata Nowak, Iterative construction of discrete Laplacians on 2D square and hexagonal lattices using BTBB and other sequences

16:00-16:25: Armen Grigoryan, Landau and Bloch theorems for planar harmonic functions

Day V: Classical/Quantum Computer Science [13th July 2023, Thursday] (10:00-18:00)

Session C1 & B (Quantum Computer Science),

10:00-10:50: K. Gnatenko, Geometric measure of entanglement of variational quantum states and its quantifying on a quantum computer

11:00-11:50: D. Kotula, K. Pomorski, Classical and Quantum Conway Game of Life

Session C2 & B (Classical/Quantum Electronics and Machine Learning),

12:30-12:55: M. Kowalik, Machine Learning in Cryogenic Experiments

13:00-13:25: M. Zubert, Z. Kulesza, M. Jankowski, A. Napieralski,

Application of Scattering Parameters to The DPL Time-Lag Parameter Estimation using MEMS Structures

Session D1 & B (Quantum Electronics),

13:30-13:55: M.Zlotkowski, Local systems management of different energy sources

Discussion panel (14:00-14:30) Chairmen: Sergiy Plaksa, Marek Danielewski, Chantal Roth

Day VI: Quantum Electronics with Physics (D+B) [14th July 2023, Friday] (10:00-18:00)

Session D2 & B (Quantum Electronics),

10:00-10:55: M. Belogolosvskii, Two new issues in superconducting electronics: transparent samples and SQUID-like Josephson junctions

11:00-11:55: J. Sosnowski, Influence of quantized vortex dynamics on superconducting electronics

12:00-12:25 V. Vachtsevanos, Emergence of topological defects in finite 2-D lattices during second phase order transitions

Session A7 (Geometrical methods in physics),

13:00-13:25 A. Paszkiewicz, Does the convex Peano curve exist and what can we say about operators? Session C3 & B & D (Computer Science in Quantum Electronics),

14:00-14:25 K. Pomorski, Theory of interface between semiconductor quantum dots and Josephson junction

14:30-15:20: B.Stojewski, K. Pomorski, Hybrid Schroedinger-GL algorithm in solving problems 15.30-15:55: L.Pluszynski, K. Pomorski, Solving Schroedinger equation with analog electonics 16:00-16:25: L.Pluszynski, Simulation and experiments with superconducting single photon detectors

Closing ceremony (16:30-16:45)

Day VII: Invitation to dr Jarek Duda Seminar [15th July 2023, Saturday] (16:00-17:00)

Sergey Rashkovskiy (RAS) Quantum Mechanics: Strange Particle Theory or Classical Field Theory?

[ us06web.zoom.us/j/81719371334?pwd=V2oxMTAxQkVKb3AzN1hmWUtlMzVTZz09 ]

(Web: th.if.uj.edu.pl/~dudaj/QMFNoT)

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