FLORENTIN SMARANDACHE & V. CHRISTIANTO NEUTROSOPHIC LOGIC, WAVE MECHANICS, AND OTHER STORIES
SELECTED WORKS 2005-2008
FLORENTIN SMARANDACHE & V. CHRISTIANTO
NEUTROSOPHIC LOGIC, WAVE MECHANICS, AND OTHER STORIES
SELECTED WORKS 2005-2008
KOGAÏON ÉDITIONS, 2009
1 The front cover image represents the first author at a conference in Indonesia in 2006 showing the journals "Progress in Physics" and "Infinite Energy".
KOGAÏON ÉDITIONS, 2009 Division of CASA GROUP OF COMPANIES SRL ,Κcoala Her۷str۷u street, code 014153 ,62 Bucharest - Romania, Europe Website: www.kogaion-editions.com e-mail: [email protected] Tél./fax: 0040-21-317.01.14
Copyright 2009 by Kogaïon Éditions and the Authors.
Plenty of ebooks can be downloaded from the following Science Digital Library: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm
Peer Reviewers: (a) Dr. M. Khoshnevisan -- School of Accounting and Finance, Griffith University, Gold Coast, Queensland 9726, Australia (b) Mark Alford, AFRL/RIEA, 525 Brooks Road, Rome, NY 13441-4503, USA (c) Alok Dhital, Math & Sciences Dept., University of New Mexico, Gallup, NM 87301, USA
(ISBN-13): 978-973-7609-31-1 (EAN): 9789737609311 Published in Romania
2 FOREWORD
There is beginning for anything; we used to hear that phrase. The same wisdom word applies to us too. What began in 2005 as a short email on some ideas related to interpretation of the Wave Mechanics results in a number of papers and books up to now. Some of these papers can be found in Progress in Physics or elsewhere.
It is often recognized that when a mathematician meets a physics-inclined mind then the result is either a series of endless debates or publication. In our story, we prefer to publish rather than perish.
Therefore, our purpose with this book is to present a selection of published papers in a compilation which enable the readers to find some coherent ideas which appear in those articles. For this reason, the ordering of the papers here is based on categories of ideas.
While some of these articles have been published in book format elsewhere, we hope that reading this book will give the readers an impression of the progress of our thoughts. A few other papers are not yet published elsewhere, or being planned to publish in other journal.
We wish to extend our sincere gratitude to plenty of colleagues, friends and relatives all around the world for sharing their ideas, insightful discussions etc. Special thanks to D. Rabounski, S. Crothers, L. Borissova for their great service in Progress in Physics journal.
One of these authors (VC) would like to thank to Profs. A. Yefremov and M. Fil’chenkov for all hospitality extended to him in the Institute of Gravitation and Cosmology of PFUR, where this book is prepared. Discussions with Prof. V.V. Kassandrov, Prof. V. Ivashchuk, & Prof. Yu P. Rybakov are appreciated. Many thanks also to Dr. S. Trihandaru and others from UKSW, Central
3 Java,Indonesia. Sincere thanks to good friends in PFUR, especially to D. Kermite, Y. Umniyati, Anastasia Golubtsova & Serguey– all other friends are of course worth mentioning here, but the margin of this book is quite limited to mention all of you.
And to all other scientist colleagues, allow us to say: Full speed ahead!
FS & VC, March 2009
4 NEUTROSOPHIC LOGIC, WAVE MECHANICS, AND OTHER STORIES: SELECTED WORKS 2005-2008
CONTENT
Foreword 3 Content 5
A. Unmatter and the Elementary Particles
1. “A new form of Matter-Unmatter, composed of Particles and Anti- Particles,” – F. Smarandache (PIP, Vol. 1, 2005) 8 2. “Verifying Unmatter by Experiments, More types of Unmatter, and a Quantum Chromodynamics formula,” – F. Smarandache (PIP, Vol. 2, 2005) 11 3. “Unmatter Entities inside Nuclei, predicted by the Brightsen Nucleon Cluster Model,” – F. Smarandache & D. Rabounski (PIP, Vol. 1, 2006) 15 4. “On Emergent Physics, Unparticles and Exotic Unmatter States,” – E. Goldfain & F. Smarandache, (PIP Vol. 4, 2008) 20 5. “Thirty Unsolved problems in the physics of elementary particles,” - V. Christianto & F. Smarandache (PIP Vol.4, 2007) 26
B. Entanglement, Neutrosophic Logic, and applications
6. “There is no speed barrier for a wave phase nor for Entangled Particles,” – F. Smarandache (PIP, Vol. 1, 2005) 29 7. “Entangled states and Quantum Causality Threshold in the General Theory of Relativity,” – D. Rabounski, L. Borissova, & F. Smarandache (PIP, Vol. 2, 2005) 31 8. “Quantum Quasi-Paradoxes and Quantum Sorites Paradox,” – F. Smarandache (PIP, Vol. 1, 2005) 38 9. “The Neutrosophic Logic view to Schrödinger’s cat paradox,” – F. Smarandache & V. Christianto (PIP, Vol. 2, 2006) 40 10. “The Neutrosophic Logic view to Schrödinger’s cat paradox, Revisited,” – V. Christianto & F. Smarandache (PIP, Vol. 3, 2008) 45 11. “A note on Geometric and Information Fusion Interpretation of Bell’s theorem and Quantum Measurement,”– F. Smarandache & V. Christianto (PIP, Vol. 4, 2006) 50
5 12. “Extension of Inagaki General Weighted Operators and a new fusion Rule Class,”– F. Smarandache (June 2008) 55 13. “n-ary Fuzzy Logic and Neutrosophic Logic Operators,”– F. Smarandache & V. Christianto (Octogon, to appear 2009) 59
C. Wave Mechanics, Gravitation and Quantization in Astrophysics
14. “Schrödinger equation and the Quantization of Celestial systems,”– F. Smarandache & V. Christianto (PIP, Vol. 2, 2006) 74 15. “Plausible explanation of quantization of intrinsic redshift from Hall effect and Weyl Quantization,”– F. Smarandache & V. Christianto (PIP, Vol. 4, 2006) 79 16. “Numerical Solution of Time-dependent Gravitational Schrödinger equation,”– V. Christianto & F. Smarandache (PIP, Vol. 2, 2007) 83 17. “Kaluza-Klein-Carmeli metric from Quaternion-Clifford-space, Lorentz’s force and some observables,”– V. Christianto & F. Smarandache (PIP, Vol. 2, 2008) 88 18. “A note of Extended Proca equations and Superconductivity,”– V. Christianto & F. Smarandache (PIP, Vol. 1, 2009) 95
D. Other topics: Quaternion space and their applications
19. “Less mundane explanation of Pioneer anomaly from Q- relativity,”– F. Smarandache & V. Christianto (PIP, Vol. 1, 2007) 99 20. “Notes on Pioneer anomaly acceleration by Satellite-Shift formula of Quaternion Relativity,”– A.P. Yefremov (PIP, Vol. 2, 2007) 103 21. “Reply to Notes on Pioneer anomaly acceleration by Satellite-Shift formula of Quaternion Relativity,”–V. Christianto & F. Smarandache (PIP, Vol. 3, 2007) 107 22. “Yang-Mills field from Quaternion Space Geometry, and its Klein- Gordon representation,” – A.P. Yefremov, F. Smarandache , V. Christianto (PIP, Vol. 4, 2007) 110 23. “Numerical solution of biquaternion Klein-Gordon equation,”–V. Christianto & F. Smarandache (PIP, Vol. 1, 2008) 119 24. “On PT-symmetric periodic potential, Quark Confinement,” –V. Christianto & F. Smarandache (PIP, Vol. 1, 2009) 121
Pictures at an exhibition (2006) 124
6
FLORENTIN SMARANDACHE & V. CHRISTIANTO
NEUTROSOPHIC LOGIC, WAVE MECHANICS, AND OTHER STORIES
SELECTED WORKS 2005-2008
7 April, 2005 PROGRESS IN PHYSICS Volume 1
A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
Besides matter and antimatter there must exist unmatter (as a new form of matter) in accordance with the neutrosophy theory that between an entity and its opposite
1 Introduction Etymologically “un-matter” comes from [ME
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artificially made tritium (3H) which has two neutrons have We can collide unmatter 1 with unmatter 2, or unmatter corresponding unmatter atoms of both types, un-deuterium 1 with antimatter, or unmatter 2 with matter. and un-tritium respectively. The isotopes of an element X When two, three, or four of them (unmatter 1, unmatter 2, differ in the number of neutrons, thus their nuclear mass is matter, antimatter) collide together, they annihilate and turn different, but their nuclear charges are the same. into energy which can materialize at high energy into new For all other matter atom X, there is corresponding an particles and antiparticles. antimatter atom and two unmatter atoms The unmatter atoms are also neutral for the same reason 4 Existence of unmatter that either the number of electrons is equal to the number of protons in the first type, or the number of antielectrons is The existence of unmatter in the universe has a similar chance equal to the number of antiprotons in the second type. to that of the antimatter, and its production also difficult for If antimatter exists then a higher probability would be for present technologies. At CERN it will be possible to test the the unmatter to exist, and reciprocally. production of unmatter. Unmatter atoms of the same type stick together form If antimatter exists then a higher probability would be for an unmatter molecule (we call it unmolecule), and so on. the unmatter to exist, and reciprocally. Similarly one has two types of unmatter molecules. The 1998 Alpha Magnetic Spectrometer (AMS) flown on The isotopes of an atom or element X have the same the International Space Station orbiting the Earth would be atomic number (same number of protons in the nucleus) able to detect, besides cosmic antimatter, unmatter if any. but different atomic masses because the different number of neutrons. Therefore, similarly the un-isotopes of type 1 of X will 5 Experiments be formed by electrons, protons, and antineutrons, while the un-isotopes of type 2 of X will be formed by antielectrons, Besides colliding electrons, or protons, would be interesting antiprotons, and neutrons. in colliding neutrons. Also, colliding a neutron with an anti- An ion is an atom (or group of atoms) X which has neutron in accelerators. last one or more electrons (and as a consequence carries a We think it might be easier to produce in an experiment negative charge, called anion, or has gained one or more an unmatter atom of deuterium (we can call it un-deuterium electrons (and as a consequence carries a positive charge, of type 1). The deuterium, which is an isotope of the ordinary called cation). hydrogen, has an electron, a proton, and a neutron. The Similarly to isotopes, the un-ion of type 1 (also called idea would be to convert/transform in a deuterium atom the un-anion 1 or un-cation 1 if resulted from a negatively neutron into an antineutron, then study the properties of the or respectively positive charge ion) of X will be formed resulting un-deuterium 1. by electrons, protons, and antineutrons, while the un-ion of Or, similarly for un-deuterium 2, to convert/transform in type 2 of X (also called un-anion 2 or un-cation 2 if resulted a deuterium atom the electron into an antielectron, and the from a negatively or respectively positive charge ion) will be proton into an antiproton (we can call it un-deuterium of formed by antielectrons, antiprotons, and neutrons. type 2). The ion and the un-ion of type 1 have the same charges, Or maybe choose another chemical element for which while the ion and un-ion of type 2 have opposite charges. any of the previous conversions/transformations might be possible. D. Nonmatter means what is not matter, therefore non- matter actually comprises antimatter and unmatter. Similarly one defines a nonnucleus. 6 Neutrons and antineutrons Hadrons consist of baryons and mesons and interact via 3 Unmatter propulsion strong force. Protons, neutrons, and many other hadrons are composed We think (as a prediction or supposition) it could be possible from quarks, which are a class of fermions that possess at using unmatter as fuel for space rockets or for weapons a fractional electric charge. For each type of quark there platforms because, in a similar way as antimatter is presup- exists a corresponding antiquark. Quarks are characterized posed to do [4, 5], its mass converted into energy will be by properties such as flavor (up, down, charm, strange, top, fuel for propulsion. or bottom) and color (red, blue, or green). It seems to be a little easier to build unmatter than A neutron is made up of quarks, while an antineutron is antimatter because we need say antielectrons and antiprotons made up of antiquarks. only (no need for antineutrons), but the resulting energy A neutron (see [9]) has one Up quark (with the charge 19 2 × × might be less than in matter-antimatter collision. of + 3 1.606 10 C) and two Down quarks (each with the
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9 April, 2005 PROGRESS IN PHYSICS Volume 1
19 1 × × charge of − 3 1.606 10 C), while an antineutron has one 3. Webster’s New World Dictionary. Third College Edition, 2 19 × × Simon and Schuster Inc., 1988. anti Up quark (with the charge of − 3 1.606 10 C) and 1 × × two anti Down quarks (each with the charge of + 3 1.606 4. Mondardini R. The history of antimatter. CERN Laboratory, 19 ×10 C). Geneve,` on-line http://livefromcern.web.cern.ch/livefromcern/ An antineutron has also a neutral charge, through it is antimatter/history/AM-history00.html. opposite to a neutron, and they annihilate each other when 5. De Rujula´ A. and Landua R. Antimatter — frequently asked meeting. questions. CERN Laboratory, Geneve,` http://livefromcern.web. Both, the neutron and the antineutron, are neither attract- cern.ch/livefromcern/antimatter/FAQ1.html. ed to nor repelling from charges particles. 6. Pompos A. Inquiring minds — questions about physics. Fermi- lab, see on-line http://www.fnal.gov/pub/inquiring/questions/ antineuron.html. 7 Characteristics of unmatter
Unmatter should look identical to antimatter and matter, also the gravitation should similarly act on all three of them. Unmatter may have, analogously to antimatter, utility in medicine and may be stored in vacuum in traps which have the required configuration of electric and magnetic fields for several months.
8 Open Questions
8.a Can a matter atom and an unmatter atom of first type stick together to form a molecule? 8.b Can an antimatter atom and an unmatter atom of sec- ond type stick together to form a molecule? 8.c There might be not only a You and an anti-You, but some versions of an un-You in between You and anti- You. There might exist un-planets, un-stars, un- galaxies? There might be, besides our universe, an anti-universe, and more un-universes? 8.d Could this unmatter explain why we see such an im- balance between matter and antimatter in our corner of the universe? (Jeff Farinacci) 8.e If matter is thought to create gravity, is there any way that antimatter or unmatter can create antigravity or ungravity? (Mike Shafer from Cornell University) I assume that since the magnetic field or the gravitons generate gravitation for the matter, then for antimatter and unmatter the corresponding magnetic fields or gravitons would look different since the charges of subatomic particles are different. . . I wonder how would the universal law of attraction be for antimmater and unmatter?
References
1. Smarandache F. Unmatter, mss., 1980, Archives Valcea.ˆ 2. Smarandache F. A unifying field in logics: neutrosophic logic. Neutrosophy, Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth, 2002, 144 p. (see it in e-print: http://www.gallup.unm.edu/∼smarandache).
F. Smarandache. A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles 11
10 July, 2005 PROGRESS IN PHYSICS Volume 2
Verifying Unmatter by Experiments, More Types of Unmatter, and a Quantum Chromodynamics Formula
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mails: [email protected]; [email protected]
As shown, experiments registered unmatter: a new kind of matter whose atoms include both nucleons and anti-nucleons, while their life span was very short, no more than −20 10 sec. Stable states of unmatter can be built on quarks and anti-quarks: applying the unmatter principle here it is obtained a quantum chromodynamics formula that gives many combinations of unmatter built on quarks and anti-quarks.
In the last time, before the apparition of my articles defining (d) an antimatter atom such that one or more (but not all) “matter, antimatter, and unmatter” [1, 2], and Dr. S. Chubb’s of the antielectrons and/or antiprotons are replaced by pertinent comment [3] on unmatter, new development has matter particles of the same corresponding charges. been made to the unmatter topic. In a more composed system we can substitute a particle by an unmatter particle and form an unmatter atom. 1 Definition of Unmatter Of course, not all of these combinations are stable, semi- In short, unmatter is formed by matter and antimatter that stable, or quasi-stable, especially when their time to bind bind together [1, 2]. The building blocks (most elementary together might be longer than their lifespan. particles known today) are 6 quarks and 6 leptons; their 12 antiparticles also exist. Then unmatter will be formed by 4 Examples of unmatter at least a building block and at least an antibuilding block During 1970-1975 numerous pure experimental verifications which can bind together. were obtained proving that “atom-like” systems built on 2 Exotic atom nucleons (protons and neutrons) and anti-nucleons (anti- protons and anti-neutrons) are real. Such “atoms”, where If in an atom we substitute one or more particles by other nucleon and anti-nucleon are moving at the opposite sides of particles of the same charge (constituents) we obtain an the same orbit around the common centre of mass, are very exotic atom whose particles are held together due to the unstable, their life span is no more than 10−20 sec. Then electric charge. For example, we can substitute in an ordinary nucleon and anti-nucleon annihilate into gamma-quanta and atom one or more electrons by other negative particles (say − − − − − more light particles (pions) which can not be connected with π , anti-ρ-meson, D ,Ds - muon, τ, Ω , Δ , etc., generally one another, see [6, 7, 8]. The experiments were done in clusters of quarks and antiquarks whose total charge is neg- mainly Brookhaven National Laboratory (USA) and, partial- ative), or the positively charged nucleus replaced by other ly, CERN (Switzerland), where “proton — anti-proton” and positive particle (say clusters of quarks and antiquarks whose “anti-proton — neutron” atoms were observed, called them total charge is positive, etc). ¯pp and ¯pn respectively, see Fig. 1 and Fig. 2. After the experiments were done, the life span of such 3 Unmatter atom “atoms” was calculated in theoretical way in Chapiro’s works It is possible to define the unmatter in a more general way, [9, 10, 11]. His main idea was that nuclear forces, acting using the exotic atom. The classical unmatter atoms were between nucleon and anti-nucleon, can keep them far way formed by particles like: from each other, hindering their annihilation. For instance, (a) electrons, protons, and antineutrons, or a proton and anti-proton are located at the opposite sides in the same orbit and they are moved around the orbit (b) antielectrons, antiprotons, and neutrons. centre. If the diameter of their orbit is much more than In a more general definition, an unmatter atom is a system the diameter of “annihilation area”, they can be kept out of of particles as above, or such that one or more particles annihilation (see Fig. 3). But because the orbit, according to are replaces by other particles of the same charge. Other Quantum Mechanics, is an actual cloud spreading far around categories would be: the average radius, at any radius between the proton and (c) a matter atom with where one or more (but not all) of the anti-proton there is a probability that they can meet one the electrons and/or protons are replaced by antimatter another at the annihilation distance. Therefore “nucleon — particles of the same corresponding charges, and anti-nucleon” system annihilates in any case, this system
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Fig. 1: Spectra of proton impulses in the reaction p+d¯ → (¯pn)+p. The upper arc — annihilation of ¯pn into even number of pions, the lower arc — its annihilation into odd number of pions. The observed maximum points out that there is a connected system¯pn. Abscissa axis represents the proton impulse in GeV/sec (and the connection energy of the system ¯pn). Ordinate axis — the number of events. Cited from [6]. is unstable by definition having life span no more than σ ¯p p d −20 Fig. 2: Probability of interaction between , and deutrons 10 sec. (cited from [7]). The presence of maximum stands out the existence Unfortunately, the researchers limited the research to the of the resonance state of “nucleon — anti-nucleon”. consideration of ¯pp and ¯pn “atoms” only. The reason was − that they, in the absence of a theory, considered ¯pp and ¯pn found in one experiment [4] for other pentaquarks, Ξs + “atoms” as only a rare exception, which gives no classes of (ddssuˆ) and Ξs (uussdˆ). matter. In order for the paper to be self-contained let’s recall Despite Benn Tannenbaum’s and Randall J. Scalise’s re- that the pionium is formed by a π+ and π− mesons, the jections of unmatter and Scalise’s personal attack on me in positronium is formed by an antielectron (positron) and an a true Ancient Inquisitionist style under MadSci moderator electron in a semi-stable arrangement, the protonium is John Link’s tolerance (MadSci web site, June-July 2005), formed by a proton and an antiproton also semi-stable, the the unmatter does exists, for example some messons and antiprotonic helium is formed by an antiproton and electron antimessons, through for a trifling of a second lifetime, so together with the helium nucleus (semi-stable), and muonium ∗ + − the pions are unmatter , the kaon K (usˆ), K (uˆs), Phi is formed by a positive muon and an electron. Also, the + 0 + − 0 (ssˆ), D (cdˆ), D (cuˆ), Ds (csˆ), J/Psi (ccˆ), B (buˆ), B mesonic atom is an ordinary atom with one or more of its 0 † (dbˆ), Bs (sbˆ), Upsilon (bbˆ), etc. are unmatter too . electrons replaced by negative mesons. The strange matter + + Also, the pentaquark theta-plus Θ , of charge 1, uuddsˆ is a ultra-dense matter formed by a big number of strange (i. e. two quarks up, two quarks down, and one anti-strange quarks bounded together with an electron atmosphere (this quark), at a mass of 1.54 GeV and a narrow width of 22 strange matter is hypothetical). MeV, is unmatter, observed in 2003 at the Jefferson Lab in From the exotic atom, the pionium, positronium, pro- Newport News, Virginia, in the experiments that involved tonium, antiprotonic helium, and muonium are unmatter. The multi-GeV photons impacting a deuterium target. Similar mesonic atom is unmatter if the electron(s) are replaced by pentaquark evidence was obtained by Takashi Nakano of negatively-charged antimessons. Also we can define a me- Osaka University in 2002, by researchers at the ELSA accel- sonic antiatom as an ordinary antiatomic nucleous with one erator in Bonn in 1997-1998, and by researchers at ITEP in or more of its antielectrons replaced by positively-charged Moscow in 1986. Besides theta-plus, evidence has been mesons. Hence, this mesonic antiatom is unmatter if the antielectron(s) are replaced by positively-charged messons. ∗ uˆd udˆ uˆ Which have the composition and , where by we mean The strange matter can be unmatter if these exists at least anti-up quark, d=down quark, and analogously u=up quark and dˆ = anti-down quark, while by ˆ we mean “anti”. an antiquark together with so many quarks in the nucleous. †Here c=charm quark, s=strange quark, b=bottom quark. Also, we can define the strange antimatter as formed by
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Strangeˆ, Charmˆ}. Hence, for combinations of n quarks and antiquarks, n 2, prevailing the colorless, we have the fol- lowing possibilities: • if n=2, we have: qa (biquark — for example the me- sons and antimessons); • if n=3, we have qqq, aaa (triquark — for example the baryons and antibaryons); • if n=4, we have qqaa (tetraquark); • if n=5, we have qqqqa, aaaaq (pentaquark); • if n=6, we have qqqaaa, qqqqqq, aaaaaa (hexaquark); Fig. 3: Annihilation area and the probability arc in “nucleon — • anti-nucleon” system (cited from [11]). if n=7, we have qqqqqaa, qqaaaaa (septiquark); • if n=8, we have qqqqaaaa, qqqqqqaa, qqaaaaaa a large number of antiquarks bound together with an anti- (octoquark); electron around them. Similarly, the strange antimatter can • be unmatter if there exists at least one quark together with if n=9, we have qqqqqqqqq, qqqqqqaaa, so many antiquarks in its nucleous. qqqaaaaaa, aaaaaaaaa (nonaquark); The bosons and antibosons help in the decay of unmatter. • if n=10, we have qqqqqaaaaa, qqqqqqqqaa, There are 13 + 1 (Higgs boson) known bosons and 14 anti- qqaaaaaaaa (decaquark); etc. bosons in present. 7 Unmatter combinations 5 Quantum Chromodynamics formula From the above general case we extract the unmatter combi- In order to save the colorless combinations prevailed in the nations: Theory of Quantum Chromodynamics (QCD) of quarks and • antiquarks in their combinations when binding, we devise For combinations of 2 we have: qa (unmatter biquark), the following formula: mesons and antimesons; the number of all possible unmatter combinations will be 6×6 = 36, but not all Q − A ∈±M3 , (1) of them will bind together. It is possible to combine an entity with its mirror opposite where M3 means multiple of three, i. e. ±M3={3k| k∈Z}= and still bound them, such as: uuˆ, ddˆ, ssˆ, ccˆ, bbˆ which ={...,−12, −9, −6, −3, 0, 3, 6, 9, 12,...}, and Q=number form mesons. It is possible to combine, unmatter + unmatter of quarks, A = number of antiquarks. But (1) is equivalent to = unmatter, as in udˆ + usˆ = uudˆsˆ (of course if they bind Q ≡ A(mod 3) (2) together). • For combinations of 3 (unmatter triquark) we can not (Q is congruent to A modulo 3). form unmatter since the colorless can not hold. To justify this formula we mention that 3 quarks form • a colorless combination, and any multiple of three (M3) For combinations of 4 we have: qqaa (unmatter tetra- quark); the number of all possible unmatter combina- combination of quarks too, i. e. 6, 9, 12, etc. quarks. In 2 2 a similar way, 3 antiquarks form a colorless combination, tions will be 6 ×6 = 1,296, but not all of them will and any multiple of three (M3) combination of antiquarks bind together. too, i. e. 6, 9, 12, etc. antiquarks. Hence, when we have • For combinations of 5 we have: qqqqa,oraaaaq hybrid combinations of quarks and antiquarks, a quark and (unmatter pentaquarks); the number of all possible 4 4 an antiquark will annihilate their colors and, therefore, what’s unmatter combinations will be 6 ×6 + 6 ×6 = 15,552, left should be a multiple of three number of quarks (in the but not all of them will bind together. case when the number of quarks is bigger, and the difference • For combinations of 6 we have: qqqaaa (unmatter in the formula is positive), or a multiple of three number hexaquarks); the number of all possible unmatter com- 3 3 of antiquarks (in the case when the number of antiquarks is binations will be 6 ×6 = 46,656, but not all of them bigger, and the difference in the formula is negative). will bind together. • 6 Quark-antiquark combinations For combinations of 7 we have: qqqqqaa, qqaaaaa (unmatter septiquarks); the number of all possible un- 5 2 2 5 Let’s note by q=quark ∈{Up, Down, Top, Bottom, Strange, matter combinations will be 6 ×6 +6 ×6 = 559,872, Charm}, and by a=antiquark ∈{Upˆ, Downˆ, Topˆ, Bottomˆ, but not all of them will bind together.
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• For combinations of 8 we have: qqqqaaaa, qqqqqqqa, we want), and negative force (repulsive, opposed to the pre- qaaaaaaa (unmatter octoquarks); the number of all the vious). There could be a combination of both positive and 4 4 7 1 1 7 unmatter combinations will be 6 ×6 +6 ×6 +6 ×6 = negative forces in the same time, or alternating positive and = 5,038,848, but not all of them will bind together. negative, etc. • For combinations of 9 we have types: qqqqqqaaa, Unenergy would similarly be a combination between qqqaaaaaa (unmatter nonaquarks); the number of all positive and negative energies (as the alternating current, 6 3 3 6 9 the unmatter combinations will be 6 ×6 +6 ×6 =2×6 a. c., which periodically reverses its direction in a circuit and = 20,155,392, but not all of them will bind together. whose frequency, f, is independent of the circuit’s constants). Would it be possible to construct an alternating-energy gen- • For combinations of 10 we have types: qqqqqqqqaa, erator? qqqqqaaaaa, qqaaaaaaaa (unmatter decaquarks); To conclusion: According to the Universal Dialectic the the number of all the unmatter combinations will be 10 unity is manifested in duality and the duality in unity. “Thus, 3×6 = 181,398,528, but not all of them will bind to- Unmatter (unity) is experienced as duality (matter vs anti- gether. Etc. matter). Ungravity (unity) as duality (gravity vs antigravity). I wonder if it is possible to make infinitely many co- Unenergy (unity) as duality (positive energy vs negative en- mbinations of quarks/antiquarks and leptons/antileptons. . . ergy) and thus also . . . between duality of being (existence) Unmatter can combine with matter and/or antimatter and the vs nothingness (antiexistence) must be ‘unexistence’ (or pure result may be any of these three. Some unmatter could be in unity)” (R. Davic). the strong force, hence part of hadrons. References 8 Unmatter charge The charge of unmatter may be positive as in the pentaquark 1. Smarandache F. A new form of matter — unmatter, composed of particles and anti-particles. Progr. in Phys., 2005, v. 1, 9–11. theta-plus, 0 (as in positronium), or negative as in anti-ρ- meson (uˆd) (M. Jordan). 2. Smarandache F. Matter, antimatter, and unmatter. Infinite Energy, v. 11, No. 62, 50–51, (July/August 2005). 9 Containment 3. Chubb S. Breaking through editorial. Infinite Energy, v. 11, No. 62, 6–7 (July/August 2005). I think for the containment of antimatter and unmatter it 4. Alt C. et al., (NA49 Collaboration). Phys. Rev. Lett., 2004, would be possible to use electromagnetic fields (a container v. 92, 042003. whose walls are electromagnetic fields). But its duration is 5. Carman D. S., Experimental evidence for the pentaquark. Eur. unknown. Phys. A, 2005, v. 24, 15–20. 6. Gray L., Hagerty P., Kalogeropoulos T. E. Evidence for the 10 Further research existence of a narrow p-barn bound state. Phys. Rev. Lett., Let’s start from neutrosophy [13], which is a generalization 1971, v. 26, 1491–1494. of dialectics, i. e. not only the opposites are combined but also 7. Carrol A. S. et al. Observation of structure in ¯pp and ¯pd total the neutralities. Why? Because when an idea is launched, a cross sections below 1.1 GeV/s. Phys. Rev. Lett., 1974, v.32, category of people will accept it, others will reject it, and 247–250. a third one will ignore it (don’t care). But the dynamics 8. Kalogeropoulos T. E., Vayaki A., Grammatikakis G., Tsilimi- between these three categories changes, so somebody ac- gras T., Simopoulou E. Observation of excessive and direct ¯pd cepting it might later reject or ignore it, or an ignorant will gamma production in annihilations at rest. Phys. Rev. Lett., accept it or reject it, and so on. Similarly the dynamicity 1974, v.33, 1635–1637. of ,
116 F. Smarandache. More Types of Unmatter and a Quantum Chromodynamics Formula
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Unmatter Entities inside Nuclei, Predicted by the Brightsen Nucleon Cluster Model
Florentin Smarandache and Dmitri Rabounski Dept. of Mathematics and Science, Univ. of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
Applying the R. A. Brightsen Nucleon Cluster Model of the atomic nucleus we discuss how unmatter entities (the conjugations of matter and antimatter) may be formed as clusters inside a nucleus. The model supports a hypothesis that antimatter nucleon clusters are present as a parton (sensu Feynman) superposition within the spatial 1 1 confinement of the proton ( H1), the neutron, and the deuteron ( H2). If model predictions can be confirmed both mathematically and experimentally, a new physics is suggested. A proposed experiment is connected to othopositronium annihilation anomalies, which, being related to one of known unmatter entity, orthopositronium (built on electron and positron), opens a way to expand the Standard Model.
1 Introduction ium has been verified experimentally as a weak resonance between a proton and antiproton using a Skyrme-type model According to Smarandache [1, 2, 3], following neutrosophy potential. Further evidence that neutral entities derive from theory in philosophy and set theory in mathematics, the union union of opposites comes from the spin induced magnetic of matter and its antimatter opposite
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Matter [NP] [NPN] [PNP] [NN] [PP] [NNN] [PPP] Clusters −→ Deuteron Triton Helium-3 Di-Neutron Di-Proton Tri-Neutron Tri-Proton Antimatter −→ i j k l m n o Clusters Stable Beta-unstable Stable
ˆ ˆ [N P ] [N] [P] Pions Pions [N] [P] a |NP| |NˆPˆ| |NP| |NˆPˆ| (q qˆ) (q qˆ) |NN| |NˆPˆ| |NˆPˆ| |PP| Stable ˆ ˆ ˆ [N P N ] [Nˆ] Pions [Pˆ] [Nˆ] Pions Tetraquarks b |NP| |NˆPˆ| (q qˆ) |NN| |NˆNˆ| |NˆPˆ| |PP| (q qˆ) (qqqˆqˆ) Beta-unstable ˆ ˆ ˆ [P N P ] [Pˆ] Pions [Pˆ] [Nˆ] Tetraquarks Pions c |NP| |NˆPˆ| (q qˆ) |NˆPˆ| |NN| |PP| |PˆPˆ| (qqqˆqˆ) (q qˆ) Stable
[NˆNˆ] Pions [N] [P] Tetraquarks [N] [P] d (q qˆ) |NN| |NˆNˆ| |NP| |NˆNˆ| (qqqˆqˆ) |NN| |NˆNˆ| |PP| |NˆNˆ|
[PˆPˆ] Pions [N] [P] Tetraquarks [N] [P] e (q qˆ) |NP| |PˆPˆ| |NP| |PˆPˆ| (qqqˆqˆ) |PˆPˆ| |NN| |PP| |PˆPˆ|
[NˆNˆNˆ] [Nˆ] Pions Tetraquarks [Nˆ] [Nˆ] Hexaquarks f |NP| |NˆNˆ| (q qˆ) (qqqˆqˆ) |NN| |NˆNˆ| |NˆNˆ| |PP| (qqqqˆqˆqˆ)
[PˆPˆPˆ] [Pˆ] Tetraquarks Pions [Pˆ] [Pˆ] Hexaquarks g |NP| |PˆPˆ| (qqqˆqˆ) (q qˆ) |PˆPˆ| |NN| |PˆPˆ| |PP| (qqqqˆqˆqˆ)
Table 1: Unmatter entities (stable, quasi-stable, unstable) created from union of matter and antimatter nucleon clusters as predicted by the gravity-antigravity formalism of the Brightsen Nucleon Cluster Model. Shaded cells represent interactions that result in annihilation of mirror opposite two- and three- body clusters. Top nucleons within cells show superposed state comprised of three valance quarks; bottom structures show superposed state of hidden unmatter in the form of nucleon clusters. Unstable pions, tetraquarks, and hexaquark unmatter are predicted from union of mass symmetrical clusters that are not mirror opposites. The symbol ˆ= antimatter, N = neutron, P = proton, q = quark. (Communication with R. D. Davic). fermions within nuclear shells, or (2) that nucleons interact confined within the formalism of 6-quark bags [NP] and collectively in the form of a liquid-drop. Compromise models 9-quark bags ([PNP] and [NPN]), as opposed to valance attempt to cluster nucleons into interacting [NP] boson pairs quarks forming free nucleons. The experimental evidence in (e.g., Interacting Boson Model-IBM), or, as in the case of support of nucleons interacting as boson-fermion clusters is the Interacting Boson-Fermion Model (IBFM), link boson now extensive and well reviewed. clusters [NP] with un-paired and independent nucleons [P] One novel nucleon cluster model is that of R. A. Bright- and [N] acting as fermions. sen, which was derived from the identification of mass- However, an alternative view, at least since the 1937 charge symmetry systems of isotopes along the Z-N Serge Resonating Group Method of Wheeler, and the 1965 Close- plot. According to Brightsen, all beta-stable matter and anti- Packed Spheron Model of Pauling, holds that the macro- matter isotopes are formed by potential combinations of scopic structure of atomic nuclei is best described as being two- and three nucleon clusters; e.g., ([NP], [PNP], [NPN], composed of a small number of interacting boson-fermion [NN], [PP], [NNN], [PPP], and/or their mirror antimatter nucleon “clusters” (e. g., helium-3 [PNP], triton [NPN], deu- clusters [NˆPˆ], [PˆNˆPˆ], [NˆPˆNˆ], [NˆNˆ], [PˆPˆ], [PˆPˆPˆ], teron [NP]), as opposed to independent [N] and [P] nucleons [NˆNˆNˆ], where the symbol ˆ here is used to denote anti- acting as fermions, either independently or collectively. matter. A unique prediction of the Brightsen model is that a Mathematically, such clusters represent a spatially localized stable union must result between interaction of mass asym- mass-charge-spin subsystem composed of strongly correlated metrical matter (positive mass) and antimatter (negative nucleons, for which realistic two- and three body wave funct- mass) nucleon clusters to form protons and neutrons, for ions can be written. In this view, quark-gluon dynamics are example the interaction between matter [PNP] + antimatter
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[NˆPˆ]. Why union and not annihilation of mass asymmetrical into gas quite soon annihilate with free electrons and elec- matter and antimatter entities? As explained by Brightsen, trons in the container’s walls. Other positrons capture elec- independent (unbound) neutron and protons do not exist in trons from gas atoms thus producing orthopositronium and nuclear shells, and the nature of the mathematical series of parapositronium (in 3:1 statistical ratio). Time spectrum of cluster interactions (3 [NP] clusters = 1[NPN] cluster + 1 positrons (number of positrons vs. life span) is the basic [PNP] cluster), makes it impossible for matter and antimatter characteristic of their annihilation in matter. clusters of identical mass to coexist in stable isotopes. Thus, In inert gases the time spectrum of annihilation of free annihilation cannot take place between mass asymmetrical positrons generally reminds of exponential curve with a two- and three matter and antimatter nucleon clusters, only plateau in its central part, known as “shoulder” [29, 30]. In strong bonding (attraction). 1965 Osmon published [29] pictures of observed time spectra Here is the Table that tells how unmatter may be formed of annihilation of positrons in inert gases (He, Ne, Ar, Kr, from nucleon clusters according to the Brightsen model. Xe). In his experiments he used 22NaCl as a source of β+- decay positrons. Analyzing the results of the experiments, 3 A proposed experimental test Levin noted that the spectrum in neon was peculiar compared to those in other one-atom gases: in neon points in the As known, Standard Model of Quantum Electrodynamics curve were so widely scattered, that presence of a “shoulder” explains all known phenomena with high precision, aside was unsure. Repeated measurements of temporal spectra of for anomalies in orthopositronium annihilation, discovered annihilation of positrons in He, Ne, and Ar, later accomplish- in 1987. ed by Levin [31, 32], have proven existence of anomaly in The Brightsen model, like many other models (see Ref- neon. Specific feature of the experiments done by Osmon, erences), is outside the Standard Model. They all pretend to Levin and some other researchers in the UK, Canada, and expand the Standard Model in one or another way. Therefore Japan is that the source of positrons was 22Na, while the today, in order to judge the alternative models as true or false, moment of birth of positron was registered according to γn- we should compare their predictions to orthopositronium quantum of decay of excited 22∗Ne annihilation anomalies, the solely unexplained by the Stand- 22∗ 22 ard Model. Of those models the Brightsen model has a chance Ne → Ne + γn , to be tested in such way, because it includes unmatter entities + 22∗ (the conjugations of particles and anti-particles) inside an from one of products of β -decay of Na. atomic nucleus that could produce effect in the forming of In his experiments [33, 34] Levin discovered that the orthopositronium by β+-decay positrons and its annihilation. peculiarity of annihilation spectrum in neon (abnormally wide scattered points) is linked to presence in natural neon of In brief, the anomalies in orthopositronium annihilation 22 are as follows. substantial quantity of its isotope Ne (around 9%). Levin Positronium is an atom-like orbital system that includes called this effect isotope anomaly. Temporal spectra were measured in neon environments of two isotopic composit- an electron and its anti-particle, positron, coupled by electro- 20 21 static forces. There are two kinds of that: parapositronium ions: (1) natural neon (90.88% of Ne, 0.26% of Ne, S and 8.86% of 22Ne); (2) neon with reduced content of 22Ne Ps, in which the spins of electron and positron are oppositely 20 21 22 directed and the summary spin is zero, and orthopositronium (94.83% of Ne, 0.22% of Ne, and 4.91% of Ne). TPs, in which the spins are co-directed and the summary spin Comparison of temporal spectra of positron decay revealed: in natural neon (the 1st composition) the shoulder is fuzzy, is one. Because a particle-antiparticle (unmatter) system is 22 unstable, life span of positronium is rather small. In vacuum, while in neon poor with Ne (the 2nd composition) the −10 parapositronium decays in τ 1.25×10 s, while ortho- shoulder is always clearly pronounced. In the part of spectr- −7 T positronium is τ 1.4×10 s after the birth. In a medium um, to which Ps-decay mostly contributes, the ratio between the life span is even shorter because positronium tends to intensity of decay in poor neon and that in natural neon (with 22 ± annihilate with electrons of the media. much isotope Ne) is 1.85 0.1 [34]. In laboratory environment positronium can be obtained Another anomaly is substantially higher measured rate of by placing a source of free positrons into a matter, for annihilation of orthopositronium (the value reciprocal to its instance, one-atom gas. The source of positrons is β+-decay, life span) compared to that predicted by QED. self-triggered decays of protons in neutron-deficient atoms∗ Measurement of orthopositronium annihilation rate is among the main tests aimed to experimental verification of + p → n + e + νe. QED laws of conservation. In 1987 thanks to new precision
+ technology a group of researchers based in the University of Some of free positrons released from β -decay source Michigan (Ann Arbor) made a breakthrough in this area. The ∗It is also known as positron β+-decay. During β−-decay in nucleus obtained results showed substantial gap between experiment − neutron decays n → p + e +˜νe. and theory. The anomaly that the Michigan group revealed
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was that measured rates of annihilation at λT(exp) = 7.0514± monly Salam, Glashow, Weinberg, etc. Today’s physicists −1 −1 ±0.0014 μs and λT(exp) = 7.0482 ± 0.0016 μs (with attempt to join electroweak interaction and strong interaction unseen-before precision of 0.02% and 0.023% using vacuum (unfinished yet). They follow an intuitive idea that forces, and gas methods [35–38]) were much higher compared to connecting electrons and a nucleus, and forces, connecting −1 λT(theor) = 7.00383 ± 0.00005 μs as predicted by QED nucleons inside a nucleus, are particular cases of a common [39–42]. The effect was later called λT-anomaly [43]. interaction. That is the basis of our claim. If that is true, our Theorists foresaw possible annihilation rate anomaly not claim is that orthopositronium atoms born in neon of different long before the first experiments were accomplished in Mi- isotope contents (22Ne, 21Ne, 20Ne) should be different from chigan. In 1986 Holdom [44] suggested that “mixed type” each other. There should be an effect of “inner” structure particles may exist, which being in the state of oscillation of neon nuclei if built by the Brightsen scheme, because stay for some time in our world and for some time in the the different proton-neutron contents built by different com- mirror Universe, possessing negative masses and energies. In positions of nucleon pairs. As soon as a free positron drags the same year Glashow [45] gave further development to the an electron from a neon atom, the potential of electro-weak idea and showed that in case of 3-photon annihilation TPs interactions have changed in the atom. Accordingly, there will “mix up” with its mirror twin thus producing two effects: in the nucleus itself should be re-distribution of strong inter- (1) higher annihilation rate due to additional mode of decay actions, than could be once as the re-building of the Brightsen TPs → nothing, because products of decay passed into the pairs of nucleons there. So, lost electron of 22Ne should have mirror Universe can not be detected; (2) the ratio between a different “inner” structure than that of 21Ne or 20Ne. Then orthopositronium and parapositronium numbers will decrease the life span of orthopositronium built on such electrons from TPs : SPs = 3:1 to 1.5 : 1. But at that time (in 1986) should be as well different. Glashow concluded that no interaction is possible between Of course, we can only qualitatively predict that dif- our-world and mirror-world particles. ference, because we have no exact picture of what really On the other hand, by the early 1990’s these theoretic happens inside a “structurized” nucleus. Yet only principal studies encouraged many researchers worldwide for experi- predictions are possible there. However even in such case mental search of various “exotic” (unexplained in QED) mo- we vote for continuation of “isotope anomaly” experiments des of TPs-decay, which could lit some light on abnormally with orthopositronium in neon of different isotope contents. high rate of decay. These were, to name just a few, search If further experiments will be positive, it could be considered for TPs → nothing mode [46], check of possible contribution as one more auxiliary proof that the Brightsen model is true. from 2-photon mode [47–49] or from other exotic modes [50–52]. As a result it has been shown that no exotic modes Acknowledgements can contribute to the anomaly, while contribution of TPs → −4 nothing mode is limited to 5.8×10 of the regular decay. We very much appreciate Dr. Robert Davic, the nephew of The absence of theoretical explanation of λT-anomaly R. A. Brightsen, for comments about the Brightsen model. encouraged Adkins et al. [53] to suggest experiments made in Japan [54] in 1995 as an alternative to the basic Michigan experiments. No doubt, high statistical accuracy of the Japan- References ese measurements puts them on the same level with the 1. Smarandache F. Matter, antimatter, and unmatter. Infinite basic experiments [35–38]. But all Michigan measurements Energy, 2005, v. 11, issue 62, 50–51. possessed the property of a “full experiment”, which in this 2. Smarandache F. A new form of matter — unmatter, composed particular case means no external influence could affect wave of particles and anti-particles. Progress in Physics, 2005, v. 1, function of positronium. Such influence is inevitable due to 9–11. electrodynamic nature of positronium and can be avoided 3. Smarandache F. Verifying unmatter by experiments, more only using special technique. In Japanese measurements [54] types of unmatter, and a quantum chromodynamics formula. this was not taken into account and thus they do not possess Progress in Physics, 2005, v. 2, 113–116. property of “full experiment”. Latest experiments of the Michigans [55], so-called Resolution of Orthopositronium- UNMATTER BASIS EXPERIMENTS Lifetime Pussle, as well do not possess property of “full 4. Gray L., Hagerty P., Kalogeropoulos T. E. Phys. Rev. Lett., experiment”, because the qualitative another statement in- 1971, v. 26, 1491–1494. cluded external influence of electromagnetic field [56, 57]. 5. Carrol A.S., Chiang I.-H., Kucia T. F., Li K. K., Mazur P.O., As early as in 1993 Karshenboim [58] showed that QED Michael D. N., Mockett P., Rahm D. C., Rubinstein R. Phys. had actually run out of any of its theoretical capabilities to Rev. Lett., 1974, v. 32, 247–250. explain orthopositronium anomaly. 6. Kalogeropoulos T. E., Vayaki A., Grammatikakis G., Tsilimig- Electric interactions and weak interactions were joined ras T., Simopoulou E. Phys. Rev. Lett., 1974, v. 33, 1635–1637. into a common electroweak interaction in the 1960’s by com- 7. Chapiro I. S. Physics-Uspekhi, 1973, v. 109, 431.
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8. Bogdanova L. N., Dalkarov O. D., Chapiro I. S. Annals of 31. Levin B. M. and Shantarovich V.P. High Energy Chemistry, Physics, 1974, v. 84, 261–284. 1977, v. 11(4), 322–323. 9. Cassidy D. B., Deng S. H. M., Greaves R. G., Maruo T., Nishi- 32. Levin B. M. Soviet J. Nucl. Physics, 1981, v. 34(6), 917–918. yama N., Snyder J. B., Tanaka H. K. M., Mills A. P.Jr. Phys. 33. Levin B. M. and Shantarovich V.P. Soviet J. Nucl. Physics, Rev. Lett., 2005, v. 95, No. 19, 195006. 1984, v. 39(6), 855–856. 10. Ball Ph. News Nature, 22 November 2005. 34. Levin B. M., Kochenda L. M., Markov A. A., and Shantarov- THE BRIGHTSEN MODEL ich V.P. Soviet J. Nucl. Physics, 1987, v. 45(6), 1119–1120. 11. Brightsen R. A. Nucleon cluster structures in beta-stable 35. Gidley D. W., Rich A., Sweetman E., and West D. Physical nuclides. Infinite Energy, 1995, v. 1, no. 4, 55–56. Review Letters, 1982, v. 49, 525–528. 12. Brightsen R. A. Correspondence of the Nucleon Cluster Model 36. Westbrook C. I., Gidley D. W., Conti R. S., and Rich A. with the Periodic Table of Elements. Infinite Energy, 1995/96, Physical Review Letters, 1987, v. 58, 1328–1331. v. 1(5/6), 73–74. 37. Westbrook C. I., Gidley D. W., Conti R. S., and Rich A. 13. Brightsen R. A. Correspondence of the Nucleon Cluster Model Physical Review A, 1989, v. 40, 5489–5499. with the Classical Periodic Table of Elements. J. New Energy, 38. Nico J. S., Gidley D. W., Rich A., and Zitzewitz P.W. Physical 1996, v. 1(1), 75–78. Review Letters, 1990, v. 65, 1344–1347. 14. Brightsen R. A. The Nucleon Cluster Model and the Periodic 39. Caswell W. E. and Lepage G. P. Phys. Rev. A, 1979, v. 20, 36. Table of Beta-Stable Nuclides. 1996 (available online at 40. Adkins G. S. Ann. Phys. (N.Y.), 1983, v. 146, 78. http://www.brightsenmodel.phoenixrising-web.net). 41. Adkins G. S., Salahuddin A. A., and Schalm K. E. Physical 15. Brightsen R. A. The nucleon cluster model and thermal neutron Review A, 1992, v. 45, 3333–3335. fission. Infinite Energy, 2000, v. 6(31), 55–63. 42. Adkins G. S., Salahuddin A. A., and Schalm K. E. Physical 16. Brightsen R. A., Davis R. Appl. of the Nucleon Cluster Model Review A, 1992, v. 45, 7774–7780. to experimental results. Infinite Energy, 1995, v. 1(3), 13–15. 43. Levin B. M. Physics of Atomic Nuclei, 1995, v. 58(2), 332–334. 17. Bass R. W. Experimental evidence favoring Brightsen’s nuc- leon cluster model. Infinite Energy, 1996, v. 2(11), 78–79. 44. Holdom B. Physics Letters B, 1986, v. 166, 196–198. 45. Glashow S. L. Physics Letters B, 1986, v. 167, 35–36. SOME REVIEW PAPERS ON CLUSTER MODELS 46. Atoyan G. S., Gninenko S. N., Razin V.I., and Ryabov Yu.V. 18. Buck B., Merchant A. C., Perez S. M. Phys. Rev. C, 2005, Physics Letters B, 1989, v. 220, 317–320. v. 71(1), 014311–15. 47. Asai S., Orito S., Sanuki T., Yasuda M., and Yokoi T. Physical 19. Akimune H., Yamagata T., Nakayama S., Fujiwara M., Fushi- Review Letters, 1991, v. 66, 1298–1301. mi K., Hara K., Hara K. Y., Ichihara K., Kawase K., Matsui K., Nakanishi K., Shiokawa A., Tanaka M., Utsunomiya H., and 48. Gidley D. W., Nico J. S., and Skalsey M. Physical Review Yosoi M. Physics of Atomic Nuclei, 2004, v. 67(9), 1721–1725. Letters, 1991, v. 66, 1302–1305. 20. Clustering Aspects of Nuclear Structure and Dynamics, 49. Al-Ramadhan A. H. and Gidley D. W. Physical Review Letters, Cluster’99, Proc. of the 7th Intern. Conf., 1999, Rab (Croatia). 1994, v. 72, 1632–1635. 21. Wuosmaa A. H., Betts R. R., Freer M., and Fulton B. R. Annual 50. Orito S., Yoshimura K., Haga T., Minowa M., and Tsuchiaki M. Review of Nuclear and Particle Science. 1995, v. 45, 89–131. Physical Review Letters, 1989, v. 63, 597–600. 22. Bromley D. A. Clust. Aspects of Nucl. Structure, Proc. of the 51. Mitsui T., Fujimoto R., Ishisaki Y., Ueda Y., Yamazaki Y., 4th Intern. Conf., Chester (UK), D. Reidel Publ., Dordrecht. Asai S., and Orito S. Phys. Rev. Lett., 1993, v. 70, 2265–2268. 23. Horiuchi H. and Ikeda K. Cluster Models and Other Topics, 52. Skalsey M. and Conti R.S. Phys. Rev. A, 1997, v. 55(2), 984. Intern. Rev. of Nucl. Physics, 1986, v. 4, World Scientific, 53. Adkins G.S., Melnikov K., and Yelkhovsky A. Phys. Rev. A, Singapore, 1–259. 1999, v. 60(4), 3306–3307.
CLUSTER MODELS THAT THE BRIGHTSEN MODEL BUILDS ON 54. Asai S., Orito S., and Shinohara N. Physics Letters B, 1995, v. 357, 475–480. 24. Wheeler J. A. Phys. Rev., 1937, v. 52, 1083. 55. Vallery R. S., Zitzewitz P.W., and Gidley D. W. Phys. Rev. 25. Wheeler J. A. Phys. Rev., 1937, v. 52, 1107. Lett., 2003, v. 90, 203402. 26. Pauling L. Proc. Natl. Acad. Sci. USA, 1965, v. 54, no. 4, 989. 56. Levin B. M. arXiv: quant-ph/0303166. 27. Pauling L. Science, 1965, v. 150, no. 3694, 297. 57. Levin B. M. CERN EXT-2004-016. 28. Pauling L. Revue Roumain de Physique, 1966, v. 11, no. 9/10, 58. Karshenboim S. G. Yadern. Fizika, 1993, v. 56(12), 155–171. 825–833.
ANOMALIES OF ORTHOPOSITRONIUM ANNIHILATION 29. Osmon P.E. Physical Review B, 1965, v. 138, 216. 30. Tao S. J., Bell J., and Green J. H. Proceedings of the Physical Society, 1964, v. 83, 453.
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On Emergent Physics, “Unparticles” and Exotic “Unmatter” States
Ervin Goldfain and Florentin SmarandacheO Photonics CoE, Welch Allyn Inc., Skaneateles Falls, NY 13153, USA E-mail: [email protected] OChair of Math & Sc. Depart., University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] Emergent physics refers to the formation and evolution of collective patterns in systems that are nonlinear and out-of-equilibrium. This type of large-scale behavior often de- velops as a result of simple interactions at the component level and involves a dynamic interplay between order and randomness. On account of its universality, there are credi- ble hints that emergence may play a leading role in the Tera-ElectronVolt(TeV) sector of particle physics. Following this path, we examine the possibility of hypothetical high- energy states that have fractional number of quanta per state and consist of arbitrary mixtures of particles and antiparticles. These states are similar to “un-particles”, mass- less fields of non-integral scaling dimensions that were recently conjectured to emerge in the TeV sector of particle physics. They are also linked to “unmatter”, exotic clusters of matter and antimatter introduced few years ago in the context of Neutrosophy.
1 Introduction long as " 1. Full scale invariance and equilibrium field theory are asymptotically recovered in the limit of physical space-time (d =4)as " ! 0 or !1[11, 12]. Quantum Field Theory (QFT) is a framework whose meth- ods and ideas have found numerous applications in various domains, from particle physics and condensed matter to cos- 2 Definitions mology, statistical physics and critical phenomena [1, 2]. As We use below the Riemann-Liouville definition for the one- successful synthesis of Quantum Mechanics and Special Rel- dimensional left and right fractal operators [13]. Consider for ativity, QFT represents a collection of equilibrium field theo- simplicity a space-independent scalar field '(t). Taking the ries and forms the foundation for the Standard Model (SM), time coordinate to be the representative variable, one writes a body of knowledge that describes the behavior of all known particles and their interactions, except gravity. Many broken Zt 1 d symmetries in QFT, such as violation of parity and CP in- DL '(t)= (t ) '( )d ; (1) variance, are linked to either the electroweak interaction or (1 ) dt the physics beyond SM [3–5]. This observation suggests that unitary evolution postulated by QFT no longer holds near or Z 1 d above the energy scale of electroweak interaction DR '(t)= ( ) ( t) '( )d : (2) 300 (1 ) dt ( GeV) [6,7]. It also suggests that progress on the the- t oretical front requires a framework that can properly handle 0 <<1 non-unitary evolution of phenomena beyond SM. We believe Here, fractional dimension denotes the order ff that fractional dynamics naturally fits this description. It op- of fractional di erentiation. In general, it can be shown that erates with derivatives of non-integer order called fractal op- is linearly dependent on the dimensionality of the space-time erators and is suitable for analyzing many complex processes support [8]. By definition, assumes a continuous spectrum with long-range interactions [6–9]. Building on the current of values on fractal supports [11]. understanding of fractal operators, we take the dimensional parameter of the regularization program " =4d to represent 3 Fractional dynamics and ‘unparticle’ physics the order of fractional differentiation in physical space-time (alternatively, " =1 d in one-dimensional space) [10, 11]. The classical Lagrangian for the free scalar field theory in + It can be shown that " is related to the reciprocal of the cutoff 3 1 dimensions reads [1–2, 14] scale " ( /), where stands for a finite and arbitrary L = @ '@' m ' ; (3) reference mass and is the cutoff energy scale. Under these " circumstances, may be thought as an infinitesimal param- and yields the following expression for the field momentum eter that can be continuously tuned and drives the departure @L @' from equilibrium. The approach to scale invariance demands = = : @' @t (4) that the choice of this parameter is completely arbitrary, as @( @t )
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It is known that the standard technique of canonical quan- 4 Mixing properties of fractal operators tization promotes a classical field theory to a quantum field theory by converting the field and momentum variables into Left and right fractal operators (L/R) are natural analogues of operators. To gain full physical insight with minimal com- chiral components associated with the structure of quantum plications in formalism, we work below in 0+1 dimensions. fields [8, 9]. The goal of this section is to show that there is an Ignoring the left/right labels for the time being, we define the inherent mixing of (L/R) operators induced by the fractional field and momentum operators as dynamics, as described below. An equivalent representation of (1) is given by ' '> = '; (5) Z0 1 d @ 0DL '(t)= ( ) [( t)] '( ) d ; (14) > = i iD : (1) dt @ j'j (6) t or Without the loss of generality, we set m =1in (3). The Z0 (1) d Hamiltonian becomes D '(t)= ( t) '( ) d = 0 L (1 ) dt 1 1 1 t > H ! H = D + ' = (> + ' ) : (7) 2 2 2 =(1) 0DR '(t) ; (15) By analogy with the standard treatment of harmonic oscil- lator in quantum mechanics, it is convenient to work with the 0DR =(1) 0DL = exp(i) 0DL : (16) destruction and creation operators defined through [1–2, 14] Starting from (2) instead, we find : 1 >a = p ['> + i> ] ; (8) D =(1) D = exp(i) D : 2 0 L 0 R 0 R (17)
: 1 Consider now the one-dimensional case d =1, take >a+ = p ['> i > ] : (9) = " =1 d " 2 and recall that continuous tuning of does not impact the physics as a consequence of scale invariance. Let Straightforward algebra shows that these operators satisfy us iterate (16) and (17) a finite number of times (n 1) under the following commutation rules the assumption that n" 1. It follows that the fractal opera- tor of any infinitesimal order may be only defined up to an ar- [>a; >a]=[>a+ ; >a+ ]=0; (10) bitrary dimensional factor exp(in") 1+(in")=1i"e, + ( 1) that is, [>a ; >a ]=i [';> > ]= > : (11) " 0 0DL;R '(t) 0DL;R i"e '(t) (18) The second relation of these leads to or " 0 1 i0D '(t)= i 0D + "e '(t) ; (19) H> = >a+ >a + >( 1): L;R L;R 2 (12) where " lim DL;R '(t)='(t) : (20) In the limit =1we recover the quantum mechanics of "0 the harmonic oscillator, namely Relations (18–20) indicate that fractional dimension "ein- 1 duces: (a) a new type of mixing between chiral components H> = >a+>a + : 2 (13) of the field and (b) an ambiguity in the very definition of the field, fundamentally different from measurement uncertain- It was shown in [6] that the fractional Hamiltonian (12) ties associated with Heisenberg principle. Both effects are leads to a continuous spectrum of states having non-integer irreversible (since fractional dynamics describes irreversible numbers of quanta per state. These unusual flavors of par- processes) and of topological nature (being based on the con- ticles and antiparticles emerging as fractional objects were cept of continuous dimension). They do not have a counter- named “complexons”. Similar conclusions have recently sur- part in conventional QFT. faced in a number of papers where the possibility of a scale- invariant “hidden” sector of particle physics extending be- 5 Emergence of “unmatter” states yond SM has been investigated. A direct consequence of this setting is a continuous spectrum of massless fields having Using the operator language of QFT and taking into account non-integral scaling dimensions called “un-particles”. The (6), (18) can be presented as reader is directed to [15–21] for an in-depth discussion of “un-particle” physics. b"'(t)=b"'(t) "e'b(t) : (21)
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Relation (21) shows that the fractional momentum op- positive particle (say clusters of quarks and antiquarks whose erator > and the field operator '>(t)='(t) are no longer total charge is positive, etc.). independent entities but linearly coupled through fractional dimension "A. From (11) it follows that the destruction and 8 Unmatter atom creation operators are also coupled to each other. As a re- sult, particles and antiparticles can no longer exist as linearly It is possible to define the unmatter in a more general way, "A independent objects. Because is continuous, they emerge using the exotic atom. as an infinite spectrum of mixed states. This surprising find- The classical unmatter atoms were formed by particles ing is counterintuitive as it does not have an equivalent in like (a) electrons, protons, and antineutrons, or (b) antielec- conventional QFT. Moreover, arbitrary mixtures of particles trons, antiprotons, and neutrons. and antiparticles may be regarded as a manifestation of “un- In a more general definition, an unmatter atom is a system matter”, a concept launched in the context of Neutrosophic of particles as above, or such that one or more particles are Logic [22–24]. replaces by other particles of the same charge. Other categories would be (c) a matter atom with where 6 Definition of unmatter one or more (but not all) of the electrons and/or protons are replaced by antimatter particles of the same corresponding In short, unmatter is formed by matter and antimatter that charges, and (d) an antimatter atom such that one or more (but bind together [23, 24]. not all) of the antielectrons and/or antiprotons are replaced by The building blocks (most elementary particles known to- matter particles of the same corresponding charges. day) are 6 quarks and 6 leptons; their 12 antiparticles also In a more composed system we can substitute a particle exist. by an unmatter particle and form an unmatter atom. Then unmatter will be formed by at least a building block Of course, not all of these combinations are stable, semi- and at least an antibuilding block which can bind together. stable, or quasi-stable, especially when their time to bind to- Let’s start from neutrosophy [22], which is a generaliza- gether might be longer than their lifespan. tion of dialectics, i.e. not only the opposites are combined but also the neutralities. Why? Because when an idea is launched, a category of people will accept it, others will reject 9 Examples of unmatter it, and a third one will ignore it (don’t care). But the dynamics between these three categories changes, so somebody accept- During 1970–1975 numerous pure experimental verifications ing it might later reject or ignore it, or an ignorant will accept were obtained proving that “atom-like” systems built on nu- it or reject it, and so on. Similarly the dynamicity of , cleons (protons and neutrons) and anti-nucleons (anti-protons
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that they can meet one another at the annihilation distance. antiatom is unmatter if the antielectron(s) are replaced by Therefore nucleon—anti-nucleon system annihilates in any positively-charged messons. case, this system is unstable by definition having life span no The strange matter can be unmatter if these exists at least more than 10 sec. an antiquark together with so many quarks in the nucleous. Unfortunately, the researchers limited the research to the Also, we can define the strange antimatter as formed by a consideration of pp and pn nuclei only. The reason was large number of antiquarks bound together with an antielec- that they, in the absence of a theory, considered pp and pn tron around them. Similarly, the strange antimatter can be “atoms” as only a rare exception, which gives no classes of unmatter if there exists at least one quark together with so matter. many antiquarks in its nucleous. The unmatter does exists, for example some messons and The bosons and antibosons help in the decay of unmatter. antimessons, through for a trifling of a second lifetime, so the There are 13+1 (Higgs boson) known bosons and 14 anti- pions are unmatter (which have the composition uˆd and udˆ, bosons in present. where by uˆ we mean anti-up quark, d = down quark, and analogously u = up quark and dˆ = anti-down quark, while 10 Chromodynamics formula by ˆ means anti), the kaon K+ (usˆ), K (uˆs), Phi (ssˆ), D+ 0 + 0 0 (cdˆ), D (cuˆ), DI (csˆ), J/Psi (ccˆ), B (buˆ), B (dbˆ), BI In order to save the colorless combinations prevailed in the (sbˆ), Upsilon (bbˆ), where c = charm quark, s = strange Theory of Quantum Chromodynamics (QCD) of quarks and quark, b = bottom quark, etc. are unmatter too. antiquarks in their combinations when binding, we devise the + + Also, the pentaquark Theta-plus ( ), of charge 1, following formula: uuddsˆ (i.e. two quarks up, two quarks down, and one anti- Q A 2M ; strange quark), at a mass of 1.54 GeV and a narrow width of 3 (22) ff 22 MeV, is unmatter, observed in 2003 at the Je erson Lab where M3 means multiple of three, i.e. M3=f3kjk2Zg= in Newport News, Virginia, in the experiments that involved f:::; 12; 9; 6; 3; 0; 3; 6; 9; 12;:::g, and Q = number multi-GeV photons impacting a deuterium target. Similar of quarks, A = number of antiquarks. pentaquark evidence was obtained by Takashi Nakano of Os- But (22) is equivalent to: aka University in 2002, by researchers at the ELSA acceler- ator in Bonn in 1997–1998, and by researchers at ITEP in Q A(mod3) (23) Moscow in 1986. Q A Besides Theta-plus, evidence has been found in one ( is congruent to modulo 3). To justify this formula we mention that 3 quarks form a experiment [25] for other pentaquarks, 5 (ddssuˆ) and + colorless combination, and any multiple of three (M3) com- 5 (uussdˆ). D. S. Carman [26] has reviewed the positive and null ev- bination of quarks too, i.e. 6, 9, 12, etc. quarks. In a similar idence for these pentaquarks and their existence is still under way, 3 antiquarks form a colorless combination, and any mul- M investigation. tiple of three ( 3) combination of antiquarks too, i.e. 6, 9, In order for the paper to be self-contained let’s recall that 12, etc. antiquarks. Hence, when we have hybrid combina- the pionium is formed by a + and mesons, the positro- tions of quarks and antiquarks, a quark and an antiquark will nium is formed by an antielectron (positron) and an electron annihilate their colors and, therefore, what’s left should be in a semi-stable arrangement, the protonium is formed by a a multiple of three number of quarks (in the case when the ff proton and an antiproton also semi-stable, the antiprotonic number of quarks is bigger, and the di erence in the formula helium is formed by an antiproton and electron together with is positive), or a multiple of three number of antiquarks (in the helium nucleus (semi-stable), and muonium is formed by the case when the number of antiquarks is bigger, and the ff a positive muon and an electron. di erence in the formula is negative). Also, the mesonic atom is an ordinary atom with one or more of its electrons replaced by negative mesons. 11 Quantum chromodynamics unmatter formula The strange matter is a ultra-dense matter formed by a big number of strange quarks bounded together with an electron In order to save the colorless combinations prevailed in the atmosphere (this strange matter is hypothetical). Theory of Quantum Chromodynamics (QCD) of quarks and From the exotic atom, the pionium, positronium, proto- antiquarks in their combinations when binding, we devise the nium, antiprotonic helium, and muonium are unmatter. following formula: The mesonic atom is unmatter if the electron(s) are re- Q A 2M3 ; (24) placed by negatively-charged antimessons. Also we can define a mesonic antiatom as an ordinary where M3 means multiple of three, i.e. M3=f3kjk2Zg= antiatomic nucleous with one or more of its antielectrons re- f:::; 12; 9; 6; 3; 0; 3; 6; 9; 12;:::g, and Q = number placed by positively-charged mesons. Hence, this mesonic of quarks, A = number of antiquarks, with Q 1 and A 1.
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But (24) is equivalent to: — For combinations of 5 we have: qqqqa, or aaaaq (un- matter pentaquarks); the number of all possible unmat- Q A(mod3) (25) ter combinations will be 6" 6+6"6 = 15,552, but not all of them will bind together; (Q is congruent to A modulo 3), and also Q 1 and A 1. — For combinations of 6 we have: qqqaaa (unmatter hex- aquarks); the number of all possible unmatter combi- 12 Quark-antiquark combinations nations will be 6! 6! = 46,656, but not all of them will bind together; Let’s note by q = quark {Up, Down, Top, Bottom, Strange, Charm}, and by a = antiquark {Up, Down, Top, Bottom, — For combinations of 7 we have: qqqqqaa, qqaaaaa (un- Strange, Charm}. matter septiquarks); the number of all possible unmat- # + # = Hence, for combinations of n quarks and antiquarks, ter combinations will be 6 6 6 6 559,872, n 2, prevailing the colorless, we have the following pos- but not all of them will bind together; sibilities: — For combinations of 8 we have: qqqqaaaa, qqqqqqqa, qaaaaaaa (unmatter octoquarks); the number of all pos- — if n = 2, we have: qa (biquark — for example the sible unmatter combinations will be 6" 6" + 6% 6 mesons and antimessons); + 6 6% = 5,038,848, but not all of them will bind = — if n 3, we have qqq, aaa (triquark — for example the together; baryons and antibaryons); — For combinations of 9 we have: qqqqqqaaa, qqqaaaaaa = — if n 4, we have qqaa (tetraquark); (unmatter nonaquarks); the number of all possible un- — if n = 5, we have qqqqa, aaaaq (pentaquark); matter combinations will be 6$6! + 6! 6$ = 26' = — if n = 6, we have qqqaaa, qqqqqq, aaaaaa (hexaquark); 20,155,392, but not all of them will bind together; — if n = 7, we have qqqqqaa, qqaaaaa (septiquark); — For combinations of 10: qqqqqqqqaa, qqqqqaaaaa, qqaaaaaaaa (unmatter decaquarks); the number of — if n = 8, we have qqqqaaaa, qqqqqqaa, qqaaaaaa (oc- all possible unmatter combinations will be 36 = toquark); 181,398,528, but not all of them will bind together; — if n = 9, we have qqqqqqqqq, qqqqqqaaa, qqqaaaaaa, — etc. aaaaaaaaa (nonaquark); I wonder if it is possible to make infinitely many combina- — if n = 10, obtain qqqqqaaaaa, qqqqqqqqaa, qqaaaaaaaa tions of quarks/antiquarks and leptons/antileptons. . . Unmat- (decaquark); ter can combine with matter and/or antimatter and the result — etc. may be any of these three. Some unmatter could be in the strong force, hence part of 13 Unmatter combinations hadrons.
From the above general case we extract the unmatter combi- 14 Unmatter charge nations: — For combinations of 2 we have: qa (unmatter biquark), The charge of unmatter may be positive as in the pentaquark (mesons and antimesons); the number of all possible Theta-plus, 0 (as in positronium), or negative as in anti-Rho unmatter combinations will be 66 = 36, but not all of meson, i.e. uˆd, (M. Jordan). them will bind together. It is possible to combine an entity with its mirror oppo- 15 Containment site and still bound them, such as: uuˆ, ddˆ, ssˆ, ccˆ, bbˆ I think for the containment of antimatter and unmatter it which form mesons. would be possible to use electromagnetic fields (a container It is possible to combine, unmatter + unmatter = un- whose walls are electromagnetic fields). But its duration is matter,asinudˆ+ usˆ = uudˆsˆ (of course if they bind unknown. together); — For combinations of 3 (unmatter triquark) we can not 16 Summary and conclusions form unmatter since the colorless can not hold. — For combinations of 4 we have: qqaa (unmatter tetra- It is apparent from these considerations that, in general, both quark); the number of all possible unmatter combina- “unmatter” and “unparticles” are non-trivial states that may tions will be 6 6 = 1,296, but not all of them will become possible under conditions that substantially deviate bind together; from our current laboratory settings. Unmatter can be thought
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as arbitrary clusters of ordinary matter and antimatter, unpar- 22. Smarandache F. A unifying field in logics, neutrosophic logic. ticles contain fractional numbers of quanta per state and carry Neutrosophy, neutrosophic set, neutrosophic probability. Am. arbitrary spin [6]. They both display a much richer dynamics Res. Press, 1998. than conventional SM doublets, for example mesons (quark- 23. Smarandache F. Matter, antimatter, and unmatter. CDS-CERN, antiquark states) or lepton pairs (electron-electron antineu- EXT-2005-142, 2004; Bulletin of Pure and Applied Sciences, trino). Due to their unusual properties, “unmatter” and “un- 2004, v. 23D, no. 2, 173–177. particles” are presumed to be highly unstable and may lead 24. Smarandache F. Verifying unmatter by experiments, more types to a wide range of symmetry breaking scenarios. In particu- of unmatter, and a quantum chromodynamics formula. Progress lar, they may violate well established conservation principles in Physics, 2005, v. 2, 113–116; Infinite Energy, 2006, v. 12, such as electric charge, weak isospin and color. Future obser- no. 67, 36–39. vational evidence and analytic studies are needed to confirm, 25. Chubb S. Breaking through editorial. Infinite Energy, 2005, expand or falsify these tentative findings. v. 11, no. 62, 6–7. 26. Carman D.S. Experimental evidence for the pentaquark. Euro / Submitted on June 26, 2008 Accepted on July 19, 2008 Phys. A, 2005, v. 24, 15–20. 27. Gray L., Hagerty P., and Kalogeropoulos T.E. Evidence for the References existence of a narrow p-barn bound state. Phys. Rev. Lett., 1971, v. 26, 1491–1494. 1. Weinberg S. The quantum theory of fields. Volume 1: founda- 28. Carrol A.S., Chiang I.-H., Kucia T.F., Li K.K., Mazur P.O., tions. Cambridge University Press, London, 1995. Michael D.N., Mockett P., Rahm D.C., and Rubinstein R. Ob- 2. Zinn-Justin J. Quantum field theory and critical phenomena. servation of structure in p-barp and p-bard total cross cections Oxford University Press, Oxford, 2002. below 1.1 GeV/c. Phys. Rev. Lett., 1974, v. 32, 247–250. 3. Hagiwara K. et al. Phys. Rev. D, 2002, v. 66, 010001-1; 29. Kalogeropoulos T.E., Vayaki A., Grammatikakis G., Tsilimi- Nir Y. CP violation and beyond the Standard Model. XXVII gras T., and Simopoulou E. Observation of excessive and direct SLAC Summer Institute, 1999; Rogalyov R.N. arXiv: hep- gamma production in p-bard annihilations at rest. Phys. Rev. ph/0204099. Lett., 1974, v. 33, 1635–1637. 4. Bigi I.I. and Sanda A.I. CP violation. Cambridge University 30. Chapiro I.S. Physics-Uspekhi (Uspekhi Fizicheskikh Nauk), Press, 2000; Yao W.M. et al. Journal of Phys. G, 2006, v. 33(1). 1973, v. 109, 431. 5. Donoghue J.F. et al.. Dynamics of the Standard Model. Cam- 31. Bogdanova L.N., Dalkarov O.D., and Chapiro I.S. Quasinu- bridge University Press, 1996. clear systems of nucleons and antinucleons. Annals of Physics, 6. Goldfain E. Chaos, Solitons and Fractals, 2006, v. 28(4), 913. 1974, v. 84, 261–284. 7. Goldfain E. Intern. Journal of Nonlinear Sciences and Numer- 32. Chapiro I.S. New “nuclei” built on nucleons and anti-nucleons. ical Simulation, 2005, v. 6(3), 223. Nature-USSR, 1975, no. 12, 68–73. 8. Goldfain E. Comm. in Nonlinear Science and Numer. Simula- 33. Barmin V.V. et al. (DIANA Collaboration). Phys. Atom. Nucl., tion, 2008, v. 13, 666. 2003, v. 66, 1715. 9. Goldfain E. Comm. in Nonlinear Science and Numer. Simula- 34. Ostrick M. (SAPHIR Collaboration). Pentaquark 2003 Work- tion, 2008, v. 13, 1397. shop, Newport News, VA, Nov. 6–8, 2003. 10. Ballhausen H. et al. Phys. Lett. B, 2004, v. 582, 144. 35. Smith T.P. Hidden worlds, hunting for quarks in ordinary mat- ter. Princeton University Press, 2003. 11. Goldfain E. Chaos, Solitons and Fractals, 2008, v. 38(4), 928. 36. Wang M.J. (CDF Collaboration). In: Quarks and Nuclear 12. Goldfain E. Intern. Journal of Nonlinear Science, 2007, v. 3(3), Physics 2004, Bloomington, IN, May 23–28, 2004. 170. 13. Samko S.G. et al. Fractional integrals and derivatives. Theory and applications. Gordon and Breach, New York, 1993. 14. Ryder L. Quantum field theory. Cambridge University Press, 1996. 15. Georgi H. Phys. Rev. Lett., 2007, v. 98, 221601. 16. Georgi H. Phys. Lett. B, 2007, v. 650, 275. 17. Cheung K. et al. Phys. Rev. D, 2007, v. 76, 055003. 18. Bander M. et al. Phys. Rev. D, 2007, v. 76, 115002. 19. Zwicky R. Phys. Rev. D, 2008, v. 77, 036004. 20. Kikuchi T. et al. Phys. Rev. D, 2008, v. 77, 094012. 21. Goldfain E. A brief note on “un-particle” physics. Progress in Physics, 2008, v. 28, v. 3, 93.
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Thirty Unsolved Problems in the Physics of Elementary Particles
Vic Christianto and Florentin SmarandacheO Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] ODepartment of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected]
Unlike what some physicists and graduate students used to think, that physics science has come to the point that the only improvement needed is merely like adding more numbers in decimal place for the masses of elementary particles or gravitational con- stant, there is a number of unsolved problems in this field that may require that the whole theory shall be reassessed. In the present article we discuss thirty of those un- solved problems and their likely implications. In the first section we will discuss some well-known problems in cosmology and particle physics, and then other unsolved prob- lems will be discussed in next section.
1 Unsolved problems related to cosmology 5. Antimatter hydrogen observation. Is it possible to find isolated antimatter hydrogen (antihydrogen) in astro- In the present article we discuss some unsolved problems physics (stellar or galaxies) observation? Is there anti- in the physics of elementary particles, and their likely im- hydrogen star in our galaxy? plications. In the first section we will discuss some well- Now we are going to discuss other seemingly interesting known problems in cosmology and particle physics, and then problems in the physics of elementary particles, in particu- other unsolved problems will be discussed in next section. lar those questions which may be related to the New Energy Some of these problems were inspired by and expanded from science. Ginzburg’s paper [1]. The problems are:
1. The problem of the three origins. According to Mar- 2 Unsolved problems in the physics of elementary par- celo Gleiser (Darthmouth College) there are three un- ticles solved questions which are likely to play significant role in 21st-century science: the origin of the universe, We discuss first unsolved problems in the Standard Model the origin of life, and the origin of mind; of elementary particles. Despite the fact that Standard Model 2. The problem of symmetry and antimatter observation. apparently comply with most experimental data up to this day, This could be one of the biggest puzzle in cosmology: the majority of particle physicists feel that SM is not a com- If it’s true according to theoretical physics (Dirac equa- plete framework. E. Goldfain has listed some of the most tion etc.) that there should be equal amounts of matter cited reasons for this belief [5], as follows: and antimatter in the universe, then why our observa- 6. The neutrino mass problem. Some recent discovery in- tion only display vast amounts of matter and very little dicates that neutrino oscillates which implies that neu- antimatter? trino has mass, while QM theories since Pauli predict 3. The problem of dark matter in cosmology model. Do that neutrino should have no mass [6]. Furthermore it we need to introduce dark matter to describe galaxy is not yet clear that neutrino (oscillation) phenomena rotation curves? Or do we need a revised method in correspond to Dirac or Majorana neutrino [7]; our cosmology model? Is it possible to develop a new 7. SM does not include the contribution of gravity and theory of galaxy rotation which agrees with observa- gravitational corrections to both quantum field theory tions but without invoking dark matter? For example and renormalization group (RG) equations; ff of such a new theory without dark matter, see Mo at 8. SM does not fix the large number of parameters that en- and Brownstein [2, 3]; ter the theory (in particular the spectra of masses, gauge 4. Cosmological constant problem. This problem repre- couplings, and fermion mixing angles). Some physi- sents one of the major unresolved issues in contempo- cists have also expressed their objections that in the rary physics. It is presumed that a presently unknown QCD scheme the number of quarks have increased to symmetry operates in such a way to enable a vanish- more than 30 particles, therefore they assert that QCD- ingly small constant while remaining consistent with quark model cease to be a useful model for elementary all accepted field theoretic principles [4]; particles;
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9. SM has a gauge hierarchy problem, which requires fine a direct reflection of the rate of radioactive decays in tuning. Another known fine-tuning problem in SM is the Earth and so of the associated energy production, “strong CP problem” [8, p. 18]; which is presumably the main source of Earth’s 10. SM postulates that the origin of electroweak symmetry heat [14]; breaking is the Higgs mechanism. Unfortunately Higgs 23. Is it possible to explain the origin of electroweak sym- particle has never been found; therefore recently some metry breaking without the Higgs mechanism or Higgs physicists feel they ought to introduce more speculative particles? For an example of such alternative theory to theories in order to save their Higgs mechanism [9]; derive boson masses of electroweak interaction without 11. SM does not clarify the origin of its gauge group introducing Higgs particles, see E. Goldfain [16]; SU(3) SU(2) U(1) and why quarks and lepton occur 24. Is it possible to write quaternionic formulation [17] of as representations of this group; quantum Hall effect? If yes, then how? 12. SM does not explain why (only) the electroweak inter- 25. Orthopositronium problem [18]. What is the dynamics actions are chiral (parity-violating) [8, p. 16]; behind orthopositronium observation? 13. Charge quantization problem. SM does not explain an- 26. Is it possible to conceive New Energy generation other fundamental fact in nature, i.e. why all particles method from orthopositronium-based reaction? If yes, have charges which are multiples of e=! [8, p. 16]. then how? Other than the known problems with SM as described 27. Muonium problem. Muonium is atom consisting of above, there are other quite fundamental problems related to muon and electron, discovered by a team led by Ver- the physics of elementary particles and mathematical physics non Hughes in 1960 [19]. What is the dynamics behind in general, for instance [10]: muonium observation? 14. Is there dynamical explanation of quark confinement 28. Is it possible to conceive New Energy generation problem? This problem corresponds to the fact that method from muonium-based reaction? If yes, then quarks cannot be isolated. See also homepage by Clay how? Institute on this problem; 29. Antihydrogen problem [20]. Is it possible to conceive 15. What is the dynamical mechanism behind Koide’s mix- New Energy generation method from antihydrogen- ing matrix of the lepton mass formula [11]? based reaction? If yes, then how? 16. Does neutrino mass correspond to the Koide mixing 30. Unmatter problem [21]. Would unmatter be more use- matrix [12]? ful to conceiving New Energy than antimatter? If yes, then how? 17. Does Dirac’s new electron theory in 1951 reconcile the quantum mechanical view with the classical electrody- It is our hope that perhaps some of these questions may namics view of the electron [13]? be found interesting to motivate further study of elementary 18. Is it possible to explain anomalous ultraviolet hydrogen particles. spectrum? 19. Is there quaternion-type symmetry to describe neutrino Acknowledgment masses? VC would like to dedicate this article for RFF. 20. Is it possible to describe neutrino oscillation dynam- ics with Bogoliubov-deGennes theory, in lieu of using Submitted on September 24, 2007 standard Schrodinger-type¨ wave equation [6]? Accepted on September 26, 2007 21. Solar neutrino problem — i.e. the seeming deficit of observed solar neutrinos [14]. The Sun through fusion, References send us neutrinos, and the Earth through fission, an- 1. Ginzburg V.L. What problems of physics and astrophysics tineutrinos. But observation in SuperKamiokande etc. seem now to be especially important and interesting (thirty discovers that the observed solar neutrinos are not as years later, already on the verge of XXI century)? Physics- expected. In SuperKamiokande Lab, it is found that the Uspekhi, 1999, v. 42(2), 353–373. number of electron neutrinos which is observed is 0.46 2. Moffat J. W. Scalar-Tensor-Vector gravity theory. To be pub- that which is expected [15]. One proposed explanation lished in J. Cosmol. Astropart. Phys., 2006; preprint arXiv: gr- for the lack of electron neutrinos is that they may have qc/0506021. oscillated into muon neutrinos; 3. Moffat J. W. Spectrum of cosmic microwave fluctuations and 22. Neutrino geology problem. Is it possible to observe the formation of galaxies in a modified gravity theory. arXiv: terrestrial neutrino? The flux of terrestrial neutrino is astro-ph/0602607.
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4. Goldfain E. Dynamics of neutrino oscillations and the cosmo- logical constant problem. To appear at Far East J. Dynamical Systems, 2007. 5. Goldfain E. Fractional dynamics in the Standard Model for par- ticle physics. To appear at Comm. Nonlin. Science and Numer. Simul., 2007; see preprint in http://www.sciencedirect.com. 6. Giunti C. Theory of neutrino oscillations. arXiv: hep-ph/ 0401244. 7. Singh D., et al. Can gravity distinguish between Dirac and Ma- jorana neutrinos? arXiv: gr-qc/0605133. 8. Langacker P. Structure of the Standard Model. arXiv: hep- ph/0304186, p.16. 9. Djouadi A., et al. Higgs particles. arXiv: hep-ph/9605437. 10. Smarandache F., Christianto V., Fu Yuhua, Khrapko R., and Hutchison J. In: Unfolding Labyrinth: Open Problems in Phys- ics, Mathematics, Astrophysics and Other Areas of Science, Phoenix (AZ), Hexis, 2006, p. 8–9; arxiv: math/0609238. 11. Koide Y. arXiv: hep-ph/0506247; hep-ph/0303256. 12. Krolikowski W. Towards a realistics neutrino mass formula. arXiv: hep-ph/0609187. 13. deHaas P.J. A renewed theory of electrodynamics in the frame- work of Dirac ether. London PIRT Conference 2004. 14. Stodolsky L. Neutrino and dark matter detection at low temper- ature. Physics-Today, August 1991, p. 3. 15. Jaffe R. L. Two state systems in QM: applications to neutrino oscillations and neutral kaons. MIT Quantum Theory Notes, Supplementary Notes for MIT’s Quantum Theory Sequence, (August 2006), p. 26–28. 16. Goldfain E. Derivation of gauge boson masses from the dy- namics of Levy flows. Nonlin. Phenomena in Complex Systems, 2005, v. 8, no. 4. 17. Balatsky A. V. Quaternion generalization of Laughlin state and the three dimensional fractional QHE. arXiv: cond-mat/ 9205006. 18. Kotov B. A., Levin B. M. and Sokolov V.I. et al. On the possi- bility of nuclear synthesis during orthopositronium formation. Progress in Physics, 2007, v. 3. 19. Jungmann K. Past, present and future of muonium. arXiv: nucl- ex/040401. 20. Voronin A. and Carbonell J. Antihydrogen-hydrogen annihila- tion at sub-kelvin temperatures. arXiv: physics/0209044. 21. Smarandache F. Matter, antimatter, and unmatter. Infinite En- ergy, 2005, v. 11, issue 62, 50–51.
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28 April, 2005 PROGRESS IN PHYSICS Volume 1
There Is No Speed Barrier for a Wave Phase Nor for Entangled Particles
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
In this short paper, as an extension and consequence of Einstein-Podolski-Rosen paradox and Bell’s inequality, one promotes the hypothesis (it has been called the Smarandache Hypothesis [1, 2, 3]) that: There is no speed barrier in the Universe and one can construct arbitrary speeds, and also one asks if it is possible to have an infinite speed (instantaneous transmission)? Future research: to study the composition of faster-than-light velocities and what happens with the laws of physics at faster-than- light velocities?
This is the new version of an early article. That early version, sert that the light speed is not a speed barrier in the based on a 1972 paper [4], was presented at the Universidad Universe. de Blumenau, Brazil, May–June 1993, in the Conference Such results were also obtained by: Nicolas Gisin at on “Paradoxism in Literature and Science”; and at the Uni- the University of Geneva, Switzerland, who successfully versity of Kishinev, in December 1994. See that early ver- teleported quantum bits, or qubits, between two labs over sion in [5]. 2 km of coiled cable. But the actual distance between the two labs was about 55 m; researchers from the University of Vienna and the Austrian Academy of Science (Rupert Ursin 1 Introduction et al. have carried out successful teleportation with particles What is new in science (physics)? of light over a distance of 600 m across the River Danube in According to researchers from the common group of the Austria); researchers from Australia National University and University of Innsbruck in Austria and US National Institute many others [6, 7, 8]. of Standards and Technology (starting from December 1997, Rainer Blatt, David Wineland et al.): 2 Scientific hypothesis • Photon is a bit of light, the quantum of electromagnetic radiation (quantum is the smallest amount of energy We even promote the hypothesis that: that a system can gain or lose); There is no speed barrier in the Universe, which would theoretically be proved by increasing, in the previous • Polarization refers to the direction and characteristics example, the distance between particles A and B as of the light wave vibration; much as the Universe allows it, and then measuring • If one uses the entanglement phenomenon, in order to particle A. transfer the polarization between two photons, then: It has been called the Smarandache Hypotesis [1,2,3]. whatever happens to one is the opposite of what hap- pens to the other; hence, their polarizations are oppos- 3 An open question now ite of each other; • In quantum mechanics, objects such as subatomic par- If the space is infinite, is the maximum speed infinite? ticles do not have specific, fixed characteristic at any “This Smarandache hypothesis is controversially inter- given instant in time until they are measured; preted by scientists. Some say that it violates the theory of • Suppose a certain physical process produces a pair relativity and the principle of causality, others support the of entangled particles A and B (having opposite or ideas that this hypothesis works for particles with no mass complementary characteristics), which fly off into spa- or imaginary mass, in non-locality, through tunneling effect, ce in the opposite direction and, when they are billions or in other (extra-) dimension(s).” Kamla John, [9]. of miles apart, one measures particle A; because B is Scott Owens’ answer [10] to Hans Gunter in an e-mail the opposite, the act of measuring A instantaneously from January 22, 2001 (the last one forwarded it to the tells B what to be; therefore those instructions would author): “It appears that the only things the Smarandache somehow have to travel between A and B faster than hypothesis can be applied to are entities that do not have real the speed of light; hence, one can extend the Einstein- mass or energy or information. The best example I can come Podolsky-Rosen paradox and Bell’s inequality and as- up with is the difference between the wavefront velocity of
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a photon and the phase velocity. It is common for the phase 9. Kamla J. Private communications. 2001. velocity to exceed the wavefront velocity c, but that does 10. Owens S. Private communications. 2001. not mean that any real energy is traveling faster than c.So, while it is possible to construct arbitrary speeds from zero in infinite, the superluminal speeds can only apply to purely imaginary entities or components.” Would be possible to accelerate a photon (or another particle traveling at, say, 0.99c and thus to get speed greater than c (where c is the speed of light)?
4 Future possible research
It would be interesting to study the composition of two velocities v and u in the cases when: v
References
1. Motta L. Smarandache Hypothesis: Evidences, Implications, and Applications. Proceedings of the Second Internation- al Conference on Smarandache Type Notions In Mathem- atics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi- bin/amca/caft-03). 2. Motta L. and Niculescu G., editors. Proceedings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, American Research Press, 2000. 3. Weisstein E. W. Smarandache Hypothesis. The Encyclopedia of Physics, Wolfram Research (http://scienceworld.wolfram. com/physics/SmarandacheHypothesis.htm). 4. Smarandache F. Collected Papers. Vol.III, Abaddaba Publ. Hse., Oradea, Romania, 2000, 158. 5. Smarandache F. There is no speed barrier in the Universe. Bulletin of Pure and Applied Sciences, Delhi, India, v. 17D (Physics), No. 1, 1998, 61. 6. Rincon P. Teleportation breakthrough made. BBC News On- line, 2004/06/16. 7. Rincon P. Teleportation goes long distance. BBC News Online, 2004/08/18. 8. Whitehouse D. Australian teleport breakthrough. BBC News Online, 2002/06/17.
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30 July, 2005 PROGRESS IN PHYSICS Volume 2
Entangled States and Quantum Causality Threshold in the General Theory of Relativity
Dmitri Rabounski, Larissa Borissova, Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]; [email protected]
This article shows, Synge-Weber’s classical problem statement about two particles interacting by a signal can be reduced to the case where the same particle is located in two different points A and B of the basic space-time in the same moment of time, so the states A and B are entangled. This particle, being actual two particles in the entangled states A and B, can interact with itself radiating a photon (signal) in the point A and absorbing it in the point B. That is our goal, to introduce entangled states into General Relativity. Under specific physical conditions the entangled particles in General Relativity can reach a state where neither particle A nor particle B can be the cause of future events. We call this specific state Quantum Causality Threshold.
1 Disentangled and entangled particles in General Rel- — particles whose states are entangled, they build a whole ativity. Problem statement system so that if the state of one particle changes the state of the other particles changes immediately as they are far In his article of 2000, dedicated to the 100th anniversary of located one from the other. the discovery of quanta, Belavkin [1] generalizes definitions In particular, because of the permission for entangled assumed de facto in Quantum Mechanics for entangled and states, Quantum Mechanics permits quantum teleportation — disentangled particles. He writes: the experimentally discovered phenomenon. The term “The only distinction of the classical theory from “quantum teleportation” had been introduced into theory quantum is that the prior mixed states cannot be in 1993 [2]. First experiment teleporting massless particles dynamically achieved from pure initial states without (quantum teleportation of photons) was done five years later, a procedure of either statistical or chaotic mixing. In in 1998 [3]. Experiments teleporting mass-bearing particles quantum theory, however, the mixed, or decoherent (atoms as a whole) were done in 2004 by two independ- states can be dynamically induced on a subsystem from the initial pure disentangled states of a composed ent groups of scientists: quantum teleportation of the ion of system simply by a unitary transformation. Calcium atom [4] and of the ion of Beryllium atom [5]. Motivated by Eintein-Podolsky-Rosen paper, in There are many followers who continue experiments with ∗ 1935 Schrodinger¨ published a three part essay on The quantum teleportation, see [6–16] for instance. Present Situation in Quantum Mechanics.Heturns It should be noted, the experimental statement on quan- to EPR paradox and analyses completeness of the tum teleportation has two channels in which information (the description by the wave function for the entangled quantum state) transfers between two entangled particles: parts of the system. (The word entangled was intro- “teleportation channel” where information is transferred in- duced by Schrodinger¨ for the description of nonse- stantly, and “synchronization channel” — classical channel parable states.) He notes that if one has pure states where information is transferred in regular way at the light ψ(σ) and χ(υ) for each of two completely separat- speed or lower of it (the classical channel is targeted to inform ed bodies, one has maximal knowledge, ψ1(σ, υ)= = ψ(σ)χ(υ), for two taken together. But the con- the receiving particle about the initial state of the first one). verse is not true for the entangled bodies, described by After teleportation the state of the first particle destroys, so a non-separable wave function ψ1(σ, υ)=ψ(σ)χ(υ): there is data transfer (not data copying). Maximal knowledge of a total system does not necess- General Relativity draws another picture of data transfer: ary imply maximal knowledge of all its parts, not the particles are considered as point-masses or waves, not even when these are completely separated one from stochastic clouds. This statement is true for both mass-bearing another, and at the time can not influence one another particles and massless ones (photons). Data transfer between at all.” any two particles is realized as well by point-mass particles, In other word, because Quantum Mechanics considers so in General Relativity this process is not of stochastic particles as stochastic clouds, there can be entangled particles origin. ∗Schrodinger¨ E. Naturwissenschaften, 1935, Band 23, 807–812, 823– In the classical problem statement accepted in General 828, 844–849. Relativity [17, 18, 19], two mass-bearing particles are con-
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sidered which are moved along neighbour world-lines, a degenerated g =0 under specific physical conditions. This signal is transferred between them by a photon. One of the space is one of Smarandache geometry spaces [22–28], be- particles radiates the photon at the other, where the photon cause its geometry is partially Riemannian, partially not. is absorbed realizing data transfer between the particles. Of As it was shown in [29, 30] (Borissova and Rabounski, course, the signal can as well be carried by a mass-bearing 2001), when General Relativity’s basic space-time degene- particle. rates physical conditions can imply observable teleportation If there are two free mass-bering particles, they fall freely of both a mass-bearing and massless particle — its instant along neighbour geodesic lines in a gravitational field. This displacement from one point of the space to another, although classical problem has been developed in Synge’s book [20] it moves no faster than light in the degenerated space-time where he has deduced the geodesic lines deviation equation area, outside the basic space-time. In the generalized space- (Synge’s equation, 1950’s). If these are two particles con- time the Synge-Weber problem statement about two particles nected by a non-gravitational force (for instance, by a spring), interacting by a signal (see fig. 1) can be reduced to the case they are moved along neighbour non-geodesic world-lines. where the same particle is located in two different points This classical statement has been developed a few years later A and B of the basic space-time in the same moment of by Weber [21], who has obtained the world-lines deviation time, so the states A and B are entangled (see fig. 2). This equation (Synge-Weber’s equation). particle, being actual two particles in the entangled states A Anyway in this classical problem of General Relativity and B, can interact with itself radiating a photon (signal) in two interacting particles moved along both neighbour geo- the point A and absorbing it in the point B. That is our goal, desic and non-geodesic world-lines are disentangled. This to introduce entangled states into General Relativity. happens, because of two reasons: Moreover, as we will see, under specific physical con- 1. In this problem statement a signal moves between two ditions the entangled particles in General Relativity can reach interacting particles at the velocity no faster than light, a state where neither particle A nor particle B can be the so their states are absolutely separated — these are cause of future events. We call this specific state Quantum disentangled states; Causality Threshold. 2. Any particle, being considered in General Relativity’s 2 Introducing entangled states into General Relativity space-time, has its own four-dimensional trajectory (world-line) which is the set of the particle’s states In the classical problem statement, Synge [20] considered from its birth to decay. Two different particles can not two free-particles (fig. 1) moving along neighbour geodesic occupy the same world-line, so they are in absolutely world-lines Γ(v) and Γ(v + dv), where v is a parameter separated states — they are disentangled particles. along the direction orthogonal to the geodesics (it is taken in The second reason is much stronger than the first one. In the plane normal to the geodesics). There is v = const along particular, the second reason leads to the fact that, in General each the geodesic line. Relativity, entangled are only neighbour states of the same particle along its own world-line — its own states separated world-line A world-line B in time, not in the three-dimensional space. No two different particles could be entangled. Any two different particles, both mass-bearing and massless ones, are disentangled in General α α Relativity. U U On the other hand, experiments on teleportation evident that entanglement is really an existing state that happens with ηα ηα particles if they reach specific physical conditions. This is the A B A B fact, that should be taken into account by General Relativity. Therefore our task in this research is to introduce en- tangled states into General Relativity. Of course, because of the above reasons, two particles can not be in entangled Fig. 1 Fig. 2 state if they are located in the basic space-time of General Relativity — the four-dimensional pseudo-Riemannian space Motion of the particles is determined by the well-known with sign-alternating label (+−−−) or (−+++). Its metric is geodesic equation strictly non-degenerated as of any space of Riemannian space dU α dxν family, namely — there the determinant g = det g of +Γα U μ =0, (1) αβ ds μν ds the fundamental metric tensor gαβ is strictly negative g<0. α We expand the Synge-Weber problem statement, considering which is the actual fact that the absolute differential DU = α α μ ν α it in a generalized space-time whose metric can become = dU +Γμν U dx of a tangential vector U (the velocity
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α world-vector U α = dx , in this case), transferred along that are filled in non-separable states if the observable time ds † geodesic line to where it is tangential, is zero. Here s is interval dτ between linked events in the particles is an invariant parameter along the geodesic (we assume it the zero dτ =0. If only dτ =0, the states become non- α separated one from the other, so the particles A and B space-time interval), and Γμν are Christoffel’s symbols of the 2nd kind. Greek α =0, 1, 2, 3 sign for four-dimensional become entangled. (space-time) indices. So we will refer to dτ =0 as the entanglement condition in The parameter v is different for the neighbour geodesics, General Relativity. the difference is dv. Therefore, in order to study relative dis- Let us consider the entanglement condition dτ =0 in placements of two geodesics Γ(v) and Γ(v + dv),weshall connection with the world-lines deviation equations. study the vector of their infinitesimal relative displacement In General Relativity, the interval of physical observable ∂xα time dτ between two events distant at dxi one from the other ηα = dv , (2) ∂v is determined through components of the fundamental metric As Synge had deduced, a deviation of the geodesic line tensor as √ g0i i dτ = g00 dt + √ dx , (7) Γ(v + dv) from the geodesic line Γ(v) can be found as the c g00 solution of his obtained equation 2 α see §84 in the well-known The Classical Theory of Fields D η α ··· β δ γ + R· U U η =0, (3) by Landau and Lifshitz [19]. The mathematical apparatus of ds2 βγδ physical observable quantities (Zelmanov’s theory of chro- that describes relative accelerations of two neighbour free- nometric invariants [31, 32], see also the brief account in α ··· particles (R· βγδ is Riemann-Chrostoffel’s curvature tensor). [30, 29]) transforms this formula to This formula is known as the geodesic lines deviation equa- tion or the Synge equation. − w − 1 i dτ = 1 2 dt 2 vidx , (8) In Weber’s statement [21] the difference is that he con- c c 2 √ siders two particles connected by a non-gravitational force where w=c (1− g00) is the gravitational potential of an α Φ , a spring for instance. So their world-trajectories are non- √g0i acting gravitational field, and vi =−c is the linear geodesic, they are determined by the equation g00 α ν α velocity of the space rotation. dU α μ dx Φ So, following the theory of physical observable quanti- +Γμν U = 2 , (4) ds ds m0c ties, in real observations where the observer accompanies his 2 α β which is different from the geodesic equation in that the right references the space-time interval ds = gαβ dx dx is part in not zero here. His deduced improved equation of the ds2 = c2dτ 2 − dσ2, (9) world lines deviation 2 α α 2 g0 g0 D η ··· 1 DΦ − i k i k α β δ γ where dσ = gik + 00 dx dx is a three-dimensional 2 + R· βγδ U U η = 2 dv , (5) g ds m0c dv (spatial) invariant, built on the metric three-dimensional ob- − g0ig0k describes relative accelerations of two particles (of the same servable tensor hik = gik + g00 . This metric observable rest-mass m0), connected by a spring. His deviation equation tensor, in real observations where the observer accompanies is that of Synge, except of that non-gravitational force Φα his references, is the same that the analogous built general 2 i k in the right part. This formula is known as the Synge-Weber covariant tensor hαβ.So,dσ = hik dx dx is the spatial equation. In this case the angle between the vectors U α and observable interval for any observer who accompanies his ηα does not remain unchanged along the trajectories references. As it is easy to see from (9), there are two possible cases ∂ α 1 α (Uαη )= 2 Φαη . (6) where the entanglement condition dτ =0 occurs: ∂s m0c (1) ds =0 and dσ =0, Now, proceeding from this problem statement, we are 2 − 2 going to introduce entangled states into General Relativity. (2) ds = dσ =0,sodσ becomes imaginary, At first we determine such states in the space-time of General we will refer to them as the 1st kind and 2nd kind entangle- Relativity, then we find specific physical conditions under ment auxiliary conditions. which two particles reach a state to be entangled. Let us get back to the Synge equation and the Synge- Definition Two particles A and B, located in the same Weber equation. spatial section∗ at the distance dxi =0 from each other, According to Zelmanov’s theory of physical observable quantities [31, 32], if an observer accompanies his references ∗A three-dimensional section of the four-dimensional space-time, placed in a given point in the time line. In the space-time there are infinitely †Such linked events in the particles A and B can be radiation of a signal many spatial sections, one of which is our three-dimensional space. in one and its absorbtion in the other, for instance.
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the projection of a general covariant quantity on the observ- Taking this one into account, we transform the classical er’s spatial section is its spatial observable projection. Synge and Weber problem statement into another. In our Following this way, Borissova has deduced (see eqs. statement the world-line of a particle, being entangled to 7.16–7.28 in [33]) that the spatial observable projection of itself by definition, splits into two different world-lines under the Synge equation is∗ teleportation conditions. In other word, as soon as the tele- portation conditions occur in a research laboratory, the world- 2 i k d η i ·i dη line of a teleported particle breaks in one world-point A 2 +2 Dk + Ak· =0, (10) dτ dτ and immediately starts in the other world-point B (fig. 2). she called it the Synge equation in chronometrically invariant Both particles A and B, being actually two different states form. The Weber equation is different in its right part con- of the same teleported particle at a remote distance one from taining the non-gravitational force that connects the particles the other, are in entangled states. So, in this statement, the (of course, the force should be filled in the spatially projected particles A and B themselves are entangled. form). For this reason, conclusions obtained for the Synge Of course, this entanglement exists in only the moment equation will be the same that for the Weber one. of the teleportation when the particle exists in two different In order to make the results of General Relativity ap- states simultaneously. As soon as the teleportation process plicable to practice, we should consider tensor quantities has been finished, only one particle of them remains so the and equations designed in chronometrically invariant form, entanglement disappears. because in such way they contain only chronometrically It should be noted, it follows from the entanglement invariant quantities — physical quantities and geometrical conditions, that only substantial particles can reach entangled properties of space, measurable in real experiment [31, 32]. states in the basic space-time of General Relativity — the Let us look at our problem under consideration from this four-dimensional pseudo-Riemannian space. Not photons. viewpoint. Here is why. 2 α β As it is known, the interval ds = gαβ dx dx can not As it easy to see, the Synge equation in its chronometric- ‡ ally invariant form (10) under the entanglement condition be fully degenerated in a Riemannian space : the condition dτ =0 becomes nonsense. The Weber equation becomes is that the determinant of the metric fundamental tensor gαβ nonsense as well. So, the classical problem statement be- must be strictly negative g = det gαβ < 0 by definition of comes senseless as soon as particles reach entangled states. Riemannian spaces. In other word, in the basic space-time At the same time, in the recent theoretical research [29] of General Relativity the fundamental metric tensor must be strictly non-degenerated as g<0. two authors of the paper (Borissova and Rabounski, 2005) 2 have found two groups of physical conditions under which The observable three-dimensional (spatial) interval dσ = i k = hik dx dx is positive determined [31, 32], proceeding particles can be teleported in non-quantum way. They have 2 been called the teleportation conditions: from physical sense. It fully degenerates dσ =0 if only the space compresses into point (the senseless case) or the (1) dτ =0 {ds =0,dσ=0}, the conditions of photon te- determinant of the metric observable tensor becomes zero leportation; h = det h =0. { 2 − 2 } ik (2) dτ =0 ds = dσ =0 , the conditions of substant- As it was shown by Zelmanov [31, 32], in real observ- ial (mass-bearing) particles teleportation. ations where an observer accompanies his references, the There also were theoretically deduced physical conditions†, determinant of the metric observable tensor is connected with which should be reached in a laboratory in order to teleport the determinant of the fundamental one by the relationship − g particles in the non-quantum way [29]. h = g00 . From here we see, if the three-dimensional ob- As it is easy to see the non-quantum teleportation con- servable metric fully degenerates h =0, the four-dimensional dition is identical to introduce here the entanglement main metric degenerates as well g =0. condition dτ =0 in couple with the 1st kind and 2nd kind We have obtained that states of two substantial particles auxiliary entanglement conditions! can be entangled, if dτ =0 {ds2 = −dσ2 =0 } in the space ∗In this formula, according to Zelmanov’s mathematical apparatus of neighbourhood. So h>0 and g<0 in the neighbourhood, ∗ 1 ∂hik 1 ∂hik hence the four-dimensional pseudo-Riemannian space is not physical observable quantities [31, 32], Dik = = √ is 2 ∂t 2 g00 ∂t degenerated. the three-dimensional symmetric tensor of the space deformation observable 1 ∂vk ∂vi 1 Conclusion Substantial particles can reach entangled states rate while Aik = − + Fivk − Fkvi is the three- 2 ∂xi ∂xk 2c2 in the basic space-time of General Relativity (the four- dimensional antisymmetric tensor of the space rotation observable angular dimensional pseudo-Riemannian space) under specific velocities, which indices can be lifted/lowered by the metric observable i im ·i im tensor so that Dk =h Dkm and Ak· = h Akm. See brief account of conditions in the neighbourhood. the Zelmanov mathematical apparatus in also [30, 33, 34, 35]. †A specific correlation between the gravitational potential w, the space ‡It can only be partially degenerated. For instance, a four-dimensional i rotation linear velocity vi and the teleported particle’s velocity u . Riemannian space can be degenerated into a three-dimensional one.
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Although ds2 = −dσ2 in the neighbourhood (dσ should where the space metric fully degenerates (there h =0 and, be imaginary), the substantial particles remain in regular sub- hence g =0). light area, they do not become super-light tachyons. It is easy Conclusion Massless particles (photons) can reach entan- to see, from the definition of physical observable time (8), gled states, only if the basic space-time fully dege- the entanglement condition dτ =0occurs only if the specific nerates g = det gαβ =0 in the neighbourhood. It is relationship holds permitted in the generalized four-dimensional space- i 2 w+viu = c (11) time which metric can be fully degenerated g 0 in that area where the degeneration conditions occur. The between the gravitational potential w, the space rotation generalized space-time is attributed to Smarandache i i linear velocity vi and the particles’ true velocity u = dx /dt geometry spaces, because its geometry is partially Rie- in the observer’s laboratory. For this reason, in the neighbour- mannian, partially not. hood the space-time metric is So, entangled states have been introduced into General Rel- w 2 ativity for both substantial particles and photons. ds2 = −dσ2 = − 1 − c2dt2 + g dxidxk, (12) c2 ik 3 Quantum Causality Threshold in General Relativity so the substantial particles can become entangled if the space (+−−−) (−+++) initial signature becomes inverted in the This term was introduced by one of the authors two years i neighbourhood, while the particles’ velocities u remain no ago (Smarandache, 2003) in our common correspondence faster than light. [36] on the theme: Another case — massless particles (photons). States of Definition Considering two particles A and B located in two phonos can be entangled, only if there is in the space the same spatial section, Quantum Causality Threshold neighbourhood dτ =0 {ds =0,dσ=0}. In this case the was introduced as a special state in which neither A determinant of the metric observable tensor becomes h =0, nor B can be the cause of events located “over” the so the space-time metric as well degenerates g =−g00 h =0. spatial section on the Minkowski diagram. This is not the four-dimensional pseudo-Riemannian space. Where is that area? In the previous works (Borissova The term Quantum has been added to the Causality and Rabounski, 2001 [30, 29]) a generalization to the basic Threshold, because in this problem statement an interaction space-time of General Relativity was introduced — the four- is considered between two infinitely far away particles (in dimensional space which, having General Relativity’s sign- infinitesimal vicinities of each particle) so this statement is alternating label (+−−−), permits the space-time metric to be applicable to only quantum scale interactions that occur in fully degenerated so that there is g 0. the scale of elementary particles. As it was shown in those works, as soon as the specific Now, we are going to find physical conditions under i 2 condition w+viu = c occurs, the space-time metric be- which particles can reach the threshold in the space-time of comes fully degenerated: there are ds =0, dσ =0, dτ =0 General Relativity. (it can be easy derived from the above definition for the Because in this problem statement we look at causal quantities) and, hence h =0and g =0. Therefore, in a space- relations in General Relativity’s space-time from “outside”, i 2 time where the degeneration condition w+viu = c is per- it is required to use an “outer viewpoint” — a point of view mitted the determinant of the fundamental metric tensor is located outside the space-time. g 0. This case includes both Riemannian geometry case We introduce a such point of outlook in an Euclidean g<0 and non-Riemannian, fully degenerated one g =0.For flat space, which is tangential to our’s in that world-point, this reason a such space is one of Smarandache geometry where the observer is located. In this problem statement we spaces [22–28], because its geometry is partially Riemannian, have a possibility to compare the absolute cause relations in partially not∗. In the such generalized space-time the 1st kind that tangential flat space with those in ours. As a matter, a entanglement conditions dτ =0 {ds =0,dσ=0} (the en- tangential Euclidean flat space can be introduced at any point tanglement conditions for photons) are permitted in that area of the pseudo-Riemannian space. At the same time, according to Zelmanov [31, 32], within ∗ In foundations of geometry it is known the S-denying of an axiom infinitesimal vicinities of any point located in the pseudo- [22–25], i. e. in the same space an “axiom is false in at least two dif- ferent ways, or is false and also true. Such axiom is said to be Smaran- Riemannian space a locally geodesic reference frame can be dachely denied, or S-denied for short” [26]. As a result, it is possible to introduced. In a such reference frame, within infinitesimal introduce geometries, which have common points bearing mixed properties vicinities of the point, components of the metric fundamental of Euclidean, Lobachevsky-Bolyai-Gauss, and Riemann geometry in the tensor (marked by tilde) same time. Such geometries has been called paradoxist geometries or Smarandache geometries. For instance, Iseri in his book Smarandache 2 Manifolds [26] and articles [27, 28] introduced manifolds that support 1 ∂ g˜αβ μ μ ν ν g˜αβ = gαβ + (˜x −x )(˜x −x )+... (13) particular cases of such geometries. 2 ∂x˜μ∂x˜ν
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are different from those g at the point of reflection to within αβ time line t only the higher order terms, which can be neglected. So, in x i = const a locally geodesic reference frame the fundamental metric tensor can be accepted constant, while its first derivatives (Christoffel’s symbols) are zeroes. The fundamental metric tensor of an Euclidean space is as well a constant, so values of g˜μν , taken in the vicinities of a point of the pseudo- Riemannian space, converge to values of g in the flat μν light hypercone, the asymptote space tangential at this point. Actually, we have a system of coordinate transformations of the flat space’s basic vectors e(α) tangential to curved coordinate lines of the pseudo-Riemannian space. Coordinate e lines in Riemannian spaces are curved, inhomogeneous, and 0 are not orthogonal to each other (the latest is true if the space e i rotates). Therefore the lengths of the basic vectors may be i x spatial section , x 0 const very different from the unit. A = Fig. 3 Writing the world-vector of an infinitesimal displacement 0 1 2 3 α as d r =(dx ,dx ,dx ,dx ), we obtain d r = e(α)dx , where As it is easy to see from formula (16), this ultimate case the components of the basic vectors e(α) tangential to the co- occurs as soon as the space rotation velocity vi reaches the 0 1 ordinate lines are e(0)={e(0), 0, 0, 0}, e(1)={0,e(1), 0, 0}, light velocity. If particles A and B are located in the space 2 2 filled into this ultimate state, neither A nor B can be the cause e(2)= {0, 0,e(2), 0}, e(3)= {0, 0, 0,e(3)}. Scalar product of 2 2 of events located “over” the spatial section in the Minkowski d r with itself is d rd r = ds or, in another ds = g dxαdxβ, αβ diagrams we use in the pictures. So, in this ultimate case so g = e e = e e cos (xα; xβ). We obtain αβ (α) (β) (α) (β) the space-time is filled into a special state called Quantum 2 0 i Causality Threshold. g00 = e(0) ,g0i = e(0)e(i) cos (x ; x ) , (14) Conclusion Particles, located in General Relativity’s space- i k time, reach Quantum Causality Threshold as soon as gik = e(i)e(k) cos (x ; x ) ,i,k=1, 2, 3 . (15) the space rotation reaches the light velocity. Quantum Then, substituting g00 and g0i from formulas that det- Causality Threshold is impossible if the space does not 2 √ ermine the gravitational potential w=c (1− g00) and the rotate (holonomic space), or if it rotates at a sub-light √g0i space rotation linear velocity vi =−c , we obtain speed. g00 So, Quantum Causality Threshold has been introduced into − 0 i vi = ce(i) cos (x ; x ) , (16) General Relativity. 0 i 0 k i k hik =e( )e( ) cos(x ; x )cos(x ; x )−cos(x ; x ) . (17) i k References From here we see: if the pseudo-Riemannian space is free 1. Belavkin V.P. Quantum causality, decoherence, trajectories and of rotation, cos (x0; xi)=0 so the observer’s spatial section information. arXiv: quant-ph/0208087, 76 pages. is strictly orthogonal to time lines. As soon as the space starts to do rotation, the cosine becomes different from zero 2. Bennett C. H., Brassard G., Crepeau C., Jozsa R., Peres A., and Wootters W. K. Teleporting an unknown quantum state so the spatial section becomes non-orthogonal to time lines via dual classical and Einstein-Podolsky-Rosen channels. Phys. (fig. 3). Having this process, the light hypercone inclines Rev. Lett., 1993, v. 70, 1895–1899. with the time line to the spatial section. In this inclination the 3. Boschi D., Branca S., De Martini F., Hardy L., and Popescu S. light hypercone does not remain unchanged, it “compresses” Experimental realization of teleporting an unknown pure because of hyperbolic transformations in pseudo-Riemannian quantum state via dual classical and Einstein-Podolsky-Rosen space. The more the light hypercone inclines, the more it Channels. Phys. Rev. Lett., 1998, v. 80, 1121–1125. symmetrically “compresses” because the space-time’s geo- 4. Riebe M., Haffner¨ H., Roos C. F., Hansel¨ W., Benhelm J., metrical structure changes according to the inclination. Lancaster G. P.T., Korber T. W., Becher C., Schmidt-Kaler F., In the ultimate case, where the cosine reach the ultimate James D. F. V., and Blatt R. Deterministic quantum tele- 0 i value cos (x ; x )=1, time lines coincide the spatial section: portation with atoms. Nature, 2004, v. 429 (June, 17), 734–736. time “has fallen” into the three-dimensional space. Of course, 5. Barrett M. D., Chiaverini J., Schaetz T., Britton J., Itano W. M., in this case the light hypercone overflows time lines and the Jost J. D., Knill E., Langer C., Leibfried D., Ozeri R., Wine- spatial section: the light hypercone “has as well fallen” into land D. J. Deterministic quantum teleportation of atomic qubits. the three-dimensional space. Nature, 2004, v. 429 (June, 17), 737–739.
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6. Pan J.-W., Bouwmeester D., Daniell M., Weinfurter H., 23. Ashbacher C. Smarandache geometries. Smarandache Notions, Zeilinger A. Experimental test of quantum nonlocality in three- book series, v. 8, ed. by C. Dumitrescu and V.Seleacu, Ameri- photon Greenberger-Horne-Zeilinger entanglement. Nature, can Research Press, Rehoboth, 1997, 212–215. 2000, v. 403 (03 Feb 2000), 515–519. 24. Chimienti S. P., Bencze M. Smarandache paradoxist geometry. 7. Mair A., Vaziri A., Weihs G., Zeilinger A. Entanglement of Bulletin of Pure and Applied Sciences, 1998, v. 17E, No. 1, the orbital angular momentum states of photons. Nature, v. 412 123–124. See also Smarandache Notions, book series, v. 9, ed. (19 July 2001), 313–316. by C. Dumitrescu and V.Seleacu, American Research Press, Rehoboth, 1998, 42–43. 8. Lukin M. D., Imamoglu A. Controlling photons using elec- tromagnetically induced transparency Nature, v. 413 (20 Sep 25. Kuciuk L. and Antholy M. An introduction to Smarandache 2001), 273–276. geometries. New Zealand Math. Coll., Massey Univ., Palmer- ston North, New Zealand, Dec 3–6, 2001 (on-line http://atlas- 9. Julsgaard B., Kozhekin A., Polzik E. S. Experimental long- conferences.com/c/a/h/f/09.htm). lived entanglement of two macroscopic objects. Nature, v. 413 (27 Sep 2001), 400–403. 26. Iseri H. Smarandache manifolds. American Research Press, Rehoboth, 2002. 10. Duan L.-M., Lukin M. D., Cirac J. I., Zoller P. Long-distance quantum communication with atomic ensembles and linear 27. Iseri H. Partially paradoxist Smarandache geometry. Sma- optics. Nature, v. 414 (22 Nov 2001), 413–418. randache Notions, book series, v. 13, ed. by J. Allen, F. Liu, D. Costantinescu, Am. Res. Press, Rehoboth, 2002, 5–12. 11. Yamamoto T., Koashi M., Ozdemir¨ S.K., ¸ Imoto N. Experim- 28. Iseri H. A finitely hyperbolic point in a smooth manifold. JP ental extraction of an entangled photon pair from two identi- Journal on Geometry and Topology, 2002, v. 2 (3), 245–257. cally decohered pairs. Nature, v. 421 (23 Jan 2003), 343–346. 29. Borissova L. B. and Rabounski D. D. On the possibility of 12. Pan J.-W., Gasparoni S., Aspelmeyer M., Jennewein T., instant displacements in the space-time of General Relativity. Zeilinger A. Experimental realization of freely propagating Progress in Physics, 2005, v. 1, 17–19. teleported qubits. Nature, v. 421 (13 Feb 2003), 721–725. 30. Borissova L. B. and Rabounski D. D. Fields, vacuum, and the 13. Pan J.-W., Gasparoni S., Ursin R., Weihs G., Zeilinger A. mirror Universe. Editorial URSS, Moscow, 2001, 272 pages Experimental entanglement purification of arbitrary unknown (the 2nd revised ed.: CERN, EXT-2003-025). states. Nature, v. 423 (22 May 2003), 417–422. 31. Zelmanov A. L. Chronometric invariants. Dissertation, 1944. 14. Zhao Zhi, Chen Yu-Ao, Zhang An-Ning, Yang T., Briegel H. J., First published: CERN, EXT-2004-117, 236 pages. Pan J.-W. Experimental demonstration of five-photon en- 32. Zelmanov A. L. Chronometric invariants and co-moving co- tanglement and open-destination teleportation. Nature, v. 430 ordinates in the general relativity theory. Doklady Acad. Nauk (01 July 2004), 54–58. USSR, 1956, v. 107 (6), 815–818. 15. Blinov B. B., Moehring D. L., Duan L.-M., Monroe C. Ob- 33. Borissova L. Gravitational waves and gravitational inertial servation of entanglement between a single trapped atom and waves in the General Theory of Relativity: A theory and a single photon. Nature, v. 428 (11 Mar 2004), 153–157. experiments. Progress in Physics, 2005, v. 2, 30–62. 16. Ursin R., Jennewein T., Aspelmeyer M., Kaltenbaek R., 34. Rabounski D. A new method to measure the speed of grav- Lindenthal M., Walther P., Zeilinger A. Communications: itation. Progress in Physics, 2005, v. 1, 3–6. Quantum teleportation across the Danube. Nature, v. 430 (19 Aug 2004), 849–849. 35. Rabounski D. A theory of gravity like electrodynamics. Progress in Physics, 2005, v. 2, 15–29. 17. Pauli W. Relativitatstheorie.¨ Encyclopaedie¨ der mathemati- 36. Smarandache F. Private communications with D. Rabounski schen Wissenschaften, Band V, Heft IV, Art. 19, 1921 (Pauli W. and L. Borissova, May 2005. Theory of Relativity. Pergamon Press, 1958). 18. Eddington A. S. The mathematical theory of relativity. Cambridge University Press, Cambridge, 1924 (referred with the 3rd expanded edition, GTTI, Moscow, 1934, 508 pages). 19. Landau L. D. and Lifshitz E. M. The classical theory of fields. GITTL, Moscow, 1939 (ref. with the 4th final exp. edition, Butterworth–Heinemann, 1980). 20. Synge J. L. Relativity: the General Theory. North Holland, Am- sterdam, 1960 (referred with the 2nd expanded edition, Foreign Literature, Moscow, 1963, 432 pages). 21. Weber J. General Relativity and gravitational waves. R. Mar- shak, New York, 1961 (referred with the 2nd edition, Foreign Literature, Moscow, 1962, 271 pages). 22. Smarandache F. Paradoxist mathematics. Collected papers, v. II, Kishinev University Press, Kishinev, 1997, 5–29.
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Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
There can be generated many paradoxes or quasi-paradoxes that may occur from the combination of quantum and non-quantum worlds in physics. Even the passage from the micro-cosmos to the macro-cosmos, and reciprocally, can generate unsolved questions or counter-intuitive ideas. We define a quasi-paradox as a statement which has a prima facie self-contradictory support or an explicit contradiction, but which is not completely proven as a paradox. We present herein four elementary quantum quasi-paradoxes and their corresponding quantum Sorites paradoxes, which form a class of quantum quasi-paradoxes.
1 Introduction stable/unstable, long time living/short time living, as well as on the fact that there is not a clear separation between these According to the Dictionary of Mathematics (Borowski and pairs of antinomies. Borwein, 1991 [1]), the paradox is “an apparently absurd or self-contradictory statement for which there is prima facie 2.1.1 Invisible Quasi-Paradox: Our visible world is com- support, or an explicit contradiction derived from apparently posed of a totality of invisible particles. unexceptionable premises”. Some paradoxes require the revi- 2.1.2 Invisible Sorites Paradox: There is not a clear frontier sion of their intuitive conception (Russell’s paradox, Cantor’s between visible matter and invisible matter. paradox), others depend on the inadmissibility of their de- (a) An invisible particle does not form a visible ob- scription (Grelling’s paradox), others show counter-intuitive ject, nor do two invisible particles, three invisible features of formal theories (Material implication paradox, particles, etc. However, at some point, the collec- Skolem Paradox), others are self-contradictory — Smarand- tion of invisible particles becomes large enough ache Paradox: “All is the
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living” matter is formed by very “short-time-living” elementary particles. 2.4.2 Short-Time-Living Sorites Paradox: Similarly, there is not a clear frontier between the “long-time-living” matter and the “short-time-living” matter.
3 Conclusion
“More such quantum quasi-paradoxes and paradoxes can be designed, all of them forming a class of Smarandache quantum quasi-paradoxes.” (Dr. M. Khoshnevisan, Griffith University, Gold Coast, Queensland, Australia [3])
References
1. Borowski E. J. and Borwein J. M. The Harper Collins Diction- ary of Mathematics. Harper Perennial, A Division of Harper Collins Publishers, New York, 1991. 2. Weisstein E. W. Smarandache Paradox. CRC Concise Encyclo- paedia of Mathematics, CRC Press, Boca Raton, Florida, 1998, 1661, (see the e-print version of this article in http://mathworld. wolfram.com/SmarandacheParadox.html). 3. Khoshnevisan M. Private communications, 1997. 4. Motta L. A Look at the Smarandache Sorites Paradox. Proceed- ings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi-bin/amca/caft-04). 5. Niculescu G. On Quantum Smarandache Paradoxes. Proceed- ings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi-bin/amca/caft-20). 6. Smarandache F. Invisible paradox. Neutrosophy, Neutrosophic Probability, Set, and Logic, American Research Press, Reho- both, 1998 (see the third e-print edition of the book in the web site http://www.gallup.unm.edu/∼smarandache). 7. Smarandache F. Sorites paradoxes. Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry (edited by M. L. Perez), Xiquan Pub- lishing House, Phoenix, 2000.
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The Neutrosophic Logic View to Schrodinger’s¨ Cat Paradox
Florentin Smarandache∗ and Vic Christianto† ∗Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] †Sciprint.org—aFreeScientificElectronicPreprintServer,http://www.sciprint.org E-mail: [email protected]
This article discusses Neutrosophic Logic interpretation of the Schrodinger’s cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic Logic could take into consideration, whereas other methods including Fuzzy Logic could not. For a balanced discussion, other interpretations have also been discussed.
1 Schrodinger¨ equation • The square of wavefunction of Schrodinger¨ equation as the vorticity distribution (including topological vorti- As already known, Schrodinger¨ equation is the most used city defects) in the fluid [6]; equation to describe non-relativistic quantum systems. Its re- • The wavefunction in Schrodinger¨ equation represents lativistic version was developed by Klein-Gordon and Dirac, tendency to make structures; but Schrodinger¨ equation has wide applicability in particular • because it resembles classical wave dynamics. For intro- The wavemechanics can also be described in terms duction to non-relativistic quantum mechanics, see [1]. of topological Aharonov effect, which then it could Schrodinger¨ equation begins with definition of total en- be related to the notion of topological quantization ergy E = p 2/2m. Then, by using a substitution [7, 8]. Aharonov himself apparently argues in favour of “realistic” meaning of Schrodinger¨ wave equation, ∂ E = i ,P= ∇, (1) whose interpretation perhaps could also be related to ∂t i Kron’s work [9]. one gets [2] ∂ ∇¯ 2 So forth we will discuss solution of this paradox. i + − U (x) ψ =0 (2) ∂t 2m or 2 Solution to Schrodinger’s¨ cat paradox i∂ ψ = Hψ . (3) ∂t 2.1 Standard interpretation While this equation seems quite clear to represent quan- It is known that Quantum Mechanics could be regarded more tum dynamics, the physical meaning of the wavefunction as a “mathematical theory” rather than a physical theory [1, itself is not so clear. Soon thereafter Born came up with hy- p. 2]. It is wave mechanics allowing a corpuscular duality. pothesis that the square of the wavefunction has the meaning Already here one could find problematic difficulties: i. e. of chance to find the electron in the region defined by dx while the quantity of wavefunction itself could be computed, (Copenhagen School). While so far his idea was quickly the physical meaning of wavefunction itself remains inde- adopted as “standard interpretation”, his original “guiding finable [1]. Furthermore, this notion of wavefunction corres- field” interpretation has been dropped after criticism by Hei- ponds to another fundamental indefinable in Euclidean geo- senberg over its physical meaning [3]. Nonetheless, a de- metry: the point [1, p. 2]. It is always a baffling question for finition of “Copenhagen interpretation” is that it gives the decades, whether the electron could be regarded as wave, wavefunction a role in the actions of something else, namely a point, or we should introduce a non-zero finite entity [4]. of certain macroscopic objects, called “measurement appa- Attempts have been made to describe wave equation in such ratus”, therefore it could be related to phenomenological non-zero entity but the question of the physical meaning of formalism [3]. wavefunction itself remains mystery. Nonetheless, we should also note here that there are other The standard Copenhagen interpretation advertised by approaches different from Born hypothesis, including: Bohr and colleagues (see DeBroglie, Einstein, Schrodinger¨ • The square of the wavefunction represents a measure who advocated “realistic” interpretation) asserts that it is of the density of matter in region defined by dx (De- practically impossible to know what really happens in quan- terminism school [3, 4, 5]). Schrodinger¨ apparently tum scale. The quantum measurement itself only represents preferred this argument, albeit his attempt to demon- reading in measurement apparatus, and therefore it is difficult strate this idea has proven to be unfruitful; to separate the object to be measured and the measurement
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apparatus itself. Bohr’s phenomenological viewpoint perhaps According to Uncertainty Principle, any measurement could be regarded as pragmatic approach, starting with the of a system must disturb the system under investigation, request not to attribute a deep meaning to the wave function with a resulting lack of precision in the measurement. Soon but immediately go over to statistical likelihood [10]. Con- after reading Einstein-Podolsky-Rosen’s paper discussing in- sequently, how the process of “wave collapse” could happen completeness of Quantum Mechanics, Schrodinger¨ in 1935 remains mystery. came up with a series of papers in which he used the “cat Heisenberg himself once emphasized this viewpoint paradox” to give an illustration of the problem of viewing when asked directly the question: Is there a fundamental these particles in a “thought experiment” [15, 17]: level of reality? He replied as follows: “One can even set up quite ridiculous cases. A cat “This is just the point: I do not know what the words is penned up in a steel chamber, along with the follow- fundamental reality mean. They are taken from our ing diabolical device (which must be secured against daily life situation where they have a good meaning, direct interference by the cat): in a Geiger counter but when we use such terms we are usually extrapol- there is a bit of radioactive substance, so small, ating from our daily lives into an area very remote that perhaps in the course of one hour one of the from it, where we cannot expect the words to have atoms decays, but also, with equal probability, per- a meaning. This is perhaps one of the fundamental haps none; if it happens, the counter tube discharges difficulties of philosophy: that our thinking hangs in and through a relay releases a hammer which shatters the language. Anyway, we are forced to use the words a small flask of hydrocyanic acid. If one has left so far as we can; we try to extend their use to the this entire system to itself for an hour, one would utmost, and then we get into situations in which they say that the cat still lives if meanwhile no atom has have no meaning” [11]. decayed. The first atomic decay would have poison- ed it. The wave-function of the entire system would A modern version of this interpretation suggests that at express this by having in it the living and the dead the time of measurement, the wave collapses instantaneously cat (pardon the expression) mixed or smeared into into certain localized object corresponding to the action of equal parts.” measurement. In other words, the measurement processes In principle, Schrodinger’s¨ thought experiment asks define how the wave should define itself. At this point, the whether the cat is dead or alive after an hour. The most wave ceases to become coherent, and the process is known as logical solution would be to wait an hour, open the box, and “decoherence”. Decoherence may be thought of as a way of see if the cat is still alive. However once you open the box making real for an observer in the large scale world only one to determine the state of the cat you have viewed and hence possible history of the universe which has a likelihood that disturbed the system and introduced a level of uncertainty it will occur. Each possible history must in addition obey the into the results. The answer, in quantum mechanical terms, laws of logic of this large-scale world. The existence of the is that before you open the box the cat is in a state of being phenomenon of decoherence is now supported by laboratory half-dead and half-alive. experiments [12]. It is worthnoting here, that there are also Of course, at this point one could ask whether it is other versions of decoherence hypothesis, for instance by possible to find out the state of the cat without having to Tegmark [13] and Vitiello [14]. disturb its wavefunction via action of “observation”. In the meantime, the “standard” Copenhagen interpreta- If the meaning of word “observation” here is defined tion emphasizes the role of observer where the “decoherence by to open the box and see the cat, and then it seems that viewpoint” may not. The problem becomes more adverse we could argue whether it is possible to propose another because the axioms of standard statistical theory themselves equally possible experiment where we introduce a pair of are not fixed forever [15, 16]. And here is perhaps the twin cats, instead of only one. A cat is put in the box while source of numerous debates concerning the interpretation another cat is located in a separate distance, let say 1 meter and philosophical questions implied by Quantum Mechanics. from the box. If the state of the cat inside the box altered From this viewpoint, Neutrosophic Logic offers a new view- because of poison reaction, it is likely that we could also point to problems where indeterminacy exists. We will dis- observe its effect to its twin, perhaps something like “sixth cuss this subsequently. For a sense of balance, we also sense” test (perhaps via monitoring frequency of the twin discuss a number of alternative interpretations. Nonetheless cat’s brain). this article will not discuss all existing interpretations of the This plausible experiment could be viewed as an alter- quantum wavefunction in the literature. native “thought experiment” of well-known Bell-Aspect-type experiment. One could also consider an entangled pair of 2.2 Schrodinger’s¨ cat paradox photons instead of twin cats to conduct this “modified” cat paradox. Of course, for this case then one would get a bit To make the viewpoint on this paradox a bit clearer, let us complicated problem because now he/she should consider reformulate the paradox in its original form. two probable state: the decaying atom and the photon pair.
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We could also say that using this alternative configurat- that there is “something else” which should be included in ion, we know exact information about the Cat outside, while the Quantum Mechanical equations in order to explain thor- indeterminate information about the Cat inside. However, oughly all quantum phenomena. Sometimes this assertion can because both Cats are entangled (twin) we are sure of all be formulated in question form [19]: Can Quantum Mech- the properties of the Cat inside “knows” the state of the Cat anics be considered complete? Interestingly, however, the outside the box, via a kind of “spooky action at distance” meaning of “complete” itself remains quite abstract (fuzzy). reason (in Einstein’s own word)∗. Therefore, for experimental purpose, perhaps it would be useful to simplify the problem by using “modified” Aspect- type experiment [16]. Here it is proposed to consider a de- caying atom of Cesium which emits two correlated photons, whose polarization is then measured by Alice (A) on the Figure 1: Aspect-type experiment left and by Bob (B) on the right (see Fig. 1). To include the probable state as in the original cat paradox, we will use a switch instead of Alice A. If a photon comes to this switch, then it will turn on a coffee-maker machine, therefore the observer will get a cup of coffee†. Another switch and coffee-maker set also replace Bob position (see Fig. 2). Then we encapsulate the whole system of decaying atom, switch, and coffee-maker at A, while keeping the system at B side open. Now we can be sure, that by the time the decaying atom of Cesium emits photon to B side and triggers the switch at Figure 2: Aspect-type experiment in box this side which then turns on the coffee-maker, it is “likely” that we could also observe the same cup of coffee at A side, An interpretation of this cat paradox suggests that the even if we do not open the box. problem arises because we mix up the macroscopic systems We use term “likely” here because now we encounter a (observer’s wavefunction and apparatus’ wavefunction) from “quasi-deterministic” state where there is also small chance microscopic system to be observed. In order to clarify this, that the photon is shifted different from −0.0116, which is it is proposed that “. . . the measurement apparatus should indeed what the Aspect, Dalibard and Roger experiment de- be described by a classical model in our approach, and the monstrated in 1982 using a system of two correlated photons physical system eventually by a quantum model” [20]. [16]. At this “shifted” phase, it could be that the switch will not turn on the coffee-maker at all, so when an observer 2.4 Hydrodynamic viewpoint and diffusion interpre- opens the box at A side he will not get a cup of coffee. tation If this hypothetical experiment could be verified in real world, then it would result in some wonderful implications, In attempt to clarify the meaning of wave collapse and deco- like prediction of ensembles of multi-particles system, — or herence phenomenon, one could consider the process from a colony of cats. (dissipative) hydrodynamic viewpoint [21]. Historically, the Another version of this cat paradox is known as GHZ pa- hydrodynamic/diffusion viewpoint of Quantum Mechanics radox: “The Greenberger-Horne-Zeilinger paradox exhibits has been considered by some physicists since the early years some of the most surprising aspects of multiparticle entangle- of wave mechanics. Already in 1933, Fuerth showed that ment” [18]. But we limit our discussion here on the original Schrodinger¨ equation could be written as a diffusion equation cat paradox. with an imaginary diffusion coefficient [1] i Dqm = . (4) 2.3 Hidden-variable hypothesis 2m But the notion of imaginary diffusion is quite difficult It would be incomplete to discuss quantum paradoxes, in to comprehend. Alternatively, one could consider a classical particular Schrodinger’s¨ cat paradox, without mentioning Markov process of diffusion type to consider wave mechan- hidden-variable hypothesis. There are various versions of ics equation. Consider a continuity equation this argument, but it could be summarised as an assertion ∂ρ = −∇ (ρv) , (5) ∗The authors are grateful to Dmitri Rabounski for his valuable com- ∂t ments discussing a case of entangled twin Cats. † where v = v0 = D∇ lnρ (see [1]), which is a Fokker-Planck The “coffee-maker” analogue came to mind after a quote: “A math- ematician is a device for turning coffee into theorems” — Alfred´ Renyi,´ a equation. Then the expectation value for the energy of par- Hungarian mathematician, 1921–1970. (As quoted by Christopher J. Mark.) ticle can be written as [1]
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2 2 mv D m 2 3 normally means y is not for sure in A); or z (0, 1, 0) belongs
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11. Buckley P., Peat F. D. A question of physics: conversations in physics and biology. Routledge and Kegan Paul, London and Henley, 1979. 12. Zurek W. Physics Today, 1999, v. 44, 36; Los Alamos Science, 2002, No. 27; arXiv: quant-ph/0306072; Rev. Mod. Phys., 2003, v. 75, 715; Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies, Addison-Wesley, 1990; Haroche S. Physics Today, 1998, v. 36. 13. Tegmark M. arXiv: quant-ph/9907009. 14. Vitiello G. arXiv: hep-th/9503135. 15. Schrodinger¨ E. Die gegenwartige Situation in der Quanten- mechanik. Naturwissenschaften, 1935, Bd. 23; English transl. in: Quantum Mechanics and Measurement, ed. J. A. Wheeler and W. Zurek, Princeton UP, 1983. 16. Meglicki Z. Two and three photons: Bell’s inequality. http:// beige.ucs.indiana.edu/M743/node70.html. 17. Edwards P.M. 75 years of Schrodinger¨ wave equation in 2001. http://ublib.buffalo.edu/libraries/units/sel/exhibits/schrodinger/ e_schro.html. 18. Exploring quantum physics. http://xqp.physik.uni-muenchen. de/explore/prog.html. 19. Vaidman L. E. arXiv: hep-th/9310176. 20. Aerts D. Int. J. Theor. Phys., 1998, v. 37, 291–304. 21. Na K., Wyatt R. E. Decoherence demystified: the hydrodyn- amic viewpoint. arXiv: quant-ph/0201108. 22. Smarandache F. Neutrosophy / neutrosophic probability, set, and logic. American Research Press, Rehoboth, 1998; A uni- fying field in logics: neutrosophic logic. Neutrosophy, neu- trosophic set, neutrosophic probability. 3rd ed., American Re- search Press, 2003. 23. Smarandache F. A unifying field in logics: neutrosophic logic. Multiple Valued Logic / An International Journal, 2002, v. 8(3), 385–438. 24. Smarandache F. Definitions derived from neutrosophics. Multiple Valued Logic / An International Journal, 2002, v. 8 (5–6), 591–604. 25. Smarandache F. An introduction to the neutrosophic probabi- lity applied in quantum physics. Bull. Pure and Appl. Sciences, Ser. D (Physics), 2003, v. 22D, No. 1, 13–25.
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The Neutrosophic Logic View to Schrodinger’s¨ Cat Paradox, Revisited
Florentin Smarandache and Vic ChristiantoO Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] OSciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected]
The present article discusses Neutrosophic logic view to Schrodinger’s¨ cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic logic can take into consideration, whereas other methods including Fuzzy logic cannot. To make this proposition clear, we revisit our previous paper by offering an illustration using modified coin tossing problem, known as Parrondo’s game.
1 Introduction where pN, represents momentum at x direction, and ratio- nalised Planck constants respectively. The present article discusses Neutrosophic logic view to By introducing kinetic energy of the moving particle, T , Schrodinger’s¨ cat paradox. In this article we argue that this and wavefunction, as follows [9]: paradox involves some degree of indeterminacy (unknown)
mv pN which Neutrosophic logic can take into consideration, T = = = k ; (3) whereas other methods including Fuzzy logic cannot. 2 2m 2m In the preceding article we have discussed how Neutro- and sophic logic view can offer an alternative method to solve the (x) = exp(ikx) : (4) well-known problem in Quantum Mechanics, i.e. the Schro-¨ Then one has the time-independent Schrodinger¨ equation dinger’s cat paradox [1, 2], by introducing indeterminacy of from [1, 3, 4]: the outcome of the observation. @ In other article we also discuss possible re-interpretation (x)=T (x) : 2m @x (5) of quantum measurement using Unification of Fusion Theo- ries as generalization of Information Fusion [3, 4, 5], which It is interesting to remark here that by convention physi- results in proposition that one can expect to neglect the prin- cists assert that “the wavefunction is simply the mathematical ciple of “excluded middle”; therefore Bell’s theorem can be function that describes the wave” [9]. Therefore, unlike the considered as merely tautological. [6] This alternative view wave equation in electromagnetic fields, one should not con- of Quantum mechanics as Information Fusion has also been sider that equation [5] has any physical meaning. Born sug- proposed by G. Chapline [7]. Furthermore this Information gested that the square of wavefunction represents the prob- Fusion interpretation is quite consistent with measurement ability to observe the electron at given location [9, p.56]. theory of Quantum Mechanics, where the action of measure- Although Heisenberg rejected this interpretation, apparently ment implies information exchange [8]. Born’s interpretation prevails until today. In the first section we will discuss basic propositions of Nonetheless the founding fathers of Quantum Mechanics Neutrosophic probability and Neutrosophic logic. Then we (Einstein, De Broglie, Schrodinger¨ himself) were dissatisfied discuss solution to Schrodinger’s¨ cat paradox. In subsequent with the theory until the end of their lives. We can summarize section we discuss an illustration using modified coin tossing the situation by quoting as follows [9, p.13]: problem, and discuss its plausible link to quantum game. “The interpretation of Schrodinger’s¨ wave function While it is known that derivation of Schrodinger’s¨ equa- (and of quantum theory generally) remains a matter of tion is heuristic in the sense that we know the answer to which continuing concern and controversy among scientists the algebra and logic leads, but it is interesting that Schro-¨ who cling to philosophical belief that the natural world dinger’s equation follows logically from de Broglie’s grande is basically logical and deterministic.” loi de la Nature [9, p.14]. The simplest method to derive Schrodinger’s¨ equation is by using simple wave as [9]: Furthermore, the “pragmatic” view of Bohr asserts that for a given quantum measurement [9, p.42]: @ exp(ikx)= k exp(ikx) : @x (1) “A system does not possess objective values of its phys- ical properties until a measurement of one of them is By deriving twice the wave and defining: made; the act of measurement is asserted to force the 2mv mv pN system into an eigenstate of the quantity being mea- k = = = ; (2) h sured.”
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In 1935, Einstein-Podolsky-Rosen argued that the axiomatic theory, the definitive and unambiguous assignment of an el- basis of Quantum Mechanics is incomplete, and subsequently ement of the set {0,1}, and so the assignment of an informa- Schrodinger¨ was inspired to write his well-known cat para- tion content of the photon itself is fraught with the same dif- dox. We will discuss solution of his cat paradox in subsequent ficulties [8]. Similarly, the problem becomes more adverse section. because the fundamental basis of conventional statistical the- ories is the same classical set {0,1}. 2 Cat paradox and imposition of boundary conditions For example the Schrodinger’s¨ cat paradox says that the quantum state of a photon can basically be in more than one As we know, Schrodinger’s¨ deep disagreement with the Born place in the same time which, translated to the neutrosophic interpretation of Quantum Mechanics is represented by his set, means that an element (quantum state) belongs and does cat paradox, which essentially questioning the “statistical” in- not belong to a set (a place) in the same time; or an ele- terpretation of the wavefunction (and by doing so, denying ment (quantum state) belongs to two diff erent sets (two dif- the physical meaning of the wavefunction). The cat paradox ferent places) in the same time. It is a question of “alternative has been written elsewhere [1, 2], but the essence seems quite worlds” theory very well represented by the neutrosophic set similar to coin tossing problem: theory. In Schrodinger’s¨ equation on the behavior of electro- “Given p = 0.5 for each side of coin to pop up, we magnetic waves and “matter waves” in quantum theory, the will never know the state of coin before we open our wave function, which describes the superposition of possible palm from it; unless we know beforehand the “state” states may be simulated by a neutrosophic function, i.e. a of the coin (under our palm) using ESP-like phenom- function whose values are not unique for each argument from ena. Prop. (1).” the domain of definition (the vertical line test fails, intersect- ing the graph in more points). The only difference here is that Schrodinger¨ asserts that the Therefore the question can be summarized as follows [1]: state of the cat is half alive and half dead, whereas in the coin problem above, we can only say that we don’t know the state “How to describe a particle in the infinite micro- of coin until we open our palm; i.e. the state of coin is inde- universe that belongs to two distinct places P1 and P2 terminate until we open our palm. We will discuss the solu- in the same time? 2 P1 and 2:P1 is a true con- tion of this problem in subsequent section, but first of all we tradiction, with respect to Quantum Concept described shall remark here a basic principle in Quantum Mechanics, above.” i.e. [9, p.45]: Now we will discuss some basic propositions in Neutrosophic “Quantum Concept: The first derivative of the wave- logic [1]. function of Schrodinger’s¨ wave equation must be single-valued everywhere. As a consequence, the 3a Non-standard real number and subsets wavefunction itself must be single-valued everywhere.” Let T,I,F be standard or non-standard real subsets ] 0, 1+[, The above assertion corresponds to quantum logic, which can = = be defined as follows [10, p.30; 11]: with sup T t sup, inf T t inf, sup I = i sup, inf I = i inf, P _ Q = P + Q PQ: (6) sup F = f sup, inf F = f inf, As we will see, it is easier to resolve this cat paradox and n sup = t sup + i sup + f sup, by releasing the aforementioned constraint of “single- = + + valuedness” of the wavefunction and its first derivative. In n inf t inf i inf f inf. + fact, nonlinear fluid interpretation of Schrodinger’s¨ equation Obviously, t sup, i sup, f sup 1 ; and t inf, i inf, f inf 0, + (using the level set function) also indicates that the physical whereas n sup 3 and n inf 0. The subsets T, I, F are not meaning of wavefunction includes the notion of multivalued- necessarily intervals, but may be any real subsets: discrete or ness [12]. In other words, one can say that observation of continuous; single element; finite or infinite; union or inter- spin-half electron at location x does not exclude its possibility section of various subsets etc. They may also overlap. These to pop up somewhere else. This counter-intuitive proposition real subsets could represent the relative errors in determining will be described in subsequent section. t, i, f (in the case where T, I, F are reduced to points). For interpretation of this proposition, we can use modal 3 Neutrosophic solution of the Schrodinger¨ cat paradox logic [10]. We can use the notion of “world” in modal logic, which is semantic device of what the world might have been In the context of physical theory of information [8], Barrett like. Then, one says that the neutrosophic truth-value of a + has noted that “there ought to be a set theoretic language statement A, NLJ(A)=1 if A is “true in all possible which applies directly to all quantum interactions”. This is worlds.” (syntagme first used by Leibniz) and all conjunc- because the idea of a bit is itself straight out of classical set tures, that one may call “absolute truth” (in the modal logic
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it was named necessary truth, as opposed to possible truth), normally means y is not for sure in A); or z (0, 1, 0) belongs whereas NLJ(A)=1if A is true in at least one world at some to A (which means one does know absolutely nothing about conjuncture, we call this “relative truth” because it is related z’s affiliation with A). to a “specific” world and a specific conjuncture (in the modal More general, x ((0.2–0.3), (0.40–0.45) [ [0.50–0.51], logic it was named possible truth). Because each “world” is {0.2, 0.24, 0.28}) belongs to the set A, which means: dynamic, depending on an ensemble of parameters, we in- — with a probability in between 20-30% particle x is in troduce the sub-category “conjuncture” within it to reflect a a position of A (one cannot find an exact approximate particular state of the world. because of various sources used); In a formal way, let’s consider the world W as being gen- — with a probability of 20% or 24% or 28% x is not in A; erated by the formal system FS. One says that statement A x belongs to the world W if A is a well-formed formula (wff ) — the indeterminacy related to the appurtenance of to in W, i.e. a string of symbols from the alphabet of W that A is in between 40–45% or between 50–51% (limits conforms to the grammar of the formal language endowing included). W. The grammar is conceived as a set of functions (formation The subsets representing the appurtenance, indeterminacy, rules) whose inputs are symbols strings and outputs “yes” or and falsity may overlap, and n sup = 30% + 51% + 28% > “no”. A formal system comprises a formal language (alpha- 100% in this case. bet and grammar) and a deductive apparatus (axioms and/or To summarize our proposition [1, 2], given the Schro-¨ rules of inference). In a formal system the rules of inference dinger’s cat paradox is defined as a state where the cat can be are syntactically and typographically formal in nature, with- dead, or can be alive, or it is undecided (i.e. we don’t know out reference to the meaning of the strings they manipulate. if it is dead or alive), then herein the Neutrosophic logic, Similarly for the Neutrosophic falsehood-value, based on three components, truth component, falsehood com- + NLB (A)=1 if the statement A is false in all possible ponent, indeterminacy component (T, I, F), works very well. worlds, we call it “absolute falsehood”, whereas NLB (A)=1 In Schrodinger’s¨ cat problem the Neutrosophic logic offers if the statement A is false in at least one world, we call it the possibility of considering the cat neither dead nor alive, “relative falsehood”. Also, the Neutrosophic indeterminacy but undecided, while the fuzzy logic does not do this. Nor- value NLE(A)=1if the statement A is indeterminate in all mally indeterminacy (I) is split into uncertainty (U) and para- possible worlds, we call it “absolute indeterminacy”, whereas dox (conflicting) (P). NLE(A)=1if the statement A is indeterminate in at least We have described Neutrosophic solution of the Schro-¨ one world, we call it “relative indeterminacy”. dinger’s cat paradox. Alternatively, one may hypothesize four-valued logic to describe Schrodinger’s¨ cat paradox, see 3b Neutrosophic probability definition Rauscher et al. [13, 14]. In the subsequent section we will discuss how this Neu- Neutrosophic probability is defined as: “Is a generalization trosophic solution involving “possible truth” and “indetermi- of the classical probability in which the chance that an event nacy” can be interpreted in terms of coin tossing problem A occurs is t% true — where t varies in the subset T, i% in- (albeit in modified form), known as Parrondo’s game. This determinate — where i varies in the subset I, and f% false approach seems quite consistent with new mathematical for- — where f varies in the subset F. One notes that NP(A) = mulation of game theory [20]. (T, I, F)”. It is also a generalization of the imprecise probabil- ity, which is an interval-valued distribution function. 4 An alternative interpretation using coin toss problem The universal set, endowed with a Neutrosophic probabil- ity defined for each of its subset, forms a Neutrosophic prob- Apart from the aforementioned pure mathematics-logical ap- ability space. proach to Schrodinger’s¨ cat paradox, one can use a well- known neat link between Schrodinger’s¨ equation and Fokker- 3c Solution of the Schrodinger’s¨ cat paradox Planck equation [18]: @2p @ @p @p D p =0: Let’s consider a neutrosophic set a collection of possible lo- @z2 @z @z @t (7) cations (positions) of particle x. And let A and B be two neutrosophic sets. One can say, by language abuse, that any A quite similar link can be found between relativistic clas- particle x neutrosophically belongs to any set, due to the per- sical field equation and non-relativistic equation, for it is centages of truth/indeterminacy/falsity involved, which varies known that the time-independent Helmholtz equation and between 0 and 1+. For example: x (0.5, 0.2, 0.3) belongs Schrodinger¨ equation is formally identical [15]. From this to A (which means, with a probability of 50% particle x is in reasoning one can argue that it is possible to explain Aharo- a position of A, with a probability of 30% x is not in A, and nov effect from pure electromagnetic field theory; and there- the rest is undecidable); or y (0, 0, 1) belongs to A (which fore it seems also possible to describe quantum mechan-
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ical phenomena without postulating the decisive role of plied external field. “observer” as Bohr asserted. [16, 17]. In idiomatic form, one With respect to the aforementioned Neutrosophic solu- can expect that quantum mechanics does not have to mean tion to Schrodinger’s¨ cat paradox, one can introduce a new that “the Moon is not there when nobody looks at”. “indeterminacy” parameter to represent conditions where the With respect to the aforementioned neat link between outcome may be affected by other issues (let say, apparatus Schrodinger’s¨ equation and Fokker-Planck equation, it is in- setting of Geiger counter). Therefore equation (14) can be teresting to note here that one can introduce “finite differ- written as: ence” approach to Fokker-Planck equation as follows. First, 1 p = " p + we can define local coordinates, expanded locally about a i;j 2 i 1;j 1 z t ) z;t point ( , we can map points between a real space ( ) 1 i; j + a p + + " p ; and an integer or discrete space ( ). Therefore we can sam- 0 i;j 1 2 i+1;j 1 (17) ple the space using linear relationship [19]: where unlike the bias parameter (1/200), the indeterminacy (z;t)=(z + i; t + j ) ; (8) parameter can be quite large depending on the system in ques- tion. For instance in the Neutrosophic example given above, where is the sampling length and is the sampling time. we can write that: Using a set of finite difference approximations for the Fokker- d 1 t Planck PDE: = k = k : 0.2 0.3 t d 0.50 (18) @p p (z0 + ; t0 ) p (z0 ; t0 ) = A1 = ; (9) @z 2 The only problem here is that in original coin tossing, one @2p cannot assert an “intermediate” outcome (where the outcome =2A = @z2 2 is neither A nor B). Therefore one shall introduce modal logic p (z ; t ) 2p (z ;t )+p (z +; t ) definition of “possibility” into this model. Fortunately, we = 0 0 0 0 0 0 ; 2 (10) can introduce this possibility of intermediate outcome into Parrondo’s game, so equation (17) shall be rewritten as: and @p p (z0;t0) p (z0;t0 ) 1 = B1 = : (11) p = " p + @t i;j 2 i 1;j 1
We can apply the same procedure to obtain: 1 +(2) pi;j 1 + + " pi+1;j 1 ; (19) @ (z0 +; t0 ) (z0 ; t0 ) 2 = A1 = : (12) @z 2 For instance, by setting 0.25, then one gets the finite Equations (9–12) can be substituted into equation (7) to difference equation: yield the required finite partial differential equation [19]: pi;j =(0.25 ") pi 1;j 1 +(0.5) pi;j 1 + p (z0;t0)=a 1 p (z0 ; t0 ) a0 p (z0;t0 )+ +(0.25 + ") pi+1;j 1 ; (20) + a p (z + ; t ) : +1 0 0 (13) which will yield more or less the same result compared with This equation can be written in terms of discrete space by Neutrosophic method described in the preceding section. using [8], so we have: For this reason, we propose to call this equation (19): Neutrosophic-modified Parrondo’s game. A generalized ex- pi;j = a 1 pi 1;j 1 + a0 pi;j 1 + a+1 pi+1;j 1 : (14) pression of equation [19] is:
Equation (14) is precisely the form required for Parron- pi;j =(p0 " ) pi 1;j 1 +(z) pi;j 1 + do’s game. The meaning of Parrondo’s game can be described +(p + " ) p ; in simplest way as follows [19]. Consider a coin tossing prob- 0 i+1;j 1 (21) lem with a biased coin: where p0, z represents the probable outcome in standard coin 1 p = "; tossing, and a real number, respectively. For the practical head 2 (15) meaning of , one can think (by analogy) of this indetermi- where " is an external bias that the game has to “overcome”. nacy parameter as a variable that is inversely proportional to This bias is typically a small number, for instance 1/200. Now the “thickness ratio” (d=t) of the coin in question. There- we can express equation (15) in finite difference equation (14) fore using equation (18), by assuming k = 0.2, coin thick- as follows: ness = 1.0 mm, and coin diameter d = 50 mm, then we get 1 1 d=t = = ( ) 1 = pi;j = " pi 1;j 1 +0pi;j 1 + + " pi+1;j 1 : (16) 50, or 0.2 50 0.004, which is negligible. But 2 2 if we use a thick coin (for instance by gluing 100 coins alto- Furthermore, the bias parameter can be related to an ap- gether), then by assuming k = 0.2, coin thickness = 100 mm,
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and coin diameter d = 50 mm, we get d=t = 0.5, or 3. Smarandache F. and Christianto V. A note on geometric and in- = 0.2(0.5) = 0.4, which indicates that chance to get out- formation fusion interpretation of Bell’s theorem and quantum come neither A nor B is quite large. And so forth. measurement. Progress in Physics, 2006, no. 4. It is worth noting here that in the language of “modal 4. Smarandache F. and Christianto V. Multivalued logic, neutros- logic” [10, p.54], the “intermediate” outcome described here ophy and Schrodinger¨ equation. Hexis, Phoenix (AZ), 2006, is given name ‘possible true’, written }A, meaning that “it is p.52–54. not necessarily true that not-A is true”. In other word, given 5. Smarandache F. and Dezert J. Advances and applications of that the cat cannot be found in location x, does not have to DSmT for information fusion. American Research Press, Re- mean that it shall be in y. hoboth (NM), 2004. Using this result (21), we can say that our proposition in 6. Kracklauer A. La theorie de Bell, est-elle la plus grande the beginning of this paper (Prop. 1) has sufficient reason- meprise de l’histoire de la physique? Ann. Fond. Louis de ing; i.e. it is possible to establish link from Schrodinger¨ wave Broglie, 2000, v. 25, 193. / / equation to simple coin toss problem, albeit in modified form. 7. Chapline G. arXiv: adap-org 9906002; quant-ph 9912019; / Furthermore, this alternative interpretation, differs apprecia- Granik A. and Chapline G. arXiv: quant-ph 0302013. bly from conventional Copenhagen interpretation. 8. Zurek W. (ed.) Complexity, entropy and the physics of infor- mation. Addison-Wesley Publ., 1990, p.378. It is perhaps more interesting to remark here that Heisen- berg himself apparently has proposed similar thought on this 9. Hunter G. Quantum chemistry: wave mechanics applied to problem, by introducing “potentia”, which means “a world atoms and molecules. Lecture notes. Chapter 1: Genesis of Quantum Mechanics. 2001, p.14, 26, 42. devoid of single-valued actuality but teeming with unreal- ized possibility” [4, p.52]. In Heisenberg’s view an atom is 10. deVries A. Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. arXiv: math.LO/0707.2161, p.30, 54. certainly real, but its attributes dwell in an existential limbo “halfway between an idea and a fact”, a quivering state of 11. Aerts D. Description of many separated physical entities with- out the paradoxes encountered in Quantum Mechanics. Found. attenuated existence. Interestingly, experiments carried out Phys., 1982, v. 12, no. 12, p.1142, 1149–1155. by J. Hutchison seem to support this view, that a piece of 12. Jin S., Liu H., Osher S., and Tsai R. Computing multival- metal can come in and out from existence [23]. ued physical observables for the semiclassical limits of the In this section we discuss a plausible way to represent the Schrodinger¨ equation. J. Comp. Phys., 2005, v. 205, 222–241. Neutrosophic solution of cat paradox in terms of Parrondo’s 13. Rauscher E.A. and Targ R. The speed of thought: investigation game. Further observation and theoretical study is recom- of complex spacetime metric to describe psychic phenomena. mended to explore more implications of this plausible link. J. Scientific Exploration, 2001, v. 15, no. 3, 344–354. 14. Rauscher E.A. and Amoroso R. The physical implications 5 Concluding remarks of multidimensional geometries and measurement. Intern. J. Comp. Anticipatory Systems, D. Dubois (ed.), 2006. In the present paper we revisit the Neutrosophic logic view of 15. Lu J., Greenleaf J., and Recami E. Limited diffraction solutions Schrodinger’s¨ cat paradox. We also discuss a plausible way to Maxwell and Schrodinger¨ equation. arXiv: physics/9610012. to represent the Neutrosophic solution of cat paradox in terms 16. Aharonov Y., et al. arXiv: quant-ph/0311155. of Parrondo’s game. 17. Goldstein S. Quantum theory without observers — part one. It is recommended to conduct further experiments in order Physics Today, March 1998, 42–46. to verify and explore various implications of this new propo- 18. Ho C.-L. and Sasaki R. Deformed Fokker Planck equations. sition, including perhaps for the quantum computation theory. arXiv: cond-mat/0612318. 19. Allison A., et al. State space visualization and fractal proper- / / Acknowledgment ties of Parrondo’s game. arXiv: cond-mat 0205536; cond-mat 0208470. The writers would like to thank to D. Rabounski for early 20. Wu J. A new mathematical representation of game theory. discussion concerning interpretation of this cat paradox. arXiv: quant-ph/0404159. 21. Smarandache F. Unification of fusion theories (UFT). Intern. J. Submitted on March 31, 2008 / Accepted on April 10, 2008 Appl. Math. and Stat., 2004, v. 2, 1–14. 22. Smarandache F. An in-depth look at information fusion rules References and unification of fusion theories. Invited speech at NASA Lan- gley Research Center, Hampton, VA, USA, November 5, 2004. 1. Smarandache F. An introduction to the Neutrosophic proba- 23. Smarandache F., Christianto V., Khrapko R., Yuhua F., Hutchi- bility applied in quantum physics. Bull. Pure and Appl. Sci., son J. Unfolding labyrinth: open problems in physics, mathe- Physics, 2003, v. 22D, no. 1, 13–25. matics, astrophysics, and other areas of science. Hexis, Phoenix 2. Smarandache F. and Christianto V. The Neutrosophic logic (AZ), 2006. view to Schrodinger’s¨ cat paradox. Progr. in Phys., 2006, no. 2.
20 F. Smarandache and V.Christianto. The Neutrosophic Logic View to Schrodinger’s¨ Cat Paradox, Revisited
49 October, 2006 PROGRESS IN PHYSICS Volume 4
A Note on Geometric and Information Fusion Interpretation of Bell’s Theorem and Quantum Measurement∗
Florentin Smarandache† and Vic Christianto‡ † Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] ‡ Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected]
In this paper we presentfour possible extensions ofBell’s Theorem: Bayesian and Fuzzy Bayesian intrepretation, Information Fusion interpretation, Geometric interpre- tation, andthe viewpoint of photon fluid as mediumfor quantum interaction.
1 Introduction quantum interaction. This proposition leadsusto Gross-Piteavskii equation which is commonly usedto It is generally acceptedthatBell’s theorem [1] is quiteexact describe bose condensation phenomena. In turn we to describe the linear hidden-variable interpretation ofquan- providearoute where Maxwell equations and Schro¨di- tummeasurement,and hence “quantum reality”. Therefore nger equation could be deducedfrom Gross-Pitaevskii null result ofthis proposition implies that no hidden-variable equation by using known algebra involving bi-quater- theory could provide good explanation of “quantum reality”. nion number. In our opinion, this new proposition pro- Nonetheless, after further thought we can findthatBell’s vides us a physical mechanism ofquantum interaction, theorem is nothing more than another kind of abstraction beyond conventional “quantum algebra” which hides ofquantum observation based on a set of assumptions and causal explanation. propositions [7]. Therefore, one should be careful before By discussing these various approaches, we use an ex- making further generalization on the null resultfrom exper- panded logic beyond “yes” or “no” type logic [3]. In other iments which are “supposed” to verify Bell’s theorem.For words, there could be new possibilities to describe quantum example, the most blatant assumption ofBell’s theorem is interaction: “both can be wrong”, or “both can be right”, as that ittakes into consideration only the classical statistical described in Table 1 below. problem of chance of outcome A or outcome B, as result of In Belnap’s four-valued logic there are, besides Truth (T) adoption of Von Neumann’s definition of “quantum logic”. and Falsehood (F), also Uncertainty (U) andContradiction Another critic will be discussed here, i. e. thatBell’s theorem (C)butthey are inter-related [30]. Belnap’s logic is a parti- is only a reformulation of statistical definition of correlation; cular case of Neutrosophic Logic (which considers three therefore it is merely tautological [5]. components: Truth, Falsehood,and Indeterminacy (I)) when Therefore in the present paper we will discuss a few indeterminacy is split into UncertaintyandContradiction. In plausible extension ofBell’s theorem: our article we have: Yes (Y), No (N), and Indeterminacy (a) Bayesian and Fuzzy Bayesian interpretation. (I, which means: neither Yes nor No), but Indeterminacy is (b) Information Fusion interpretation. In particular, we split into“both can be wrong” and “both can be right”. propose a modified version ofBell’s theore m, which It could be expectedthat acombined interpretation re- takes into consideration this multivalued outcome, in presents multiple-facetsofquantum reality. And hopefully it particular using the information fusion Dezert- could bring better understanding on the physical mechanism Smarandache Theory (DSmT) [2, 3, 4]. We suppose beneath quantummeasurement, beyond simple algebraic no- that in quantum reality the outcomeof P (A ∪ B) and tions. Further experiment s are of course recommended in also P (A ∩ B) shall also be taken into consideration. order to verifyorrefute this proposition. This is where DSmTand Unification of Fusion Theor- ies (UFT) could be found useful [2, 17]. 2 Bell’s theorem. Bayesian andfuzzy Bayesian inter- (c) Geometric interpretation, using a known theorem con- pretation necting geometry and imaginary plane. In turn, this Despitewidespread belief of its ability to describe hidden- leadsusto8-dimensional extended-Minkowski metric. variables ofquantum reality[1],it shall be notedthatBell’s (d)Asanalternative to this geometric interpretation, we theorem startswithaset of assumptions inherent in its for- submitthe viewpoint of photon fluid as mediumfor mulation. It is assumedtha t each pair of particles possesses ∗Note: The notion “hronir wave” introduced here was inspiredfrom aparticular value of λ,and we define quantity p (λ) so that Borges’ Tlon, Uqbar, Orbis Tertius. probabilityof a pair being produced between λ and λ + dλ
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Alternative Bell’s theorem Implications Special relativity
QM is nonlocal InvalidCausality breaks down; Observer Is not always applicable determines the outcome QM is local withhidden ValidCausality preserved;Themoon No interaction can exceedthe speed of variable is there even without observer light Both can be right Valid,butthere is a way to QM, special relativityand Max- Can be expanded using 8-dimensional explain QM without violat- well electromagnetic theory can Minkowski metric withimaginary ing Special Relativity be unified. New worldview shall plane be used Both can be wrong Invalid,and so Special Rel- New nonlocal QM theory is re- Is not always applicable ativity is. Weneed anew quired, involving quantum po- theory tential
Table 1: Going beyond classical logic view of QM is p (λ)dλ.It is also assu medthatthis is normalized so that: Nonetheless, it should also be noted here thatthere is shortcoming ofthis Bayesian approach. As Kracklauer points p (λ) dλ =1. (1) out, Bell’s theorem is nothing but areformulation of statist- Further analysis shows thatthe integral thatmeasures the ical definition of correlation [5]: correlation between two spin components that are at an angle |AB|−AB of (δ − φ) with each other, is therefore equal to C (δ − φ). Corr (A, B)= . (6) A2B2 We can therefore write: 2 2 |C (φ) − C (δ)|−C (δ − φ) 1 (2) When A or B equals to zero and A B =1 then equation (6) reduces to Bell’s theorem. Thereforeassuchit which is known as Bell’s theorem,and it was supposedto could be considered as merely tautological [5]. represent any local hidden-variable theorem. But it shall be notedthat actually this theorem cannot be tested completely 3Information fusion interpretation of Bell’s theorem. because it assumes that all particle pairs have been detected. DSmT modification In other words, we findthat ahidden assumption behind Bell’s theorem is that it uses classical probability assertion In the context of physical theory of information [8], Barrett [12], which may or may be not applicable to describe Quan- has notedthat “there oughttobeasettheoretic language tum Measurement. which applies directly to all quantum interactions”. This is It is wothnoting here thatthe standard interpretation of because the idea of abit is itself straight out of classical Bell’s theorem includes the use ofBayesian posterior proba- set theory, the definitive and unambiguous assignment of bility [13]: an element ofthe set {0, 1},and so the assignment of an p (α) p (x|α) P (α|x)= . (3) information content ofthe photon itself is fraught with the p (β) p (x|β) β same difficulties [8]. Similarly, the problem becomes more As we know Bayesian method is based on classical two- adverse because the fundamental basis of conventional stat- valued logic. In the meantime, it is known thatthe restriction istal theories is the same classical set {0, 1}. of classical propositional calculus toatwo-valued logic has Not only that, there is also criticism over the use of created someinteresting paradoxes. For example, the Barber Bayesian approach, i. e.: [13] of Seville has a rule that all and only those men who donot (a) In real world,neither class probabilities nor class den- shave themselves are shaved by the barber.Itturns outthat sities are precisely known; the only way for this paradoxtoworkisifthe statement is both true and false simultaneously [14]. This brings us to (b) This implies that one should adopt a parametric model fuzzy Bayesian approach [14]asanextension of (3): for the class probabilities and class densities, andthen use empirical data. p (M |si) p (si) P (si|M)= , (4) (c) Therefore, in the context where multiple sensors can p (M) be used,information fusion approach could beabetter where [14, p. 339]: alternative to Bayes approach. r In other words, we shouldfind an extension tostandard p (M |si)= p (xk |si) μM (xk) . (5) k=1 propositioninstatistical theory [8, p. 388]:
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P (AB |C)=P (A|BC)P (B |C) (7) geometry and algebra withimaginary plane [18]: | | = P (B AC)P (A C) (8) x + iy = ρeiφ. (14) P (A|B)+P (A¯|B)=1. (9) Therefore one could expecttocomeupwithgeometrical Such an extension is already known in the area of infor- explanation ofquantum interaction, provided we could gen- mation fusion [2], known as Dempster-Shafer theory: eralize the metric using imaginary plane: m(A)+m(B)+m(A ∪ B)=1. (10) X + iX = ρeiφ . (15) Interestingly, Chapline [13] notedthat neither Bayesian Interestingly, Amoroso and Rauscher [19] have proposed theory nor Dempster-Shafer could offer insight on how to exactly the sameidea, i. e. generalizing Minkowski metric to minimize overall energy usage in the network. In the mean- become8-dimensional metric which can be represented as: time, Dezert-Smarandache (DSmT) [2] introducedfurther Zμ = Xμ + iXμ = ρeiφ . (16) improvement of Dempster-Shafer theory by taking into con- re im sideration chance to observe intersection between A and B: A characteristic result ofthis 8-dimensional metric is that “space separation” vanishes, andquantum-type interaction m(A)+m(B)+m(A ∪ B)+m(A ∩ B)=1. (11) could happen in no time. Therefore, introducing this extension from equation (11) Another viewpoint could be introduced in this regard, intoequation (2), one findsamodified version ofBell’s the- i. e. thatthe wave nature of photon arises from “photon fluid” orem in the form: medium, which serves to enable photon-photon interaction. It has been arguedthatthis photon-fluidmedium could be |C (φ) − C (δ)|− described using Gross-Pitaevskii equation[20].Inturns, we (12) − C (δ − φ)+C (δ ∪ φ)+C (δ ∩ φ) 1 , could expectto“derive” Schro¨dinger wave equation from which could be called as modifiedBell’s theorem according the Gross-Pitaevskii equation. to Dezert-Smarandache (DSmT) theory [2]. Its direct impli- It will be shown, that we couldderive Schro¨dinger wave cations suggestthat it could be useful to include more sen- equation from Gross-Pitaevskii equation. Interestingly, sors in order to capture various possibilities beyond simple anewterm similar toeq uation (14) arises here, which then {0, 1} result, which is typical in Bell’s theorem. we propose to call it “hronir wave”. Therefore one could Further generalization of DSmT theory (11) is known as expectthatthis “hronir wave” plays the role of “invisible Unification of Fusion Theories [15, 16, 17]: light”aspostulated by Maxwell long-time ago. Consider the well-known Gross-Pitaevskii equation in m(A)+m(B)+m(A ∪ B)+m(A ∩ B)+ the context of superfluidity or superconductivity [21]: (13) ¯ ¯ ¯ ∪ ¯ ¯ ∩ ¯ 2 + m(A)+m(B)+m(A B)+m(A B)=1, ∂Ψ ¯h −1 i¯h = − ΔΨ + V (x) − γ |Ψ|p Ψ, (17) where A¯ is the complement of A and B¯ is the complement ∂t 2m of B (if we consider the settheory). where p<2N/(N −2) if N 3. In physical problems, the (But if we consider the logical theory then A¯ is the equation for p =3 is known as Gross-Pitaevskii equation. negation of A and B¯ is the negation of B.Thesettheory and This equation (17) has standing wave solution quitesimilar logical theory in this example are equivalent, hence doesn’t to Schro¨dinger equation, in the form: matter which one we use fromthem.) In equation (13) above −iEt/¯h · we have a complement/negation for A. We mightdefine the Ψ(x, t)=e u(x) . (18) ¯ A as the entangle of particle A. Hence we could expect Substituting equation (18) intoequation (17) yields: to further extendBell’s inequality considering UFT; non- 2 ¯h −1 etheless we leave this further generalization for the reader. − Δu + V (x) − E u = |u|p u, (19) Of course, new experimental design is recommended in 2m order to verifyandto find various implications ofthis new which is nothing buttime-independent linear form of Schro-¨ | |p−1 proposition. dinger equation, exceptfor term u [21]. In case the right-hand sideofthis equation is negligible, equation (19) reduces tostandard Schro¨dinger equation. Using Maclaurin 4 An alternative geometric interpretation of Bell-type series expansion, we getfor (18): measurement. Gross-Pitaevskii equation and the 2 3 “hronir wave” iEt iEt − iEt Ψ(x, t)= 1− + ¯h + ¯h + ... ·u(x) . (20) Apartfromthe aforementionedBayesian interpretation of ¯h 2! 3! Bell’s theorem, we can consider the problemfrom purely Therefore we can say that standing wave solution of geometric viewpoint. As we know, there is linkage between Gross-Pitaevskii equation (18) is similar tostanding wave
F. Smarandache, V. Christianto. A NoteonFusionInterpretation ofBell’s Theorem and Quantum Measurement 29
52 Vo lume 4PROGRESS IN PHYSICSOctober, 2006 solution of Schro¨dinger equation (u),exceptfor nonlinear Interestingly, equation (28) is equivalentto Maxwell eq- term which comes from Maclaurin series expansion (20). uations. [22] Now we can generalize our resultfromthe By neglecting third and other higher order termsof equation preceding section, in the form ofthe following conjecture: (20), one gets an approximation: Conjecture 1 Given a particular solution of Schro¨dinger Ψ(x, t)= 1 − iEt/¯h · u(x) . (21) equation (24), then the approximate solution of Gross- Pitaevskii equation (17) reduces to the first order equation: Note thatthis equation (21) is very near to hyperbolic form z = x + iy [18]. Therefore one could conclude that 1 − iEt/¯h D + M h F =0. (30) standing wave solution of Gross-Pitaevskii equation is mere- Therefore we can conclude here thatthere is neat linkage ly an extension from ordinary solution of Schro¨dinger equa- between Schro¨dinger equation, Maxwell equation, Riccati tion into Cauchy (imaginary) plane. In other words, there equation via biquaternion expression [22, 23, 24]. And ap- shall be “hronir wave” part of Schro¨dinger equationinorder proximate solution of Gross-Pitaevskii equation is similar to to describe Gross-Pitaevskii equation. We will use this result solution of Schro¨dinger equation, exceptthat it exhibitsa in the subsequent section, butfirst we consider how to derive new term called here “the hronir wave” (30). bi-quaternion from Schro¨dinger equation. Our proposition is that considering equation (30) has im- It is known that solutions of Riccatiequation are loga- aginary plane wave, therefore it could be expectedtopro- rithmic derivatives of solutions of Schro¨dinger equation, and vided “physical mechanism”ofquantum interaction, in the vice versa [22]: u + vu =0. (22) same sense of equation (14). Further experiments are of course recommended in order to verifyorrefute this Bi-quaternion ofdifferentiable function of x =(x1,x2,x3) is defined as [22]: 5 Some astrophysical implications of Gross-Pitaevskii Dq = −div(q) + grad(q0) + rot(q) . (23) description Byusingalternative representation of Schro¨dinger equa- Interestingly, Moffat [25, p. 9] has also used Gross-Pitaevskii tion [22]: in his “phion condensatefluid” to describe CMB spectrum. −Δ+u f =0, (24) Therefore we could expectthatthis equation will also yield interesting resultsincosmological scale. where f is twice differentiable, and introducing quaternion Furthermore, it is well-known that Gross-Pitaevskii equa- equation: 2 Dq + q = −u. (25) tion could exhibit topologically non-trivial vortex solutions [26, 27], which can be expressed as quantized vortices: Then we couldfind q, where q is purely vectorial diffe- rentiable bi-quaternion valuedfunction [22]. p • dr = Nv 2π¯h. (31) Wenote that solutions of (24) are relatedto (25) as fol- lows [22]: Thereforeanimplication of Gross-Pitaevskii equation • For any nonvanishing solution f of (24), its logarithm- [25] is thattopologically quantized vortex could exhibit in ic derivative: astrophysical scale. In this context we submitthe viewpoint Df q = , (26) thatthis proposition indeed has been observed in the form f of Tifft’s quantization [28, 29]. The following description isasolution of equation(25),and vice versa [22]. supports this assertion oftopological quantized vortices in astrophysical scale. Furthermore, we also note thatfor an arbitrary scalar Westart withstandarddefinition of Hubble law [28]: twice differentiable function f, the following equality is per- mitted [22]: δλ Hr z = = (32) λ c −Δ+u f = D + M h D − M h f, (27) or c r = z. (33) provided h is solution of equation (25). H Therefore we can summarize that given a particular solu- Now we suppose thatthe major partsof redshift data tion of Schro¨dinger equation (24), the general solution redu- could be explained via Doppler shift effect, therefore [28]: ces to the first order equation [22, p. 9]: δλ v z = = . (34) D + M h F =0, (28) λ c where √ D ε In order tointerpret Tifft’s observation ofquantized red- h = . (29) ε shift corresponding to quantized velocity 36.6 km/sec and
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72.2 km/sec, then we could write from equation (34): 5. Kracklauer A. La theorie de Bell, est-elle la plus grande meprise de l’histoire de la physique? Annales de la Fondation δv δλ = δz = δ . (35) Louis de Broglie, v. 25, 2000, 193. c λ 6. Aharonov Y. et al. arXiv: quant-ph/0311155. Or from equation (33) we get: 7. Rosu H. C. arXiv: gr-qc/9411035. c 8. Zurek W.(ed.), Complexity, entropy andthe physics of infor- δr = δz . (36) mation. SantaFeInst.Studies, Addison-Wesley, 1990. H 9. Schrieffer J. R. Macroscopic quantum phenomena from pairing In other words, we submitthe viewpointthat Tifft’s ob- in superconductors. Lecture, December 11, 1972. servation ofquantized redshift implies a quantizeddistance 10. Anandan J. S. In: Quantum Coherence and Reality. Columbia between galaxies [28], which could be expressed in the form: SC, World Sci., 1994; arXiv: gr-qc/9504002. 11. Goldstein S. Quantumtheory without observers — part one. rn = r0 + n (δr) . (35a) Physics Today, March 1998, 42–46. It is proposed here thatthis equation ofquantizeddistan- 12. Pitowski I. arXiv: quant-ph/0510095. ce (5) is resultedfromtopological quantized vortices (31), 13. Chapline G. arXiv: adap-org/9906002; quant-ph/9912019; and agrees with Gross-Pitaevskii (quantum phion condensa- Granik A. andChapline G. arXiv: quant-ph/0302013; http:// te) description ofCMB spectrum [25]. Nonetheless, further www.whatsnextnetwork.com/technology/index.php/2006/03/. observation is recommended in order to verify the above 14. Ross T. J. fuzzy logic with engineering applications. McGraw- proposition. Hill, 1995, 196–197, 334–341. 15. Smarandache F. Unification of Fusion Theories (UFT). Intern. Concluding remarks J. of Applied Math. & Statistics, 2004,v.2,1–14. In the present paper we review a few extension ofBell’s 16. SmarandacheF.Anin-depth look at information fusion rules theorem which couldtake into consideration chance to ob- and unification offusion theories. Invited speech at NASA Langley Research Center, Hampton, VA, USA, Nov 5, 2004. serve outcome beyond classical statistical theory, in parti- cular using the information fusion theory. A new geometrical 17. Smarandache F. Unification ofthe fusion theory (UFT). interpretation ofquantum interaction has been considered, Invitedat speech NATO Advance StudyInstitute, Albena, Bulgaria, May 16–27, 2005. using Gross-Pitaevskii equation. Interestingly, Moffat [25] also consideredtequahis tion in the context of cosmology. 18. Gogberashvili M. arXiv: hep-th/0212251. It is recommendedtoconductfurther experimentsin 19. Rauscher E. A. and Amoroso R. Intern. J. of Comp. Anticipa- order to verifyand also toexplore various implications of tory Systems, 2006. this new proposition, including perhaps for the quantum com- 20. Chiao R. et al.. arXiv: physics/0309065. putation theory [8, 13]. 21. Dinu T. L. arXiv: math.AP/0511184. 22. Kravchenko V. arXiv: math.AP/0408172. Acknowledgment 23. Lipavsky P. et al. arXiv: cond-mat/0111214. The writers would like to thank to Profs. C. Castro, J. Dezert, 24. De Haas E. P. Proc. of the Intern. Conf. PIRT-2005, Moscow, P. Vallin, T. Love, D. Rabounski, and A. Kaivarainen for MGTU Publ., 2005. valuable discussions. The new term “hronir wave”introduced 25. Moffat J. arXiv: astro-ph/0602607. here was inspiredfromBorges’ Tlon, Uqbar, Orbis Tertius. 26. Smarandache F. andChristiantoV. Progress in Physics, 2006, Hronir wave is defined here as “almost symmetrical mirror” v. 2, 63–67. of Schro¨dinger-type wave. 27. Fischer U. arXiv: cond-mat/9907 457; cond-mat/ 0004339. 28. Russell Humphreys D. Our galaxyisthe centre ofthe universe, References “quantized”red shifts show. TJ Archive, 2002, v. 16(2), 95– 104;http://answersingenesis.org/tj/v16/i2/galaxy.asp. 1. Shimony A. http://plato.stanford.edu/entries/bell-theorem/; 29. SetterfieldB.http://www.journaloftheoretics.com. http://plato.stanford.edu/entries/kochen-specker/. 30. Belnap N. A useful four-valued logic, modern uses of 2. Smarandache F. and Dezert J. Advances and applications of multiple-valued logic. (D. Reidel, editor), 8–37, 1977. of DSmT for information fusion. American Research Press, Rehoboth, 2004. 3. Smarandache F. Bulletin of Pure and Applied Sciences, Ser. Physics, 2003, v. 22D, No. 1, 13–25. 4.Smarandache F. andChristiantoV.Multivalued logic, neutro- sophy and Schro¨dinger equation. Hexis, Phoenix, 2005.
F. Smarandache, V. Christianto. A NoteonFusionInterpretation ofBell’s Theorem and Quantum Measurement 31
54 Extension of Inagaki General Weighted Operators
and
A New Fusion Rule Class of Proportional Redistribution of Intersection Masses
Florentin Smarandache Chair of Math & Sciences Depart. University of New Mexico, Gallup, USA
Abstract. In this paper we extend Inagaki Weighted Operators fusion rule (WO) in information fusion by doing redistribution of not only the conflicting mass, but also of masses of non-empty intersections, that we call Double Weighted Operators (DWO). Then we propose a new fusion rule Class of Proportional Redistribution of Intersection Masses (CPRIM), which generates many interesting particular fusion rules in information fusion. Both formulas are presented for 2 and for n 3 sources.
1. Introduction. θθθθ= { } n ≥ Sθ =∪∩()θτ Let 12, ,..., n , for 2 , be the frame of discernment, and ,,, its super-power set, where (x) means complement of x with respect to the total ignorance.
Let It = total ignorance = 1 2… n. θ c S = 2 ^ refined = 2^(2^ ) = D , when refinement is possible, where c = { ( 1), ( 2), …, ( n)}. We consider the general case when the domain is Sθ , but Sθ can be replaced by Dθ = θ ( ,,) or by 2 = ( ,) in all formulas from below. mm⋅⋅ Sθ [ ] Let 12() and () be two normalized masses defined from to 0,1 . m ⋅ m ⋅ We use the conjunction rule to first combine 1() with 2 () and then we redistribute the mass of mX() Y ≠ 0 , when XY =Φ. mA=⊕() m mA = mXmY Let’s denote 212 () ()B 12 ( ) () using the conjunction rule. θ XY, ∈ S ()XY = A Let’s note the set of intersections by:
1
55 IYXSX∈=θθ| yz , where yzS , ∈Φ \{ } , LL SX = JZ is in a canonical form, and . (1) LL K[X contains at least an symbol in its formula
In conclusion, S is a set of formulas formed with singletons (elements from the frame of discernment), such that each formula contains at least an intersection symbol , and each formula is in a canonical form (easiest form).
For example: AAS ∉ since AA is not a canonical form, and AAA = . Also,
()ABB is not in a canonical form but ()ABBABS=∈ . Let Φ S = the set of all empty intersections from S , and S nonΦ = S nonΦ ,r {the set of all non-empty intersections from whose masses are redistributed to other sets, which actually depends on the sub-model of each application}.
2. Extension of Inagaki General Weighted Operators (WO).
Inagaki general weighted operator ()WO is defined for two sources as: ∀∈A θ { Φ} mA=+ mXmYWAm⋅Φ 2\ , ()WO ()B 1 ( ) 2 ()m ()2 (), (2) θ XY,2∈ ()XY = A where
B WXm ()1= and all Wm ()⋅∈[ 0,1]. (3) θ X∈2 So, the conflicting mass is redistributed to non-empty sets according to these weights
Wm ()⋅ . In the extension of this WO , which we call the Double Weighted Operator ()DWO , we m Φ redistribute not only the conflicting mass 2 () but also the mass of some (or all) non-empty S nonΦ Sθ intersections, i.e. those from the set ,r , to non-empty sets from according to some weights
Wm ()⋅ for the conflicting mass (as in WO), and respectively according to the weights Vm(.) for S nonΦ the non-conflicting mass of the elements from the set ,r : ∀∈ASSθ nonΦ {Φ} mA = mXmYWAm+ ⋅ Φ+ VA ⋅ mz () \,1rD \ , WO()BB ( )2 ()m ()2 () m ()2 (), XY∈∈ Sθ z SnonΦ , ,r ()XY = A (4) where
B WXm ()1= and all Wm ()⋅∈[ 0,1], as in (3) XS∈ θ
2
56 and
B Vzm ()= 1 and all Vm ()⋅∈[ 0,1]. (5) zS∈ nonΦ ,r S nonΦ In the free and hybrid modes, if no non-empty intersection is redistributed, i.e. ,r contains no elements, DWO coincides with WO . In the Shafer’s model, always DWO coincides with WO . For s ≥ 2 sources, we have a similar formula: s ∀∈ASSθ nonΦ {}Φ mA = mXWAm+ ⋅ Φ+ VA ⋅ mz () \,rD \ , WO() BB∏ i(imsms ) () () () () XX X∈∈ Sθ i= zSnonΦ 12, ,..., n 1 ,r s XAi = i=1 (6) with the same restrictions on Wm (⋅ ) and Vm (⋅ ) .
3. A Fusion Rule Class of Proportional Redistribution of Intersection Masses
A∈ΦSSθ nonΦ { I} For ()\\,,rt for two sources we have: mXmY() () mAmAfA=+⋅ 12 CPR\2 M () () () B , (7) θ f z XY , ∈ S B () {}Φ=XY and AM ⊆ zM⊆ Φ≠XYS ∈nonΦ AN ⊆ or {},r and where f ()X is a function directly proportional to XfS, :θ →∞[ 0, ] . (8) f XmX= For example, ()2 (), or (9) f ()XcardX= (), or card() X fX()= (ratio of cardinals), or card() M f XmXcardX=+ ()2 () (), etc.; and M is a subset of Sθ , for example: (10) MXY=τ () , or MXY= () , or M is a subset of X Y , etc., where N is a subset of Sθ , for example: (11) NXY= , or N is a subset of X Y , etc.
And mImI=+ mXmY CPR\2 M() t ()t B 12() (). (12) θ XY , ∈ S LLIY JZXY =Φ and ( M =Φ or B fz ( )= 0) K[LLzM⊆ 3
57 s ≥ mm⋅⋅ m ⋅ These formulas are easily extended for any 2 sources 12( ), ( ),...,s ( ) . Let’s denote, using the conjunctive rule: s mA=⊕⊕⊕() m m mA = mxii ss () 12... ( ) B ∏ () (13) XX12, ,..., XSs∈Θ= ^ i 1 s XAi= i=1
s
∏ mXii() i=1 mAs ()=⋅ ms ∩ () A +f(A) B (14) XX X∈ Sθ B fz()≠ 0 12 , ,..., n LLIYs zM⊆ JZΦ= XAMi and ⊆ K[LLi=1 LLIYs Φ≠XS ∈nonΦ AN ⊆ or JZ ir, and K[LLi=1
where f (),⋅ MN, and are similar to the above where instead of X Y (for two sources) we X XX mX mX take 12... s (for s sources), and instead of 2 () for two sources we take s ( ) for s sources.
This new fusion rule Class of Proportional Redistribution of Intersection Masses (CPRIM) generates many interesting particular fusion rules in information fusion.
References:
[1] T. Inagaki, Independence Between Safety-Control Policy and Multiple-Sensors Schemes via Dempster-Shafer Theory, IEEE Transaction on Reliability, 40, 182-188, 1991.
[2] E. Lefèbvre, O. Colot, P. Vannoorenberghe, Belief Function Combination and Conflict Management, Information Fusion 3, 149-162, 2002.
June 2008
4
58
n-ary Fuzzy Logic and Neutrosophic Logic Operators
Florentin Smarandache, Chair of Math & Sc. Dept. University of New Mexico, Gallup, NM 87301, USA, [email protected]
and V. Christianto, SciPrint.org administrator, [email protected]
Abstract.
We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic operators using the smarandache codification of the Venn diagram and a defined vector neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.
Keywords: binary/trinary/n-ary fuzzy logic operators, T-norm, T-conorm, binary/trinary/n-ary neutrosophic logic operators, N-norm, N-conorm
Introduction.
For the beginning let’s consider the Venn Diagram of two variables x and y , for each possible operator, as in Knuth’s table, but we adjust this table to the Fuzzy Logic (FL). Let’s denote the fuzzy logic values of these variables as FL x t f () (,11 ) where t x 1 = truth value of variable , f 1 = falsehood value of variable x, tf t f with 011 , 1 1and 1 ; 1 and similarly for y : FL y t f () (,22 ) tf t f with the same 022 , 1 and 2 2 1. We can define all 16 Fuzzy Logical Operators with respect to two FL operators: FL conjunction ()FLC and FL negation ()FLN . Since in FL the falsehood value is equal to 1- truth value , we can deal with only one component: the truth value. The Venn Diagram for two sets X and Y
O
1 2 12
1
59
has 242 disjoint parts: 0 = the part that does not belong to any set (the complement or negation) 1 = the part that belongs to 1st set only; 2 = the part that belongs to 2nd set only; 12 = the part that belongs to 1st and 2nd set only; {called Smarandache’s codification [1]}.
Shading none, one, two, three, or four parts in all possible combinations will make 2 2242 16 possible binary operators. We can start using a T norm and the negation operator. Let’s take the binary conjunction or intersection (which is a T norm ) denoted as cxyF (, ): 2 cF : 0,1 0,1 0,1 0,1
and unary negation operator denoted as nxF (), with:
nF : 0,1 0,1 0,1 0,1
O cnxnyFF((),()
1 2
12
cxnyFF ,() cnxy cxyF (, ) FF(),
The fuzzy logic value of each part is: P12 part 12 intersection of x and y ; so FL( P 12) cF ( x , y ) .
P11 part intersection of x and negation of y ; FL(1) P cFF (, x n ()) y .
Ppart22 intersection of negation of x and y ; FL(2) P cFF ( n (),) x y .
P00 part intersection of negation of x and the negation of y ; FL ( P 0) cFF ( n ( x ), n F ( y )) , and for normalization we set the condition:
cxycFF(, ) ((), nxy c FFFFF xny , () cnxny (),() (1,0).
2 60
Then consider a binary T conorm (disjunction or union), denoted by dxyF (, ): 2 dF : 0,1 0,1 0,1 0,1 dxyttf f F ,,11212 x y tt if and are disjoint and 121. This fuzzy disjunction operator of disjoint variables allows us to add the fuzzy truth- values of disjoint parts of a shaded area in the below table. When the truth-value increases, the dxxk x false value decreases. More general, Fk12, ,..., , as a k-ary disjunction (or union), for k 2 , is defined as: k k dF : 0,1 0,1 0,1 0,1 dxxk x tt tf f f k Fk12, ,..., 1 2... k ,1 2 ... k 1 x tt t if all i are disjoint two by two and 12...k 1. As a particular case let’s take as a binary fuzzy conjunction: cxy ttf f ff F ,,12 1 2 1 2 and as unary fuzzy negation: nx t f ft F () 111 ,1 11 , , where FL x t f tf tf () (,11 ), with 111, and 0,11 1; FL y t f tf tf () (,22 ), with 221, and 0,22 1. whence: FLP tt f f ff (12)1 21 , 2 12 FL P t f f t f t (1)12 , 1 2 12 FLP ft t f tf (2)12 , 1 2 1 2 FL P f f t t t t (0)12 , 1 2 12 The Venn Diagram for n 2 and considering only the truth values, becomes:
O tttt 1 1212
1 2
12 ttt ttt 112 tt 212 12
3 61 since tf t t t tt 12 1(1 2 ) 1 12 f tttttt 12(1 1 ) 2 2 12 f ftttttt(1 )(1 ) 1 12 1 2 1 2 12. We now use: dPk PPP tttttt tt tt tt F 12, 1, 2, 0 12 1 12 2 12 1 1 2 12 , ffffftfttftftttt 31,0 1212121212121212 . FL 4 So, the whole fuzzy space is normalized under ( ) . For the neurosophic logic, we consider NL x T I F TIF () (11 , , 1 ), with 0,,111 1 ; NL y T I F TIF () (22 , , 2 ), with 0,,122 2 ; if the sum of components is 1 as in Atanassov’s intuitionist fuzzy logic, i.e. TIFii i 1, they are considered normalized; otherwise non-normalized, i.e. the sum of the components is <1 (sub- normalized) or >1 (over-normalized). We define a binary neutrosophic conjunction (intersection) operator, which is a particular case of an N-norm (neutrosophic norm, a generalization of the fuzzy t-norm): 2 cN : 0,1 0,1 0,1 0,1 0,1 0,1 cxy TTIIITTIFFFIFTFTFI N (, )12 , 12 12 12, 1 2 12 12 21 21 .
The neutrosophic conjunction (intersection) operator x 6N y component truth, indeterminacy, and falsehood values result from the multiplication TIF 4 TI F 11 1 2 2 2 since we consider in a prudent way TIF , where “ ” means “weaker”, i.e. the products
TIij will go to I , TFij will go to F , and IijF will go to F (or reciprocally we can say that F prevails in front of I and of T ,
(T1 I1 F1) (T1 I1 F1) (T1 I1 F1)
(T2 I2 F2) (T2 I2 F2) (T2 I2 F2)
TT I IITTI So, the truth value is 12, the indeterminacy value is 12 12 12 and the false value is
FF FI FT FT FI x . Ny TIFTI T F x 12 12 12 21 21. The norm of is 11 1 2 2 2. Thus, if and y are normalized, then x.N y is also normalized. Of course, the reader can redefine the neutrosophic conjunction operator, depending on application, in a different way, for example in a more optimistic way, i.e. I TF or T prevails with respect to I , then we get: cxyTTTITIIIFFFIFTFTFIITF N (, ) 12 12 21,, 12 1 2 12 12 21 21 . Or, the reader can consider the order TFI, etc. Let’s also define the unary neutrosophic negation operator:
nN : 0,1 0,1 0,1 0,1 0,1 0,1
4 62
nTIFN ,, FIT ,, by interchanging the truth T and falsehood F vector components. Then: NLPTTIIITITFFFIFTFTFI ( 12)12 , 12 12 21, 1 2 12 12 21 21 NLPTFIIIFITFTFIFFTTTI (1)12 , 12 12 21, 12 12 12 21 21 NLPFTIIITIFTFTITTFFFI (2)12 , 12 12 2 1, 1 2 12 12 2 1 21 NLP FFII IF IFTT TI TF TF TI (0)12 , 12 12 21, 12 12 12 21 21 Similarly as in our above fuzzy logic work, we now define a binary N conorm (disjunction or union), i.e. neutrosophic conform. 2 dN : 0,1 0,1 0,1 0,1 0,1 0,1 9H TT TT I dxy: T TI T I 12 FFT 12I N (, ): , , I ;J: 1212I IFF1 IIFFI 12 1 2 12 1 2 x y TT x/ y if and are disjoint, and 121 where is the neutrosophic norm of N , i.e. TTIFTI F 11 1 2 2 2. x NL x T I F We consider as neutrosophic norm of , where () 11 1, the sum of its TIF components: 11 1, which in many cases is 1, but can also be positive <1 or >1. TT When the truth value increases 12 is the above definition, the indeterminacy and I I falsehood values decrease proportionally with respect to their sums 12 and respectively FF 12. This neutrosophic disjunction operator of disjoint variables allows us to add neutrosophic truth values of disjoint parts of a shaded area in a Venn Diagram.
Now, we complete Donald E. Knuth’s Table of the Sixteen Logical Operators on two variables with Fuzzy Logical operators on two variables with Fuzzy Logic truth values, and Neutrosophic Logic truth/indeterminacy/false values (for the case TIF ).
5 63
Table 1 Fuzzy Logic Venn Notations Operator Name(s) Truth Values Diagram symbol Contradiction, 0 0 falsehood; constant 0 tt xyx. yx y . Conjunction; and 12 , , & Nonimplication; ttt x.yx y x y x y difference, but not 112 ,,[], t x L Left projection 1 Converse ttt x .#yx,,[], y x y y x # nonimplication; 212 not…but t y R Right projection 2 Exclusive disjunction; tt tt x yx yxy. nonequivalence; “xor” 122 12 ,, (Inclusive) disjunction; tttt x / yx y / or; and/or 1212 ,| Nondisjunction, joint tt tt / denial, neither…nor 1 1212 x .//Oyx,,, yx yx y Equivalence; if and tt tt x 0yx,, yx y only if 1212 12 Right t yyyy- R complementation 1 2 ,,!, x /#1yx,, yx y , Converse implication ttt y # if 1 212 [],x yx Left t x -xx x L complementation 1 1 ,,!, x /yx,, yxV y , Implication; only if; ttt x if..then 1 112 [],x yy
6 64
Nonconjunction, not tt . both…and; “nand” 1 12 x /..yx,,,| yx yx y Affirmation; validity; 1 1 T tautology; constant 1
7 65
Table 2
Venn Diagram Neutrosophic Logic Values
(0,0,1) TT I I IT F F FI FT , where ITITIT similarly FI FT 12,, 12 1 2 12 21 , ; 9H : I :TF II IT F F F I FTI : 12,, 12 yyy y yI : I ;J: IF PP12 TIF 11,, 1 9H : I :FT I I IT F F F I F TI : 12,, 12 xxx x xI ;J: IFI PP22 TIF 22,, 2 9 TF TF H : TF IT I FF T I : ,,PP PP I :; 12I IFF 1 2IIFF IJ PP12 P 1 P 2 PP12 P 1 P 2 Where TTIFT I Fwhich is the neutrosophic norm 11 1 2 2 2 TT TI TF I I IF FF 12 ,,12 1 2 FF II IFTT TI TF 12,, 12 12 9 TF TF H : F TFITITFI :PP ,,PP I ;: 12II FF 12II FF JI PP12 P 1 P 2 PP12 P 1 P 2 FIT 222,, FF FI FT II ITFT xx x x,,12 x 12 FIT 111,, F FFIFTIIITTF yy y y,,12 y 1 2
8 66
F FFIFTIIITTT 12 ,,12 12 (1,0,0)
These 16 neutrosophic binary operators are approximated, since the binary N-conorm gives an approximation because of ‘indeterminacy’ component.
Tri-nary Fuzzy Logic and Neutrosophic Logic Operators
In a more general way, for k 2: d k k N : 0,1 0,1 0,1 0,1 0,1 0,1 , 9Hkk : 88TTI : kk9Hki 9k H kiI k : ::IIii11I dxxNkii, ,..., xT:88 T ,:: III, 8Fi TI 12 : ;J::IIkk ; J I : ii1188I FIFi1 I ;J: ii iiI ii11 k x if all i are disjoint two by two, and 8Ti 1. i1 We can extend Knuth’s Table from binary operators to tri-nary operators (and we get 3 n 22 256 tri-nary operators) and in general to n-ary operators (and we get 22 n-ary operators). Let’s present the tri-nary Venn Diagram, with 3 variables x,,yz
O
1 12 2
123
23 13
3
using the name Smarandache codification. This has 283 disjoint parts, and if we shade none, one, two, …, or eight of them and consider all possible combinations we get 22568 tri-nary operators in the above tri-nary Venn Diagram. For n=3 we have:
9 67
Pcxyz123 F ( , , )
Pcxynz12 FF , , ( )
Pcxnyz13 FF , ( ),
Pcnxyz23 FF ( ), ,
Pcxnynz1,(),() FF F
Pcnxynz2(),,() FF F
Pcnxnyz3(),(), FF F
P0(),(),() cnxnynzFF F F Let FL x t f tf tf () (,11 ), with 111, 0 11 , 1 , FL y t f tf tf () (,22 ), with 221, 0 22 , 1 , FL z t f tf tf () (,33 ), with 331, 0 33 , 1 . We consider the particular case defined by tri-nary conjunction fuzzy operator: 3 cF : 0,1 0,1 0,1 0,1 c xyz ttt f f f ff ff ff fff F (, ,)123 , 1 2 3 1 2 2 3 3 1 1 2 3 because tf.. . tf tf ttf f ff tf 11,,FF 22 33 , 121 , 2 12F 33 , ttt f f f ff ff ff fff 123, 1 2 3 1 2 2 3 3 1 1 2 3 and the unary negation operator:
nF : 0,1 0,1 0,1 0,1
nxF () (1 t ,1 f ) ( ft , ) 11 11. We define the function: L 1 : 0,1 0,1 0,1 0,1 L TT 1(,,) and the function L 2 : 0,1 0,1 0,1 0,1 L 2 (,,) then:
10 68 FL P L t t t L f f f (123)1123 (,,),( 2 1 , 2 , 3 ) FL P L t t f L f f t (12)112 (,, 3 ),( 2 1 , 23 ,) FL P L t f t L f t f (13)1123 (, ,),( 2123 ,, ) FL P L f t t L t f f (23)1123 ( ,,),(, 2123 , ) FL P L t f f L f t t (1)11 (, 2 , 3 ),( 2 123 ,,) FL P L f t f L t f t (2)1123 ( ,, ),(, 2123 ,) FL PLfftLttf ( 3)1123 ( , , ), 2123 ( , , ) FL P L f f f L t t t (0)1123 ( , , ),(,,) 2123 We thus get the fuzzy truth-values as follows: FL P t t t t ( 123) 123 FL P t t t t t t t t t ( 12)12 (1 3 ) 12 123 FL P t t t t t t t t t (13)12313123 (1 ) FLP t tt tt ttt t (23)(1)123 23 123 FL P t t t t t t t t t t t t (1)12 (1 )(1 311213123 ) FL P t t t t t t t t t t t t (2)(1)(1)1 2 3 2 12 23 123 FL P ttttttttttt t (3) (112331323123 )(1 )
FLPt (0)(1)(1 t t )(1) t 1 tttttttttttt 1 2 3 123121323123. dxyt tf f x We, then, consider the same disjunction or union operator F (, )1212 , 1, if and y tt are disjoint, and 121 allowing us to add the fuzzy truth values of each part of a shaded area.
Neutrophic Composition Law
Let’s consider k 2 neutrophic variables, xiiiiTIF,, , for all ik%%&1,2,..., . Let denote TTT 1,..., k I II 1,..., k
FFF ,..., k 1 . We now define a neutrosophic composition law oN in the following way:
oTIFN :,,%& 0,1 k zTIF% %&zz + z If ,, then oiN . i1 If zw,,,% %& TIF then
11 69
k 1 zwoo wz 8 ziijj... zw ... w NN 11rrk r 1 %&iij j%& k 11 ,...,rr , ,..., k 1,2,..., iiC% r k 1 ,...,r 1,2,..., jjC% kr k rk1 ,..., 1,2,..., r where Ck1,2,..., means the set of combinations of the elements %&1,2,..., k taken by r . kr [Similarly for Ck1,2,..., ]. zw z In other words, oN is the sum of all possible products of the components of vectors and w , such that each product has at least a zi factor and at least wj factor, and each product has exactly k factors where each factor is a different vector component of z or of w . Similarly if we multiply three vectors: k 2 TIF 8 TIFF ooNN iiuj j l uv ... l k 1 ... uuv1... 1 uvk , , u v 1 %&iij jl l%& k 11 ,...,u , u ,..., uv , uv 1 ,..., k 1,2,..., iiC%%u kj j 11 ,...,uu 1,2,..., , ,..., uv %%CkllCkvk uv 1,2,..., ,uv1 ,..., k 1,2,..., Let’s see an example for k 3. x TIF 1111,, x TIF 2222,, x TIF 3333,,
TToo TTTI, I III, F o F FFF NN123 123 N1 2 3
To I TII ITI IIT TTI TIT ITT N 123 123 123 123 123 123
To F TFF FTF FFT TTF TFT FTT N 123 123 123 123 123 123
Io F IFF FIF FFI IIF IFI FII N 123 123 123 123 123 123
TIFoo TIF TFI ITF IFTFIT FTI NN 12 3 1 23 12 3 1 23 123 123 For the case when indeterminacy I is not decomposed in subcomponents {as for example I PU where P =paradox (true and false simultaneously) and U =uncertainty (true or false, not sure which one)}, the previous formulas can be easily written using only three components as:
TIFoo 8 TIFijr NN ijr,,%P (1,2,3) where P(1, 2, 3) means the set of permutations of (1, 2, 3) i.e. %&(1,2,3),(1,3,2),(2,1,3),(2,3,1,),(3,1,2),(3,2,1) 3 zwoi8 zwwjj wzzijr N r i1 (,ijr , ) (1,2,3) (,)jr%P 2 (1,2,3) This neurotrophic law is associative and commutative.
Neutrophic Logic Operators
12 70
Let’s consider the neutrophic logic cricy values of variables x,,yz (so, for NL( x ) T , I , F with 0 T , I , F 1 n 3) 11 1 11 1 NL y T I F T I F ()22 , , 2 with 022 , , 2 1 NL z T I F T I F ()33 , , 3 with 033 , , 3 1 In neutrosophic logic it is not necessary to have the sum of components equals to 1, as in intuitionist fuzzy logic, i.e. TIFkk k is not necessary 1, for 1k 3 As a particular case, we define the tri-nary conjunction neutrosophic operator: 3 cN : 0,1 0,1 0,1 0,1 0,1 0,1
cxyTTIIITFFFIFTNoooooo(, ) , , NN N N N N
If x or y are normalized, then cxyN ( , ) is also normalized.
If x or y are non-normalized then cxyN (, )T xy where T means norm.
cN is an N-norm (neutrosophic norm, i.e. generalization of the fuzzy t-norm). Again, as a particular case, we define the unary negation neutrosophic operator:
nN : 0,1 0,1 0,1 0,1 0,1 0,1 nx nTIF FIT NN()11 ,, 1 111 ,, . We take the same Venn Diagram for n 3. So, NL x T I F ()11 , , 1 NL y T I F ()22 , , 2 NL z T I F ()33 , , 3. Vectors 9HT 9HI 9HF : 1 I : 1 I : 1 I : I : I : I T= :T I , I= :I I and F= :F I . : 2 I : 2 I : 2 I ;J:T I ;J:I I ;J:F I 3 3 3 9HF 9HT 9HT 9HF : 1I : 1 I : 1 I : 1 I : I : I : I : I We note T=x :T I , T=y :F I , T=z :T I, T=xy :F I, etc. : 2 I : 2 I : 2 I : 2 I ;J:T I ;J:T I ;J:F I ;J:T I 3 3 3 3 and similarly 9HT 9HF 9HT : 1 I : 1 I : 1 I : I : I : I F=x :F I, FTy = : I, F=xz :F I, etc. : 2 I : 2 I : 2 I ;J:F I ;J:F I ;J:T I 3 3 3 zw zw zwv zwv For shorter and easier notations let’s denote oN and respectively ooNN for the vector neutrosophic law defined previously. Then
NL P123 cN ( x , y ) TT , II IT , FF FI FT FIT
13 71 TTT III IIT ITI TII ITT TIT TTI 1 23,, 123 123 123 123 123 123 123 FFF FFI FI F I FF FI I I FI I I F 1 2 3 1 23 12 3 1 2 3 123 1 23 12 3 FFT FTF TFF FTT TFT TTF 123123123123123123 TI F TFI IFT ITF FIT FT I 12 3 1 23 1 23 12 3 123 12 3
NLP12 cNN x , yn , ( z ) TTzz , II IT z , FF z z FI z FT zz FIT z z
NL P13 cNN x , n ( y ), z Tyy T , II IT y , F y F y F y I F yy T F y IT y
NLP23 cNN n ( x ), yz , TTxx , II IT x , FF x x FI x FT xx FIT x y
NLPcxnynz1,(),(),NNN TTIIITFFFIFTFITyzyz yzyzyz , yz yzyz yzyz
NLP2(),,(),, cnxynzNNN TTIIITFFxzxz xzxzxz FIFT xz xzxz FIT xzxz
NLP0(),(),(),, cNNNN n xn ynz TTxyz xyz IIITFFxyz xyz xyz FI xyz FT xyz xyz FIT xyz xyz FFII,, IFTT TI TF TIF .
n-ary Fuzzy Logic and Neutrosophic Logic Operators
We can generalize for any integer n 2 . n 2 Pi i j j The Venn Diagram has 2 disjoint parts. Each part has the form 11...kk ... n , where kn nk n%&ii k 0 , and of course 0 ; 1,..., k is a combination of elements of the set %&n %&j j nk %&j jnii %&%& 1,2,..., , while kn1,..., the elements left, i.e. kk11,..., 1,2,..., \ ,..., k. %&ii %&n %&j j 1,..., k are replaced by the corresponding numbers from 1,2,..., , while kn1,..., are replaced by blanks. For example, when n 3, Pi i j P%&%& i i 12 3 13 if 1 , 2 1,3 , Pi j j P%& i %& 123 1 if 1 1 . Hence, for fuzzy logic we have:
Piij...kk ... j n cx Fi ,..., xnx iFj , ,..., nx Fj 11 11kk n whence 9H9Hkn9H FL Pi i j j :I: tI: tI f f f I ...kk ... n ::++iI: 1j I , ,..., nI 11 :;J: rsI;J: I 12 I ;Jrsk11 n where :0,1 0,1, n SS Snl11 S S 12, ,..., nn1 2 3 ... ( 1)8 ( 1) l l1 where
14 72
n S 1 8 i i1 S 2 8 ij 1ijn ......
Sli 8 i... i 12 l ii in 1...12 l ...... S TTT nn12... And for neutrosophic logic we have:
Piij...kk ... j n cx Ni ,..., xnx iNj , ,..., nx Nj 11 11kk n whence: NL Pi i j j T I F 1...kk 1 ... n12... n , 12... n , 12... n, where 9Hkn : I TTTnxxxxT:++ T iI F j. 12... jjjj...nn ... rI s kk11;J: rsk11
IIIITnxx , 12... jj... n k1
FFFnxxxxxxxxxxxxxx FIFT FIT 12... jjjj...nn ... jj... n jjjj... nn ... jjjj... n... n kk11 k 1 kk 11 k 1 k 1
Conclusion:
A generalization of Knuth’s Boolean binary operations is presented in this paper, i.e. we present n-ary Fuzzy Logic Operators and Neutrosophic Logic Operators based on Smarandache’s codification of the Venn Diagram and on a defined vector neutrosophic law which helps in calculating fuzzy and neutrosophic operators. Better neutrosophic operators than in [2] are proposed herein.
References:
[1] F. Smarandache & J. Dezert, Advances and Applications of DSmt for Information Fusion, Am. Res. Press, 2004. [2] F. Smarandache, A unifying field in logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, 1998, 2001, 2003. [3] Knuth, Art of Computer Programming, The, Volumes 1-3 Boxed Set (2nd Edition), Addison-Wesley Professional, 1998. [4] Zadeh, Fuzzy Sets, Information and Control, Vol. 8, 338-353, 1965. [5] Driankov, Dimiter; Hellendoorn, Hans; and Reinfrank, Michael: An Introduction to Fuzzy Control. Springer, Berlin/Heidelberg, 1993. [6] Atanassov, K., Stoyanova, D., Remarks on the intuitionistic fuzzy sets. II, Notes on Intuitionistic Fuzzy Sets, Vol. 1, No. 2, 85 – 86, 1995.
15 73 April, 2006 PROGRESS IN PHYSICS Volume 2
Schrodinger¨ Equation and the Quantization of Celestial Systems
Florentin Smarandache∗ and Vic Christianto† ∗Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] †Sciprint.org—aFreeScientificElectronicPreprintServer,http://www.sciprint.org E-mail: [email protected]
In the present article, we argue that it is possible to generalize Schrodinger¨ equation to describe quantization of celestial systems. While this hypothesis has been described by some authors, including Nottale, here we argue that such a macroquantization was formed by topological superfluid vortice. We also provide derivation of Schrodinger¨ equation from Gross-Pitaevskii-Ginzburg equation, which supports this superfluid dynamics interpretation.
1 Introduction Interestingly, Nottale’s scale-relativistic method [2, 3] was also based on generalization of Schrodinger¨ equation In the present article, we argue that it is possible to generalize to describe quantization of celestial systems. It is known Schrodinger¨ equation to describe quantization of celestial that Nottale-Schumacher’s method [6] could predict new systems, based on logarithmic nature of Schrodinger¨ equa- exoplanets in good agreement with observed data. Nottale’s tion, and also its exact mapping to Navier-Stokes equa- scale-relativistic method is essentially based on the use of tions [1]. first-order scale-differentiation method defined as follows [2] While this notion of macro-quantization is not widely ac- ∂V cepted yet, as we will see the logarithmic nature of Schrodin-¨ = β (V )=a + bV + ... . (5) ger equation could be viewed as a support of its applicability ∂(lnδt) to larger systems. After all, the use of Schrodinger¨ equation Now it seems clear that the natural-logarithmic derivat- has proved itself to help in finding new objects known as ion, which is essential in Nottale’s scale-relativity approach, extrasolar planets [2, 3]. And we could be sure that new also has been described properly in Schrodinger’s¨ original extrasolar planets are to be found in the near future. As an equation [5]. In other words, its logarithmic form ensures alternative, we will also discuss an outline for how to derive applicability of Schrodinger¨ equation to describe macro- Schrodinger¨ equation from simplification of Ginzburg- quantization of celestial systems. [7, 8] Landau equation. It is known that Ginzburg-Landau equation exhibits fractal character, which implies that quantization 2 Quantization of celestial systems and topological could happen at any scale, supporting topological interpret- quantized vortices ation of quantized vortices [4]. First, let us rewrite Schrodinger¨ equation in its common In order to emphasize this assertion of the possibility to de- form [5] scribe quantization of celestial systems, let us quote Fischer’s ∂ ∇¯ 2 i + − U (x) ψ = 0 (1) description [4] of relativistic momentum from superfluid ∂t 2m dynamics. Fischer [4] argues that the circulation is in the or relativistic dense superfluid, defined as the integral of the ∂ψ i = Hψ. (2) momentum ∂t μ γs = pμ dx =2πNv , (6) Now, it is worth noting here that Englman and Yahalom [5] argues that this equation exhibits logarithmic character and is quantized into multiples of Planck’s quantum of action. This equation is the covariant Bohr-Sommerfeld quantization ln ψ(x, t)=ln |ψ(x, t)| + i arg ψ(x, t) . (3) of γs. And then Fischer [4] concludes that the Maxwell Schrodinger¨ already knew this expression in 1926, which equations of ordinary electromagnetism can be written in then he used it to propose his equation called “eigentliche the form of conservation equations of relativistic perfect fluid Wellengleichung” [5]. Therefore equation (1) can be re- hydrodynamics [9]. Furthermore, the topological character of written as follows equation (6) corresponds to the notion of topological elect- ronic liquid, where compressible electronic liquid represents ∂ ln|ψ| superfluidity [25]. For the plausible linkage between super- 2m +2∇¯ ln |ψ|∇¯ arg ψ +∇¯ ∇¯ arg ψ =0. (4) ∂t fluid dynamics and cosmological phenomena, see [16–24].
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It is worth noting here, because vortices could be defined for the orbit radius for any quantum number of the form [28] as elementary objects in the form of stable topological exci- 2 2 tations [4], then equation (6) could be interpreted as Bohr- n g r = 2 2 , (10) Sommerfeld-type quantization from topological quantized 4π GMm vortices. Fischer [4] also remarks that equation (6) is quite which can be rewritten in the known form of gravitational interesting for the study of superfluid rotation in the context Bohr-type radius [2, 7, 8] of gravitation. Interestingly, application of Bohr-Sommerfeld quantization for celestial systems is known in literature [7, 8], n2GM which here in the context of Fischer’s arguments it has r = 2 , (11) v0 special meaning, i. e. it suggests that quantization of celestial systems actually corresponds to superfluid-quantized vortices where r, n, G, M, v0 represents orbit radii, quantum number at large-scale [4]. In our opinion, this result supports known (n = 1, 2, 3,...), Newton gravitation constant, and mass of experiments suggesting neat correspondence between con- the nucleus of orbit, and specific velocity, respectively. In densed matter physics and various cosmology phen- this equation (11), we denote [28] omena [16–24]. 2π v0 = GMm . (12) To make the conclusion that quantization of celestial g systems actually corresponds to superfluid-quantized vortices at large-scale a bit conceivable, let us consider the problem The value of m is an adjustable parameter (similar to g) of quantization of celestial orbits in solar system. [7, 8]. In accordance with Nottale, we assert that the specific In order to obtain planetary orbit prediction from this velocity v0 is 144 km/sec for planetary systems. By noting hypothesis we could begin with the Bohr-Sommerfeld’s con- that m is meant to be mass of celestial body in question, then jecture of quantization of angular momentum. This con- we could find g parameter (see also [28] and references cited jecture may originate from the fact that according to BCS therein). theory, superconductivity can exhibit macroquantum phen- Using this equation (11), we could predict quantization of omena [26, 27]. In principle, this hypothesis starts with celestial orbits in the solar system, where for Jovian planets observation that in quantum fluid systems like superfluidity we use least-square method and use M in terms of reduced (M1 + M2) [28]; it is known that such vortexes are subject to quantization mass μ = M1M2 . From this viewpoint the result is shown condition of integer multiples of 2π,or vs dl = 2πn/m. in Table 1 below [28]. As we know, for the wavefunction to be well defined and For comparison purpose, we also include some recent unique, the momenta must satisfy Bohr-Sommerfeld’s quant- observation by Brown-Trujillo team from Caltech [29–32]. ization condition [28] It is known that Brown et al. have reported not less than four new planetoids in the outer side of Pluto orbit, including pdx=2πn (6a) 2003EL61 (at 52 AU), 2005FY9 (at 52 AU), 2003VB12 (at Γ 76 AU, dubbed as Sedna). And recently Brown-Trujillo team for any closed classical orbit Γ. For the free particle of unit reported a new planetoid finding, called 2003UB31 (97 AU). mass on the unit sphere the left-hand side is [28] This is not to include their previous finding, Quaoar (42 AU), which has orbit distance more or less near Pluto (39.5 AU), T therefore this object is excluded from our discussion. It is v2dτ = ω2T =2πω , (7) 0 interesting to remark here that all of those new “planetoids” are within 8% bound from our prediction of celestial quant- where T = 2π/ω is the period of the orbit. Hence the quantiz- ization based on the above Bohr-Sommerfeld quantization ation rule amounts to quantization of the rotation frequency hypothesis (Table 1). While this prediction is not so precise (the angular momentum): ω = n. Then we can write the compared to the observed data, one could argue that the force balance relation of Newton’s equation of motion [28] 8% bound limit also corresponds to the remaining planets, including inner planets. Therefore this 8% uncertainty could GMm mv2 2 = . (8) be attributed to macroquantum uncertainty and other local r r factors. Using Bohr-Sommerfeld’s hypothesis of quantization of While our previous prediction only limits new planet angular momentum, a new constant g was introduced [28] finding until n = 9 of Jovian planets (outer solar system), it seems that there are sufficient reasons to suppose that ng mvr = . (9) more planetoids in the Oort Cloud will be found in the near 2π future. Therefore it is recommended to extend further the Just like in the elementary Bohr theory (before Schrodin-¨ same quantization method to larger n values. For prediction ger), this pair of equations yields a known simple solution purpose, we include in Table 1 new expected orbits based
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At this point, it seems worth noting that some criticism Δ Object No. Titius Nottale CSV Observ. , % arises concerning the use of quantization method for de- 1 0.4 0.43 scribing the motion of celestial systems. These criticism 2 1.7 1.71 proponents usually argue that quantization method (wave mechanics) is oversimplifying the problem, and therefore Mercury 3 4 3.9 3.85 3.87 0.52 cannot explain other phenomena, for instance planetary mig- Venus 4 7 6.8 6.84 7.32 6.50 ration etc. While we recognize that there are phenomena Earth 5 10 10.7 10.70 10.00 −6.95 which do not correspond to quantum mechanical process, at Mars 6 16 15.4 15.4 15.24 −1.05 least we can argue further as follows: Hungarias 7 21.0 20.96 20.99 0.14 1. Using quantization method like Nottale-Schumacher Asteroid 8 27.4 27.38 27.0 1.40 did, one can expect to predict new exoplanets (extra- solar planets) with remarkable result [2, 3]; Camilla 9 34.7 34.6 31.5 −10.00 2. The “conventional” theories explaining planetary mig- Jupiter 2 52 45.52 52.03 12.51 ration normally use fluid theory involving diffusion − Saturn 3 100 102.4 95.39 7.38 process; Uranus 4 196 182.1 191.9 5.11 3. Alternatively, it has been shown by Gibson et al. [35] Neptune 5 284.5 301 5.48 that these migration phenomena could be described via Pluto 6 388 409.7 395 −3.72 Navier-Stokes approach; 2003EL61 7 557.7 520 −7.24 4. As we have shown above, Kiehn’s argument was based Sedna 8 722 728.4 760 4.16 on exact-mapping between Schrodinger¨ equation and 2003UB31 9 921.8 970 4.96 Navier-Stokes equations [1]; Unobserv. 10 1138.1 5. Based on Kiehn’s vorticity interpretation one these authors published prediction of some new planets in Unobserv. 11 1377.1 2004 [28]; which seems to be in good agreement with Brown-Trujillo’s finding (March 2004, July 2005) of Table 1: Comparison of prediction and observed orbit distance of planetoids in the Kuiper belt; planets in Solar system (in 0.1AU unit) [28]. 6. To conclude: while our method as described herein may be interpreted as an oversimplification of the real on the same quantization procedure we outlined before. For planetary migration process which took place some- Jovian planets corresponding to quantum number n = 10 and time in the past, at least it could provide us with useful n = 11, our method suggests that it is likely to find new tool for prediction; orbits around 113.81 AU and 137.71 AU, respectively. It is recommended therefore, to find new planetoids around these 7. Now we also provide new prediction of other planet- predicted orbits. oids which are likely to be observed in the near future (around 113.8 AU and 137.7 AU). It is recommended As an interesting alternative method supporting this pro- to use this prediction as guide to finding new objects position of quantization from superfluid-quantized vortices (in the inner Oort Cloud); (6), it is worth noting here that Kiehn has argued in favor of re-interpreting the square of the wavefunction of Schrodinger¨ 8. There are of course other theories which have been equation as the vorticity distribution (including topological developed to explain planetoids and exoplanets [36]. vorticity defects) in the fluid [1]. From this viewpoint, Kiehn Therefore quantization method could be seen as merely suggests that there is exact mapping from Schrodinger¨ equa- a “plausible” theory between others. tion to Navier-Stokes equation, using the notion of quantum All in all, what we would like to emphasize here is vorticity [1]. Interestingly, de Andrade and Sivaram [33] also that the quantization method does not have to be the true suggest that there exists formal analogy between Schrodinger¨ description of reality with regards to celestial phenomena. equation and the Navier-Stokes viscous dissipation equation: As always this method could explain some phenomena, while perhaps lacks explanation for other phenomena. But at least ∂V = ν ∇2V, (13) it can be used to predict something quantitatively, i. e. mea- ∂t surable (exoplanets, and new planetoids in the outer solar where ν is the kinematic viscosity. Their argument was based system etc.). on propagation torsion model for quantized vortices [23]. In the meantime, it seems also interesting here to consider While Kiehn’s argument was intended for ordinary fluid, a plausible generalization of Schrodinger¨ equation in partic- nonetheless the neat linkage between Navier-Stokes equation ular in the context of viscous dissipation method [1]. First, and superfluid turbulence is known in literature [34, 24]. we could write Schrodinger¨ equation for a charged particle
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interacting with an external electromagnetic field [1] in the This equation (18) has a pair of exact solutions. It could form of Ulrych’s unified wave equation [14] be shown that for small values of δ, the Hulthen potential (18) approximates the effective Coulomb potential, in particular (−i∇−qA) (−i∇−qA)μ ψ = μ for large radius (14) 2 2 ∂ eff −e ( +1) = −i2m +2mU(x) ψ. VCoulomb = + . (19) ∂t r 2mr2 By inserting (19), equation (17) could be rewritten as In the presence of electromagnetic potential, one could 2 2 2 include another term into the LHS of equation (14) − ∇2 −e (+1) ∂ψ ∗ ψ + + 2 ψ = i . (20) μ 2m r 2mr ∂t (−i∇−qA) (−i∇−qA) + eA0 ψ = μ ∂ (15) For large radii, second term in the square bracket of LHS =2m −i + U(x) ψ. of equation (20) reduces to zero [41], ∂t 2 ( +1) → This equation has the physical meaning of Schrodinger¨ 2 0 , (21) 2mr equation for a charged particle interacting with an external el- so we can write equation (20) as ectromagnetic field, which takes into consideration Aharonov effect [37]. Topological phase shift becomes its immediate 2 ∂ψ − ∇2 + U(x) ψ = i , (22) implication, as already considered by Kiehn [1]. 2m∗ ∂t As described above, one could also derived equation (11) from scale-relativistic Schrodinger¨ equation [2, 3]. It where Coulomb potential can be written as should be noted here, however, that Nottale’s method [2, e2 U(x)=− . (22a) 3] differs appreciably from the viscous dissipative Navier- r Stokes approach of Kiehn [1], because Nottale only considers This equation (22) is nothing but Schrodinger¨ equation his equation in the Euler-Newton limit [3]. Nonetheless, (1), except for the mass term now we get mass of Cooper it shall be noted here that in his recent papers (2004 and pairs. In other words, we conclude that it is possible to re- up), Nottale has managed to show that his scale relativistic derive Schrodinger¨ equation from simplification of (Gross- approach has linkage with Navier-Stokes equations. Pitaevskii) Ginzburg-Landau equation for superfluid dyn- amics [40], in the limit of small screening parameter, δ. 3 Schrodinger¨ equation derived from Ginzburg- Calculation shows that introducing this Hulthen effect (18) Landau equation into equation (17) will yield essentially similar result to (1), in particular for small screening parameter. Therefore, we Alternatively, in the context of the aforementioned superfluid conclude that for most celestial quantization problems the dynamics interpretation [4], one could also derive Schrodin-¨ result of TDGL-Hulthen (20) is essentially the same with the ger equation from simplification of Ginzburg-Landau equa- result derived from equation (1). Now, to derive gravitational tion. This method will be discussed subsequently. It is known Bohr-type radius equation (11) from Schrodinger¨ equation, that Ginzburg-Landau equation can be used to explain vari- one could use Nottale’s scale-relativistic method [2, 3]. ous aspects of superfluid dynamics [16, 17]. For alternative approach to describe superfluid dynamics from Schrodinger-¨ 4 Concluding remarks type equation, see [38, 39]. According to Gross, Pitaevskii, Ginzburg, wavefunction ∗ What we would emphasize here is that this derivation of of N bosons of a reduced mass m can be described as [40] Schrodinger¨ equation from (Gross-Pitaevskii) Ginzburg- 2 Landau equation is in good agreement with our previous con- 2 2 ∂ψ − ∇ ψ + κ |ψ| ψ = i . (16) jecture that equation (6) implies macroquantization corres- 2m∗ ∂t ponding to superfluid-quantized vortices. This conclusion is For some conditions, it is possible to replace the potential the main result of this paper. Furthermore, because Ginzburg- energy term in equation (16) with Hulthen potential. This Landau equation represents superfluid dynamics at low- substitution yields temperature [40], the fact that we can derive quantization of celestial systems from this equation seems to support 2 2 ∂ψ the idea of Bose-Einstein condensate cosmology [42, 43]. − ∇ ψ + VHulthenψ = i , (17) 2m∗ ∂t Nonetheless, this hypothesis of Bose-Einstein condensate where cosmology deserves discussion in another paper. −δr 2 δe Above results are part of our book Multi-Valued Logic, VHulthen = −Ze . (18) 1 − e−δr Neutrosophy, and Schrodinger¨ Equation that is in print.
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Acknowledgments 26. Schrieffer J. R. Macroscopic quantum phenomena from pairing in superconductors. Lecture, December 11th, 1972. The authors would like to thank to Profs. C. Castro, A. Rub- 27. Coles P. arXiv: astro-ph/0209576. ¨ cic, R. M. Kiehn, M. Pitkanen, E. Scholz, A. Kaivarainen and 28. Christianto V. Apeiron, 2004, v. 11(3). E. Bakhoum for valuable discussions. 29. NASA News Release (Jul 2005), http://www.nasa.gov/vision/ universe/solarsystem/newplanet-072905.html. References 30. BBC News (Oct 2004), http://news.bbc.co.uk/1/hi/sci/tech/ 4730061.stm. 1. Kiehn R. M. An interpretation of wavefunction as a measure of vorticity. http://www22.pair.com/csdc/pdf/cologne.pdf. 31. Brown M., et al. ApJ. Letters, Aug. 2004; arXiv: astro-ph/ 0404456; ApJ., forthcoming issue (2005); astro-ph/0504280. 2. Nottale L., et al., Astron. Astrophys., 1997, v. 322, 1018. 32. Brown M. (July 2005), http://www.gps.caltech.edu/∼mbrown/ 3. Nottale L. Astron. Astrophys., 1997, v. 327, 867-889. planetlila/. 4. Fischer U. W. arXiv: cond-mat/9907457 (1999). 33. de Andrade L. G. and Sivaram C. arXiv: hep-th/9811067. 5. Englman R. and Yahalom H. arXiv: physics/0406149 (2004). 34. Godfrey S. P., et al. A new interpretation of oscillating flow 6. Nottale L., Schumacher G., Levefre E. T. Astron. Astrophys., experiments in superfluid Helium II, J. Low Temp. Physics, 2000, v. 361, 379–387. Nos. 112, Oct 2001. 7. Rubcic A. and Rubcic J. The quantization of solar-like gravi- 35. Gibson C. and Schild R. arXiv: astro-ph/0306467. tational systems. Fizika B, 1998, v. 7(1), 1–13. 36. Griv E. and Gedalin M. The formation of the Solar System by 8. Agnese A. G. and Festa R. Discretization of the cosmic scale Gravitational Instability. arXiv: astro-ph/0403376. inspired from the Old Quantum Mechanics. Proc. Workshop on Modern Modified Theories of Grav. and Cosmology, 1997; 37. Anandan J. S. Quantum Coherence and Reality, Proc. Conf. arXiv: astro-ph/9807186. Fundamental Aspects of Quantum Theory, Columbia SC., ed. by J. S. Anandan and J. L. Safko, World Scientific, 1994; arXiv: 9. Fischer U. W. arXiv: cond-mat/9907457. gr-qc/9504002. 10. Aharonov Y., et al. arXiv: quant-ph/0311155. 38. Varma C. M. arXiv: cond-mat/0109049. 11. Hofer W. A. arXiv: physics/9611009; quant-ph/9801044. 39. Lipavsky P., et al. arXiv: cond-mat/0111214. 12. Hooft G. arXiv: quant-ph/0212095. 40. Infeld E., et al. arXiv: cond-mat/0104073. 13. Blasone M., et al. arXiv: quant-ph/0301031. 41. Pitkanen¨ M. http://www.physics.helsinki.fi/∼matpitka/articles/ 14. Rosu H. C. arXiv: gr-qr-qc/9411035. nottale.pdf. 15. Oudet X. The quantum state and the doublets. Annales de la 42. Trucks M. arXiv: gr-qc/9811043. Fondation Louis de Broglie, 2000, v. 25(1). 43. Castro C., et al. arXiv: hep-th/0004152. 16. Zurek W. Cosmological experiments in superfluids and super- conductors. Proc. Euroconference Formation and Interaction of Topological Defects, ed. A. Davis and R. Brandenberger, Plenum, 1995; arXiv: cond-mat/9502119. 17. Volovik G. Superfluid analogies of cosmological phenomena. arXiv: gr-qc/0005091. 18. Volovik G. Links between gravity and dynamics of quantum liquids. Int. Conf. Cosmology. Relativ. Astrophysics. Cosmo- particle Physics (COSMION-99); arXiv: gr-qc/0004049. 19. Volovik G. arXiv: gr-qc/0104046. 20. Nozieres P. and Pines D. The theory of quantum liquids: Superfluid Bose Liquid. Addison-Wesley, 1990, 116–124. 21. Winterberg F. Z. Naturforsch., 2002, v. 57a, 202–204; presented at 9th Canadian Conf. on General Relativity and Relativ. Astrophysics, Edmonton, May 24–26, 2001. 22. Winterberg F. Maxwell’s aether, the Planck aether hypo- thesis, and Sommerfeld’s fine structure constant. http://www. znaturforsch.com/56a/56a0681.pdf. 23. Kaivarainen A. arXiv: physics/020702. 24. Kaivarainen A. Hierarchic models of turbulence, superfluidity and superconductivity. arXiv: physics/0003108. 25. Wiegmann P. arXiv: cond-mat/9808004.
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Plausible Explanation of Quantization of Intrinsic Redshift from Hall Effect and Weyl Quantization
Florentin Smarandache∗ and Vic Christianto† ∗ Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] † Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] Using phion condensate model as described by Moffat [1], we consider a plausible explanation of (Tifft)intrinsic redshift quantization as described by Bell [6] as result of Hall effect in rotating frame. We also discuss another alternative toexplain redshift quantization fromthe viewpoint ofWeyl quantization, which could yieldBohr- Sommerfeldquantization.
1 Introduction describe CMB spectrumdata. Therefore we could expect thatthis equation will also yield interesting results in gala- In a recent paper by Moffat [1] it is shown thatquantum xies scale. See also [1b, 1c, 13] for other implications of phion condensate model with Gross-Pitaevskii equation low-energy phion fluidmodel. yields an approximate fit to data corresponding to CMB Interestingly,it could be shown, that we couldderive spectrum,and it also yieldsamodified Newtonian accelera- (approximately) Schro¨dinger wave equation from Gross- tion law which is in good agreement with galaxyrotation Pitaevskii equation. We consider the well-known Gross- curve data. It seems therefore interesting toextendfurther Pitaevskii equation in the context of superfluidity or super- this hypothesis toexplain quantization of redshift, as shown conductivity[14]: by Tifft et al. [2,6,7].We also argue in other paper that 2 this redshift quantization could be explained as signature ∂Ψ ¯h −1 i¯h = − ΔΨ + V (x) − γ |Ψ|p Ψ, (1) oftopological quantized vortices, which also agrees with ∂t 2m Gross-Pitaevskiian description [3, 5]. p<2N/(N − 2) N 3 Nonetheless, there is remaining question in this quantiz- where if . In physical problems, the p =3 ed vortices interpretation, i. e. how to provideexplanation equation for is known as Gross-Pitaevskii equation. of “intrinsic redshift”argument by Bell [6]. In the present This equation (1) has standing wave solution quitesimilar to paper, we argue that it sounds reasonable tointerpretthe solution of Schro¨dinger equation, in the form: − ¯ intrinsic redshift data fromthe viewpoint of rotating Hall Ψ(x, t)=e iEt/h · u(x) (2) effect,i.e.rotational motion of clusters of galaxies exhibit quantum Hall effect which can be observed in the form Substituting equation (2) intoequation (1) yields: of “intrinsic redshift”. While this hypothesis is very new, 2 ¯h −1 it could be expectedthat we can draw someprediction, − Δu + V (x) − E u = |u|p u, (3) including possibility to observe small “blue-shift” effect ge- 2m nerated by antivortex part ofthe Hall effect [5a]. which is nothing but a time-independent linear form of −1 Another possibilityistoexplain redshift quantization Schro¨dinger equation, exceptfor term |u|p [14]. Ifthe f romthe viewpoint ofWeyl-Moyal quantization theory [25]. right-hand sideofthis equation is negligible, equation (3) It is shown that Schro¨dinger equation can be derivedfrom reduces tostandard Schro¨dinger equation. Weyl approach [8], therefore quantization in this sense comes Now it is worthnoting here thatfrom Nottale et al. we from “graph”-type quantization. In large scale phenomena can derive a gravitational equivalent ofBohr radius from ge- like galaxyredshift quantization one couldthen ask whether neralized Schro¨dinger equation [4]. Therefore we could also there is possibilityof “super-graph” quantization. expect a slightdeviation ofthis gravitational Bohr radius in Further observation is of course recommend ed in order we consider Gross-Pitaevskii equation instead of generalized to verifyorrefute the propositions outlined herein. Schro¨dinger equation. According to Moffat, the phion condensate model im- 2 Interpreting quantized redshift from Hall effect. plies a modification of Newtonian acceleration law tobe- Cosmic String come [1, p. 11]:
In a recent paper, Moffat [1, p. 9] has used Gross-Pitaevskii G∞ M exp (−μ r) a(r)=− + K φ (1 + μ r) , (4) in conjunction withhisphion condensate fluid model to r2 r2 φ
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where In other words, we submitthe viewpointthat Tifft’s ob- M0 servation ofquantized redshift implies a quantizeddistance G∞ = G 1+ . (5) M between galaxies [2, 5], which could be expressed in the form: Therefore we can conclude thatthe use of phion con- rn = r0 + n(δr) , (12) densate model implies a modification of Newton gravitation- al constant, G, to become (5). Plugging in this new equation where n is integer (1,2,3, . . . ) similar to quantum number. (5) intoaNottale’s gravitational Bohr radius equation [4] Because it can be shown using standarddefinition of Hubble law that redshift quantization implies quantizeddistance yields: between galaxies in the same cluster, then one could say ≈ 2 GM M0 ≈ · 2 GM rn n 2 1+ χ n 2 , (6) thatthis equation ofquantizeddistance (11) is a result of v0 M v0 topological quantized vortices (9) in astrophysical scale [5]; where n is integer (1,2,3 . . . ) and: and it agrees with Gross-Pitaevskii (quantum phion condens- ate) description ofCMB spectrum [1]. It is perhaps more M0 χ = 1+ . (7) interesting if we note here, thatfrom (11) then we also get M an equivalent expression of (12): Thereforeweconclude that — providedthe higher order c c c zn = z0 + n δz (13) Yukawa term of equation (4) could be neglected — one has H H H a modified gravitational Bohr-radius in the form of (6). It or can be shown (elsewhere) that using similar argument one zn = z0 + n(δz) (14) could expecttoexplain a puzzling phenomenon of receding or Moon at a constant rateof ±1.5 per year. Andfromthis δz observedfact one could get an estimateofthis χ factor. It zn = z0 1+n . (15) z0 is more interesting tonote here, that anumber of coral reef data also seems to supportthe sameidea ofmodification Nonetheless, there is a problem here, i. e. how toexplain factor in equation(5),butdiscussion ofthis subjectdeserves intrinsic redshift relatedtoTifftquantization as observed in another paper. Fundamental Plane clusters and also from various quasars Asomewhat similar idea has been putforward by Mas- data [6, 6a]: − reliez [18] using the metric: ziQ = zf N 0.1MN (16) 2 2 2 2 2 ds = eαβ dx + dy + dz − (icdt) . (8) where zf =0.62 is assumedtobeafundamental redshift con- stant,and N (=1, 2, 3 ...), and M is function of N [6a]. Another alternative ofthis metric has been proposed by Meanwhile, it is interesting tonote here similaritybetween Socoloff and Starobinski [19] using multi-connected hyper- equation (15) and (16). Here, the number M seems to play surface metric: aroleˆ similar to secondquantum number in quantum
2 2 −2x 2 2 physics [7]. ds = dx + e (dy + dz ) (9) Now we will putforward an argumentthat intrinsic red- with boundaries: e−x =Λ. shift quantization (16) could come from rotating quantum Therefore one can conclude thatthe use of phion con- Hall effect [5a]. densate model has led us toaform of expanding metric, It is argued by Fischer [5a] that “Hall quantization is which has been discussed by a few authors. of necessity derivable from a topological quantum number Furthermore, it is well-known that Gross-Pitaevskii eq- relatedto this (quantum) coherence”. He usedtotal particle uation could exhibit topologically non-trivial vortex solu- momentum [5a]: tions [4, 5], which also corresponds to quantized vortices: p = mv + mΩ × r + qA. (17) The uniqueness condition ofthe collective phase repre- p · dr = Nv 2π¯h. (10) sented in (9) then leads, if we takeapathinthe bulk of el- Therefore an implication of Gross-Pitaevskii equation ectron liquid, for which the integral of mv can be neglected, [1] is thattopologically quantized vortex could exhibit in to the quantization ofthe sum of a Sagnac flux,andthe magnetic flux [5a]: astrophysical scale. In this context we submitthe viewpoint thatthis proposition indeed has been observed in the form · × · of Tifft’s redshift quantization [2, 6]: Φ=q A dr + m Ω r dr = (18) c δr = δz . (11) = B · dS = Nv 2π¯h. H
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This fluxquantization rule corresponds to the factthat a andWeyl metric: vortex is fundamentally characterised by the winding number − − k − − k N alone [5a]. In this regardthe vortex couldtake the form of RWeyl =(d 1)(d 2) AkA 2(d 1) ∂k A . (20) cosmic string [22]. Now it is clear from (15) thatquantized vortices could be formed by different source of flux. Therefore one could expecttoexplain astrophysical After a few more reasonable assumptions one could quantization using Weyl method in lieu of using generalised obtain a generalised Faraday law, which in rotating frame Schro¨dinger equation as Nottale did [4]. To our knowledge will give in a non-dissipative Hall state the quantization of this possibility has never been explored before elsewhere. Hall conductivity [5a]. For instance, it can be shown that one can obtain Bohr- Therefore one could observe that it is quitenatural to Sommerfeldtype quantization rule fromWeyl approach [24, interpretthe quantizeddistance between galaxies (11) as an p. 12], which for kinetic plus potential energy will take the form: implication ofquantum Hall effect in rotating frame (15). ∞ j While this proposition requires further observation, one 2πN¯h = ¯h Sj (E) , (21) couldthink of it in particular using known analogy between j=0 condensedmatter physics and cosmology phenomena [10, k 22]. Ifthis proposition corresponds to the facts, then one which can be solved by expressing E = ¯h Ek as power couldthink that redshift quantizationisanimprint of gene- series in ¯h [24]. Now equation (10) could be rewritten as ralizedquantization in various scales frommicrophysics to follows: macrophysics, just as Tifft once put it [2]: ∞ j “The redshift has imprinted on it apattern that appears p · dr = Nv 2π¯h = ¯h Sj (E) . (22) tohaveits origin in microscopic quantum physics, yet j=0 it carries this imprint across cosmological boundaries”. Or if we consider quantum Hall effect, then equation (18) In the present paper, Tifft’s remark representsnatural im- can be used instead of equation (10), which yields: plication oftopological quantization, which could be formed at any scale [5]. We will explore further this proposition in · × · the subsequent section, using Weyl quantization. Φ=q A dr + m Ω r dr = Furthermore, while this hypothesis is new, it could be ex- ∞ (23) j pectedthat we can draw some new prediction, for instance, = B · dS = ¯h Sj (E) . like possibility to observe small “blue-shift” effect generated j=0 by the Hall effectfrom antivortex-galaxies [23]. Of course, in order to observe such a “blue-shift” one shall first exclude The above method is known as “graph kinematic” [25] other anomalous effectsof redshift phenomena [6]. (For in- or Weyl-Moyal’s quantization [26]. We could also expectto stance: one could argue that perhaps Pioneer spacecraft ano- find Hall effectquantization fromthis deformation quantiza- maly’s blue-shifting of Doppler frequency may originate tion method. fromthe same effect as described herein.) Consideraharmonic oscillator, which equation can be One could expectthatfurther observationinparticular expressed in the form ofdeformation quantization instead of in the area of low-energy neutrino will shed some light on Schro¨dinger equation [26]: this issue [20]. In this regard, one could view thatthe Sun i¯h 2 i¯h 2 is merely a remnant of aneutron star in the past, therefore x + ∂p + p − ∂x − 2E f (x, p)=0. (24) it could be expectedthat it also emitsneutrino similar to 2 2 neutron star [21]. This equation could be separatedto become two simple PDEs. For imaginary part one gets [26]: 3 An alternative interpretation of astrophysical quanti- zation from Weyl quantization. Graph andquanti- (x∂p − p∂x) f =0. (25) zation Now, considering Hall effect, one can introduce our defi- An alternative way tointerpretthe above proposition con- nition oftotal particle momentum (17), therefore equation cerning topological quantum number andtopological quan- (25) may be written: tization [5a], is by using Weyl quantization. In this regards, Castro [8, p. 5] has shown recently that x∂p − (mv + mΩ × r + qA) ∂x f =0. (26) one couldderive Schro¨dinger equation fromWeyl geometry using continuityequation: Our proposition here is that in the context ofdeformation ∂ρ 1 √ quantization it is possible to findquantization solution of + √ ∂ gρvi (19) ∂t g i harmonic oscillator without Schro¨dinger equation. And
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because it corresponds to graph kinematic [25], generalized 9. Schrieffer J. R. Macroscopic quantum phenomena from pairing Bohr-Sommerfeldquantization rule for quantized vortices in superconductors. Lecture, December 11, 1972. (22) in astrophysical scale could be viewed as signature of 10. Zurek W.H. Cosmological experiments in superfluidsand “super-graph”quantization. superconductors. In: Proc. Euroconference in Formation and This proposition, however, deserves further theoretical Interaction of Topological Defects, PlenumPress, 1995; arXiv: considerations. Further experiments are also recommended cond-mat/9502119. in order to verifyand explore further this proposition. 11. Anandan J. S. In: Quantum Coherence and Reality, Proc. Conf. Fundamental Aspects of Quantum Theory, Columbia SC., edited by J. S. Anandan and J. L. Safko, World Scientific, 1994; Concluding remarks arXiv: gr-qc/9504002. In a recent paper, Moffat [1] has used Gross-Pitaevskii in his 12. Rauscher E. A. and Amoroso R. The physical implications of “phion condensatefluid” to describe CMB spectrumdata. multidimensional geometries andmeasurement. Intern. J. of Weextendthis proposition toexplain Tifft redshift quanti- Comp. Anticipatory Systems, 2006. zation fromthe viewpoint oftopological quantized vortices. 13. Chiao R. et al. arXiv: physics/0309065. In effect we consider thatthe intrinsic redshift quantization 14. Dinu T. L. arXiv: math.AP/0511184. could be interpreted as result of Hall effect in rotating frame. 15. Kravchenko V. arXiv: math.AP/0408172. Another alternative toexplain redshift quantization is 16. Lipavsky P. et al. arxiv: cond-mat/0111214. to consider quantized vortices fromthe viewpoint ofWeyl 17. De Haas E. P. Proc. of the Intern. Conf. PIRT-2005, Moscow, quantization (which could yieldBohr-Sommerfeldquanti- MGTU Publ., 2005. zation). 18. Masreliez J. Apeiron, 2005, v. 12. It is recommendedtoconductfurther observation in order to verifyand also toexplore various implications of 19. Marc L.-R. and Luminet J.-P. arXiv: hep-th/9605010. our propositions as described herein. 20. Lanou R. arXiv: hep-ex/9808033. 21. Yakovlev D. et al. arXiv: astro-ph/0012122. Acknowledgment 22. Volovik G. arXiv: cond-mat/0507454. 23. BalentsL.et al. arXiv: cond-mat/9903294. The writers would like to thank to Profs. C. Castro,T.Love, 24. Gracia-Saz A. Ann. Inst. Fourier, Grenoble, 2005, v. 55(5), E. Scholz, D. Rabounski, and A. Kaivarainen for valuable 1001–1008. discussions. 25. Asribekov V.L. arXiv: physics/0110026. 26. Zachos C. arXiv: hep-th/0110114. References
1. Moffat J. arXiv: astro-ph/0602607; [1a] Consoli M. arXiv: hep-ph/0109215; [1b] Consoli M. et al. arXiv: physics/ 0306094. 2. Russell Humphreys D. Our galaxyisthe centre ofthe universe, “quantized”red shifts show. TJ Archive, 2002, v. 16(2), 95– 104;http://answersingenesis.org/tj/v16/i2/galaxy.asp. 3. Smarandache F. andChristiantoV.Anoteongeometric and information fusion interpretation ofBell theorem andquantum measurement. Progress in Physics, 2006, v. 4, 27–31. 4.Smarandache F. andChristiantoV.Schro¨dinger equation and the quantization of celestial systems. Progress in Physics, 2006, v. 2, 63–67. 5. Fischer U. arXiv: cond-mat/9907457; [5a] arXiv: cond-mat/ 0004339. 6. Bell M. B. arXiv: astro-ph/0111123; [6a] arXiv: astro-ph/ 0305112; [6b] arXiv: astro-ph/0305060. 7. SetterfieldB.http://www.journaloftheoretics.com;http://www. setterfield.org. 8. Castro C.and Mahecha J. On nonlinear Quantum Mechanics, Brownian motion, Weyl geometry, and Fisher information. Progress in Physics, 2006, v. 1, 38–45.
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Numerical Solution of Time-Dependent Gravitational Schro¨dinger Equation
Vic Christianto∗, Diego L. Rapoport† and Florentin Smarandache‡ ∗ Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] † Dept. of Sciences and Technology, Universidad Nacional de Quilmes, Bernal, Argentina E-mail: [email protected] ‡ Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected]
In recent years, there are attempts to describe quantization of planetary distance based on time-independent gravitational Schro¨dinger equation, including Rubcic & Rubcic’s method and also Nottale’s Scale Relativity method. Nonetheless, there is no solution yetfor time-dependent gravitational Schro¨dinger equation (TDGSE). In the present paper, a numerical solution oftime-dependent gravitational Schro¨dinger equation is presented, apparently for the firsttime. These numerical solutions leadto gravitational Bohr-radius, as expected.Inthe subsequent section, we also discuss plausible extension ofthis gravitational Schro¨dinger equation to include the effect of phion condensate via Gross-Pitaevskii equation, as described recently by Moffat.Alternatively one can consider this condensate fromthe viewpoint ofBogoliubov-deGennes theory, which can be approximated with coupledtime- independent gravitational Schro¨dinger equation. Further observation is of course recommended in order torefuteorverify this proposition.
1 Introduction effect of phion condensate via Gross-Pitaevskii equation, as described recently by Moffat [5]. Alternatively one can In the pastfew years, there have been some hypotheses sug- consider this phion condensate model fromthe viewpoint of gesting thatquantization of planetary distance can be derived Bogoliubov-deGennes theory, which can be approximated from a gravitational Schro¨dinger equation, such as Rubcic with coupledtime-independent gravitational Schro¨dinger & Rubcic and also Nottale’s scale relativity method [1, 3]. equation. To our knowledge this proposition of coupledtime- Interestingly, the gravitational Bohr radius derivedfromthis independent gravitational Schro¨dinger equation has never gravitational Schro¨dinger equation yieldsprediction of new been considered before elsewhere. type of astronomical observation in recent years, i.e. extra- Further observation is of course recommended in order solar planets, with unprecedented precision [2]. to verifyorrefute the propositions outlined herein. Furthermore, as we discuss in preceding paper [4], using All numerical computation was performed using Maple. similar assumption based on gravitational Bohr radius, one Please note that in all conditions considered here, we use could predict new planetoidsinthe outer orbitsofPluto only gravitational Schreo¨qduainger tion as described in Rub- which are apparently in good agreement with recent observa- cic & Rubcic [3], therefore we neglectthe scale relativistic tional finding.. Therefore one could induce fromthis observ- effectfor clarity. ation thatthe gravitational Schro¨dinger equation (and gravi- tational Bohr radius) deserves further consideration. 2 Numerical solution of time-independent gravitational In the meantime, it is known that all presenttheories Schro¨dinger equation and time-dependent gravita- discussing gravitational Schro¨dinger equation only take its tional Schro¨dinger equation time-independent limit. Therefore it seems worth to find out the solution and implication oftime-dependent gravitational First we write down the time-independent gravitational Schro¨dinger equation (TDGSE). This is what we will discuss Schro¨dinger radial wave equation in accordance with Rubcic in the present paper. & Rubcic [3]: First we will find out numerical solution oftime-inde- d2R 2 dR 8πm2E pendent gravitational Schro¨dinger equation which shall yield + + R + dr2 r dr H2 gravitational Bohr radius as expected [1, 2, 3]. Then we ex- 2 2 (1) tend our discussion to the problem oftime-dependent grav- 2 4π GMm − ( + 1) + 2 R 2 R =0. itational Schro¨dinger equation. r H r In the subsequent section, we also discuss plausible ex- When H, V , E represents gravitational Planck constant, tension ofthis gravitational Schro¨dinger equation to include the Newtonian potential, andthe energy per unitmass ofthe
56 V. Christianto, D. L. Rapoport and F. Smarandache. Numerical Solution of Time-Dependent Gravitational Schro¨dinger Equation
83 April, 2007 PROGRESS IN PHYSICS Volume 2 orbiting body, respectively, and [3]: equation is more or less similar with the above steps, except that we shall take into consideration the right hand side Mmn of Schro¨dinger equation and also assuming time dependent H = h 2πf 2 , (2) m0 form of r: − ( ) R = e αr t . (12) GMm V (r)=− , (3) r Therefore the gravitational Schro¨dinger equation now reads: E E = . (4) d2R 2 dR 8πm2E m + + R + dr2 r dr H2 By assuming that R takes the form: (13) 2 4π2GMm2 ( + 1) dR −αr + R − R = H , R = e (5) r H2 r2 dt and substituting it intoequation(1),and using simplified or by using Leibniz chain rule, we can rewriteequation terms only of equation (1), one gets: (15) as: −α r 2 −α r 2 2 2 − 2αe 8πGMm e dR dr(t) d R 2 dR 8πm E Ψ=α e αr − + . (6) −H + + + R + r rH2 dr(t) dt dr2 r dr H2 (14) After factoring this equation (7) and solving it by equat- 2 4π2GMm2 ( + 1) + R − R =0. ing the factor with zero, yields: r H2 r2 2 2 The remaining steps are similar with the aforementioned 2 4πGMm − H α − RR = 2 2 =0, (7) procedures for time-independent case, exceptthat now one α H getsanadditional termfor RR: or 2 − 2 d RR =4πGMm H α =0, (8) RR = H3α r(t) r(t) − α2r(t)H2 + dt (15) and solving for α, one gets: +8πGMm2 − 2H2α =0. 4π2GMm2 d a = 2 . (9) Atthis point one shall assign a value for r(t) term, H dt because otherwise the equation cannot be solved. We choose Gravitational Bohr radius is defined as inverse ofthis d dt r(t)=1for simplicity, then equation (15) can be rewritten solution of α, then one finds (in accordance with Rubcic & as follows: Rubcic [3]): 2 3 2 2 H rH α rH α 2 2 − 2 r1 = , (10) RR := + +4π GMm H α =0. (16) 4π2GMm2 2 2 The rootsofthis equation (16) can be found as follows: and by substituting back equation (2) into (10), one gets [3]: √ 2 − 2 +2 + 4 4−4 3 +4 2−32 2 2 2π f a1:= r H H r H H r H rGMm π , r1 = GM . (11) 2rH αc √ (17) − 2 +2 − 4 4−4 3 +4 2−32 2 2 a2:= r H H r H H r H rGMm π . Equation (11) can be rewritten as follows: 2rH GM Therefore one can conclude thatthere is time-dependent r1 = 2 , (11a) ν0 modification factor to conventional gravitational Bohr radius (10). For complete Maple session ofthese steps, see Ap- where the “specific velocity” for the system in question can pendix 2. be defined as: −1 2π f ν0 = = α c. (11b) 3 Gross-Pitaevskii effect. Bogoliubov-deGennes appro- αc g ximation and coupled time-independent gravitational The equations (11a)-(11b) are equivalent withNottale’s Schro¨dinger equation result [1, 2], especially when we introduce the quantization 2 Atthis point it seemsworthwhile to take into consideration a number: rn = r1 n [3]. For complete Maple session ofthese all steps, see Appendix 1. Furthermore, equation (11a) may propositionbyMoffat, regarding modification of Newtonian be generalisedfurther to include multiple nuclei, by rewrit- acceleration law due to phion condensate medium, to include 2 2 ing itto become: r1 =(GM)/v ⇒ r1 =(G ΣM)/v , where Yukawa type potential [5, 6]: ΣM represents the sum of central masses. G∞ M exp (−μ r) a(r)=− + K φ (1 + μ r) . (18) Solution oftime-dependent gravitational Schro¨dinger r2 r2 φ
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Therefore equation (1) can be rewritten to become: withequation 1): ⎛ ⎞ 2 2 8πGMm2e−α r 2αe−α r d R 2 dR 8πm E α2 e−αr− 2 + + 2 R + ⎜ rH2 r ⎟ dr r dr H A = ⎝ ⎠ (25) −α r 2 −α r 2 2 2 −αr 2αe 8πGMm e 2 4π GM − K exp(−μφ r)(1 + μφ r) m α e − − + R − (19) r rH2 r H2 and − ( + 1) f (r) 2 R =0, Ψ = . (26) r g (r) or by assuming μ =2μ0 = μ0 r for the exponential term, Numerical solution ofthis matrixdifferential equation equation (19) can be rewritten as: can be found in the same way with the previous methods, d2R 2 dR 8πm2E however we leave this problem as an exercise for the readers. + + R + dr2 r dr H2 It is clear here, however, thatBogoliubov-deGennes ap- (20) 2 −2μ0 2 proximation of gravitational Schro¨dinger equation, taking 4π GM−Ke (1+μ0r) m 2 −(+1) + 2 R 2 R =0. into consideration phion condensate medium will yield non- r H r linear effect, because it requires solution ofmatrixdifferen- ∗ Then instead of equation (8), one gets: tial equation (21) rather than standard ODE in conventional Schro¨dinger equation (or time-dependentPDE in 3D- 2− 2 − 2 2 −μ0 RR =8πGMm 2H α 8π m Ke (1+μ)=0. (21) condition). This perhaps may explain complicated structures Solving this equation will yield a modified gravitational beyond Jovian Planets, such as Kuiper Belt, inner and outer Bohr radius which includes Yukawa effect: OortCloud etc. which of course these structures cannot be 2 predicted by simple gravitational Schro¨dinger eq uation. In H turn, fromthe solution of (21) one could expectthatthere are r1 = 2 −2 2 (22) 4π (GM − Ke μ0 )m numerous undiscovered celestial objectsinthe OortCloud. andthe modification factor can be expressed as ratio between Further observation is also recommended in order to equation (22) and (10): verifyand explore further this proposition. GM χ = . (23) 4 Concluding remarks (GM − Ke−2μ0 ) In the present paper, a numerical solution oftime-dependent (For complete Maple session ofthese steps, see Appendix 3.) gravitational Schro¨dinger equation is presented, apparently A careful reader may note thatthis “Yukawa potential for the firsttime. This numerical solution leads to gravita- effect”asshowninequation (20) could be usedtoexplain ± tional Bohr-radius, as expected. the small discrepancy (around 8%)between the “observed In the subsequent section, we also discuss plausible ex- distance” andthe computeddistance based on gravitational tension ofthis gravitational Schro¨dinger equation to include Bohr radius [4, 6a]. Nonetheless, in our opinion such an the effect of phion condensate via Gross-Pitaevskii equation, interpretation remains an open question, therefore itmay be as described recently by Moffat.Alternatively one can con- worth toexplore further. sider this condensate fromthe viewpoint ofBogoliubov- There is, however, an alternative way to consider phion deGennes theory, which can be approximated with coupled condensate medium i.e. by introducing coupled Schro¨dinger time-independent gravitational Schro¨dinger equation. equation, which is known as Bogoliubov-deGennes theory It is recommendedtoconductfurther observationinorder [7]. This method can be interpreted also as generalisation of to verifyand also toexplore various implications of our pro- assumption by Rubcic-Rubcic [3] of subquantum structure positions as described herein. composed of positive-negative Planck mass. Therefore, taking this proposition seriously, then one comes to hypo- Acknowledgment thesis thatthere shall be coupled Newtonian potential, in- stead of only equation (3). The writers would like to thank to Profs. C. Castro for valu- To simplify Bogoliubov-deGennes equation, we neglect able discussions. the time-dependent case, therefore the wave equation can be Submitted on January 08, 2007 written in matrixform [7, p.4]: Accepted on January 30, 2007 Corrected after revision on March 06, 2007 [A][Ψ]=0, (24) ∗For recent articles discussing analytical solution ofmatrixdifferential equations, the reader is referredto Electronic Journal of Differential Equa- where [A] is 2×2 matrix and [Ψ] is 2×1 matrix, respectively, tions (free access online on many mirrors as http://ejde.math.txstate.edu, which can be represented as follows (using similar notation http://ejde.math.unt.edu, http://www.emis.de/journals/EJDE etc.).
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References > SCHEQ3:= 4*piˆ2*G*M*mˆ2−hˆ2*alpha=0;
3:=4 2 2 − 2 =0 1. Nottale L. et al. Astron. & Astrophys., 1997, v. 322, 1018. SCHEQ π GMm H α 2. Nottale L., Schumacher G. and Levefre E. T. Astron. & Astro- > a:= solve (SCHEQ3, alpha); 2 2 4π GMm phys., 2000, v. 361, 379–389; accessed online on http://daec. a := obspm.fr/users/nottale. H2 3. Rubcic A. and Rubcic J. The quantization of solar like > #Gravitational Bohr radius is defined as inverse of alpha: > gravBohrradius:=1/a; 2 gravitational systems. Fizika, B-7, 1998, v. 1, No. 1–13. H rgravBohr := 4.Smarandache F. andChristiantoV.Progress in Physics, 2006, 4π2GMm2 v. 2, 63–67. 5. Moffat J. arXiv: astro-ph/0602607. Appendix 2 Time-dependent gravitational Schro¨dinger equation 6. Smarandache F. andChristiantoV.Progress in Physics, 2006, > #Solution of gravitational Schrodinger equation (Rubcic, Fizika 1998); v. 4, 37–40; [6a] ChristiantoV.EJTP, 2006, v. 3, No. 12, 117– > restart; 144; accessed onle on http://www.ejtp.com. > #with time evolution (Hagendorn’s paper); > S:=r(t); R:=exp(−(alpha*S)); R1:=exp(−(alpha*r)); 7. Lundin N. I. Microwave induced enhancement ofthe Joseph- son DC. Chalmers University of Technology & Gotterborg S := r(t)
University Report,p.4–7. − R := e αr 8. Griffin M. arXiv: cond-mat/9911419. > D4R:=diff(S,t); D1R:=−alpha*exp(−(alpha*S)); D2R:=−alphaˆ2* 9. Tuszynski J. et al. Physica A, 2003, v. 325, 455–476; accessed exp(−(alpha*S)); D5R:=D1R*D4R; onle on http://sciencedirect.com. d D4R := r(t) 10. Toussaint M. arXiv: cs.SC/0105033. dt
− ( ) 11. Fischer U. arXiv: cond-mat/9907457; [8a] arXiv: cond-mat/ D1R := −αe αr t 0004339. 2 − ( ) D2R := −α e αr t
12. Zurek W.(ed.) In: Proc. Euroconference in Formation and − ( ) d D1R := −αe αr t r(t) Interaction of Topological Defects, PlenumPress, 1995; dt accessed online: arXiv: cond-mat/9502119. > #Using simplifiedterms only from equation (A*8) 13. Volovik G. arXiv: cond-mat/0507454. > SCHEQ3:=−h*D5R+D2R+D1R*2/S+8*piˆ2*G*M*mˆ2*R/(S*hˆ2); > XX2:=factor(SCHEQ3); −αr(t) 3 dr(t) ( ) − 2 ( ) 2− 2 2 +8 2 2 e H α dt r t α r t H H α π GMm XX2:= =0 r(t)H2 Appendix 1 Time-independent gravitational Schro¨dinger equation > #From standard solution of gravitational Schrodinger equation, we know (Rubcic, Fizika 1998): > restart; > SCHEQ4:=4*piˆ2*G*M*mˆ2−hˆ2*alpha; > with (linalg); = − > R: exp( (alpha*r)); 2 2 2 − 4:=4 − R := e αr SCHEQ π GMm H α > D1R:=diff (R,r); D2R:=diff (D1R,r); > #Therefore time-dependent solution of Schrodinger equation may introduce new termto this gravitational Bohr radius. − D1R := −αe αr > SCHEQ5:=(XX2*(S*hˆ2)/(exp(−(alpha*S))))−2*SCHEQ4;
2 −αr D2R := −α e 3 dr(t) 2 2 ODESCHEQ5:=H α r(t) − α r(t)H dt > SCHEQ1:=D2R+D1R*2/r+8*piˆ2*m*E*R/hˆ2+8*piˆ2*G*M*mˆ2*R/(r*hˆ2)− l*(l+1)*R/rˆ2=0; > #Then we shall assume for simplicity by assigning value to d[r(t)]/dt: > XX1:=factor (SCHEQ1); > D4R:=1; > #Using simplifiedterms only from equation (A*8, of Rubcic & Rubcic, 1998) > Therefore SCHEQ5 can be rewritten as: > ODESCHEQ:=D2R+D1R*2/r+8*piˆ2*G*M*mˆ2*R/(r*hˆ2)=0; > SCHEQ5:= Hˆ3*alpha*r/2+alphaˆ2*r*Hˆ2/2−4*piˆ2*G*M*mˆ2−Hˆ2*alpha=0;
−α r 2 2 −α r 3 2 2 2 − 2αe 8π GMm e rH α rH α 2 2 2 ODESCHEQ := α e αr − + =0 SCHEQ5:= + +4π GMm − H α =0 r rH2 2 2
> XX2:=factor (SCHEQ2); > Then we can solve again SCHEQ5 similar to solution of SCHEQ4: > a1:=solve(SCHEQ5,alpha); − 2 2 2 2 2 e αr α rH − 2H α +8π GMm 2:= =0 2 4 4 3 2 2 2 XX 2 −r H +2H + r H − 4H r +4H − 32rGMm π rH a1:= 2rH > RR:= solve (XX2, r); 2 −r H +2H − r4H4 − 4H3r +4H2 − 32rGMm2π2 2 2 2 2:= 2(4π GMm − H α) a RR := − 2rH α2H2 > #Therefore one could expectthatthere is time-dependent change of gravitational > #Then solving for RR=0, yields: Bohr radius.
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Appendix 3 Time-independent gravitational Schro¨dinger equation with Yukawa potential [5]
> #Extension of gravitational Schrodinger equation (Rubcic, Fizika 1998); > restart; > #departure from Newton potential; > R:=exp(−(alpha*r)); − R := e αr
> D1R:= diff (R,r); D2R:= diff (D1R,r);
− D1R := −αe αr 2 − D2R := −α e αr
> SCHEQ2:=D2R+D1R*2/r+8*piˆ2*(G*M−K*exp(−2*mu)*(1+mu*r))*mˆ2*R/ (r*hˆ2)=0; −α r 2 − 2αe ODESCHEQ := α e αr − + r 2 −2 2 − 8 (GM − Ke μ(1 + μr))m e α r + π =0 rH2 > XX2:=factor(SCHEQ2); > RR1:=solve(XX2,r);
2 2 2 2 2 −2 2(−H α +4π GMm − 4π m Ke μ) RR1:=− −α2H2 +8π2m2Ke−2μ > #from standard gravitational Schrodinger equation we know: > SCHEQ3:=4*piˆ2*G*M*mˆ2−hˆ2*alpha=0; > a:=solve(SCHEQ3, alpha); > #Gravitational Bohr radius is defined as inverse of alpha: > gravBohrradius:=1/a; 2 H rgravBohr := 4π2GMm2 > #Thereforeweconclude thatthe new termsof RR shall yield new terms(YY)into this gravitational Bohr radius: > PI:= (RR*(alphaˆ2*hˆ2)−(−8*piˆ2*G*M*mˆ2+2*hˆ2*alpha)); > #This new term induced by pion condensation via Gross-Pitaevskii equation may be observed in the form of long-range potential effect. (see Moffat J., arXiv: astro- ph/0602607, 2006; also Smarandache F. andChristiantoV. Progress in Physics, v. 2, 2006, & v. 1, 2007, www.ptep-online.com) > #We can also solve directly: > SCHEQ5:=RR*(alphaˆ2*hˆ2)/2;
2 2 2 2 2 2 2 −2 α H (−H α +4π GMm − 4π m Ke μ) SCHEQ5:= −α2H2 +8π2m2Ke−2μ > a1:=solve(SCHEQ5, alpha);
2 2 −2 4π m (GM − Ke μ) a1:=0, 0, H2 > #Then one finds modified gravitational Bohr radius in the form: > modifgravBohrradius:=1/(4*piˆ2*(G*M−K*exp(−2*mu))*mˆ2/hˆ2); 2 H rmodified.gravBohr := 4π2m2(GM − Ke−2μ)
> #This modification can be expressed in chi-factor: > chi:=modifgravBohrradius/gravBohrradius;
GM χ := GM − Ke−2μ
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Kaluza-Klein-Carmeli Metric from Quaternion-Clifford Space, Lorentz’ Force, and Some Observables
Vic Christianto and Florentin SmarandacheO Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] ODepartment of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] It was known for quite long time that a quaternion space can be generalized to a Clifford space, and vice versa; but how to find its neat link with more convenient metric form in the General Relativity theory, has not been explored extensively. We begin with a representation of group with non-zero quaternions to derive closed FLRW metric [1], and from there obtains Carmeli metric, which can be extended further to become 5D and 6D metric (which we propose to call Kaluza-Klein-Carmeli metric). Thereafter we discuss some plausible implications of this metric, beyond describing a galaxy’s spiraling motion and redshift data as these have been done by Carmeli and Hartnett [4, 5, 6]. In subsequent section we explain Podkletnov’s rotating disc experiment. We also note possible implications to quantum gravity. Further observations are of course recommended in order to refute or verify this proposition.
1 Introduction approach is more rigorous than (5) in order to describe neat link between quaternion space and FLRW metric. It was known for quite long time that a quaternion space can We begin with a representation of group with non-zero be generalized to a Clifford space, and vice versa; but how to quaternions to derive closed FLRW metric [1], and from there find its neat link to more convenient metric form in the Gen- we argue that one can obtain Carmeli 5D metric [4] from this eral Relativity theory, has not been explored extensively [2]. group with non-zero quaternions. The resulting metric can First it is worth to remark here that it is possible to find be extended further to become 5D and 6D metric (which we a flat space representation of quaternion group, using its al- propose to call Kaluza-Klein-Carmeli metric). gebraic isomorphism with the ring division algebra [3, p.3]: Thereafter we discuss some plausible implications of this metric, beyond describing a galaxy’s spiraling motion and EiEj = ij + fijkEk : (1) redshift data as these have been done by Carmeli and Hartnett dim Working for R , we get the following metric [3]: [4–7]. Possible implications to the Earth geochronometrics and possible link to coral growth data are discussed. In the ds = dxdx ; (2) subsequent Section we explain Podkletnov’s rotating disc ex- imposing the condition: periment. We also note a possible near link between Kaluza- xx = R : (3) Klein-Carmeli and Yefremov’s Q-Relativity, and also possi- ble implications to quantum gravity. This rather elementary definition is noted here because it The reasons to consider this Carmeli metric instead of the was based on the choice to use the square of the radius to conventional FLRW are as follows: represent the distance (x), meanwhile as Riemann argued • One of the most remarkable discovery from WMAP long-time ago it can also been represented otherwise as the is that it reveals that our Universe seems to obey Eu- square of the square of the radius [3a]. clidean metric (see Carroll’s article in Nature, 2003); Starting with the complex n = 1, then we get [3]: • In this regards, to explain this observed fact, most ar- q = x + x E + x E + x!E! : (4) guments (based on General Relativity) seem to agree that in the edge of Universe, the metric will follow Eu- With this special choice of x we can introduce the spe- clidean, because the matter density tends to approach- cial metric [3]: ing zero. But such a proposition is of course in contra- ds = R ( ij @i@j ): (5) diction with the basic “assumption” in GTR itself, i.e. This is apparently most direct link to describe a flat metric that the Universe is homogenous isotropic everywhere, from the ring division algebra. In the meantime, it seems very meaning that the matter density should be the same too interesting to note that Trifonov has shown that the geometry in the edge of the universe. In other words, we need of the group of nonzero quaternions belongs to closed FLRW a new metric to describe the inhomogeneous isotropic metric. [1] As we will show in the subsequent Section, this spacetime.
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()( R_ )2 00 0 * R + * + * 0 ()0 0 + g = * + : (6) ( 00 () sin2( )0) 00 0 () sin2( ) sin2(#)
• Furthermore, from astrophysics one knows that spiral then equation (6) reduces to closed FLRW metric [1, p.5]. galaxies do not follow Newtonian potential exactly. Therefore one can say that closed FLRW metric is neatly as- Some people have invoked MOND or modified (Post-) sociated to the group of nonzero quaternions. Newton potential to describe that deviation from New- Now consider equation (7), which can be rewritten as: tonian potential [8, 9]. Carmeli metric is another pos- ()(R_ )2 = R2: sible choice [4], and it agrees with spiral galaxies, and (8) also with the redshift data [5–7]. Since we choose (8), then the radial distance can be ex- • Meanwhile it is known, that General Relativity is strict- pressed as: ly related to Newtonian potential (Poisson’s equation). dR2 = dz2 + dy2 + dx2: (9) All of this seems to indicate that General Relativity is only applicable for some limited conditions, but it may Therefore we can rewrite equation (8) in terms of (9): not be able to represent the rotational aspects of gravi- 2 2 2 2 2 ()(dR_ ) =(dR) = dz + dy + dx ; (10) tational phenomena. Of course, there were already ex- tensive research in this area of the generalized gravita- and by defining tion theory, for instance by introducing a torsion term, 1 1 ()= 2 = = : which vanishes in GTR [10]. 2 2 n (11) H0 () (H0 ) Therefore, in order to explain spiral galaxies’ rotation curve and corresponding “dark matter”, one can come up with Then we can rewrite equation (10) in the form: adifferent route instead of invoking a kind of strange matter. 2 2 2 2 2 2 ()(dR_ ) = (dv) = dz + dy + dx ; (12) In this regards, one can consider dark matter as a property of the metric of the spacetime, just like the precession of the first or 2(dv)2 + dz2 + dy2 + dx2 =0; planet is a property of the spacetime in General Relativity. (13) Of course, there are other methods to describe the inho- which is nothing but an original Carmeli metric [4, p.3, equa- mogeneous spacetime, see [15, 16], for instance in [16] a tion (4)] and [6, p.1], where H0 represents Hubble constant new differential operator was introduced: = 1 1 , which Ho c t (by setting =n=1, while in [12] it is supposed that =1:2, seems at first glance as quite similar to Carmeli method. But n =1). Further extension is obviously possible, where equa- to our present knowledge Carmeli metric is the most con- tion (13) can be generalized to include the (icdt) component sistent metric corresponding to generalized FLRW (derived in the conventional Minkowski metric, to become (Kaluza- from a quaternion group). Klein)-Carmeli 5D metric [5, p.1]: Further observations are of course recommended in order to refute or verify this proposition. 2(dv)2 + dz2 + dy2 + dx2 +(icdt)2 =0: (14)
2 FLRW metric associated to the group of non-zero Or if we introduce equation (13) in the general relativistic quaternions setting [4, 6], then one obtains: 2 2 2 2 2 2 2 2 The quaternion algebra is one of the most important and well- ds = (dv) e dr R (d# + sin # d ): (15) studied objects in mathematics and physics; and it has natural The solution for (15) is given by [6, p.3]: Hermitian form which induces Euclidean metric [1]. Mean- while, Hermitian symmetry has been considered as a method dr = exp ; to generalize the gravitation theory (GTR), see Einstein paper dv 2 (16) in Ann. Math. (1945). which can be written as: In this regards, Trifonov has obtained that a natural exten- sion of the structure tensors using nonzero quaternion bases dr_ dv = = 1 exp : (17) will yield formula (6). (See [1, p.4].) dr dr 2 Interestingly, by assuming that [1]: This result implies that there shall be a metric deforma- _ 2 R tion, which may be associated with astrophysics observation, () =1; (7) R such as the possible AU differences [11, 12].
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_ Furthermore, this proposition seems to correspond neatly R = () ; (22) to the Expanding Earth hypothesis, because [13]: R “In order for expansion to occur, the moment of inertia or R R F constraints must be overcome. An expanding Earth would = = () : _ _ (23) necessarily rotate more slowly than a smaller diameter planet R R so that angular momentum would be conserved.” (Q.1) Therefore, one can say that there is a direct proportion- ff We will discuss these e ects in the subsequent Sections. ality between the spacevelocity expansion of, let say, Virgo We note however, that in the original Carmeli metric, galaxy and the Earth geochronometry. Table 1 displays the equation (14) can be generalized to include the potentials to calculation of the Earth’s radial expansion using the formula be determined, to become [5, p.1]: represented above [17]:
Therefore, the Earth’s radius increases at the order of ds = 1+ (dv) dr + 1+ c dt ; c (18) 0.166 cm/year, which may correspond to the decreasing angular velocity (Q.1). This number, albeit very minute, may where also correspond to the Continental Drift hypothesis of A. We-
dr = dz + dy + dx : (19) gener [13, 17]. Nonetheless the reader may note that our cal- culation was based on Kaluza-Klein-Carmeli’s phase-space The line element represents a spherically symmetric inho- spacevelocity metric. mogeneous isotropic universe, and the expansion is a result of Interestingly, there is a quite extensive literature suggest- the spacevelocity component. In this regards, metric (18) de- ing that our Earth experiences a continuous deceleration rate. scribes funfbein (“five-legs”) similar to the standard Kaluza- For instance, J. Wells [14] described a increasing day-length Klein metric, for this reason we propose the name Kaluza- of the Earth [14]: Klein-Carmeli for all possible metrics which can be derived “It thus appears that the length of the day has been in- or extended from equations (8) and (10). creasing throughout geological time and that the number of To observe the expansion at a definite time, the (icdt) days in the year has been decreasing. At the beginning of the term in equation (14) has been ignored; therefore the met- D Cambrian the length of the day would have been 21 .” (Q.2) ric becomes “phase-space” Minkowskian. [5, p.1]. (A simi- Similar remarks have been made, for instance by lar phase-space Minkowskian has been considered in various G. Smoot [13]: places, see for instance [16] and [19].) Therefore the metric “In order for this to happen, the lunar tides would have to in (18) reduces to (by taking into consideration the isotropic slow down, which would affect the length of the lunar month. condition): . . . an Earth year of 447 days at 1.9 Ga decreasing to an Earth year of 383 days at 290 Ma to 365 days at this time. However, dr + 1+ (dv) =0: (20) the Devonian coral rings show that the day is increasing by 24 seconds every million years, which would allow for an Alternatively, one can suppose that in reality this assump- expansion rate of about 0.5% for the past 4.5 Ga, all other tion may be reasonable by setting c 0, such as by consid- factors being equal.” (Q.3) ering the metric for the phonon speed cI instead of the light Therefore, one may compare this result (Table 1) with the speed c; see Volovik, etc. Therefore (18) can be rewritten as: increasing day-length reported by J. Wells [13].
ds = 1+ (dv) dr + phonon 4 Observable B: the Receding Moon from the Earth
(21) + 1+ c dt : It is known that the Moon is receding from the Earth at a c s s constant rate of 4cm/year [17, 18]. & Using known values: G = 6.6724 10 cm /(g sec ) To summarize, in this Section we find out that not only $ ! and = 5.5 10 g/m , and the Moon’s velocity 7.9 km/sec, closed FLRW metric is associated to the group of nonzero then one can calculate using known formulas: quaternions [1], but also the same group yields Carmeli met- 4 ric. In the following Section we discuss some plausible im- Vol = (R +R)!; (24) plications of this proposition. 3 M +M =Vol ; (25) 3 Observable A: the Earth geochronometry G (M +M) r +r = ; v (26) One straightforward implication derived from equation (8) is that the ratio between the velocity and the radius is directly where r, v, M each represents the distance from the Moon to proportional, regardless of the scale of the system in question: the Earth, the Moon’s orbital velocity, and the Earth’s mass,
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Nebula Radial velocity Distance Ratio the Earth dist. Predicted the Earth exp. (mile/s) (10! kly) (10 # cm/yr) (R, km) ( R, cm/year) Virgo 750 39 2.617 6371 0.16678 Ursa Mayor 9300 485 2.610 6371 0.166299 Hydra 38000 2000 2.586 6371 0.164779 Bootes 2 86000 4500 2.601 6371 0.165742
Average 2.604 0.1659
Table 1: Calculation of the radial expansion from the Galaxy velocity/distance ratio. Source: [17]. respectively. Using this formula we obtain a prediction of the of the coordinates, and m = const, then (29) reduces [22]:
Receding Moon at the rate of 0.00497 m/year. This value is @v v / F = m H ; around 10% compared to the observed value 4 cm year. CH @t 2 (32) Therefore one can say that this calculation shall take into consideration other aspects. While perhaps we can use other which will be equivalent to equation (27) only if: reasoning to explain this discrepancy between calculation and @v v prediction, for instance using the “conformal brane” method HV = + H : @t 2 (33) by Pervushin [20], to our best knowledge this effect has neat link with the known paradox in astrophysics, i.e. the observed Further analysis of this effect to describe the Receding matter only contributes around 1–10% of all matter that is Moon from the Earth will be discussed elsewhere. In this Sec- supposed to be “there” in the Universe. tion, we discuss how the calculated expanding radius can de- An alternative way to explain this discrepancy is that there scribe (at least partially) the Receding Moon from the Earth. is another type of force different from the known Newtonian Another possible effect, in particular the deformation of the potential, i.e. by taking into consideration the expansion of surrounding medium, shall also be considered. the “surrounding medium” too. Such a hypothesis was pro- posed recently in [21]. But we will use here a simple argu- 5 Observable C: Podkletnov’s rotation disc experiment ment long-time ago discussed in [22], i.e. if there is a force other than the gravitational force acting on a body with mass, It has been discussed how gravitational force shall take into then it can be determined by this equation [22, p.1054]: consideration the full description of Newton’s law. In this d(mv ) Section, we put forth the known equivalence between New- = F + F ; dt CH (27) ton’s law (31) and Lorentz’ force [23], which can be written (supposing m to be constant) as follows: where v is the velocity of the particle relative to the absolute d d~v 1 space [22a]. The gravitational force can be defined as before: F~ = (~m~v)=m = q E~ + ~v B~ ; dt dt c (34) FCH = m HV; (28) where the relativistic factor is defined as: where the function V is solution of Poisson’s equation: H 1 = : H V =4K; (29) 1 (35) K and represents Newtonian gravitational constant. For sys- while we can expand this equation in the cylindrical coordi- tem which does not obey Poisson’s equation, see [15]. nates [23], we retain the simplest form in this analysis. In It can be shown, that the apparent gravitational force that accordance with Spohn, we define [24]: is produced by an aether flow is [22]: E = HA: @v v dm (36) FCH = m + m H mv Hv + v ; (30) @t 2 dt B = HA: (37) which is an extended form of Newton law: For Podkletnov’s experiment [26–28], it is known that d d~v d~m there in a superconductor E =0[25], and by using the mass F~ = (m~~ v)=m + v : (31) A dt dt dt m in lieu of the charge ratio ? in the right hand term of (34) called the “gravitational Lorentz force”, we get: If the surrounding medium be equivalent to Newton’s the- d~v m 1 ory, this expression shall reduce to that given in (27). Suppos- m = ~v B~ = ~p B~ : (38) ing the aether be irrotational relative to the particular system dt
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Let us suppose we conduct an experiment with the weight problem with this description is that it neglects the directions w = 700 g, the radius r = 0.2 m, and it rotates at f = 2 cps of the velocity other than against the x line. (cycle per second), then we get the velocity at the edge of Therefore, one can generalize further the metric to be- the disc as: come [1, p.5]: v = 2 fr= 2.51 m/sec; (39)
& (dvR) + dz + dy + dx =0; (44) and with known values for G = 6.67 10 , c 3 10 m/sec, " $ Mearth = 5.98 10 kg, rearth = 3 10 m, then we get: or by considering each component of the velocity vector [23]:
G ' F = Mv / : (i dv ) +(i dv ) +(i dv ) + gr 3.71 10 newton kgm sec (40) X Y Z c r (45) + dz + dy + dx =0: Because B = F=meter, then from (39), the force on the disc is given by: From this viewpoint one may consider it as a generaliza- ~ c Fdisc = Bearth ~pdisc Bearth m : (41) tion of Minkowski’s metric into biquaternion form, using the modified Q-relativity space [30, 31, 32], to become: High-precision muon experiment suggests that its speed ds =(dx + i dv ) q : can reach around 0.99 c. Let us suppose in our disc, the k k k (46) particles inside have the speed 0.982 c, then = 0.1889. Please note here that we keep using definition of Yefre- Now inserting this value into (40), yields: mov’s quaternion relativity (Q-relativity) physics [30], albeit ' & we introduce dv instead of dt in the right term. We propose Fdisc =(3.71 10 ) (0.7) (3 10 ) 0.189 = (42) to call this metric quaternionic Kaluza-Klein-Carmeli metric. = = : 0.147 newton 14.7 gr One possible further step for the generalization this equa- dt Therefore, from the viewpoint of a static observer, the tion, is by keep using the standard Q-relativistic term, to disc will get a mass reduction as large as ":% = 2.13%, which become: % ds =(dx + icdt + i dv ) q ; seems quite near with Podkletnov’s result, i.e. the disc can k k k k (47) obtain a mass reduction up to 2% of the static mass. which yields 9-Dimensional extension to the above quater- We remark here that we use a simplified analysis using nionic Kaluza-Klein-Carmeli metric. In other words, this Lorentz’ force, considering the fact that superconductivity generalized 9D KK-Carmeli metric is seemingly capable to may be considered as a relativistic form of the ordinary elec- bring the most salient features in both the standard Carmeli tromagnetic field [25]. metric and also Q-relativity metric. Its prediction includes ff Interestingly, some authors have used di erent methods to plausible time-evolution of some known celestial motion in explain this apparently bizarre result. For instance, using Taj- the solar system, including but not limited to the Earth-based = a = : =0:2 mar and deMatos’ [29] equation: .In satellites (albeit very minute). It can be compared for instance : =2% other words, it predicts a mass reduction around ':& , using Arbab’s calculation, that the Earth accelerates at rate which is quite similar to Podkletnov’s result. 3.05 arcsec/cy , and Mars at 1.6 arcsec/cy [12]. Detailed Another way to describe those rotating disc experiments calculation will be discussed elsewhere. is by using simple Newton law [33]. From equation (31) one We note here that there is quaternionic multiplication rule F =0 g = dv has (by setting and because dt ): which acquires the compact form [30–32]: dm m m = g = g; 1q = q 1=q ;qq = + " q ; dt v !R (43) k k k j k jk jkn n (48) Therefore one can expect a mass reduction given by an where kn and "jkn represent 3-dimensional symbols of Kro- angular velocity (but we’re not very how Podkletnov’s exper- necker and Levi-Civita, respectively [30]. It may also be iment can be explained using this equation). worth noting here that in 3D space Q-connectivity has clear We end this section by noting that we describe the rotating geometrical and physical treatment as movable Q-basis with disc experiment by using Lorentz’ force in a rotating system. behavior of Cartan 3-frame [30]. Further extension of this method in particular in the context In accordance with the standard Q-relativity [30, 31], it of the (extended) Q-relativity theory, will be discussed in the is also possible to write the dynamics equations of Classical subsequent Section. Mechanics for an inertial observer in the constant Q-basis, as follows: d m (x q )=F q : 6 Possible link with Q-Relativity. Extended 9D metric dt k k k k (49) In the preceding Section, we have discussed how closed Because of the antisymmetry of the connection (the gen- FLRW metric is associated to the group with nonzero quater- eralized angular velocity), the dynamics equations can be nions, and that Carmeli metric belongs to the group. The only written in vector components, by the conventional vector no-
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tation [30, 32]: on the plausible quantization of a gravitational field, which may have observable effects for instance in the search of ex- m ~a +2~ ~v + ~ ~r + ~ (~ ~r) = F;~ (50) trasolar planets [35a]. Furthermore, considering the “phonon metric” described which represents known types of classical acceleration, i.e. in (20), provided that it corresponds to the observed facts, the linear, the Coriolis, the angular, and the centripetal acce- in particular with regards to the “surrounding medium” vor- leation, respectively. tices described by (26–29), one can say that the “surrounding Interestingly, as before we can use the equivalence be- medium” is comprised of the phonon medium. This proposi- tween the inertial force and Lorentz’ force (34), therefore tion may also be related to the superfluid-interior of the Sun, equation (50) becomes: which may affect the Earth climatic changes [35b]. Therefore d~v m +2~ ~v + ~ ~r + ~ (~ ~r) = one can hypothesize that the signatures of quantum gravity, dt in the sense of the quantization in gravitational large-scale 1 (51) phenomena, are possible because the presence of the phonon = q E~ + ~v B~ ; c medium. Nonetheless, further theoretical works and observa- or tions are recommended to explore this new proposition. d~v q 1 = E~ + ~v B~ dt m c 8 Concluding remarks (52) 2~ ~v + ~ ~r + ~ (~ ~r) : In the present paper we begun with a representation of a group m with non-zero quaternions to derive closed FLRW metric [1], Please note that the variable q here denotes electric and we obtained Carmeli 5D metric [4] from this group. The charge, not quaternion number. resulting metric can be extended further to become 5D and Therefore, it is likely that one can expect a new effects 6D metric (called by us Kaluza-Klein-Carmeli metric). other than Podkletnov’s rotating disc experiment as discussed Thereafter we discussed some plausible implications of in the preceding Section. this metric. Possible implications to the Earth geochrono- Further interesting things may be expected, by using (34): metrics and possible link to the coral growth data were dis- cussed. In subsequent Section we explained Podkletnov’s d~v 1 F~ = m = q E~ + ~v B~ rotating disc experiment. We also noted possible neat link dt c between Kaluza-Klein-Carmeli metric and Yefremov’s 1 (53) m (d~v)=q E~ + ~v B~ dt : Q-Relativity, in particular we proposed a further extension c of Q-relativity to become 9D metric. Possible implications to Therefore, by introducing this Lorentz’ force instead of quantum gravity, i.e. possible observation of the quantization ff the velocity into (44), one gets directly a plausible extension e ects in gravitation phenomena was also noted. of Q-relativity: Nonetheless we do not pretend to have the last word on some issues, including quantum gravity, the structure of the q ~ 1 ~ ds = dxk + i Ek + ~vk Bk dtk qk : (54) aether (phonon) medium, and other calculations which re- m c main open. There are also different methods to describe the This equation seems to indicate how a magnetic worm- Receding Moon or Podkletnov’s experiments. What this pa- hole can be induced in 6D Q-relativity setting [16, 19]. The per attempts to do is to derive some known gravitational phe- reason to introduce this proposition is because there is known nomena, including Hubble’s constant, in a simplest way as link between magnetic field and rotation [34]. Nonetheless possible, without invoking a strange form of matter. Further- further experiments are recommended in order to refute or more, the Earth geochronometry data may enable us to verify verify this proposition. the cosmological theories with unprecedented precision. Therefore, it is recommended to conduct further observa- tions in order to verify and also to explore the implications of 7 Possible link with quantum gravity our propositions as described herein. In this Section, we remark that the above procedure to de- rive the closed FLRW-Carmeli metric from the group with Acknowledgment nonzero quaternions has an obvious advantage, i.e. one can find Quantum Mechanics directly from the quaternion frame- The writers would like to thank to Profs. C. Castro and work [35]. In other words, one can expect to put the gravita- A. Yefremov for valuable discussions. Special thanks to Prof. tional metrical (FLRW) setting and the Quantum Mechanics D. Rapoport for insightful remarks in particular concerning setting in equal footing. After all, this may be just a goal possible link between gravitation and torsion. sought in “quantum gravity” theories. See [4a] for discussion Submitted on February 25, 2008 / Accepted on March 10, 2008
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References 21. Rughede O.L. On the theory and physics of aether. Progress in Physics, 2006, v. 2. 1. Trifonov V. Geometry of the group of nonzero quaternions. 22. Kirkwood R.L. Gravitational field equations. Phys. Rev., 1954, / / arXiv: physics 0301052; [1a] arXiv: math-ph 0606007. v. 95, no. 4, 1051–1056; [22a] Cahill R. Dynamical fractal 3- 2. Carrion H.L., et al. Quaternionic and octonionic spinors. A space and the generalized Schrodinger¨ equation: equivalence classification. arXiv: hep-th/0302113. principle and vorticity effects. Progress in Physics, 2006, v. 2. 3. Abdel-Khalek K. The ring division self duality. arXiv: hep- 23. Murad P.A. Relativistic fluid dynamics and light speed travel, th/9710177; [3a] Riemann B. In which line-element may be AIAA, 1999. ff expressed as the fourth root of a quartic di erential expression. 24. Spohn H. Semiclassical limit of the Dirac equation and spin Nature, v. VIII, nos. 183 and 184, 14–17, 36, and 37. Translated precession. arXiv: quant-ph/9911065. by William K. Clifford. 25. Cui H.Y. Relativistic mechanism of superconductivity. arXiv: 4. Carmeli M. Is galaxy dark matter a property of spacetime? physics/0212059. arXiv: astro-ph/9607142; [4a] Carmeli M., et al. The SL(2,c) gauge theory of gravitation and the quantization of gravitational 26. Podkletnov E. and Nieminen R. Physica C, 1992, v. 203, 441. field. arXiv: gr-qc/9907078. 27. Podkletnov E. Weak gravitation shielding properties of com- 5. Hartnett J.G. Carmeli’s accelerating universe is spatially flat posite bulk YBa Cu!O% N superconductor below 70 K under / without dark matter. arXiv:gr-qc/0407083. e.m. field. arXiv: cond-mat 9701074. 6. Hartnett J.G. Extending the redshift-distance relation in 28. Podkletnov E. and Modanese G. Impulse gravity generator Cosmological General Relativity to higher redshifts. arXiv: based on charged YBa Cu!O% O superconductor with compos- / physics.gen-ph/0705.3097. ite crystal structure. arXiv: physics 0108005. 7. Hartnett J.G. The distance modulus determined from Carmeli’s 29. Tajmar M. and deMatos C. Induction and amplification of non- / cosmology. arXiv: astro-ph/0501526. Newtonian gravitational fields. arXiv: gr-qc 0107012; [29a] deMatos C. and Tajmar M. Gravitational pointing vector and 8. Fujii Y. Cosmological constant, quintessence and Scalar-Tensor gravitational Larmor theorem in rotating bodies with angular theories of gravity. arXiv: gr-qc/0001051. acceleration. arXiv: gr-qc/0107014. 9. Fujii Y. Quintessence, Scalar-Tensor theories, and non- 30. Yefremov A. Quaternions: algebra, geometry and physical the- Newtonian gravity. arXiv:gr-qc/9911064. ories. Hypercomplex Numbers in Geometry and Physics, 2004, 10. Hehl F.W. and Obukhov Y. arXiv: gr-qc/0711.1535, p.1–2; v. 1(1), 105. [10a] Rapoport D. In: Quantization in Astrophysics, Brownian 31. Yefremov A. In: Quantization in Astrophysics, Brownian mo- motion, and Supersymmetry, F. Smarandache and V.Christianto tion, and Supersymmetry, F. Smarandache and V.Christianto (eds), Chenai, Tamil Nadu, 2007. (eds), Chenai, Tamil Nadu, 2007. 11. Noerdlinger P. arXiv: astro-ph/0801.3807. 32. Smarandache F. and Christianto V. Less mundane explana- 12. Arbab I.A. On the planetary acceleration and the rotation of the tion of Pioneer anomaly from Q-relativity. Progress in Physics, / Earth. arXiv: astro-ph 0708.0666, p. 5–7. 2007, v. 3. 13. Smoot N.C. Earth geodynamic hypotheses updated. Jour- 33. Shipov G. Shipov on torsion. http://www.shipov.com/ nal of Scientific Exploration, 2001, v. 15, no. 3, 465–494; 200506 news03.pdf http://www.scientificexploration.org/jse/ 34. Dzhunushaliev V. Duality between magnetic field and rotation. 14. Wells J.W. Coral growth and geochronometry. Nature, 9 March arXiv: gr-qc/0406078. 1963; http://freepages.genealogy.rootsweb.com/springport/ 35. Trifonov V. Geometrical modification of quaternionic quantum geology/coral growth.html mechanics. arXiv: math-ph/0702095; [35a] Bower G.C., et 15. Christodoulou D. and Kazanas D. Exact solutions of the al. Radio astrometric detection and characterization of extra- isothermal Lane-Emdeen equations with rotation and implica- Solar planets: a white paper submitted to the NSF ExoPlanet / tions for the formation of planets and satellites. arXiv: astro-ph Task Force. arXiv: astro-ph/0704.0238; [35b] Manuel O.K., 0706.3205. et al. Superfluidity in the Solar interior: implications for Solar 16. Sussman R. A dynamical system approach to inhomogeneous eruptions and climate. J. Fusion Energy, 2002, v. 21, 192–198; dust solutions. arXiv: gr-qc/0709.1005. arXiv: astro-ph/0501441. 17. Sollanych M.J. Continental drift and the expansion of the uni- verse. http://www3.bc.sympathico.ca/moon/index.html 18. O’Brien R. In: 2002 Yearbook of Astronomy, P. Moore (ed.), Macmillan Publ. Ltd., London, 2002, 214–223, 262–263. 19. Friedman J.L., et al. Reduced phase space formalism for spher- ically symmetric geometry with a massive dust shell. arXiv: gr-qc/970605, p.19 20. Pervushin V.N., et al. Astrophysics in relative units as the theo- ry of conformal brane. arXiv: hep-th/0209070.
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A Note of Extended Proca Equations and Superconductivity
Vic Christianto , Florentin Smarandachey, and Frank Lichtenbergz Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] yChair of the Dept. of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] zBleigaesschen 4, D-86150 Augsburg, Germany E-mail: [email protected] It has been known for quite long time that the electrodynamics of Maxwell equations can be extended and generalized further into Proca equations. The implications of in- troducing Proca equations include an alternative description of superconductivity, via extending London equations. In the light of another paper suggesting that Maxwell equations can be written using quaternion numbers, then we discuss a plausible exten- sion of Proca equation using biquaternion number. Further implications and experi- ments are recommended.
1 Introduction The background argument of Proca equations can be sum- marized as follows [6]. It was based on known definition of It has been known for quite long time that the electrody- derivatives [6, p. 3]: namics of Maxwell equations can be extended and general- 9 @ @ @ @ @ > ized further into Proca equations, to become electrodynamics @ = = ; ; ; = @ ; H => @x @t @x @y @z with finite photon mass [11]. The implications of introduc- ; (3) @ > ing Proca equations include description of superconductivity, @ = = @ ; H ; by extending London equations [18]. In the light of another @x paper suggesting that Maxwell equations can be generalized @a @ a = + H~a; using quaternion numbers [3, 7], then we discuss a plausi- @t (4) ble extension of Proca equations using biquaternion number. @ @ @ @ It seems interesting to remark here that the proposed exten- @ @ = = @ H = @@ ; @t @x @y @z (5) sion of Proca equations by including quaternion differential operator is merely the next logical step considering already where H is Laplacian and @@ is d’Alembertian operator. published suggestion concerning the use of quaternion differ- For a massive vector boson (spin-1) field, the Proca equation ential operator in electromagnetic field [7, 8]. This is called can be written in the above notation [6, p. 7]: Moisil-Theodoresco operator (see also Appendix A). @@ A @ (@A )+m A = j : (6)
2 Maxwell equations and Proca equations Interestingly, there is also a neat link between Maxwell equations and quaternion numbers, in particular via the D In a series of papers, Lehnert argued that the Maxwell pic- Moisil-Theodoresco operator [7, p. 570]: ture of electrodynamics shall be extended further to include a @ @ @ D = i + i + i! : (7) more “realistic” model of the non-empty vacuum. In the pres- @x @x @x! ence of electric space charges, he suggests a general form of There are also known links between Maxwell equations the Proca-type equation [11]: and Einstein-Mayer equations [8]. Therefore, it seems plau- 1 @ sible to extend further the Maxwell-Proca equations to bi- H A = J;=1; 2; 3; 4: (1) c @t quaternion form too; see also [9, 10] for links between Proca equation and Klein-Gordon equation. For further theoretical A =(A; i =c) Here , where A and are the magnetic description on the links between biquaternion numbers, Max- vector potential and the electrostatic potential in three-space, well equations, and unified wave equation, see Appendix A. and: J =(j; ic ) : (2) 3 Proca equations and superconductivity However, in Lehnert [11], the right-hand terms of equa- tions (1) and (2) are now given a new interpretation, where In this regards, it has been shown by Sternberg [18], that the is the nonzero electric charge density in the vacuum, and j classical London equations for superconductors can be writ- stands for an associated three-space current-density. ten in differential form notation and in relativistic form, where
40 V.Christianto, F. Smarandache, F. Lichtenberg. A Note of Extended Proca Equations and Superconductivity
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they yield the Proca equations. In particular, the field itself Another way to generalize Proca equations is by using acts as its own charge carrier [18]. its standard expression. From d’Alembert wave equation we Similarly in this regards, in a recent paper Tajmar has get [6]:
shown that superconductor equations can be rewritten in 1 @ r A = J ;=1; 2; 3; 4; terms of Proca equations [19]. The basic idea of Tajmar ap- c @t (12) pears similar to Lehnert’s extended Maxwell theory, i.e. to include finite photon mass in order to explain superconduc- where the solution is Liennard-Wiechert potential. Then the tivity phenomena. As Tajmar puts forth [19]: Proca equations are [6]: “In quantum field theory, superconductivity is explain- 1 @ m c r + F A =0;=1; 2; 3; 4; ed by a massive photon, which acquired mass due to c @t (13) gauge symmetry breaking and the Higgs mechanism. The wavelength of the photon is interpreted as the Lon- where m is the photon mass, c is the speed of light, and is don penetration depth. With a nonzero photon mass, the reduced Planck constant. Equation (13) and (12) imply the usual Maxwell equations transform into the so- that photon mass can be understood as charge density:
called Proca equations which will form the basis for 1 mFc J = : (14) our assessment in superconductors and are only valid for the superconducting electrons.” Therefore the “biquaternionic” extended Proca equations Therefore the basic Proca equations for superconductor (13) become: will be [19, p. 3]:
@B mFc rE = ; }} + A =0;=1; 2; 3; 4: (15) @t (8) and 1 @E 1 The solution of equations (10) and (12) can be found us- rB = j + A: ing the same computational method as described in [2]. c @t (9) Similarly, the generalized structure of the wave equation The Meissner effect is obtained by taking curl of equation in electrodynamics — without neglecting the finite photon (9). For non-stationary superconductors, the same equation mass (Lehnert-Vigier) — can be written as follows (instead (9) above will yield second term, called London moment. of eq. 7.24 in [6]): ff Another e ects are recognized from the finite Photon m c }} + F A= = RA= ;=1; 2; 3; 4: mass, i.e. the photon wavelength is then interpreted as the (16) London penetration depth and leads to a photon mass about 1/1000 of the electron mass. This furthermore yields the It seems worth to remark here that the method as de- Meissner-Ochsenfeld effect (shielding of electromagnetic scribed in equation (15)-(16) or ref. [6] is not the only pos- fields entering the superconductor) [20]. sible way towards generalizing Maxwell equations. Other Nonetheless, the use of Proca equations have some known methods are available in literature, for instance by using topo- problems, i.e. it predicts that a charge density rotating at an- logical geometrical approach [14, 15]. gular velocity should produce huge magnetic fields, which is Nonetheless further experiments are recommended in or- not observed [20]. One solution of this problem is to recog- der to verify this proposition [23,24]. One particular implica- nize that the value of photon mass containing charge density tion resulted from the introduction of biquaternion differential is different from the one in free space. operator into the Proca equations, is that it may be related to the notion of “active time” introduced by Paine & Pensinger ff 4 Biquaternion extension of Proca equations sometime ago [13]; the only di erence here is that now the time-evolution becomes nonlinear because of the use of 8- Using the method we introduced for Klein-Gordon equation dimensional differential operator. [2], then it is possible to generalize further Proca equations (1) using biquaternion differential operator, as follows: 5 Plausible new gravitomagnetic effects from extended Proca equations (}})A J =0;=1; 2; 3; 4; (10) where (see also Appendix A): While from Proca equations one can expect to observe gravi- tational London moment [4,22] or other peculiar gravitational @ @ @ @ } = rG + irG = i + e + e + e + shielding effect unable to predict from the framework of Gen- @t @x @y ! @z eral Relativity [5, 16, 22], one can expect to derive new grav- @ @ @ @ itomagnetic effects from the proposed extended Proca equa- + i i + e + e + e : @T @X @Y ! @Z (11) tions using the biquaternion number as described above.
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Furthermore, another recent paper [1] has shown that 7 Concluding remarks given the finite photon mass, it would imply that if m is due to a Higgs effect, then the Universe is effectively simi- In this paper we argue that it is possible to extend further lar to a Superconductor. This may support De Matos’s idea Proca equations for electrodynamics of superconductivity to of dark energy arising from superconductor, in particular via biquaternion form. It has been known for quite long time that Einstein-Proca description [1, 5, 16]. the electrodynamics of Maxwell equations can be extended It is perhaps worth to mention here that there are some and generalized further into Proca equations, to become elec- indirect observations [1] relying on the effect of Proca energy trodynamics with finite photon mass. The implications of in- (assumed) on the galactic plasma, which implies the limit: troducing Proca equations include description of supercon- % ductivity, by extending London equations. Nonetheless, fur- m) = ! eV: (17) ther experiments are recommended in order to verify or refute this proposition. Interestingly, in the context of cosmology, it can be shown that Einstein field equations with cosmological constant are approximated to the second order in the perturbation to a Acknowledgement flat background metric [5]. Nonetheless, further experiments Special thanks to Prof. M. Pitkanen for comments on the draft are recommended in order to verify or refute this proposi- version of this paper. tion. Submitted on September 01, 2008 / Accepted on October 06, 2008 6 Some implications in superconductivity research
We would like to mention the Proca equation in the follow- Appendix A: Biquaternion, Maxwell equations and uni- ing context. Recently it was hypothesized that the creation of fied wave equation [3] superconductivity at room temperature may be achieved by In this section we’re going to discuss Ulrych’s method to describe a resonance-like interaction between an everywhere present unified wave equation [3], which argues that it is possible to define background field and a special material having the appropri- a unified wave equation in the form [3]: ate crystal structure and chemical composition [12]. Accord- D (x)=m (x); (A:1) ing to Global Scaling, a new knowledge and holistic approach in science, the everywhere present background field is given where unified (wave) differential operator D is defined as: by oscillations (standing waves) in the universe or physical D = (P qA) P qA : (A:2) vacuum [12]. The just mentioned hypothesis how superconductivity at To derive Maxwell equations from this unified wave equation, room temperature may come about, namely by a resonance- he uses free photon expression [3]: like interaction between an everywhere present background DA(x)=0; (A:3) field and a special material having the appropriate crystal structure and chemical composition, seems to be supported where potential A(x) is given by: by a statement from the so-called ECE Theory which is pos- A(x)=A (x)+jA (x); (A:4) sibly related to this hypothesis [12]: and with electromagnetic fields: “. . . One of the important practical consequences is that i i i a material can become a superconductor by absorption E (x)=@ A (x) @ A (x); (A:5) i ijk of the inhomogeneous and homogeneous currents of B (x)= @j Ak(x): (A:6) ECE space-time ...”[6]. Inserting these equations (A.4)-(A.6) into (A.3), one finds This is a quotation from a paper with the title “ECE Gen- Maxwell electromagnetic equation [3]: eralizations of the d’Alembert, Proca and Superconductivity HE(x) @ C(x)+ijHB(x) Wave Equations ...” [6]. In that paper the Proca equation is derived as a special case of the ECE field equations. j(HxB(x) @ E(x) HC(x)) (A:7) These considerations raises the interesting question about i(HxE(x)+@ B(x))=0: the relationship between (a possibly new type of) supercon- ductivity, space-time, an everywhere-present background For quaternion differential operator, we define quaternion Nabla operator: field, and the description of superconductivity in terms of the @ @ @ @ Proca equation, i.e. by a massive photon which acquired mass Hq c + i + j + k = @t @x @y @z by symmetry breaking. Of course, how far these suggestions (A:8) @ are related to the physical reality will be decided by further = c +~i H~ : experimental and theoretical studies. @t
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And for biquaternion differential operator, we may define a dia- 3. Christianto V. Electronic J. Theor. Physics, 2006, v. 3, no. 12. mond operator with its conjugate [3]: 4. De Matos C. J. arXiv: gr-qc/0607004; Gravio-photon, su-
( @ perconductor and hyperdrives. http://members.tripod.com/da }} ? + c i + frg~ (A:9) (J @t theoretical1/warptohyperdrives.html 5. De Matos C.J. arXiv: gr-qc/0609116. where Nabla-star-bracket operator is defined as: 6. Evans M.W. ECE generalization of the d’Alembert, Proca and @ @ frg~ + i i + superconductivity wave equations: electric power from ECE @x @X § // / / / (A:10) space-time. 7.2; http: aias.us documents uft a51stpaper.pdf @ @ @ @ + + i j + + i k: 7. Kravchenko V.V. and Oviedo H. On quaternionic formulation @y @Y @z @Z of Maxwell’s equations for chiral media and its applications. J. for Analysis and its Applications, 2003, v. 22, no. 3, 570. In other words, equation (A.9) can be rewritten as follows: 8. Kravchenko V.G. and Kravchenko V.V. arXiv: math-ph/ @ @ @ @ }} c + c i + + i i + 0511092. @t @T @x @X (A:11) 9. Jakubsky V. and Smejkal J. A positive definite scalar product @ @ @ @ + + i j + + i k: for free Proca particle. arXiv: hep-th/0610290. @y @Y @z @Z 10. Jakubsky V. Acta Polytechnica, 2007, v. 47, no. 2–3. From this definition, it shall be clear that there is neat link be- 11. Lehnert B. Photon physics of revised electromagnetics. Prog- tween equation (A.11) and the Moisil-Theodoresco D operator, i.e. ress in Physics, 2006, v. 2. [7, p. 570]: 12. Lichtenberg F. Presentation of an intended research project:
@ @ searching for room temperature superconductors. August, }} c + c i +(D + iD )= @t @t xi Xi 2008, http://www.sciprint.org; http://podtime.net/sciprint/fm/ D E uploads/files/1218979173Searching for Room Temperature @ @ @ @ @ = c +c i + i +i +i! + (A:12) Superconductors.pdf @t @T @x @x @x! D E @ @ @ 13. Paine D.A. and Pensinger W.L. Int. J. Quantum Chem., 1979, + i i + i + i : v.15, 3; http://www.geocities.com/moonhoabinh/ithapapers/ @X @X ! @X ! hydrothermo.html In order to define biquaternionic representation of Maxwell 14. Olkhov O.A. Geometrization of classical wave fields. arXiv: equations, we could extend Ulrych’s definition of unified differential 0801.3746. operator [3,17,21] to its biquaternion counterpart, by using equation 15. Olkhov O.A. Zh. Fiz. Khim., 2002, v. 21, 49; arXiv: hep-th/ (A.2) and (A.10), to become: 0201020. D E fDg fP g qfAg fPg qfAg ; (A:13) 16. Poenaru D. A. Proca (1897–1955). arXiv: physics/0508195; http://th.physik.uni-frankfurt.de/poenaru/PROCA/Proca.pdf or by definition P = ir, equation (A.13) could be written as: 17. Ulrych S. arXiv: physics/0009079. D E fDg frg~ qfAg frg~ qfAg ; (A:14) 18. Sternberg S. On the London equations. PNAS, 1992, v. 89, no. 22, 10673–10675. where each component is now defined in term of biquaternionic rep- 19. Tajmar M. Electrodynamics in superconductors explained by resentation. Therefore the biquaternionic form of the unified wave Proca equations. arXiv: 0803.3080. equation [3] takes the form: 20. Tajmar M. and De Matos C.J. arXiv: gr-qc/0603032. 21. Yefremov A., Smarandache F. and Christianto V. Yang-Mills fDg (x)=m (x) ; (A:15) field from quaternion space geometry, and its Klein-Gordon which is a wave equation for massive electrodynamics, quite similar representation. Progress in Physics, 2007, v. 3. to Proca representation. 22. Gravitational properties of superconductors. http://functional- Now, biquaternionic representation of free photon fields could materials.at/rd/rd spa gravitationalproperties de.html be written as follows: 23. Magnetism and superconductivity observed to exist in harmony. fDgA(x)=0: (A:16) Aug. 28, 2008, http://www.physorg.com/news139159195.html 24. Room temperature superconductivity. Jul. 8, 2008, http://www. References physorg.com/news134828104.html
1. Adelberger E., Dvali G., and Gruzinov A. Photon-mass bound destroyed by vortices. Phys. Rev. Lett., 2007, v. 98, 010402. 2. Christianto V. and Smarandache F. Numerical solution of ra- dial biquaternion Klein-Gordon equation. Progress in Physics, 2008, v.1.
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Less Mundane Explanation of Pioneer Anomaly from Q-Relativity
Florentin Smarandache∗ and Vic Christianto† ∗ Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] † Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected]
There have been various explanations ofPioneer blueshift anomaly in the pastfew years; nonetheless no explanation has been offeredfromthe viewpoint of Q-relativity physics. In the present paper it is arguedthatPioneer anomalous blueshift may be caused by Pioneer spacecraft experiencing angular shift induced by similar Q- relativity effect which may also affect Jupiter satellites. By taking into consideration “aether drift” effect, the proposedmethod as described herein could explain Pioneer blueshift anomaly within ∼0.26% error range, which speaks for itself.Another new proposition of redshift quantization is also proposedfrom gravitational Bohr-radius which is consistent with Bohr-Sommerfeldquantization. Further observation is of course recommended in order torefut eorverify this proposition.
1 Introduction tional phenomena. Einstein himself apparently recognizes this limitation [8a, p.61]: In the pastfew years, it is becoming well-known thatPioneer spacecraft has exhibited an anomalous Doppler frequency “. . . all bodies of reference K should be given prefer- blueshifting phenomenon which cannot be explainedfrom ence in this sense, andthey should be exactly equiva- conventional theories, including General Relativity[1,4]. lentto K for the formation of natural laws, provided Despite the nature of such anomalous blueshift remains un- thatthey are in a stateof uniform rectilinear and non- known, some people began to argue that a post-einsteinian rotary motion with respectto K.” (Italic by Einstein). gravitation theory may be in sight, which may be considered Therefore, it shall be clear thatthe restriction of non- as further generalisation of pseudo-Riemannian metric of rotary motion remains a limitation for all considerations by general relativity theory. relativity theory, albeitthe uniform rectilinear part has been Nonetheless, atthis point one may ask: Why do we re- relaxed by general relativity theory. quire a generalization of pseudo-Riemannian tensor, instead After further thought,it becomes apparentthat it is re- of using “patch-work” as usual to modify general relativity quiredto consideranewkind ofmetric which may be able theory? A possible answer is: sometimes too much path- to representthe rotational aspectsof gravitation phenomena, work doesn’t add up. For instance, let us begin witha and by doing so extends the domain of validityof general thought-experiment which forms the theoretical motivation relativity theory. behind General Relativity, an elevator was put in free-falling In this regard, the present paper will discuss the afore- motion [8a]. The passenger inside the elevator will notfeel mentionedPioneer blueshift anomaly fromthe viewpoint of any gravitational pull, which then it is interpreted as formal Q-relativity physics, which has been proposed by Yefremov analogue that “inertial acceleration equals to gravitational [2] in order tobringinto application the quaternion number. acceleration” (Equivalence Principle). More recent experi- Despite the use ofquaternion number in physical theories ments(aEfter o¨tvos)¨ suggest, however, thatthis principle is is very scarce in recent years — apart ofPauli matrix — only applicable at certain conditions. it has been argued elsewhere that using quaternion number Further problemmay arise if we ask: what ifthe elevator one could expectto unify all known equations in Quantum also experiences lateral rotation around itsvertical axis? Mechanics into the same framework, in particular via the Does itmean thatthe inertial acceleration will be slightly known isomorphism between Dirac equation and Maxwell higher or lower than gravitational pull? Similarly we observe equations [5]. that a disc rotating at high speed will exert out-of-plane Another problemthat was often neglected in mosttreat- field resemble an acceleration field. All ofthis seems to ises on Pioneer spacecraft anomaly is the plausible role of indicate thatthe thought-experiment which forms the basis aether drift effect [6]. Here it can be shown thattaking of General Relativity is only applicable for somelimited this effect in to consideration along with the aforementioned dv conditions,inparticular the F = m dt part (because General Q-relativitysatellite’s apparent shift could yield numerical Relativityisstrictly relatedto Newtonian potential), but it prediction ofPioneer blueshift within ∼0.26% error range, may not be able to representthe rotational aspectsof gravita- which speaks for itself.
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We also suggest a new kind of Doppler frequency shift Because of antisymmetry ofthe connection (generalised which can be predicted using Nottale-type gravitational Bohr- angular velocity) the dynamics equations can be written in radius, by taking into consideration varying G parameter as vector components, by conventional vector notation [2]: described by Moffat [7]. To our knowledge this proposition × × × × of new type of redshift corresponding to gravitational Bohr- m a +2Ω v + Ω r + Ω Ω r = F. (4) radius has never been considered before elsewhere. Therefore, from equation (4) one recognizes known types Further observation is of course recommended in order of classical acceleration, i.e. linear, coriolis, angular, centri- to verifyorrefute the propositions outlined herein. petal. Meanwhile it is known that General Relativityintro- duces Newton potential as rigid requirement [2a, 6b]. In 2 Some novel aspects of Q-relativity physics. Pioneer other words, we can expect — using Q-relativity—topredict blueshift anomaly new effects that cannot be explained with General Relativity. Fromthis viewpoint one may consider a generalisation In this section, first we will review some basic conceptsof of Minkowski metric intobiquaternion form [2]: quaternion number andthen discuss itsimplications to qua- ternion relativity (Q-relativity) physics [2]. Then we discuss dz =(dxk + idtk) qk , (5) Yefremov’s calculation of satellite time-shift which may be observed by precise measurement [3]. We however introduce withsome novel properties, i.e.: a new interpretation here that such a satelliteQ-timeshift is • temporal interval is defined by imaginary vector; already observed in the form ofPioneer spacecraft blueshift • space-timeofthe model appears tohavesixdimen- anomaly. sions (6D); Quaternion number belongs to the group of “very good” • algebras: of real, complex, quaternion, and octonion [2]. vector ofthe displacement ofthe particle and vector While Cayley also proposed new terms such as quantic, it of corresponding time change must always be normal is less known than the above group. Quaternion number can to each other, or: be viewed as an extension ofCauchy imaginary plane to dxkdtk =0. (6) become [2]: It is perhaps quiteinteresting tonote here that Einstein Q ≡ a + bi + cj + dk , (1) himself apparently once considered similar approach, by pro- where a, b, c, d are real numbers, and i, j, k are imaginary posing tensors with Riemannian metric with Hermitian sym- quaternion units. These Q-units can be represented either via metry [8]. Nonetheless, there is difference with Q-relativity 2×2 matrices or 4×4matrices [2]. described above, because in Einstein’s generalised Riemann- It is interesting tonote here thatthere is quaternionic ian metric it has 8-dimensions, rather than 3d-space and 3d- multiplication rule which acquires compactform: imaginary time. − One particularly interesting feature ofthis new Q-relativ- 1qk = qk1=qk ,qj qk = δjk + εjknqn , (2) ity(orrotational relativity) is thatthere is universal character where δkn and εjkn represent 3-dimensional symbols of ofmotion ofthe bodies (including non-inertial motions), Kronecker and Levi-Civita, respectively [2]. Therefore it which can be described in unifiedmanner (Hestenes also could be expectedthat Q-algebra may have neat link with considers Classical Mechanics from similar spinor language). pseudo-Riemannian metric used by General Relativity. Inte- For instance advanced perihelion of planets can be described restingly, it has been argued in this regardthat such Q-units in term of such rotational precession [2]. can be generalisedto become Finsler geometry,inparticular Inspired by this new Q-relativity physics, it can be argued with Berwald-Moor metric. It also can be shown that Finsler- thatthere should be anomalous effect in planets’ satellite Berwald-Moor metric is equivalent with pseudo-Riemannian motion. In this regard, Yefremov argues thatthere should metric, and an expression of Newtonian potential can be be a deviation ofthe planetary satellite position, due to foundfor this metric [2a]. discrepancy between calculated and observedfromthe Earth Itmay also be worthnoting here that in 3D space Q- motion magnitudes characterizing cyclic processes on this connectivity has clear geometrical and physical treatment as planet or near it. He proposes [2]: movable Q-basis with behaviour ofCartan 3-frame [2]. ωVeVp It is also possible to write the dynamics equations of Δϕ ≈ t, (7) c2 C lassical Mechanics for an inertial observer in constant Q- or basis. SO(3,R)-invariance oftwo vectors allow to represent ωV V Δϕ ≈− e p t . (8) these dynamics equations in Q-vector form [2]: c2 d2 Therefore, given a satelliteorbit radius r,its position m (x q )=F q . (3) dt2 k k k k shift is found in unitsof length Δl = r Δϕ. His calculation
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Satellites Cycle frequency ω, 1/s Angular shift Δϕ, /100 yrs Linear shift Δl,km/100 yrs Linear size a, km Phobos (Mars) 0.00023 18.2 54 20 Deimos (Mars) 0.00006 4.6 34 12 Metis (Jupiter) 0.00025 10.6 431 40 Adrastea (Jupiter) 0.00024 10.5 429 20 Amalthea (Jupiter) 0.00015 6.3 361 189
Table1:Thefollowing table gives values ofthe effectfor five fast satellites of Mars and Jupiter. Orbital linear velocities are: ofthe Earth VE = 29.8 km/s, of Mars VP = 24.1 km/s, of Jupiter VP = 13.1 km/s; the value ofthe light velocityisc = 299 793 km/s; observation period is chosen 100 years. Courtesy of A. Yefremov, 2006 [3]. for satellites of Mars and Jupiter is given in Table 1. None- Interestingly this angular shift can be explained with the theless he gave no indication as to how to observe this sameorder ofmagnitude fromthe viewpoint of Q-satellite anomalous effect. angular shift (see Table 1), in particular for Jupiter’s Adrastea In this regard,weintroduce here an alternative interpreta- (10.5 per 100 years). There is however, a large discrepancy tion ofthe aforementioned Q-satellite time-shift effect by atthe order of 50%fromthe expected angular shift. Yefremov, i.e. this effect actually has similar effect with Pio- It is proposed here that such discrepancy between Q- neer spacecraft blueshift anomaly. It is known thatPioneer satellite angular shift and expected angular shift required spacecraft exhibits this anomalous Doppler frequency while toexplain Pioneer anomaly can be reduced if we take into entering Jupiter orbit [1, 4], therefore one may argue that consideration the “aether drift” effect [6]. Interestingly we this effect is caused by Jupiter planetary gravitational effect, can use experimental result of Thorndike [6, p.9], saying which also may cause similar effecttoitssatellites. thatthe aether drift effect implies a residual apparent Earth Despite the apparent contradiction with Yefremov’s own velocityisvobs = 15 ± 4 km/sec. Therefore the effective Ve intention, one couldfindthatthe aforementioned Q-satellite in equation (8) becomes: time-shift could yield anatural explanation ofPioneer space- V = v + V = 44.8 km/sec. (12) craft blueshift anomaly. In this regard, Taylor [9] argues that e.eff obs e there is possibilityof a mundane explanation of anomal- Using this improved value for Earth velocityinequation −9 ous blueshift ofPioneer anomaly (5.99 ×10 Hz/sec). The (8), one will get larger values than Table 1, which for Adras- all-angle formulae for relativistic Doppler shift is given tea satellite yields: by [9a, p.34]: ωVe.eff Vp Ve.eff − Δϕobs = 2 t = Δϕ = 15.935 /100 yrs. (13) (1 β cos φ) c V 0 e v = v γ 2 , (9) 1 − β Using this improved prediction, the discrepancy with required angular shift only (15.894 per 100 years) becomes − 2 where β = v/c. By neglecting the 1 β term because of ∼ 0.26%, which speaks for itself. Therefore one may con- low velocity, one gets the standard expression: clude thatthis less mundane explanation ofPioneer blueshift anomaly with Q-relativity may deserve further consideration. v = v0 γ (1 − β cos φ) . (9a)
The derivative with respectto φ is: 3 A new type of redshift from gravitational Bohr radius. dv Possible observation in solar system. = v0 γβsin φ, (10) dφ In preceding paper [10, 11] we argued in favour of an alter- −9 dv × native interpretation of Tifft redshift quantization fromthe where dφ = 5.99 10 Hz/sec, i.e. the observedPioneer anomaly. Introducing thisvalueintoequation (10), one gets viewpoint ofquantizeddistance between galaxies. A method requirement of an effecttoexplain Pioneer anomaly: can be proposed as further test ofthis proposition bothat solar system scale or galaxies scale, by using the known −9 arcsin (5.99×10 Hz) −12 quantized Tifft redshift [14, 15, 16]: dφ = = 1.4×10 deg/sec. (11) v0 γβ c δr ≈ δz . (14) −9 H Therefore, we can conclude thattoexplain 5.99 ×10 Hz/sec blueshift anomaly, it is requiredto find ashift of In this regards, we use gravitational Bohr radius equation: −12 emission angle atthe order 1.4 ×10 degree/sec only (or 2 GM rn = n 2 . (15) around 15.894 per 100 years). v0
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Inserting equation (15) into(14), then one gets quantized Another new proposition of redshift quantization is also redshift expectedfrom gravitational Bohr radius: proposedfrom gravitational Bohr-radius which is consistent with Bohr-Sommerfeldquantization. It is recommendedto H 2 GM zn = n (16) conductfurther observationinorder to verifyand also to c v2 0 explore various implications of our propositions as described which can be observed either in solar system scale or galax- herein. ies scale. To our present knowledge, this effect has never been described elsewhere before. Acknowledgment Therefore, it is recommendedto observe such an accele- rated Doppler-freequency shift, which for big jovian planets The authors are would like to thank to Profs. C. Castro, this effectmay be detected.It is also worthnoting here E. Scholz, T. Love, and D. L. Rapoportfor valuable discus- that according toequation (16), this new Doppler shift is sions. Special thanks to Prof.A.Yefremov for sending his quantized. recent calculation of Jupiter satellite’s Q-time-shift effect. Atthis point one may also take into consideration a propositionbyMoffat, regarding modification of Newtonian References acceleration law to become [7]: − 1. Anderson J.D., Campbell J.K., & Nieto M.M. arXiv: astro- −G∞M exp( μφr) a(r)= 2 + K 2 (1 + μφr) (17) ph/ 0608087; [1a] NietoM.M.&Anderson J.D. arXiv: gr- r r qc/0507052. where 2. Yefremov A. Hypercomplex Numbers in Geometry and Phys- M0 ics, 2004, v. 1(1), 105; [2a] Pavlov D. G. arXiv: math-ph/ G∞ = G 1+ . (17a) M 0609009. Therefore equation (16) may be rewritten to become: 3. Yefremov A. Privatecommunication, October 2006. Email: [email protected]. ≈ H 2 GM M0 ≈ H 2 GM 4. Laemmerzahl C.&Dittus H. Clocks and gravity, fromquan- zn n 2 1+ χ n 2 (18) tumtocosmos. UCLA, 2006, http://www.physics.ucla.edu/ c v0 M c v0 quantum_to_cosmos/q2c06/Laemmerzahl.pdf where n is integer(1,2,3,...)and: 5. ChristiantoV. EJTP, 2006, v. 3, No. 12, http://www.ejtp.com. 6. Consoli M. arXiv: physics/0306094, p. 9; [6a] Consoli M. et M0 χ = 1+ . (18a) al. arXiv: gr-qc/0306105; [6b] arXiv: hep-ph/0109215. M 7. Moffat J. arXiv: astro-ph/0602607. To use the above equations, one may start by using Bell’s 8. Einstein A. Ann. Math.,1945, v. 46; [8a] Einstein A. Relati- suggestion thatthere is fundamental redshift z = 0.62 which vity: the special and general theory. Crown Trade Paperback, is typical for various galaxies andquasars [14]. Assuming New York, 1951, pp. 61, 66–70. we can use equation (16), thenbysetting n = 1, we can 9. Taylor S. arXiv: physics/0603074; [9a] Engelhardt W. Apeiron, expecttopredictthe mass ofquasar centre or galaxy centre. 2003, v. 10, No. 4,34. Then the result can be usedtocompute back how time- 10. Smarandache F. & ChristiantoV.Progress in Physics, 2006, variation parameter affectsredshift pattern in equation (18). v. 4, 27–31. In solar system scale, time-varying radius may be observed 11. Smarandache F. & ChristiantoV.Progress in Physics, 2006, in the form of changing Astronomical Unit [4]. v. 4, 37–40. This proposition, however, deserves further theoretical 12. Smarandache F. & ChristiantoV.Progress in Physics, 2006, considerations. Further observation is also recommended in v. 2, 63–67. order to verifyand explore further this proposition. 13. Fischer U. arXiv: cond-mat/9907457; [13a] arXiv: cond-mat/ 0004339. 4 Concluding remarks 14. Bell M.B. arXiv: astro-ph/0111123; [14a] Bell M.B. arXiv: astro-ph/0305112; [14b] Bell M.B. arXiv: astro-ph/0305060. In the present paper it is arguedthatPioneer anomalous 15. Humphreys R. TJ Archive, v. 16, http://answersingenesis.org. blueshift may be caused by Pioneer spacecraft experiencing 16. Munera´ H. Apeiron, 1998, v. 5, No. 3–4. angular shift induced by similar Q-relativity effect which may also affect Jupiter satellites. By taking into considera- 17. Zurek W.(ed.) Proc. Euroconference in Formation and Inter- tion aether drift effect, the proposedmethod as described action of Topological Defects, Plenum, 1995; arXiv: cond- mat/9502119. herein could predictPioneer blueshift within ∼0.26% error range, which speaks for itself.Further observa tion is of course 18. Volovik G. arXiv: cond-mat/0507454. recommended in order torefute or verify this proposition.
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Notes on Pioneer Anomaly Explanation by Sattellite-Shift Formula of Quaternion Relativity: Remarks on “Less Mundane Explanation of Pioneer Anomaly from Q-Relativity”
Alexander Yefremov Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, Miklukho-Maklaya Str. 6, Moscow 117198, Russia E-mail: [email protected]
Use of satelliteshift formula emerging in Quaternion (Q-) model of relativity theory for explanation ofPioneer anomaly [1] is critically discussed. A cinematic scheme more suitable for the case is constructed with the help of Q-model methods. An appropriate formula for apparentdeceleration resulting from existence of observer- object relative velocityisderived. Preliminary quantitative assessments madeonthe base ofPioneer 10/11 data demonstrate closure ofthe assumed “relativistic decele- ration” and observed “Doppler deceleration” values.
1 Introduction. Limits of satellite-shift formula 0.26%)with value of emission angle shift requiredtoexplain observation dataofPioneer’s signal Doppler residuals [3]. Recently [1] there was an attempttogiveanexplanation This surprisingly exact result nevertheless should not ofPioneer anomaly essentially using formula for relativistic leadtounderstanding that obtained by Q-methodmathema- shift of planet’s fast satellites observedfromthe Earth. This tical description of a specific mechanical model can bear uni- formula was derived within framework of Q-methoddeve- versal character andfitto arbitrary relativistic situation. One lopedto calculate relativistic effects using SO(1, 2) form- needs to recognize thatPioneer cinematic scheme essentially invariantquaternion square rootfrom space-timeinterval differs fromthat ofthe Earth-planet-satellite model; but if rather than the interval itself [2]; in particular this advanta- one tries toexplain the Pioneer effect using the same rela- geously permits to describe relativistic motions of any non- tivistic idea as for satelliteshift then an adequate cinematic inertial frames. The last option was usedto findmentioned scheme should be elaborated. Happily the Q-method readily formula thatdescribes cinematic situation comprising three offers compact and clear algorithmfor construction andde- Solar System objects: the Earth (with observer on it), a scripotionf any relativistic models. In Section2amodel planet,and itssatellite revolving withcomparatively large referring observedfrequency shift ofPioneer spacecraft sig- angular velocity. Due toexistence of Earth-planet relative nals to purely relativistic reasons is regarded;some quantita- velocity, not greatthough and variable but permanent, the tive assessments are made as well as conclusions on ability cycle frequency of satelliterotation (observedfromthe ofthe model toexplain the anomaly. In Section3ashort Earth) is apparently less that in realty, i.e. the “planet’s discussion is offered. clock” is slowing down, and calculation shows thatthe gap is growing linearly with time. Visually it looks thatthe satellite 2 Earth-Pioneer Q-model and signal frequency shift positiononitsorbit is apparently behind an expected place. For very fast satellites (like Jupiter’s Metis and Adrastea) Paper [3] enumeratesanuomffber actors attractedto analyze andfor sufficiently long period oftime the effect can probab- radio data receivedfromPioneer 10/11 spacecraft,among ly be experimentally detected.Same effect existsof course them gravitational planetary perturbations, radiation pres- for Mars’s satellites and it is computedthatmonthly apparent sure, interplanetary media, General Relativity∗, the Earth’s shift on itsorbit of e.g. Phobos is about 50 meters (that is precession and nutation. It is worthnoting here that one sig- by the way can be important andtaken into account when nificantfactor, time delay caused by relative probe-observer planning expedition of spacecraft closely approaching the motion, is notdistinguished in [3]. The fact is understand- moon). able: relative motion of spacecraft and observer on the Earth In paper of F. Smarandache and V. Christianto[1]the dis- is utterly non-inertial one; Special Relativityisnot at all cussedformula was usedto describe famous Pioneer effect, able to cope with the case while General Relativity methods implying thatthe last great acceleration the space probe involving specific metric and geodesic lines construction received when approached very close to Jupiter; in particular ∗ data concerning Adrastea, whose location was as close to Unfortunately paper [3] does not indicate towhatdepth General Relativityistaken into account:whether only Newtonian gravityismodi- Jupiter as the space probe, were cited in [1]. Combined with fied by Schwarzschild, Kerr (or other) metrics, or cinematic effects are ether drift effectthe formula gives good coincidence (up to regardedtoo.
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(with all curvature tensor components zero) or additional period is apparently longer than it really is. Vice versa, ob- vector transport postulates are mathematically difficult. Con- servedfrequency f =1/T is smaller than the real one f trary to this the Q-relativity method easily allows building of 1 1 f any non-inertial relativistic scheme; an example describing f = = = = f 1 − (V/c)2, (4) T T cosh ψ cosh ψ a spacecraft (probe) and an Earth’s observer is given below. Assume thatPioneer anomaly is a purely relativistic ef- or for small relative velocity fect caused by existence of Earth-Pioneer relative velocity, 2 ∼ V f = f 1 − . variable but permanent. Construct respective model using the 2c2 Q-method algorithm. Choose Q-frames. Let Σ=(q1, q2, q3) be the Earth’s frame whose Cartesian directing vectors are This means thatthere exists certain purely apparent re- lativistic shift ofthe probe’s signal detected by the Earth given by quaternion “imaginary” units qk obeying the multi- plication rule∗ observer 2 2 V Δ f V ε 1 q = q 1=q , q q = −δ + ε q . (1) Δ f = f − f = f , or = = , (5) k k k k l kl klj j 2c2 f 2c2 c2 { } Let Q-frame Σ = qk belong to a probe. Suppose for ε being the probe’s kinetic energy per unitmass computed simplicity that vectors q2, q3 are in the ecliptic plane as in a chosen frame. Contrary to pure Doppler effectthe shift well as (approximately) the probe’s trajectory. Assume that given by Eq.(5)does notdepend on the direction of relative vector q2 of Σ is always parallel to Earth-probe relative velocityof involved objects since in fact it is just another velocity V . Now one is able to writerotational equation, manifestation of relativistic delay oftime. Light coming to main relation of Q-relativity, which ties two frames observer from any relatively (and arbitrary) moving bodyis −iψ universally “more red” than originally emitted signal; as well Σ = O1 Σ , (2) all other frequencies attributedto observedmoving bodies −iψ here O1 is 3×3orthogonal matrix of rotation about axis are smaller then original ones, and namely this idea was No.1 at imaginary angle −iψ exploredfor derivation of satelliteshift formula. ⎛ ⎞ ⎛ ⎞ Experimental observation ofthe frequency change (5) cos(iψ) − sin(iψ)0 cosh ψ −i sinh ψ 0 must leadto conclusion thatthere exists respective “Doppler − iψ ⎝ sin(− ) cos( )0⎠ ⎝ sinh cosh 0 ⎠ O1 = iψ iψ = i ψ ψ velocity” VD entering formula well known from Special Re- 00 1 00 1lativity f VD thus “converting” frame Σ into Σ.Thefirst row in the f = 1 − cos β , (6) 1 − (V /c)2 c matrix equation (2) D β being angle between velocityvector and wave vector of q1 = q1 cosh ψ − q2 i sinh ψ emitted signal. If β =0and smaller relativistic correction are after straightforward algebra neglectedthen Eq. (6) can be rewritten in the form similar dt toEq.(5) 1 1− 2 ⇒ 1 1− 2 Δ f ∼ VD q = cosh ψ (q q i tanh ψ) q = (q q iVψ) = ; (7) dt f c2 with usual relativistic relations comparison of Eqs. (7) and (5) yields very simple formula V = tanh ψ, dt = dt cosh ψ (3) for calculated (and allegedly existent) “Doppler velocity” corresponding to observed relativistic frequency change acquires the form of basic cinematic space-time object of Q-relativity ∼ ε VD = . (8) idt q1 = idtq1 + drq2 , c a specific quaternion square rootfrom space-timeinterval of Estimation ofthe value of VD can be done using picture Special Relativity ofPioneer 10/11 trajectories (Fig.1) projected upon ecliptic plane (provided in NASA report [4]); other spacecraft traces ⇒ (idt q1 )(idt q1 )=(idtq1 + drq2)(idtq1 + drq2) are also shown, the Earth’s orbit radius too small tobe ⇒ dt2 = dt2 − dr2, indicated.
Schematically the cinematic situation for Pioneer 10 is dt being proper time segment ofthe probe. Eq. (3) yields shown at Fig. 2 where the trajectory looks as a straight line ratio for probe-Earth signal period (small comparedto time 2 inclined at constant angle λ toaxis q , while the Earth’s of observation) T = T cosh ψ, i.e. observedfrom Earth the positiononitsorbit is determined by angle α =Ωt, Ω= −7 −1 ∗ × Latin indices are 3-dimensional (3D), δkl is 3D Kroneker symbol, = 3.98 10 s being the Earth’s orbital angular velocity. εjkl is 3D Levi-Civitasymbol; summation convention is assumed. Vectors ofthe probe’s and Earth’s velocities in Solar Ecliptic
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Fig. 1: Spacecraft trajectories on the ecliptic plane. (After NASA Fig. 2: Earth-Pioneer 10 cinematic scheme, where the trajectory original data[4]. Used by permission.) looks as a straight line inclined at constant angle λ toaxis q2.
∗ (SE) coordinatesystem are respectively denoted as VP Below only radial componentsofthe probe’s velocity and VE; their vector subtraction gives relative Earth-probe and acceleration in Newtonian gravity are taken into account velocity V = VP − VE so that in Eq.(11);it is quite a rough assessment but it allows to 2 2 2 − − conceive order of values. The probe’s acceleration caused V VP + VE 2VP VE cos(Ωt λ) VD(t)= = , (9) by the Sun’s Newtonian gravityis 2c 2c GM and respective “Doppler acceleration” is V˙ = − , (12) P R2 aD = V˙D(t)= × −11 3 × 2 × 30 G = 6.67 10 m /kg s , M = 1.99 10 kg are respec- V V˙ −V˙ V cos(Ωt−λ)+ΩV V sin(Ωt−λ) (10) = P P P E P E . tively gravitational constant andmass ofthe Sun. NASA c data [5] show that in the very middle part (1983–1990) ofthe In Eq. (10) the firstterm in the numerator claimsexist- whole observational period ofPioneer 10 itsradial distance ∼ 12 ence of secular deceleration, since escaping fromthe Sun’s fromthe Sun changes from R = 28.8 AU = 4.31×10 mto ∼ 12 and Jupiter’s gravity the probe is permanently decelerated, R = 48.1 AU = 7.2×10 m, while year-mean radial velocity × 3 × 3 V˙p < 0; the result is thatthe frequency gap shrinks giving varies from VP = 15.18 10 m/s to VP = 12.81 10 m/s. Re- rise to pure relativistic blue shift.Other sign-changing terms spective values ofthe secular “relativistic deceleration” va- in right-hand-sideof Eq. (10) are periodic (annual) ones; lues for this period computed with the help of Eqs. (11), (12) × −10 2 × −10 they may cause blue shift as well as red shift. Thus Eq. (10) vary from aSD =−3.63 10 m/s to aSD =−1.23 10 shows that,although relative probe-Earth velocity incorpo- m/s2.It is interesting (and surprising as well) thatthese re- rates into difference between real and observedfrequency, sults are very close in order toanomalous “Doppler decele- × −10 2 nevertheless secular change ofthe difference is to be related ration” ofthe probe aP =−(8±3) 10 m/s cited in [3]. only to relative probe-Sun velocity. Distinguish this term Analogous computations for Pioneer 11, as checking temporary ignoring the annual modulations; then the secular point, show the following. Full timeof observation ofPio- deceleration formula is reduced as neer 11 is shorter so observational period is taken from 1984 ˙ to 1989, with observational data fromthe same source [5]. ∼ VP VP aSD = . (11) Radial distances for beginning and end ofthe period are c ∼ 12 ∼ 12 R =15.1AU=2.26×10 m, R =25.2AU=3.77×10 m; re- ∗The SE is a heliocentric coordinatesystem with the z-axis normal to 3 spective year-mean radial velocities are V = 11.86×10 m/s, and northwardfromthe ecliptic plane. The x-axis extends towardthe first P × 3 point of Aries (Vernal Equinox, i.e. to the Sun from Earthinthe firstday VP = 12.80 10 m/s. Computed “relativistic deceleration” × −10 2 of Spring). The y-axis completes the right handed set. values for this period are then aSD =−10.03 10 m/s ,
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× −10 2 aSD =−5.02 10 m/s : this is even in much better cor- relation (within limitsofthe cited error) withexperimental value of aP .
3 Discussion
Quantitative estimations presented above allow to conclude: additional blue shift,experimentally registered in Pioneer 10 and 11 signals, and interpreted as Sun-directed acceleration ofthe spacecraft tosomeextent, supportthe assumption of pure relativistic nature ofthe anomaly. Of course one notes that while Pioneer 11 case shows good coincidence of observed and calculated values ofdeceleration, values of aSD for Pioneer 10 constitute only (45–15)% of observed Doppler residual; moreover generally in this approach “rela- tivistic deceleration” is a steadily decreasing function, while experimentally (though notdirectly) detecteddeceleration aP is claimed nearly constant. These defects couldfind ex- planation first of all in the factthat aprimitive “Newtonian radial model” was usedfor assessments. Preliminary but more attentive reference to NASA data allows noticing that observed angular acceleration ofthe probes too could signi- ficantly incorporate to values of “relativistic deceleration”. This problem remains to be regarded elsewhere together with analysis ofthe angular acceleration itself.
Acknowledgments
The author would like to thank Profs. F. Smarandache and V. Christianto for attention tohisquaternion approach aimed to calculate relativistic effectsof non-inertial motions. Spe- cial thanks toV.Christianto for given information on Pioneer anomaly and numerous helpful discussions.
Submitted on March 01, 2007 Accepted on March 05, 2007
References
1. Smarandache F. andChristiantoV.Progr. in Phys., 2007, v. 1, 42–45. 2. Yefremov A. Grav. & Cosmol., 1996, v. 2, No. 4, 335–341. 3. Anderson J. D. et al. arXiv: gr-qc/9808081. 4.http://cohoweb.gsfc.nasa.gov/helios/book1/b1_62.html 5. http://cohoweb.gsfc.nasa.gov/helios/book2/b2_03.html
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Reply to “Notes on Pioneer Anomaly Explanation by Satellite-Shift Formula of Quaternion Relativity”
Vic Christianto∗ and Florentin Smarandache† ∗Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] †Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected]
In the present article we would like to make a few comments on a recent paper by A. Yefremov in this journal [1]. It is interesting to note here that he concludes his analysis by pointing out that using full machinery of Quaternion Relativity it is possible to explain Pioneer XI anomaly with excellent agreement compared with observed data, and explain around 45% of Pioneer X anomalous acceleration. We argue that perhaps it will be necessary to consider extension of Lorentz transformation to Finsler-Berwald metric, as discussed by a number of authors in the past few years. In this regard, it would be interesting to see if the use of extended Lorentz transformation could also elucidate the long-lasting problem known as Ehrenfest paradox. Further observation is of course recommended in order to refute or verify this proposition.
1 Introduction is ambiguous term in physical sense, it does not take into con- sideration 3-angle problem in more general problem. We are delighted to read A. Yefremov’s comments on our This is why we pointed out in our preceding paper that preceding paper [3], based on his own analysis of Pioneer apparently General Relativity inherits the same drawback anomalous “apparent acceleration” [1]. His analysis made from Special Relativity [3]. use of a method called Quaternion Relativity, which essen- Another problem of special relativistic definition of Lo- tially is based on SO(1, 2) form invariant quaternion square rentz transformation is known as “reciprocity postulate”, root from space-time interval rather than the interval itself because in Special Relativity it is assumed that: x ↔ x, [1, 2]. Nonetheless it is interesting to note here that he con- t ↔ t, v ↔−v [6]. This is why Doppler shift can be derived cludes his analysis by pointing out that using full machinery without assuming reciprocity postulate (which may be re- of Quaternion Relativity it is possible to explain Pioneer XI garded as the “third postulate” of Special Relativity) and anomaly with excellent agreement compared with observed without special relativistic argument, see [7]. Nonetheless, in data, and explain around 45% of Pioneer X anomalous acce- our opinion, Yefremov’s Quaternion Relativity is free from leration [1]. this “reciprocity” drawback because in his method there is In this regard, we would like to emphasize that our pre- difference between moving-observer and static-observer [2]. ceding paper [3] was based on initial “conjecture” that in An example of implications of this drawback of 1-angle order to explain Pioneer anomaly, it would be necessary problem of Lorentz transformation is known as Ehrenfest to generalize pseudo-Riemann metric of General Relativity paradox, which can be summarized as follows: “According theory into broader context, which may include Yefremov’s to Special Relativity, a moving rod will exhibit apparent Quaternion Relativity for instance. It is interesting to note length-reduction. This is usually understood to be an obser- here, however, that Yefremov’s analytical method keeps use vational effect, but if it is instead considered to be a real standard Lorentz transformation in the form Doppler shift effect, then there is a paradox. According to Ehrenfest, effect (Eq. 6): the perimeter of a rotating disk is like a sequence of rods. f vD So does the rotating disk shatter at the rim?” Similarly, f = 1 − cos β . (1) 2 c after some thought Klauber concludes that “The second re- 1 − vD c lativity postulate does not appear to hold for rotating While his method using relativistic Doppler shift a la systems” [8]. Special Relativity is all right for such a preliminary analysis, While it is not yet clear whether Quaternion-Relativity in our opinion this method has a drawback that it uses is free from this Ehrenfest paradox, we would like to point “standard definition of Lorentz transformation” based on 2- out that an alternative metric which is known to be nearest dimensional problem of rod-on-rail as explained in numer- to Riemann metric is available in literature, and known ous expositions of relativity theory [5]. While this method of as Finsler-Berwald metric. This metric has been discussed rod-on-rail seems sufficient to elucidate why “simultaneity” adequately by Pavlov, Asanov, Vacaru and others [9–12].
24 V.Christianto and F. Smarandache. Reply to “Notes on Pioneer Anomaly Explanation by Satellite-Shift Formula”
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2 Extended Lorentz-transformation in Finsler-Berwald In order to provide an illustration on how the transforma- metric tion keeps the Finslerian metric invariant, we can use Maple algorithm presented by Asanov [10, p.29]: It is known that Finsler-Berwald metric is subset of Finsler- = = = ian metrics which is nearest to Riemannian metric [12], > c1: cos(tau);c2: cos(psi);c3: cos(phi); > u1:=sin(tau);u2:=sin(psi);u3:=sin(phi); therefore it is possible to construct pseudo-Riemann metric > l1:=c2*c3−c1*u2*u3;l2:=−c2*u3−c1*u2*c3;l3:=u1*u2; based on Berwald-Moor geometry, as already shown by Pav- > m1:=u2*c3+c1*c2*u3;m2:=−u2*u3+c1*c2*c3;m3:=−u1*c2; lov [4]. The neat link between Berwald-Moor metric and > n1:=u1*u3; u1*c3; c1; = + + + + + + + + + Quaternion Relativity of Yefremov may also be expected > F1: (e1)ˆ((l1 m1 n1 l2 m2 n2 l3 m3 n3 1)/4)* (e2)ˆ((−l1−m1−n1+l2+m2+n2−l3−m3−n3+1)/4)* because Berwald-Moor metric is also based on analytical (e3)ˆ((l1+m1+n1−l2−m2−n2−l3−m3−n3+1)/4)* functions of the H4 variable [4]. (e4)ˆ((−l1−m1−n1−l2−m2−n2+l3+m3+n3+1)/4): More interestingly, there was an attempt in recent years > F2:=(e1)ˆ((−l1+m1−n1−l2+m2−n2−l3+m3−n3+1)/4)* − + − + − + − + + to extend 2d-Lorentz transformation in more general frame- (e2)ˆ((l1 m1 n1 l2 m2 n2 l3 m3 n3 1)/4)* (e3)ˆ((−l1+m1−n1+l2−m2+n2+l3−m3+n3+1)/4)* work on H4 of Finsler-Berwald metric, which in limiting (e4)ˆ((l1−m1+n1+l2−m2+n2−l3+m3−n3+1)/4): cases will yield standard Lorentz transformation [9, 10]. In > F3:=(e1)ˆ((l1−m1−n1+l2−m2−n2+l3−m3−n3+1)/4)* this letter we will use extension of Lorentz transformation (e2)ˆ((−l1+m1+n1+l2−m2−n2−l3+m3+n3+1)/4)* − − − + + − + + + derived by Pavlov [9]. For the case when all components (e3)ˆ((l1 m1 n1 l2 m2 n2 l3 m3 n3 1)/4)* (e4)ˆ((−l1+m1+n1−l2+m2+n2+l3−m3−n3+1)/4): but one of the velocity of the new frame in the old frame > F4:=(e1)ˆ((−l1−m1+n1−l2−m2+n2−l3−m3+n3+1)/4)* coordinates along the three special directions are equal to (e2)ˆ((l1+m1−n1−l2−m2+n2+l3+m3−n3+1)/4)* zero, then the transition to the frame moving with velocity (e3)ˆ((−l1−m1+n1+l2+m2−n2+l3+m3−n3+1)/4)* + − + + − − − + + V1 in the old coordinates can be expressed by the new frame (e4)ˆ((l1 m1 n1 l2 m2 n2 l3 m3 n3 1)/4): > a:=array(1..4,1..4): as [9, p.13]: for i from 1 to 4 ⎛ ⎞ ⎛ ⎞ x0 x0 do ⎜ ⎟ ⎜ ⎟ for j from 1 to 4 ⎜ x1 ⎟ F 0 ⎜ x1 ⎟ ⎝ ⎠ = = ⎝ ⎠ (2) do x2 0 F x2 a[i,j]:=diff(F||i,e||j); x3 x3 end do: end do: where the transformation matrix for Finsler-Berwald metric > b:=array(1..4,1..4): is written as follows [9, p.13]: for i from 1 to 4 ⎛ ⎞ do 1 V1 for j from 1 to 4 1− 2 1− 2 ⎜ V1 V1 ⎟ do F = ⎝ ⎠ (3) = || || = V1 1 b[i,j]: simplify(add(1/F k*diff(a[k,i],e j),k 1..4),symbolic); 2 2 end do: 1−V1 1−V1 end do: and > print(b); 00 0 = . (4) 00 The result is as follows: ⎡ ⎤ Or 0000 ⎢ ⎥ x 0 + Vx1 Vx 0 + x1 ⎢ 0000⎥ x0 = 2 x1 = 2 , (5) ⎣ ⎦ . 1 − V1 1 − V1 0000 0000 and x2 + Vx3 Vx2 + x3 x2 = x3 = . (6) This result showing that all the entries of the matrix are − 2 − 2 1 V1 1 V1 zeroes support the argument that the metricity condition is It shall be clear that equation (5) (x0,x1) ↔ (x0,x1) true [10]. coincides with the corresponding transformation of Special Relativity, while the transformation in equation (6) differs 3 Concluding remarks from the corresponding transformation of Special Relativity where x2 = x2, x3 = x3 [9]. In the present paper we noted that it is possible to gene- While we are not yet sure whether the above extension of ralise standard Lorentz transformation into H4 framework of Lorentz transformation could explain Pioneer anomaly better Finsler-Berwald metric. It could be expected that this ex- than recent analysis by A. Yefremov [1], at least it can be tended Lorentz transformation could shed some light not expected to see whether Finsler-Berwald metric could shed only to Pioneer anomaly, but perhaps also to the long-lasting some light on the problem of Ehrenfest paradox. This propo- problem of Ehrenfest paradox which is also problematic in sition, however, deserves further theoretical considerations. General Relativity theory, or by quoting Einstein himself:
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“...Thusallourpreviousconclusions based on gen- 10. Asanov G. S. Finslerian metric function of totally anisotropic eral relativity would appear to be called in question. type. Relativistic aspects. arXiv: math-ph/0510007; [10a] In reality we must make a subtle detour in order to Asanov G. S. arXiv: math-ph/0510049. be able to apply the postulate of general relativity 11. Vacaru S., Stavrinos P., Gaburov E. and Gonta D. Clifford exactly” [5]. and Riemann-Finsler structures in geometric mechanics and This reply is not intended to say that Yefremov’s preli- gravity. Geometry Balkan Press, 2005; arXiv: gr-qc/0508023, minary analysis is not in the right direction, instead we only p. 39. highlight a possible way to improve his results (via extend- 12. Szabo Z. Berwald metrics constructed by Chevalley’s poly- ing Lorentz transformation). Furthermore, it also does not nomials. arXiv: math.DG/0601522. mean to say that Finsler-Berwald metric could predict better 13. Weber T. Measurements on a rotating frame in relativity. Am. than Quaternion Relativity. Nonetheless, further observation J. Phys., 1997, v. 65, 946–953. is of course recommended in order to refute or verify this proposition.
Acknowledgment
The writers would like to thank to Profs. C. Castro for citing the Vacaru et al.’s book [11]. We are also grateful for dis- cussion with Profs. A. Yefremov, G. Asanov and Z. Szabo.
Submitted on April 06, 2007 Accepted on April 12, 2007
References
1. Yefremov A. Notes on Pioneer anomaly explanation by satellite-shift formula of Quaternion Relativity. Progress in Physics, 2007, v. 3, 93–96. 2. Yefremov A. Quaternions: algebra, geometry and physical theories. Hypercomplex Numbers in Geometry and Physics, 2004, v. 1(1), 105. 3. Smarandache F. and Christianto V. Less mundane explanation of Pioneer anomaly from Q-relativity. Progress in Physics, 2007, v. 1, 42–45. 4. Pavlov D. G. Construction of the pseudo Riemannian geometry on the base of the Berwald-Moor geometry. arXiv: math-ph/ 0609009. 5. Einstein A. Relativity: the special and general theory. Crown Trade Paperback, New York, 1951, 66–70 (quoted from the ebook version, p. 97: http://www.ibiblio.org/ebooks/Einstein/ Einstein_Relativity.pdf); [5a] Janssen M. http://www.tc.umn. edu/∼janss011/pdf %20files/appendix-SR.pdf; [5b] http:// www.ph.utexas.edu/∼gleeson/NotesChapter11.pdf. 6. De Lange O. L. Comment on Space-time exchange invariance: Special relativity as a symmetry principle. Am.J.Phys., 2002, v. 70, 78–79. 7. Engelhardt W. Relativistic Doppler effect and the principle of relativity. Apeiron, 2003, v. 10, No. 4, 29–49. 8. Klauber R. Toward a consistent theory of relativistic rotation. In: Relativity in Rotating Frames, Kluwer Academic, 2004; arXiv: physics/0404027. 9. Pavlov D. et al. he notions of distance and velocity modulus in the linear Finsler spaces. Hypercomplex Numbers in Geometry and Physics, 2005, v. 1(3), 13.
26 V.Christianto and F. Smarandache. Reply to “Notes on Pioneer Anomaly Explanation by Satellite-Shift Formula”
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Yang-Mills Field from Quaternion Space Geometry, and Its Klein-Gordon Representation
Alexander Yefremov∗, Florentin Smarandache† and Vic Christianto‡
∗Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, Miklukho-Maklaya Str. 6, Moscow 117198, Russia E-mail: [email protected] †Chair of the Dept. of Mathematics and Science, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] ‡Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected]
Analysis of covariant derivatives of vectors in quaternion (Q-) spaces performed using Q-unit spinor-splitting technique and use of SL(2C)-invariance of quaternion multiplication reveals close connexion of Q-geometry objects and Yang-Mills (YM) field principle characteristics. In particular, it is shown that Q-connexion (with quaternion non-metricity) and related curvature of 4 dimensional (4D) space-times with 3D Q-space sections are formally equivalent to respectively YM-field potential and strength, traditionally emerging from the minimal action assumption. Plausible links between YM field equation and Klein-Gordon equation, in particular via its known isomorphism with Duffin-Kemmer equation, are also discussed.
1 Introduction In paper [5] a unified theory of gravity and electroweak forces is built with Lagrangian as a scalar curvature of space- Traditionally YM field is treated as a gauge, “auxiliary”, field time with torsion; if trace and axial part of the torsion vanish involved to compensate local transformations of a ‘main’ the Lagrangian is shown to separate into Gilbert and YM (e.g. spinor) field to keep invariance of respective action parts. Regardless of somehow artificial character of used functional. Anyway there are a number of works where YM- models, these observations nonetheless hint that there may field features are found related to some geometric properties exist a deep link between supposedly really physical object, of space-times of different types, mainly in connexion with YM field and pure math constructions. A surprising analogy contemporary gravity theories. between main characteristics of YM field and mathematical Thus in paper [1] violation of SO(3, 1)-covariance in objects is found hidden within geometry induced by quater- gauge gravitation theory caused by distinguishing time di- nion (Q-) numbers. rection from normal space-like hyper-surfaces is regarded as In this regard, the role played by Yang-Mills field cannot spontaneous symmetry violation analogous to introduction of be overemphasized, in particular from the viewpoint of the mass in YM theory. Paper [2] shows a generic approach to Standard Model of elementary particles. While there are a formulation of a physical field evolution based on description number of attempts for describing the Standard Model of of differential manifold and its mapping onto “model” spaces hadrons and leptons from the viewpoint of classical electro- defined by characteristic groups; the group choice leads to magnetic Maxwell equations [6, 7], nonetheless this question gravity or YM theory equations. Furthermore it can be shown remains an open problem. An alternative route toward [2b] that it is possible to describe altogether gravitation in achieving this goal is by using quaternion number, as describ- a space with torsion, and electroweak interactions on 4D ed in the present paper. In fact, in Ref. [7] a somewhat similar real spacetime C2, so we have in usual spacetime with torsion approach with ours has been described, i.e. the generalized a unified theory (modulo the non treatment of the strong Cauchy-Riemann equations contain 2-spinor and C-gauge forces). structures, and their integrability conditions take the form of Somewhat different approach is suggested in paper [3] Maxwell and Yang-Mills equations. where gauge potentials and tensions are related respectively It is long ago noticed that Q-math (algebra, calculus and to connexion and curvature of principle bundle, whose base related geometry) naturally comprise many features attribut- and gauge group choice allows arriving either to YM or to ed to physical systems and laws. It is known that quaternions gravitation theory. Paper [4] dealing with gravity in Riemann- describe three “imaginary” Q-units as unit vectors directing Cartan space and Lagrangian quadratic in connexion and cur- axes of a Cartesian system of coordinates (it was initially de- vature shows possibility to interpret connexion as a mediator veloped to represent subsequent telescope motions in astro- of YM interaction. nomical observation). Maxwell used the fact to write his
42 A. Yefremov, F. Smarandache and V.Christianto. Yang-Mills Field from Quaternion Space Geometry
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(A) (A) B (A) equations in the most convenient Q-form. Decades later are tied as dX = gB dy , with Lame´ coefficients gB , Fueter discovered a formidable coincidence: a pure math functions of yA,sothatX(A) are generally non-holonomic. Cauchy-Riemann type condition endowing functions of Q- Irrespectively of properties of UN each its point may be variable with analytical properties turned out to be identical attached to the origin of a frame, in particular presented by in shape to vacuum equations of electrodynamics [9]. “imaginary” Q-units qk, this attachment accompanied by a Later on other surprising Q-math — physics coincidences rule tying values of coordinates of this point with the triad A were found. Among them: “automatic” appearance of Pauli orientation M ↔{y , Φξ}. All triads {qk} so defined on magnetic field-spin term with Bohr magneton as a coefficient UN form a sort of “tangent” manifold T (U, q), (really tangent when Hamiltonian for charged quantum mechanical particle only for the base U3). Due to presence of frame vectors was built with the help of Q-based metric [10]; possibility to qk(y) existence of metric and at least proper (quaternionic) endow “imaginary” vector Q-units with properties of not only connexion ωjkn=−ωjnk, ∂j qk= ωjknqn, is implied, hence stationary but movable triad of Cartan type and use it for a one can tell of T (U, q) as of a Q-tangent space on the base very simple description of Newtonian mechanics in rotating UN . Coordinates xk defined along triad vectors qk in T (U, q) frame of reference [11]; discovery of inherited in Q-math are tied with non-holonomic coordinates X(A) in proper (A) variant of relativity theory permitting to describe motion of tangent space TN by the transformation dxk ≡ hk(A)dX non-inertial frames [12]. Preliminary study shows that YM with hk(A) being locally depending matrices (and generally field components are also formally present in Q-math. not square) of relative e(A) ↔ qk rotation. Consider a special In Section 2 notion of Q-space is given in necessary de- case of unification U ⊕ T (U, q) with 3-dimensional base tail. Section 3 discussed neat analogy between Q-geometric space U = U3. Moreover, let quaternion specificity of T3 objects and YM field potential and strength. In Section 4 reflects property of the base itself, i.e. metric structure of U3 YM field and Klein-Gordon correspondence is discussed. inevitably requires involvement of Q-triads to initiate Car- Concluding remarks can be found in Section 5. tesian coordinates in its tangent space. Such 3-dimensional Part of our motivation for writing this paper was to space generating sets of tangent quaternionic frames in each explicate the hidden electromagnetic field origin of YM its point is named here “quaternion space” (or simply Q- fields. It is known that the Standard Model of elementary space). Main distinguishing feature of a Q-space is non- ∗ particles lack systematic description for the mechanism of symmetric form of its metric tensor gkn ≡ qkqn =− δkn + quark charges. (Let alone the question of whether quarks do + εknjqj being in fact multiplication rule of “imaginary” exist or they are mere algebraic tools, as Heisenberg once Q-units. It is easy to understand that all tangent spaces puts forth: If quarks exist, then we have redefined the word constructed on arbitrary bases as designed above are Q- “exist”.) On the other side, as described above, Maxwell spaces themselves. In most general case a Q-space can be described his theory in quaternionic language, therefore it treated as a space of affine connexion Ωjkn =Γjkn +Qjkn + seems natural to ask whether it is possible to find neat link + Sjkn + ωjnk + σjkn comprising respectively Riemann between quaternion language and YM-fields, and by doing connexion Γjkn, Cartan contorsion Qjkn, segmentary cur- so provide one step toward describing mechanism behind vature (or ordinary non-metricity) Sjkn, Q-connexion ωjnk, quark charges. and Q-non-metricity σjkn; curvature tensor is given by stand- Further experimental observation is of course recom- ard expression Rknij = ∂iΩjkn − ∂j Ωikn +ΩikmΩjmn− mended in order to verify or refute our propositions as − ΩjnmΩimk. Presence or vanishing of different parts of described herein. connexion or curvature results in multiple variants of Q- spaces classification [13]. Further on only Q-spaces with 2 Quaternion spaces pure quaternionic characteristics (Q-connexion and Q-non- metricity) will be considered. Detailed description of Q-space is given in [13]; shortly but with necessary strictness its notion can be presented as 3 Yang-Mills field from Q-space geometry following. Let UN be a manifold, a geometric object consisting of Usually Yang-Mills field ABμ is introduced as a gauge field points M ∈ UN each reciprocally and uniquely correspond- in procedure of localized transformations of certain field, e.g. ingtoasetofN numbers-coordinates {yA} : M ↔{yA}, spinor field [14, 15] β (A =1, 2 ...N). Also let the sets of coordinates be trans- ψa → U(y ) ψa . (1) formed so that the map becomes a homeomorphism of a If in the Lagrangian of the field partial derivative of ψ class C . It is known that U may be endowed with a a k N is changed to “covariant” one proper tangent manifold TN described by sets of orthogonal → ≡ − unite vectors e(A) generating in TN families of coordinate ∂β Dβ ∂β gAβ , (2) ( ) →{ A } ∗ lines M X , indices in brackets being numbers of Latin indices are 3D, Greek indices are 4D; δkn,εknj are Kronecker frames’ vectors. Differentials of coordinates in UN and TN and Levi-Civita symbols; summation convention is valid.
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Aβ ≡ iACβTC , (3) This equation is in fact equivalent to definition of the proper connexion ω . If a transformation of Q-units is where g is a real constant (parameter of the model), T are αmk C given by spinor group (leaving quaternion multiplication rule traceless matrices (Lie-group generators) commuting as invariant) −1 qk = U(y) q˜ U (y) (13) [TB, TC ]= ifBCD TD (4) k
(qk˜ are constants here) then Eq. (12) yields with structure constants fBCD , then −1 −1 −1 ∂α U qk˜ U + U qk˜ ∂α U = ωαkn U qn˜ U . (14) DβU ≡ (∂β − gAβ) U =0, (5)
But one can easily verify that each “imaginary” Q-unit qk˜ and the Lagrangian keeps invariant under the transformations can be always represented in the form of tensor product of its (1). The theory becomes “self consistent” if the gauge field eigen-functions (EF) ψ(k˜), ϕ(k˜) (no summation convention terms are added to Lagrangian for indices in brackets): αβ LYM ∼ F Fαβ , (6) ± ± qk˜ ψ(k˜) = iψ(k˜),ϕ(k˜)qk˜ = iϕ(k˜) (15) ≡ Fαβ FCαβTC . (7) having spinor structure (here only EF with positive parity μν (with sign +) are shown) The gauge field intensity FB expressed through poten- tials ABμ and structure constants as − qk˜ = i(2ψ(k˜)ϕ(k˜) 1); (16) − FCαβ = ∂α ACβ ∂β ACα + fCDE ADαAEβ. (8) this means that left-hand-side (lhs) of Eq. (14) can be equiv- alently rewritten in the form Vacuum equations of the gauge field 1 −1 −1 αβ αβ ∂ F + A ,F =0 (9) (∂αU qk˜ U + U qk˜ ∂αU )= α α 2 (17) −1 −1 are result of variation procedure of action built from Lagran- =(∂αUψ(k˜)) ϕ(k˜)U + Uψ(k˜) (ϕ(k˜)∂αU ) gian (6). Group Lie, e.g. SU(2) generators in particular can be which strongly resembles use of Eq. (1) for transformations represented by “imaginary” quaternion units given by e.g. of spinor functions. traceless 2×2-matrices in special representation (Pauli-type) Here we for the first time underline a remarkable fact: form-invariance of multiplication rule of Q-units under their iT → q˜ = −iσ (σ are Pauli matrices), B k k k spinor transformations gives expressions similar to those Then the structure constants are Levi-Civita tensor com- conventionally used to initiate introduction of gauge fields ponents f → ε , and expressions for potential and BCD knm of Yang-Mills type. intensity (strength) of the gauge field are written as: Now in order to determine mathematical analogues of 1 these “physical fields”, we will analyze in more details Eq. Aβ = g Akβ˜ qk˜ , (10) 2 (14). Its multiplication (from the right) by combination U qk˜ with contraction by index k˜ leads to the expression Fkαβ = ∂α Akβ − ∂β Akα + εkmn AmαAnβ . (11) − −1 It is worthnoting that this conventional method of intro- 3 ∂αU + U qk˜ ∂αU Uqk˜ = ωαkn U qn˜ qk˜ . (18) duction of a Yang-Mills field type essentially exploits heu- This matrix equation can be simplified with the help of ristic base of theoretical physics, first of all the postulate the always possible development of transformation matrices of minimal action and formalism of Lagrangian functions ≡ −1 − construction. But since description of the field optionally U a + bk qk˜ ,U= a bk qk˜ , (19) uses quaternion units one can assume that some of the above −1 2 relations are appropriate for Q-spaces theory and may have UU = a + bk bk =1, (20) geometric analogues. To verify this assumption we will use where a, bk are real scalar and 3D-vector functions, qk˜ are Q- an example of 4D space-time model with 3D spatial quater- units in special (Pauli-type) representation. Using Eqs. (19), nion section. the second term in lhs of Eq. (18) after some algebra is Begin with the problem of 4D space-time with 3D spatial reduced to remarkably simple expression section in the form of Q-space containing only one geometric −1 object: proper quaternion connexion. Q-covariant derivative U qk˜ ∂αU Uqk˜ = of the basic (frame) vectors qm identically vanish in this =(a + bnqn˜ ) q˜ (∂αa − ∂αbmqm˜ )(a + bl q˜) q˜ = (21) space: k l k ˜ ≡ Dα qk (δmk ∂α + ωαmk) qm =0. (12) = ∂α(a + bnqn˜ )=−∂αU
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so that altogether lhs of Eq. (18) comprises −4 ∂αU while is added to rhs of (31) then all quadratic terms in the right right-hand-side (rhs) is hand side can be given in the form
− AmαAnβ − AmβAnα − δmnAjαAjβ + δmnAjβAjα = ωαkn U qn˜ qk˜ = εknm ωαkn U qm˜ ; (22) =(δ δ − δ δ )(A A − A A )= then Eq. (18) yields mp qn mn qp pα qβ pβ qα = ε ε (A A − A A )= 1 kmq kpn pα qβ pβ qα ∂αU − εknm ωαkn U qm˜ =0. (23) − 4 = ωαknωβkm + ωβknAαkm. If now one makes the following notations Substitution of the last expression into Eq. (31) accom- 1 panied with new notation Akα≡ εknm ωαkn , (24) 2 Rmn αβ ≡ εkmnFkαβ (33) 1 leads to well-known formula: A ≡ A q˜ , (25) α 2 n n Rmn αβ = ∂αωβmn − ∂βω αmn+ then notation (25) exactly coincides with the definition (10) (34) + ω ω − ω ω . (provided g =1), and Eq. (23) turns out equivalent to Eq. (5) αnk βkm βnk αkm
← ← This is nothing else but curvature tensor of Q-space built UDα ≡ U(∂ α − Aα)=0. (26) out of proper Q-connexion components (in their turn being functions of 4D coordinates). By other words, Yang-Mills Expression for “covariant derivative” of inverse matrix field strength is mathematically (geometrically) identical to follows from the identity: quaternion space curvature tensor. But in the considered −1 −1 ∂ UU = −U∂ U . (27) case of Q-space comprising only proper Q-connexion, all α α components of the curvature tensor are identically zero. So Using Eq. (23) one easily computes Yang-Mills field in this case has potential but no intensity. The picture absolutely changes for the case of quaternion −1 1 −1 −∂ U − ε ω q ˜ U =0 (28) space with Q-connexion containing a proper part ωβkn and α 4 knm αkn m also Q-non-metricity σβkn or −1 −1 α DαU ≡ (∂α + Aα) U =0. (29) Ωβkn(y )=ωβkn + σβkn (35) Direction of action of the derivative operator is not essen- so that Q-covariant derivative of a unite Q-vector with conne- tial here, since the substitution U −1 → U и U → U −1 is xion (35) does not vanish, its result is namely the Q-non- always possible, and then Eq. (29) exactly coincides with metricity
Eq. (5). Dˆ αqk ≡ (δmk∂α +Ωαmk) qm = σαmk qk . (36) Now let us summarize first results. We have a remarkable fact: form-invariance of Q-multiplication has as a corollary For this case “covariant derivatives” of transformation “covariant constancy” of matrices of spinor transformations spinor matrices may be defined analogously to previous case of vector Q-units; moreover one notes that proper Q-conne- definitions (26) and (29) xion (contracted in skew indices by Levi-Civita tensor) plays ←ˆ ← ≡ ˆ − ˆ ˆ −1 ≡ ˆ the role of “gauge potential” of some Yang-Mills-type field. UDα U(∂ α Aα), DαU (∂α + Aα)U. (37) By the way the Q-connexion is easily expressed from Eq. (24) But here the “gauge field” is built from Q-connexion (35) ω = ε A . (30) 1 1 αkn mkn mα Aˆ ≡ ε Ω , Aˆ ≡ Aˆ q˜ . (38) kα 2 knm αkn α 2 n n Using Eq. (25) one finds expression for the gauge field It is not difficult to verify whether the definitions (37) are intensity (11) (contracted by Levi-Civita tensor for conve- consistent with non-metricity condition (36). Action of the nience) through Q-connexion “covariant derivatives” (37) onto a spinor-transformed unite
εkmnFkαβ = Q-vector ˆ → ˆ −1 ˆ −1 = εkmn(∂αAkβ − ∂βAkα)+εkmnεmlj AlαAjβ = (31) Dα qk (DαU)qk˜ ∂αU + U qk˜ (DαU )= = ∂ ω − ∂ ω + A A − A A . ← α βmn β αmn mα nβ mβ nα − 1 −1 = UDα εjnmΩαnmUq˜j qk˜ U + If identically vanishing sum 4 −1 1 −1 − + U qk˜ DαU + εjnmΩαnmq˜j U δmnAjαAjβ + δmnAjβAjα =0 (32) 4
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together with Eqs. (26) and (29) demand: Furthermore, sometime ago it has been shown that four- dimensional coordinates may be combined into a quaternion, ← −1 UDα = DαU =0 (39) and this could be useful in describing supersymmetric exten- sion of Yang-Mills field [18]. This plausible neat link be- leads to the expected results tween Klein-Gordon equation, Duffin-Kemmer equation and Yang-Mills field via quaternion number may be found useful, 1 −1 Dˆ α qk → εjnmσαnmUεjkl q˜U = because both Duffin-Kemmer equation and Yang-Mills field 2 l −1 play some kind of significant role in description of standard = σαklU q˜U = σαklql l model of particles [16]. i.e. “gauge covariant” derivative of any Q-unit results in Q- In this regards, it has been argued recently that one non-metricity in full accordance with Eq. (36). can derive standard model using Klein-Gordon equation, in Now find curvature tensor components in this Q-space; it particular using Yukawa method, without having to introduce is more convenient to calculate them using differential forms. a Higgs mass [19, 20]. Considering a notorious fact that Given Q-connexion 1-form Higgs particle has not been observed despite more than three
β decades of extensive experiments, it seems to suggest that Ω kn =Ωβkndy (40) an alternative route to standard model of particles using from the second equation of structure (quaternion) Klein-Gordon deserves further consideration. In this section we will discuss a number of approaches 1 α β Rˆknαβ dy ∧ dy = dΩkn +Ωkm ∧ Ωmn (41) by different authors to describe the (quaternion) extension 2 of Klein-Gordon equation and its implications. First we will one gets the curvature tensor component review quaternion quantum mechanics of Adler. And then we discuss how Klein-Gordon equation leads to hypothetical Rˆknαβ = ∂αΩβkn − ∂βΩαkn+ (42) imaginary mass. Thereafter we discuss an alternative route +ΩαkmΩβmn − ΩαnmΩβmk for quaternionic modification of Klein-Gordon equation, and implications to meson physics. quite analogously to Eq. (34). Skew-symmetry in 3D indices allows representing the curvature part of 3D Q-section as 3D 4.1 Quaternion Quantum Mechanics axial vector 1 Fˆ ≡ ε Rˆ (43) Adler’s method of quaternionizing Quantum Mechanics grew mαβ 2 knm knαβ out of his interest in the Harari-Shupe’s rishon model for and using Eq. (38) one readily rewrites definition (43) in composite quarks and leptons [21]. In a preceding paper [22] the form he describes that in quaternionic quantum mechanics (QQM), the Dirac transition amplitudes are quaternion valued, i.e. Fˆ = ∂ Aˆ − ∂ Aˆ + ε Aˆ Aˆ (44) mαβ α mβ β mα knm kα nβ they have the form which exactly coincides with conventional definition (11). q = r0 + r1i + r2 j + r3 k (46) QED. where r0, r1, r2, r3 are real numbers, and i, j, k are quaternion imaginary units obeying 4 Klein-Gordon representation of Yang-Mills field i2 = j2 = k2 = −1,ij= −ji = k, (47) In the meantime, it is perhaps more interesting to note here − − that such a neat linkage between Yang-Mills field and quater- jk = kj = i, ki = ik = j. nion numbers is already known, in particular using Klein- Using this QQM method, he described composite fermion Gordon representation [16]. In turn, this neat correspondence states identified with the quaternion real components [23]. between Yang-Mills field and Klein-Gordon representation can be expected, because both can be described in terms of 4.2 Hypothetical imaginary mass problem in Klein- SU(2) theory [17]. In this regards, quaternion decomposition Gordon equation of SU(2) Yang-Mills field has been discussed in [17], albeit It is argued that dynamical origin of Higgs mass implies it implies a different metric from what is described herein: that the mass of W must always be pure imaginary [19, 2 2 2 2 2 2 2 ds = dα1 + sin α1 dβ1 + dα2 + sin α2 dβ2 . (45) 20]. Therefore one may conclude that a real description for (composite) quarks and leptons shall avoid this problem, i.e. However, the O(3) non-linear sigma model appearing in by not including the problematic Higgs mass. the decomposition [17] looks quite similar (or related) to the Nonetheless, in this section we can reveal that perhaps Quaternion relativity theory (as described in the Introduction, the problem of imaginary mass in Klein-Gordon equation is there could be neat link between Q-relativity and SO(3, 1)). not completely avoidable. First we will describe an elemen-
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tary derivation of Klein-Gordon from electromagnetic wave lem comes from “mixed” interpretation of what we mean equation, and then by using Bakhoum’s assertion of total with “velocity”. In original Einstein’s paper (1905) it is energy we derive alternative expression of Klein-Gordon defined as “kinetic velocity”, which can be measured when implying the imaginary mass. standard “steel rod” has velocity approximates the speed of We can start with 1D-classical wave equation as derived light. But in quantum mechanics, we are accustomed to make from Maxwell equations [24, p.4]: use it deliberately to express “photon speed” = c. Therefore, 2 2 in special relativity 1905 paper, it should be better to interpret ∂ E − 1 ∂ E c 2 2 2 =0. (48) it as “speed of free electron”, which approximates .For ∂x c ∂t hydrogen atom with 1 electron, the electron occupies the This equation has plane wave solutions: first excitation (quantum number n =1), which implies that their speed also approximate c, which then it is quite safe i(kx−ωt) 2 E(x, t)=E0e (49) to assume E ∼ mc . But for atoms with large number of electrons occupying large quantum numbers, as Bakhoum which yields the relativistic total energy: showed that electron speed could be far less than c, therefore 2 2 2 2 2 4 it will be more exact to use E = mv , where here v should ε = p c + m c . (50) be defined as “average electron speed” [25]. Therefore we can rewrite (48) for non-zero mass particles In the first approximation of relativistic wave equation, as follows [24]: we could derive Klein-Gordon-type relativistic equation from equation (54), as follows. By introducing a new parameter: 2 2 2 2 ∂ 1 ∂ m c i ( − ) − − px Et v 2 2 2 2 Ψe =0. (51) ζ = i , (55) ∂x c ∂t c Rearranging this equation (51) we get the Klein-Gordon then we can use equation (55) in the known procedure to equation for a free particle in 3-dimensional condition: derive Klein-Gordon equation: m2c2 1 ∂2Ψ 2 2 2 2 2 4 ∇− Ψ= . (52) E = p c + ζ m0c , (56) 2 c2 ∂t2 where E = mv2. By using known substitution: It seems worthnoting here that it is more proper to use ∂ total energy definition according to Noether’s theorem in lieu E = i ,p= ∇ , (57) of standard definition of relativistic total energy. According ∂t i to Noether’s theorem [25], the total energy of the system 2 corresponding to the time translation invariance is given by: and dividing by ( c) , we get Klein-Gordon-type relativistic equation [25]: ∞ 2 cw 2 2 2 E = mc + γ 4πr dr = kμc (53) −2 ∂Ψ 2 2 −c + ∇ Ψ=k0 Ψ , (58) 2 0 ∂t where k is dimensionless function. It could be shown, that where ζm0c for low-energy state the total energy could be far less than k0 = . (59) E = mc2. Interestingly Bakhoum [25] has also argued in favor of using E = mv2 for expression of total energy, Therefore we can conclude that imaginary mass term which expression could be traced back to Leibniz. Therefore appears in the definition of coefficient k0 of this new Klein- it seems possible to argue that expression E = mv2 is more Gordon equation. generalized than the standard expression of special relativity, in particular because the total energy now depends on actual 4.3 Modified Klein-Gordon equation and meson obser- velocity [25]. vation From this new expression, it is possible to rederive Klein- As described before, quaternionic Klein-Gordon equation has Gordon equation. We start with Bakhoum’s assertion that it neat link with Yang-Mills field. Therefore it seems worth to is more appropriate to use E = mv2, instead of more con- discuss here how to quaternionize Klein-Gordon equation. venient form E = mc2. This assertion would imply [25]: It can be shown that the resulting modified Klein-Gordon 2 2 2 2 2 2 equation also exhibits imaginary mass term. H = p c − m0c v . (54) Equation (52) is normally rewritten in simpler form (by A bit remark concerning Bakhoum’s expression, it does asserting c =1): 2 2 not mean to imply or to interpret E = mv2as an assertion ∂ m ∇− Ψ= . (60) that it implies zero energy for a rest mass. Actually the prob- ∂t2 2
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Interestingly, one can write the Nabla-operator above in • For equation (66) we get: quaternionic form, as follows: ( mˆ2−D[#,t,t])&[y[x,t]]== A. Define quaternion-Nabla-operator as analog to quaternion 2 (0,2) number definition above (46), as follows [25]: m + y [x, t]=0 DSolve[%,y[x,t],{x,t}] q ∂ ∂ ∂ ∂ ∇ = −i + e1 + e2 + e3 , (61) ∂t ∂x ∂y ∂z m2t2 y[x, t] → + C[1][x]+tC[2][x] where e1, e2, e3 are quaternion imaginary units. Note that 2 equation (61) has included partial time-differentiation. One may note that this numerical solution is in quadratic 2 2 m t B. Its quaternion conjugate is defined as follows: form 2 + constant, therefore it is rather different from equation (67) in [26]. q ∂ ∂ ∂ ∂ ∇¯ = −i − e1 − e2 − e3 . (62) ∂t ∂x ∂y ∂z In the context of possible supersymetrization of Klein- Gordon equation (and also PT-symmetric extension of Klein- C. Quaternion multiplication rule yields: Gordon equation [27, 29]), one can make use biquaternion ∂2 ∂2 ∂2 ∂2 number instead of quaternion number in order to generalize ∇q ∇¯ q = − + + + . (63) ∂t2 ∂2x ∂2y ∂2z further the differential operator in equation (61): E. Define a new “diamond operator” to extend quaternion- D. Then equation (63) permits us to rewrite equation (60) in Nabla-operator to its biquaternion counterpart, according to quaternionic form as follows: the study [25]: m2 ∇q∇¯ qΨ= . (64) 2 q q ∂ ∂ ∂ ∂ ♦ = ∇ +i ∇ = −i +e1 +e2 +e3 + ∂t ∂x ∂y ∂z Alternatively, one used to assign standard value c =1and (68) also =1, therefore equation (60) may be written as: − ∂ ∂ ∂ ∂ + i i +e1 +e2 +e3 , 2 ∂T ∂X ∂Y ∂Z ∂ 2 2 −∇ + m ϕ(x, t)=0, (65) ∂t2 where e1, e2, e3 are quaternion imaginary units. Its conjugate can be defined in the same way as before. where the first two terms are often written in the form of To generalize Klein-Gordon equation, one can generalize square Nabla operator. One simplest version of this equa- its differential operator to become: tion [26]: 2 2 2 ∂S0 2 ∂ 2 ∂ 2 2 − + m =0 (66) −∇ +i −∇ ϕ(x, t)=−m ϕ(x, t), (69) ∂t ∂t2 ∂t2 yields the known solution [26]: or by using our definition in (68), one can rewrite equation (69) in compact form: S0 = ±mt + constant . (67) 2 ♦♦¯ + m ϕ(x, t)=0, (70) The equation (66) yields wave equation which describes a particle at rest with positive energy (lower sign) or with and in lieu of equation (66), now we get: negative energy (upper sign). Radial solution of equation 2 2 ∂S0 ∂S0 2 (66) yields Yukawa potential which predicts meson as obser- + i = m . (71) vables. ∂t ∂t It is interesting to note here, however, that numerical 1-D Numerical solutions for these equations were obtained in solution of equation (65), (66) and (67) each yields slightly similar way with the previous equations: different result, as follows. (All numerical computation was • performed using Mathematica [28].) For equation (70) we get: • For equation (65) we get: (−D[#,x,x]+D[#,t,t]−I*D[#,x,x]+I*D[#,t,t]+mˆ2) (−D[#,x,x]+mˆ2+D[#,t,t])&[y[x,t]]== &[y[x,t]]== 2 (0 2) (2 0) m2 + y(0,2)[x, t] − y(2,0)[x, t]=0 m +(1+i) y , [x, t] − (1 + i) y , [x, t]=0 DSolve[%,y[x,t],{x,t}] DSolve[%,y[x,t],{x,t} m2x2 1 i y[x, t] → + C[1][t − x]+C[2][t + x] y[x, t] → − m2 x2 +C[1][t − x]+C[2][t + x] 2 4 4
48 A. Yefremov, F. Smarandache and V.Christianto. Yang-Mills Field from Quaternion Space Geometry
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• For equation (71) we get: garded as a theory of pure geometric objects: Q-connexion and Q-curvature with Lagrangian quadratic in curvature (as: (−mˆ2+D[#,t,t]+I*D[#,t,t])&[y[x,t]]== Einstein’s theory of gravitation is a theory of geometrical m2 +(1+i) y(0,2)[x, t]=0 objects: Christoffel symbols and Riemann tensor, but with linear Lagrangian made of scalar curvature). DSolve[%,y[x,t],{x,t}] Presence of Q-non-metricity is essential. If Q-non- 1 i metricity vanishes, the Yang-Mills potential may still exist, y[x, t] → − m2 x2 + C[1][x]+tC[2][x] 4 4 then it includes only proper Q-connexion (in particular, com- ponents of Q-connexion physically manifest themselves as Therefore, we may conclude that introducing biquater- “forces of inertia” acting onto non-inertially moving ob- nion differential operator (in terms of “diamond operator”) server); but in this case all Yang-Mills intensity components, yield quite different solutions compared to known standard being in fact components of curvature tensor, identically are solution of Klein-Gordon equation [26]: equal to zero. 1 i 2 2 The above analysis of Yang-Mills field from Quaternion y(x, t)= − m t + constant . (72) 4 4 Space geometry may be found useful in particular if we consider its plausible neat link with Klein-Gordon equation ± 1 1 − i In other word: we can infer hat t = m y/ 4 4 , and Duffin-Kemmer equation. We discuss in particular a therefore it is likely that there is imaginary part of time biquaternionic-modification of Klein-Gordon equation. Since dimension, which supports a basic hypothesis of the afore- the potential corresponding to this biquaternionic KGE is mentioned BQ-metric in Q-relativity. neither Coulomb, Yukawa, nor Hulthen potential, then one Since the potential corresponding to this biquaternionic can expect to observe a new type of matter. Further obser- KGE is neither Coulomb, Yukawa, nor Hulthen potential, vation is recommended in order to support or refute this then one can expect to observe a new type of matter, which proposition. may be called “supersymmetric-meson”. If this new type of particles can be observed in near future, then it can be Acknowledgment regarded as early verification of the new hypothesis of PT- symmetric QM and CT-symmetric QM as considered in some Special thanks to Profs. C. Castro and D. Rapoport for numer- recent reports [27, 29]. In our opinion, its presence may be ous discussions. expected in particular in the process of breaking of Coulomb Submitted on May 26, 2007 barrier in low energy schemes. Accepted on May 29, 2007 Nonetheless, further observation is recommended in order to support or refute this proposition. References
5 Concluding remarks 1. Antonowitcz M. and Szczirba W. Geometry of canonical var- iables in gravity theories. Lett. Math. Phys., 1985, v. 9, 43–49. If 4D space-time has for its 3D spatial section a Q-space with 2. Rapoport D. and Sternberg S. On the interactions of spin with Q-connexion Ωβkn containing Q-non-metricity σβkn, then torsion. Annals of Physics, 1984, v. 158, no. 11, 447–475. the Q-connexion, geometric object, is algebraically identical MR 86e:58028; [2a] Rapoport D. and Sternberg S. Clas- to Yang-Mills potential sical Mechanics without lagrangians nor hamiltoneans. Nuovo Cimento A, 1984, v. 80, 371–383, MR 86c:58055; [2b] Rapo- 1 Aˆkα ≡ εknmΩαkn , port D. and Tilli M. Scale Fields as a simplicity principle. 2 Proceedings of the Third International Workshop on Hadronic while respective curvature tensor Rˆknαβ, also a geometric Mechanics and Nonpotential Interactions, Dept. of Physics, object, is algebraically identical to Yang-Mills “physical Patras Univ., Greece, A. Jannussis (ed.), in Hadronic J. Suppl., field” strength 1986, v. 2, no. 2, 682–778. MR 88i:81180. 1 Fˆ ≡ ε Rˆ . 3. Chan W. K. and Shin F. G. Infinitesimal gauge transformations mαβ 2 knm knαβ and current conservation in Yang theory of gravity. Meet. Thus Yang-Mills gauge field Lagrangian Frontier Phys., Singapore, 1978, v. 2. 4. Hehl F. M. and Sijacki D. To unify theory of gravity and strong ∼ ˆαβ ˆ 1 ˆαβ ˆ 1 ˆαβ ˆ interactions? Gen. Relat. and Grav., 1980, v. 12(1), 83. LYM Fk Fkαβ = εkmnεkjl RmnRjlαβ = RmnRmnαβ 4 2 5. Batakis N. A. Effect predicted by unified theory of gravitational can be geometrically interpreted as a Lagrangian of “non- and electroweak fields. Phys. Lett. B, 1985, v. 154(5–6), 382– linear” or “quadratic” gravitational theory, since it contains 392. quadratic invariant of curvature Riemann-type tensor con- 6. Kyriakos A. Electromagnetic structure of hadrons. arXiv: hep- tracted by all indices. Hence Yang-Mills theory can be re- th/0206059.
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7. Kassandrov V.V. Biquaternion electrodynamics and Weyl- 29. Castro C. The Riemann hypothesis is a consequence of CT- Cartan geometry of spacetime. Grav. and Cosmology, 1995, invariant Quantum Mechanics. Submitted to JMPA, Feb. 12, v. 1, no. 3, 216–222; arXiv: gr-qc/0007027. 2007. 8. Smarandache F. and Christianto V. Less mundane explanation of Pioneer anomaly from Q-relativity. Progress in Physics, 2007, v. 1, 42–45. 9. Fueter R. Comm. Math. Helv., 1934–1935, v. B7, 307–330. 10. Yefremov A. P. Lett. Nuovo. Cim., 1983, v. 37(8), 315–316. 11. Yefremov A. P. Grav. and Cosmology, 1996, v. 2(1), 77–83. 12. Yefremov A. P. Acta Phys. Hung., Series — Heavy Ions, 2000, v. 11(1–2), 147–153. 13. Yefremov A. P. Gravitation and Cosmology, 2003, v. 9(4), 319–324. [13a] Yefremov A. P. Quaternions and biquater- nions: algebra, geometry, and physical theories. arXiv: math- ph/0501055. 14. Ramond P. Field theory, a modern primer. The Benjamin/Cum- ming Publishing Co., ABPR Massachussetts, 1981. 15. Huang K. Quarks, leptons and gauge fields. World Scientific Publishing Co., 1982. 16. Fainberg V. and Pimentel B. M. Duffin-Kemmer-Petiau and Klein-Gordon-Fock equations for electromagnetic, Yang-Mills and external gravitational field interactions: proof of equiv- alence. arXiv: hep-th/0003283, p. 12. 17. Marsh D. The Grassmannian sigma model in SU(2) Yang-Mills theory. arXiv: hep-th/07021342. 18. Devchand Ch. and Ogievetsky V. Four dimensional integrable theories. arXiv: hep-th/9410147, p. 3. 19. Nishikawa M. Alternative to Higgs and unification. arXiv: hep- th/0207063, p. 18. 20. Nishikawa M. A derivation of the electro-weak unified and quantum gravity theory without assuming a Higgs particle. arXiv: hep-th/0407057, p. 22. 21. Adler S. L. Adventures in theoretical physics. arXiv: hep-ph/ 0505177, p. 107. 22. Adler S. L. Quaternionic quantum mechanics and Noncommu- tative dynamics. arXiv: hep-th/9607008. 23. Adler S. L. Composite leptons and quarks constructed as triply occupied quasiparticles in quaternionic quantum mechanics. arXiv: hep-th/9404134. 24. Ward D. and Volkmer S. How to derive the Schrodinger¨ equa- tion. arXiv: physics/0610121. 25. Christianto V. A new wave quantum relativistic equation from quaternionic representation of Maxwell-Dirac equation as an alternative to Barut-Dirac equation. Electronic Journal of Theoretical Physics, 2006, v. 3, no. 12. 26. Kiefer C. The semiclassical approximation of quantum gravity. arXiv: gr-qc/9312015. 27. Znojil M. PT-symmetry, supersymmetry, and Klein-Gordon equation. arXiv: hep-th/0408081, p. 7–8; [27a] arXiv: math- ph/0002017. 28. Toussaint M. Lectures on reduce and maple at UAM-I, Mexico. arXiv: cs.SC/0105033.
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Numerical Solution of Radial Biquaternion Klein-Gordon Equation
Vic Christianto and Florentin SmarandacheO Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] ODepartment of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected]
In the preceding article we argue that biquaternionic extension of Klein-Gordon equa- tion has solution containing imaginary part, which differs appreciably from known so- lution of KGE. In the present article we present numerical /computer solution of ra- dial biquaternionic KGE (radialBQKGE); which differs appreciably from conventional Yukawa potential. Further observation is of course recommended in order to refute or verify this proposition.
1 Introduction and quaternion Nabla operator is defined as [1]:
G @ @ @ @ In the preceding article [1] we argue that biquaternionic ex- H = i + e + e + e! : (5) tension of Klein-Gordon equation has solution containing @t @x @y @z ff imaginary part, which di ers appreciably from known solu- (Note that (3) and (5) included partial time-differentiation.) tion of KGE. In the present article we presented here for the In the meantime, the standard Klein-Gordon equation / first time a numerical computer solution of radial biquater- usually reads [3, 4]: nionic KGE (radialBQKGE); which differs appreciably from
conventional Yukawa potential. @ H '(x; t)=m '(x; t) : (6) This biquaternionic effect may be useful in particular to @t explore new effects in the context of low-energy reaction Now we can introduce polar coordinates by using the (LENR) [2]. Nonetheless, further observation is of course following transformation: recommended in order to refute or verify this proposition. 1 @ @ ` H = r : (7) 2 Radial biquaternionic KGE (radial BQKGE) r @r @r r
In our preceding paper [1], we argue that it is possible to Therefore, by substituting (7) into (6), the radial Klein- ff write biquaternionic extension of Klein-Gordon equation Gordon equation reads — by neglecting partial-time di eren- as follows: tiation — as follows [3, 5]:
( ( 1 @ @ `(` +1) H + E H '(x; t)= r + m '(x; t)=0; (8) r @r @r r (J (J (1) = m '(x; t); and for ` =0, then we get [5]:
or this equation can be rewritten as: 1 @ @ r + m '(x; t)=0: r @r @r (9) }} + m '(x; t)=0; (2) The same method can be applied to equation (2) for radial provided we use this definition: biquaternionic KGE (BQKGE), which for the 1-dimensional @ @ @ @ situation, one gets instead of (8): } = rG + i rG = i + e + e + e + @t @x @y ! @z @ @ @ @ @ @ @ @ i + m '(x; t)=0: (10) + i i + e + e + e ; @r @r @r @r @T @X @Y ! @Z (3) In the next Section we will discuss numerical/computer where e , e , e! are quaternion imaginary units obeying solution of equation (10) and compare it with standard solu- (with ordinary quaternion symbols: e = i, e = j, e! = k): tion of equation (9) using Maxima software package [6]. It can be shown that equation (10) yields potential which differs
i = j = k = 1 ;ij= ji = k; appreciably from standard Yukawa potential. For clarity, all (4) jk = kj = i; ki= ik = j: solutions were computed in 1-D only.
40 V.Christianto and F. Smarandache. Numerical Solution of Radial Biquaternion Klein-Gordon Equation
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3 Numerical solution of radial biquaternionic Klein- 4. Li Yang. Numerical studies of the Klein-Gordon-Schrodinger¨ Gordon equation equations. MSc thesis submitted to NUS, Singapore, 2006, p. 9 (http://www.math.nus.edu.sg/bao/thesis/Yang-li.pdf). Numerical solution of the standard radial Klein-Gordon equa- 5. Nishikawa M. A derivation of electroweak unified and quantum tion (9) is given by: gravity theory without assuming Higgs particle. arXiv: hep-th/ 0407057, p. 15. (%i1) diff(y,t,2)-’diff(y,r,2)+mˆ2*y; 6. Maxima from http://maxima.sourceforge.net (using GNU m y d y (%o1) d x Common Lisp). (%i2) ode2 (%o1,y,r); 7. http://en.wikipedia.org/wiki/Yukawa potential (%o2) y =%k % exp(mr)+%k % exp(mr) (11) 8. Christianto V. A new wave quantum relativistic equation from quaternionic representation of Maxwell-Dirac equation as an In the meantime, numerical solution of equation (10) for alternative to Barut-Dirac equation. Electronic Journal of The- radial biquaternionic KGE (BQKGE), is given by: oretical Physics, 2006, v. 3, no. 12. (%i3) diff(y,t,2)- (%i+1)*’diff(y,r,2)+mˆ2*y; 9. Gyulassy M. Searching for the next Yukawa phase of QCD. / m y (i +1) d y arXiv: nucl-th 0004064. (%o3) d r (%i4) ode2 (%o3,y,r); jmjr jmjr (%o4) y =%k sin F +%k cos F (12) i i Therefore, we conclude that numerical solution of radial biquaternionic extension of Klein-Gordon equation yields different result compared to the solution of standard Klein- Gordon equation; and it differs appreciably from the well- known Yukawa potential [3, 7]: g u(r)= e mr: r (13) Meanwhile, Comay puts forth argument that the Yukawa lagrangian density has theoretical inconsistency within itself [3]. Interestingly one can find argument that biquaternion Klein-Gordon equation is nothing more than quadratic form of (modified) Dirac equation [8], therefore BQKGE describ- ed herein, i.e. equation (12), can be considered as a plausible solution to the problem described in [3]. For other numerical solutions to KGE, see for instance [4]. Nonetheless, we recommend further observation [9] in or- der to refute or verify this proposition of new type of potential derived from biquaternion Klein-Gordon equation.
Acknowledgement
VC would like to dedicate this article for RFF.
Submitted on November 12, 2007 Accepted on November 30, 2007
References
1. Yefremov A., Smarandache F. and Christianto V. Yang-Mills field from quaternion space geometry, and its Klein-Gordon representation. Progress in Physics, 2007, v. 3, 42–50. 2. Storms E. http://www.lenr-canr.org 3. Comay E. Apeiron, 2007, v. 14, no. 1; arXiv: quant-ph/ 0603325.
V.Christianto and F. Smarandache. Numerical Solution of Radial Biquaternion Klein-Gordon Equation 41
120 January, 2009 PROGRESS IN PHYSICS Volume 1
On PT-Symmetric Periodic Potential, Quark Confinement, and Other Impossible Pursuits
Vic Christianto and Florentin SmarandacheO Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] ODepartment of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected]
As we know, it has been quite common nowadays for particle physicists to think of six impossible things before breakfast, just like what their cosmology fellows used to do. In the present paper, we discuss a number of those impossible things, including PT-symmetric periodic potential, its link with condensed matter nuclear science, and possible neat link with Quark confinement theory. In recent years, the PT-symmetry and its related periodic potential have gained considerable interests among physicists. We begin with a review of some results from a preceding paper discussing derivation of PT-symmetric periodic potential from biquaternion Klein-Gordon equation and proceed further with the remaining issues. Further observation is of course recommended in order to refute or verify this proposition.
1 Introduction 2 PT-symmetric periodic potential
As we know, it has been quite common nowadays for parti- It has been argued elsewhere that it is plausible to derive a cle physicists to think of six impossible things before break- new PT-symmetric Quantum Mechanics (PT-QM; sometimes fast [1], just like what their cosmology fellows used to do. it is called pseudo-Hermitian Quantum Mechanics [3, 9]) In the present paper, we discuss a number of those impossi- which is characterized by a PT-symmetric potential [2] ble things, including PT-symmetric periodic potential, its link V (x)=V ( x) : (1) with condensed matter nuclear science, and possible neat link with Quark Confinement theory. One particular example of such PT-symmetric potential In this regards, it is worth to remark here that there were can be found in sinusoidal-form potential some attempts in literature to generalise the notion of sym- V = sin ': (2) metries in Quantum Mechanics, for instance by introducing CPT symmetry, chiral symmetry etc. In recent years, the PT- PT-symmetric harmonic oscillator can be written accord- symmetry and its related periodic potential have gained con- ingly [3]. Znojil has argued too [2] that condition (1) will siderable interests among physicists [2, 3]. It is expected that yield Hulthen potential the discussions presented here would shed some light on these A B V ()= + : (3) issues. (1 e E) (1 e E) We begin with a review of results from our preceding pa- Interestingly, a similar periodic potential has been known pers discussing derivation of PT-symmetric periodic potential for quite a long time as Posch-Teller potential [9], although from biquaternion Klein-Gordon equation [4–6]. Thereafter it is not always related to PT-Symmetry considerations. The we discuss how this can be related with both Gribov’s theory Posch-Teller system has a unique potential in the form [9] / of Quark Confinement, and also with EQPET TSC model for condensed matter nuclear science (aka low-energy reaction U(x)= cosh x: (4) or “cold fusion”) [7]. We also highlight its plausible impli- It appears worth to note here that Posch-Teller periodic cation to the calculation of Gamow integral for the (periodic) potential can be derived from conformal D’Alembert equa- non-Coulomb potential. tions [10, p.27]. It is also known as the second Posch-Teller In other words, we would like to discuss in this paper, potential ( 1) ` (` +1) whether there is PT symmetric potential which can be ob- V()= + : (5) served in Nature, in particular in the context of condensed sinh cosh matter nuclear science (CMNS) and Quark confinement The next Section will discuss biquaternion Klein-Gordon theory. equation [4, 5] and how its radial version will yield a sinu- Nonetheless, further observation is of course recommend- soidal form potential which appears to be related to equa- ed in order to refute or verify this proposition. tion (2).
V.Christianto and F. Smarandache. On PT-Symmetric Periodic Potential, Quark Confinement, and Other Impossible Pursuits 63
121 Volume 1 PROGRESS IN PHYSICS January, 2009
3 Solution of radial biquaternion Klein-Gordon equa- Using Maxima computer package we find solution of tion and a new sinusoidal form potential equation (15) as a new potential taking the form of sinusoidal potential
In our preceding paper [4], we argue that it is possible to jmj r jmj r write biquaternionic extension of Klein-Gordon equation as y = k sin p + k cos p ; (16) i 1 i 1 follows @ @ where k and k are parameters to be determined. It appears r + i r '(x; t)= k @t @t very interesting to remark here, when is set to 0, then equa- tion (16) can be written in the form of equation (2) = m '(x; t) ; (6) V = k sin '; (17) or this equation can be rewritten as by using definition }} + m '(x; t)=0 (7) jmj r ' = sin p : (18) i 1 provided we use this definition
@ @ @ @ In retrospect, the same procedure which has been tradi- } = rG + irG = i + e + e + e + tionally used to derive the Yukawa potential, by using radial @t @x @y ! @z biquaternion Klein-Gordon potential, yields a PT-symmetric @ @ @ @ + i i + e + e + e ; periodic potential which takes the form of equation (1). @T @X @Y ! @Z (8) 4 Plausible link with Gribov’s theory of Quark Confine- where e , e , e! are quaternion imaginary units obeying ment (with ordinary quaternion symbols e = i, e = j, e! = k): i = j = k = 1 ;ij= ji = k; (9) Interestingly, and quite oddly enough, we find the solution (17) may have deep link with Gribov’s theory of Quark con- jk = kj = i; ki= ik = j; (10) finement [8, 11]. In his Third Orsay Lectures he described a and quaternion Nabla operator is defined as [4] periodic potential in the form [8, p.12] @ @ @ @ 3 sin =0: rG = i + e + e + e : (19) @t @x @y ! @z (11) By using Maxima package, the solution of equation (19) Note that equation (11) already included partial time- is given by 4 9 differentiation. F @O > x = k k F cos(y) => Thereafter one can expect to find solution of radial bi- $ quaternion Klein-Gordon Equation [5, 6]. R ; (20) F 1 @O > k1 cos(y) ; First, the standard Klein-Gordon equation reads x = k + F $ @ r '(x; t)= m '(x; t) : while Gribov argues that actually the equation shall be like @t (12) nonlinear oscillation with damping, the equation (19) indi- At this point we can introduce polar coordinate by using cates close similarity with equation (2). the following transformation Therefore one may think that PT-symmetric periodic po- tential in the form of (2) and also (17) may have neat link 1 @ @ ` r = r : (13) with the Quark Confinement processes, at least in the con- r @r @r r text of Gribov’s theory. Nonetheless, further observation is of course recommended in order to refute or verify this pro- Therefore by introducing this transformation (13) into position. (12) one gets (setting ` =0)
1 @ @ r + m '(x; t)=0: (14) 5 Implication to condensed matter nuclear science. r @r @r Comparing to EQPET/TSC model. Gamow integral By using the same method, and then one gets radial ex- In accordance with a recent paper [6], we interpret and com- pression of BQKGE (6) for 1-dimensional condition as fol- pare this result from the viewpoint of EQPET/TSC model lows [5, 6] which has been suggested by Prof. Takahashi in order to ex- 1 @ @ 1 @ @ plain some phenomena related to Condensed matter nuclear r i r +m '(x; t)=0: r @r @r r @r @r (15) Science (CMNS).
64 V.Christianto and F. Smarandache. On PT-Symmetric Periodic Potential, Quark Confinement, and Other Impossible Pursuits
122 January, 2009 PROGRESS IN PHYSICS Volume 1
Takahashi [7] has discussed key experimental results 6 Concluding remarks in condensed matter nuclear effects in the light of his EQPET/TSC model. We argue here that his potential model In recent years, the PT-symmetry and its related periodic po- with inverse barrier reversal (STTBA) may be comparable to tential have gained considerable interests among physicists. the periodic potential described above (17). In the present paper, it has been shown that one can find In [7] Takahashi reported some findings from condensed a new type of PT-symmetric periodic potential from solu- matter nuclear experiments, including intense production of tion of the radial biquaternion Klein-Gordon Equation. We helium-4, "He atoms, by electrolysis and laser irradiation ex- also have discussed its plausible link with Gribov’s theory of periments. Furthermore he [7] analyzed those experimental Quark Confinement and also with Takahashi’s EQPET/TSC results using EQPET (Electronic Quasi-Particle Expansion model for condensed matter nuclear science. All of which Theory). Formation of TSC (tetrahedral symmetric conden- seems to suggest that the Gribov’s Quark Confinement the- sate) were modeled with numerical estimations by STTBA ory may indicate similarity, or perhaps a hidden link, with the (Sudden Tall Thin Barrier Approximation). This STTBA Condensed Matter Nuclear Science (CMNS). It could also be model includes strong interaction with negative potential near expected that thorough understanding of the processes behind the center. CMNS may also require revision of the Gamow factor to take One can think that apparently to understand the physics into consideration the cluster deuterium interactions and also behind Quark Confinement, it requires fusion of different PT-symmetric periodic potential as discussed herein. fields in physics, perhaps just like what Langland program Further theoretical and experiments are therefore recom- wants to fuse different branches in mathematics. mended to verify or refute the proposed new PT symmetric Interestingly, Takahashi also described the Gamow inte- potential in Nature. gral of his STTBA model as follows [7] Submitted on November 14, 2008 / Accepted on November 20, 2008 Zb = = n =0:218 (Vb Ed) dr : (21) References r 1. http://www-groups.dcs.st-and.ac.uk/history/Quotations/ Using b =5:6 fm and r =5fm, he obtained [7] Dodgson.html 2. Znojil M. arXiv: math-ph/0002017. P =0:77; "D (22) 3. Znojil M. arXiv: math-ph/0104012, math-ph/0501058. and 4. Yefremov A.F., Smarandache F., and Christianto V. Progress V =0:257 ; B MeV (23) in Physics, 2007, v. 3, 42; also in: Hadron Models and Related which gave significant underestimate for 4D fusion rate when New Energy Issues, InfoLearnQuest, USA, 2008. rigid constraint of motion in 3D space attained. Nonetheless 5. Christianto V. and Smarandache F. Progress in Physics, 2008, v. 1, 40. by introducing different values for "D the estimate result can be improved. Therefore we may conclude that Takahashi’s 6. Christianto V. EJTP, 2006, v. 3, no. 12. STTBA potential offers a good approximation (just what the 7. Takahashi A. In: Siena Workshop on Anomalies in Metal-D/H name implies, STTBA) of the fusion rate in condensed matter Systems, Siena, May 2005; also in: J. Condensed Mat. Nucl. nuclear experiments. Sci., 2007, v. 1, 129. It shall be noted, however, that his STTBA lacks sufficient 8. Gribov V.N. arXiv: hep-ph/9905285. theoretical basis, therefore one can expect that a sinusoidal 9. Correa F., Jakubsky V., Plyushckay M. arXiv: 0809.2854. ff periodic potential such as equation (17) may o er better re- 10. De Oliveira E.C. and da Rocha R. EJTP, 2008, v. 5, no. 18, 27. sult. 11. Gribov V.N. arXiv: hep-ph/9512352. All of these seem to suggest that the cluster deuterium may yield a different inverse barrier reversal which cannot be 12. Fernandez-Garcia N. and Rosas-Ortiz O. arXiv: 0810.5597. predicted using the D-D process as in standard fusion theory. 13. Chugunov A.I., DeWitt H.E., and Yakovlev D.G. arXiv: astro- In other words, the standard procedure to derive Gamow fac- ph/0707.3500. tor should also be revised [12]. Nonetheless, it would need further research to determine the precise Gamow energy and Gamow factor for the cluster deuterium with the periodic po- tential defined by equation (17); see for instance [13]. In turn, one can expect that Takahashi’s EQPET/TSC model along with the proposed PT-symmetric periodic poten- tial (17) may offer new clues to understand both the CMNS processes and also the physics behind Quark confinement.
V.Christianto and F. Smarandache. On PT-Symmetric Periodic Potential, Quark Confinement, and Other Impossible Pursuits 65
123 FLORENTIN SMARANDACHE & V. CHRISTIANTO
Meet in person at International Airport, Jakarta, Indonesia, May 2006
124 Seminar of Neutrosophic Logic and its applications in Central Java, Indonesia, May 2006
With Dr. Peter Handoko (first from left), Dr. S. Trihandaru (second from right), May 2006
125 Seminar of Neutrosophic Logic and its applications in Central Java, Indonesia, May 2006
Discussion with Dr Harsono (lecturer of Fuzzy Logic), May 2006
126 Seminar of Neutrosophic Logic and its applications in Central Java, Indonesia, May 2006
A few audience of the seminar (including Prof. Theo Van Beusekom fourth from right and Prof .L Wilardjo second from right), May 2006
127 Seminar of Neutrosophic Logic and its applications in Central Java, Indonesia, May 2006
Audience of the seminar (mostly students from Electronics Dept.)
128 FLORENTIN SMARANDACHE
Visiting Blanco Museum, Bali Island, Indonesia, May 2006
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NEUTROSOPHIC LOGIC, WAVE MECHANICS AND OTHER STORIES: SELECTED WORKS 2005-2008
There is beginning for anything; we used to hear that phrase. The same wisdom word applies to us too. What began in 2005 as a short email on some ideas related to interpretation of the Wave Mechanics results in a number of papers and books up to now. Some of these papers can be found in Progress in Physics or elsewhere.
Our purpose here is to present a selection of those papers in a compilation which enable the readers to find some coherent ideas which appeared in those articles. For this reason, the ordering of the papers here is based on categories of ideas.
While some of these articles have been published in book format elsewhere, we hope that reading this book will give the readers impression of progression of our thoughts.
We wish to extend our sincere gratitude to plenty of colleagues, friends and relatives all around the world for sharing their ideas, insightful discussions etc. Special thanks to D. Rabounski, S. Crothers, L. Borissova for their great service in Progress in Physics journal.
FS & VC April 2009
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