Yod field statistics for tachyon interactions

Transcription

1 Yod field statistics for tachyon interactions Joe Helmick Ohio State University, College of Arts and Sciences, 5253 E. Broad St. 109, Columbus, Ohio Abstract A dynamic system of differential rotation yields operator yod that describes a solution to negative values under the square root in the time dilation equation for velocities faster than light. Statistical streams are output in deterministic packets whose source is calculated by divergence recast as a volume with Gauss law. Particle interactions within multiple asymmetric structures of yod suggest boundary layers for Tachyon traps. Keywords: time dilation, complex function, asymmetric selection 1

2 1. Introduction Derived from imaginary iota, yod [1] is a field operator accessible to measurement by an observer using values greater than c in the time dilation equation. [2] A negative value under the square root function in the denominator of the time dilation equation inhibits direct observation by separating an observer from the imaginary plane. The yod operator, derived from a composition of complex functions (Fig.1) on iota, creates statistical projections onto physical space. An observer is able to take physical measurements that satisfy the logic of observing tachyon final velocity absorption before initial velocity emission. [3] Sartori further reasons that relative velocity is possible for v > c in casualty when t is negative and c/v < v/c < 1, true for v < Field Theory For the negative branch of the operand under the square root in the denominator, 1 v 2 / c 2 < 0 (1) v 2 > c 2 (2) ± v > c (3) The one-sided definition of absolute value veils [4] the physics of real velocities as a method of convenience from an arbitrary decision by Euler that a double negative equals a positive. [5] At the same time however Euler does credit the double negative equals a negative as equally functional. In addition, Godel notes that there is no proof in Sigma that a double negative equals a positive. [6] With respect to the ( ) = branch, the absolute value function collapses in quadrant 2 and the ( ) = branch appears as a 90 rotation to the third quadrant where -x = -y or 5π/4, the bilateral edge of the yod selection matrix. For the v < c branch, Sudarshan argues the decay of a particle implies its creation. [7] If tachyon is spinless and non-interacting [3], tachyon mass is also negative, which poses intrinsic considerations of self-propagation in space-time. Yod localizes the propagation state of superluminal particles such as tachyon and introduces field measures of higher dimension that result from input continued fractions e, π, (2) 1/2, and (3) 1/2 and Pythagorean special angles as selection matrices on the unit circle in a system of differential relation dl/dθ with constraints, ( ) = and ±0 1 = [8, 9] on [dl/dθ = dl/dθ rad dθ rad /dθ ] yod. The statistics of yod null, the nucleus of all symmetric and asymmetric yod orbits, appear to hold a stable description of yod rest mass or ground state energy level. Yod null statistics may also correspond to a nuclear description of tachyon in an unstable vacuum. [10] Assume v = 2 c and T 0 = 2 years in the time dilation equation where yod = (-) ½ (Fig.1) quantizes the faster than light field. As a finite set of field statistics mixed with clustering n-tuples, the deterministic system defines a finite energy. [11] T = 2 / [1 - ( 2 c 2 ) / c 2 ] ½ (4) T = 2 / (-1) ½ (5) Composing a negative sign from under the root of iota as a composition of complex functions, yod provides a statistical field description of superluminal velocity. 2

3 T = 2 / [ (-1) ½ (1) ½ ] (6) T = 2 / [ (-) ½ (1) ½ ] (7) T = 2 / (yod) (8) Observing tachyon absorption before emission suggests a mechanism of tachyon interaction. Conjecture for tachyon condensation is to format packets of yod statistics in calibration with their known angles of occurrence using an optical encoder, angular positioner and stepper motor and focus yod s discrete wave frequencies from its electromagnetic pulses through a 5π/4 electric charged cylinder for charge density [12] ρ as in equations (9) and (12) here. Also, Fourier analysis of yod statistics plotted by probability curves expresses complex waveforms that are translatable into sound. [13] In ionic beams of electromagnetic pulses, a mechanical description of quantum and semi-classical equivalence [14] is proposed. Small violations of fundamental symmetry differences [15] for non-accelerator physics experiments may be evident in the rapidly inter-converting resonance forms of asymmetric yod molecular conformations including optical isomers. Yod fields are represented by more than one statistical quantification: angular measures, n-tuple cluster distributions, and non-clustering distributions in frequency modulations. The n-tuple clusters found in yod packets may well prove to occupy the vertex of a nose cone thereby satisfying a physical description of a wave torpedo with v > c. [12] Furthermore, angular measures enable orientation and direction plots of linear sequencing for target transmitters and receivers as well as for construction of pre-nucleic acids by supervised classification of digit clusters from Pythagorean special angles and from multiple combinations of yod 7-null selection matrices. As extensions of the angular measures, clustering n-tuple digits suggest functional groups for field interactions. From base radian through the descending orbits of yod fields, resonance boundaries of asymmetric selection also suggest interactions for tachyon detection and potential capture [16, 17]. Since an electromagnetic field is known by its statistical description, the source can be predicted using divergence recast as volume with Gauss s law, ρ as charge density, and E as electric field. [18] E da = Q enc. / ε o = ρ dv / ε o (9) = v E dv (10) = v E (11) = ρ / ε o (12) Knowing the physical source of yod field statistics may be valuable to understand new and unobvious theories of electromagnetic phenomena, for example energy transfer, associated with yod-tachyon statistical fields and system mechanics. Also, once the source is located, light curves may provide valuable data for astronomers. [13] 3. Conclusion The relation between yod statistics and tachyon is valid due to the double negative equals a negative constraint that initiates the complex composition of yod from iota. The assumed cause in constraint ±0 1 = enables yod existence, and shows a necessary and related effect as a statistical solution to negative values under the square root in the time dilation equation. Asymmetric selection of input values displays the resonance layers of yod descent to 5π/4 and convergence to the null set as transition boundaries for energy transfer [12] and tachyon traps. 3

4 Acknowledgments I would like to thank Samir Mathur, Ping-shun Chan, and Ben Dundee for their suggestions and comments. References 1. J. Helmick, Set Theory to Physics, in: Khrennikov A. Ed., Proceedings of the Conference, Quantum Theory: Reconsideration of Foundations, Vaxjo Univ. Press, Vaxjo (2002). 2. O.P.M. Bilaniuk, V.K. Deshpande & E.C.G. Sudarshan Meta particle losing energy accelerates, Am. J. Phys (1962). 3. L. Sartori Understanding Relativity, Univ. of California Press, Berkeley (1996). 4. J.L. Dobson, The Equations of Maya, in Mathews C.N. & Varghese R.A. Eds., Cosmic Beginnings and Human Ends, Open Court, Chicago 261 (1995). 5. L. Euler, Elemens d Algebra, A Petersbourg, Paris, Book (1789) in P.J. Nahin,An Imaginary Tale, Princeton Univ. Press, Princeton, 14 (1998). 6. K. Godel, In what sense is intuitionist logic constructive? in Fefferman S. et al. Eds.,Collected Works Vol. III, Oxford Univ.Press, New York, 199 (1995). 7. E.C.G. Sudarshan, Beyond Immutability: The Birth and Death of Elementary Particles, in: Mathews C.N. & Varghese R.A. Eds., Cosmic Beginnings and Human Ends, Open Court, Chicago (1995). 8. A.C. Eringen, Electrodynamics of Continua, Springer Verlag, New York 37 (1990). 9. W. Kahan, Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit, in The State of the Art in Numerical Analysis, Iserles and Powell eds. Clarendon Press, Oxford (1987). 10. W. Taylor, Tachyon Condensation in Open String Field Theory, Pacific Northwest String Seminar, PIMS, Vancouver (2000) A. Einstein, The Meaning of Relativity, Princeton Univ. Press (1955). 12. V.S. Barashenkov and W.A. Jr Rodrigues, Launching of Non-Dispersive Superluminal Beams, arxiv.org: physics/ (1997). 13. J. Lochner, Imagine the Universe, Spectral Analysis, & Lightcurves, Spectra and Images, NASA Goddard Space Flight Center (1997) G. Feinberg, Possibility of Faster-than-Light Particles, Phys Rev. Ser (1967). 4

5 15. M.L. Perl et al. Eds, Elementary Particle Physics, National Academy Press, Washington DC. 9 (1986). 16. G-Z. Li et al, Antihydrogen production in a combined trap, Hyperfine Interactions (1993). 17. R. Blumel, Chaos and order in ion traps and storage rings, In:Trapped charged particles and fundamental physics, DHE Dubin et al eds, AIP, Woodbury (1999). 18. D.J. Griffiths, Introduction to Electrodynamics, Third ed., Prentice-Hall, Upper Saddle River (1999). 5

6 Fig. 1 yod operator flowchart (-) 1/2 = yod If - -1 and -1 = i, then - = (-) 1/2 = yod f (x) = x x R - U {-, f: R - C U {yod R - = negative Real numbers g (x) = 0-1 = -1 x {-1, g:{-1 {-1 C = Complex numbers h (x) = ±0 1 = - x {T, h:{t {- T = True U = union calculate (f g) (x) f(g (x) ) = (f g) (x) F if f (-1) = -1 then -1 = i T if ±0 1 = - and (f h) (x) = f(h (x) ) where f (x) = x and h (x) = - then f (h (x) ) = f ( (-) ) = - i 2-1 and (-1) 2 = -1 f (h (x) ) = f (-) = - = (-) 1/2 =yod 6

7 Appendix I: dl/dθ algorithm in Mathematica code for 16 special angles Programming Parameters & Packages Needs [ Graphics Graphics ] Needs [ Statistics DataManipulation ] LengthofString = Digit Representations d = RealDigits [ E, 10, LengthofString ] [ [1] ] ; c = RealDigits [ Pi, 10, LengthofString ] [ [1] ] ; Digit Representations in Special Angles SpecialAngles = (Table [ {0 + 2 Pi k + 30, Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , { k, 0,.95 LengthOfString / 360 ] / / Flatten ) /. Pi 180 ; cc = Part [ c, (Table [ { Pi k + 30, Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , { k, 0,.95 LengthOfString / 360 ] / / Flatten ) /. Pi 180 ] ; dd = Part [ d, (Table [ { Pi k + 30, Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , { k, 0,.95 LengthOfString / 360 ] / / Flatten ) /. Pi 180] ; Length [ cc ] Special Angle Number ( 1 = Pi / 6, 2 = Pi / 4 ) for Matching Digit Positions Flatten [ Position [ Table [ dd [ [ k ] ] == [ [ k ] ], { k, 1, Length [ cc ] ], True] ] Matching Special Angles Part [ (Table [ { Pi k + 30, Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , Pi k , 7

8 0 + 2 Pi k , Pi k , Pi k , Pi k , { k, 0,.95 LengthOfString / 360 ] / / Flatten ) /. Pi 180, Flatten [ % ]] Matching Digit Pairs MatchingDigits = c [ [ % ] ] d [ [ %% ] ] Frequencies [ MatchingDigits ] Histogram [ MatchingDigits ] Table [ ListPlot [ Transpose [ { Drop [ MatchingDigits, k ], Drop [ MatchingDigits, - k ] ] ], { k, 1, 100, 10 ] -- 8

9 Appendix II: dl/dθ algorithm in C++ for selection angle matching and non-matching digits. #include <cstdlib> #include <iostream> #include <fstream> using namespace std; int main(int argc, char *argv[]) { string pistr, rt2str, rt3str, estr, sangstr; string s150, s180, s210, s225, s240, s270, s300; int lngth; //output streams ofstream nomatchdig ("results/non_matching_input_digits.csv"); ofstream nomatchsel ("results/non_matching_selection_string_digits.csv"); ofstream matchdig ("results/matching_input_digits.csv"); ofstream matchspec ("results/matching_input_with_special_angles.csv"); ofstream matchselpos ("results/matching_selection_angle_position.csv"); ofstream matchselposval ("results/matching_selection_angle_position_value.csv"); //input streams ifstream pi ("inputs/pi.txt"); ifstream rt2 ("inputs/rt2.txt"); ifstream rt3 ("inputs/rt3.txt"); ifstream e ("inputs/e.txt"); ifstream sangfile ("Special Angles/special_angles.txt"); ifstream f150 ("Special Angles/150.txt"); ifstream f180 ("Special Angles/180.txt"); ifstream f210 ("Special Angles/210.txt"); ifstream f225 ("Special Angles/225.txt"); ifstream f240 ("Special Angles/240.txt"); ifstream f270 ("Special Angles/270.txt"); ifstream f300 ("Special Angles/300.txt"); //set up the column names on the files nomatchdig << "Digit Place, Pi, rt2, rt3, e \n"; matchdig << "Digit Place, Pi, rt2, rt2, e \n"; matchspec <<"Digit Place, Digit % 360, Digit Value, Matching Special Angle \n"; nomatchsel << "Digit Place, 5pi/6, pi, 7pi/6, 5pi/4, 4pi/3, 3pi/2, 5pi/3 \n"; matchselpos << "Digits Place, Digit, 5pi/6, pi, 7pi/6, 5pi/4, 4pi/3, 3pi/2, 5pi/3 \n"; matchselposval << "Digits Place, Digit, 5pi/6, pi, 7pi/6, 5pi/4, 4pi/3, 3pi/2, 5pi/3 \n"; //read the special angles(7 yod angles for now) getline(sangfile, sangstr); lngth = atoi(sangstr.c_str()); int specialangles[lngth]; int i = 0; int angle; while(! sangfile.eof()) { getline (sangfile, sangstr); 9

10 angle = atoi(sangstr.c_str()); specialangles[i] = angle; i ++; bool first = true; int line_len; //total is a counter on the number of digits read from the selection angles file int total = 0; //if the files are open if (pi.is_open() & rt2.is_open() & rt3.is_open() & sangfile.is_open() & f150.is_open() & f180.is_open() & f210.is_open() & f225.is_open() & f240.is_open() & f270.is_open() & f300.is_open()) { //while not at the end of file while(! pi.eof()) { if(first) { line_len = 68; first = false; else { line_len = 70; //read a line from the files //essentiall puts one line from a file into a string getline (pi, pistr); getline (rt2, rt2str); getline (rt3, rt3str); getline (e, estr); getline(f150, s150); getline(f180, s180); getline(f210, s210); getline(f225, s225); getline(f240, s240); getline(f270, s270); getline(f300, s300); for(int j = 0; j < line_len; j ++) { if ((pistr[j] == 'x')) { j = line_len; else { //get the non-matching input digits if ((pistr[j]!= rt2str[j]) (rt2str[j]!= rt3str[j]) (estr[j]!= rt3str[j])) { //print the non-matching digits nomatchdig << total << ", " << pistr[j] << ", " << rt2str[j] << ", " << rt3str[j] << ", " << estr[j] <<"\n"; //print the digits of the selection string for non-matching inputs nomatchsel << total << ", " << s150[j] << ", " << s180[j] << ", " << s210[j] << ", " << s225[j] << ", " << s240[j] << ", " << s270[j] << ", " << s300[j] << "\n "; //get the matching input digits if ((pistr[j] == rt2str[j]) & (rt2str[j] == rt3str[j]) & (rt3str[j] == estr[j])) { 10

11 //print the matching digits matchdig << total << ", " << pistr[j] << ", " << rt2str[j] << ", " << rt3str[j] << ", " << estr[j] <<"\n"; bool anglematch = false; //print the matching special angle digit, if there is one(x otherwise) for(int i = 0; i < lngth; i ++ ){ <<"\n"; if( total % 360 == specialangles[i]) { matchspec << total << ", " <<total % 360 << ", " <<pistr[j] << ", " << specialangles[i] anglematch = true; if(anglematch == false) { matchspec << total << ", " <<total % 360 << ", " << pistr[j] << ", " << "x" <<"\n"; //print matching selection angles by position matchselpos << total <<", " << pistr[j] <<", " << s150[j] << ", " << s180[j] << ", " << s210[j] << ", " << s225[j] << ", " << s240[j] << ", " << s270[j] << ", " << s300[j] << "\n "; //print matching selection angles by position and by value matchselposval << total << ", " << pistr[j] << ", "; if(pistr[j] == s150[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; if(pistr[j] == s180[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; if(pistr[j] == s210[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; if(pistr[j] == s225[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; if(pistr[j] == s240[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; 11

12 if(pistr[j] == s270[j]) { matchselposval << pistr[j] << ", "; else { matchselposval << "x, "; if(pistr[j] == s300[j]) { matchselposval << pistr[j] << "\n "; else { matchselposval << "x \n"; //increment to the next digit total++; else cout << "Unable to open file(s) \n"; pi.close(); rt2.close(); rt3.close(); cout << "Program finished. \n"; system("pause"); return 0; 12