Scalar Science

1. Scalar Science What is it? Part III Presented by Doug BundyJanuary 17, 2007

2. Larsonian Science • New science • Based on unification of all things • New ideas call for new approach • Includes some of Newtonian science • Changes some foundational mathematics and physical concepts, but not all

3. “Whole New Science” • Smolin • “Great unifications become the founding ideas on which whole new sciences are erected.” • What are the RST’s “founding ideas?” • Redefinition of space, time, energy and matter • What is conserved? • Motion is conserved

4. Chart of Motion Change of Units Change of Positions Change of Intervals Change of Scales Three Dimensions of Magnitude 10 20 30 40 1 1 1 1 Point unit 11 21 31 41 4 4 3 1 2 Line unit 12 22 32 42 4 16 9 1 4 16 8 Area unit 12 13 23 33 43 16 16 64 27 1 8 Volume unit 16 16

5. Conservation of Motion • Four bases of motion • M1 = abstract magnitudes • M2 = change of position magnitudes • M3 = change of interval magnitudes • M4 = change of scale magnitudes • Three dimensions of motion • Mn0 = 0D magnitudes • Mn1 = 1D magnitudes • Mn2 = 2D magnitudes • Mn3 = 3D magnitudes

6. Calculus • Defines velocity in context of infinitely divisible continuum. • Velocity can vary in arbitrary distances • Thus, must take delta to the limit • Applies to M2, change of position motion • Does not apply directly to M3 and M4 motion

7. The Newtonian Concept of Force • Ultimately a quantity of change of motion • A quantity of acceleration • F = ma, or a scalar times a component vector • Has evolved to fundamental, autonomous, status • Electromagnetic “charge” • Weak nuclear “charge” • Strong nuclear “charge” • Gravitational “charge”

8. Dimensions of Force and Acceleration • Force dimensions are energy per unit space • t/s * 1/s = t/s2 • Acceleration dimensions are velocity per unit time • s/t * 1/t = s/t2 • Energy is scalar & velocity is vectorial • As “quantity of acceleration,” force should have dimensions of acceleration • Both should be scalar, not vectorial • (t/s * 1/s) = (t/s)3 + (s/t * 1/t) = t/s2, • Force = energy per scalar unit of space • Acceleration = velocity per scalar unit of time.

9. The Newtonian Concept of Work • Work divides energy into two concepts • Potential energy (no direction) • Kinetic energy (directed energy, or energy of motion (vector). • Thus, force dimensions become energy dimensions through displacement (change of position motion) • Kinetic energy is force by distance • t/s2 * s = t/s • If mass is displaced (moves), work is performed and the potential energy of force (energy per unit space) is transformed into the kinetic energy of force (energy per unit space times displacement, or length) in a given direction. • Forms basis of analysis in Newtonian science

10. Newtonian Principles of Analysis • Based on concepts of M2 motion • Vectors • Vector spaces • Functions in vector spaces • Focuses on Analysis of vectors in vector spaces • Spaces of vectors are linear spaces • Thus, in a given vector space • Vectors can be added together • Vectors can be multiplied by scalars • Need not be limited to geometric spaces (visualizable in three dimensions), but may also be abstract spaces • Represents vectors with complex numbers • Opens whole new world of possibilities • Transforms vectors into scalars!

11. The Vector Space of Transformations • Linear operators • Operator = transform of functions • Example: differential operator (f(x) -> f’(x)) • An operator is a symbol that tells you to do something with whatever follows the symbol • Linear operator • Satisfies two conditions: An operator O is said to be linear if, for every pair of functions f and g and scalar s , • O(f+g) = Of +Og and • O(sf) = sOf • In other words, distributive (ordered) functions (functions compatible with the addition and scalar multiplication)

12. Definitions and Dimensions • Vis morte, or dead force of mass (inertia) • Vis viva, or live force of motion • Became energy per Leibniz’s idea (E = mv2) in conservation of energy • Initially was momentum per Newton’s idea (p = mv) in conservation of momentum • Dimensions of momentum are energy squared • p = mv = (t/s)3 * s/t = (t/s)2 • Dimensions of energy are mass times velocity squared • E = mv2 = (t/s)3 * (s/t)2 = t/s • Dimensions of mass are momentum times energy • m = p * E = (t/s)2 * t/s = (t/s)3 • Thus, energy, momentum and mass are 1, 2, and 3 dimensional magnitudes of inverse velocity • Energy (t/s); Momentum (t/s)2, and Mass (t/s)3

13. Conservation Law Sans Force Concept • Conservation of motion • Two forms of motion • Velocity (s/t) • Inverse velocity (t/s) • Two modes of motion • Translational (unbounded) • Vibrational (bounded) • Conservation of direction • M2 is motion in one direction (line) • M3 is motion in two simultaneous directions (area) • M4 is motion in three simultaneous directions (volume) • Conservation of dimension

14. Draft Plan for Erecting New Science • Compare and contrast with previous science • View in context of chart of motions • Identify where new unification simplifies • Document findings • Identify systematic tools to use • Reciprocal System of Mathematics • Chart of Motion • World line charts • Progression Algorithms (PAs) • Synthesize effective analysis procedure • Conservation of motion analyzed in terms of form, mode, and dimension

15. Learning from Newtonian Science • Examine history of mechanical analysis • Vector analysis • Examine history of quantum mechanical analysis • Functional analysis • Examine history of mathematical development • Differential calculus • Linear analysis • Operator and group theory • Translate into lessons learned

16. Mechanical Analysis • Revisit physical concepts • Energy, momentum, mass • Force and acceleration • Conservation and invariance (symmetry) • Look for clarification of mathematics • Number systems • Discrete principles vs. continuum principles • Meaning and use of complex numbers

17. Quantum Mechanical Analysis • Examine changes in mechanical concepts • Rotation and angular momentum • Discrete energy viz-a-viz potential/kinetic concept • Role of potential energy in wave equation • Look for mathematical meaning of rotation • Complex numbers and vector spaces • Quantum phase and renormalization • Meaning of non-commutative mathematics • Master key concepts of standard model • Gauge principle • Lie groups and Lie algebra • Higgs potential and Higgs Boson

18. What We’ve Learned So Far • Potential, kinetic, & total energy of pendulum • Total energy conserved as potential, kinetic energy transform in SHM • Rotation is special case of pendulum SHM • With no gravitational field, rotation is analog of pendulum SHM • Wave equation maps rotation to pendulum dynamics • Same dynamics without reversals of direction • Key is complex number solution to wave equation (WE) • WE reduces to d2ψ(x)/dx2 +/- k2ψ(x) = 0, with + and - solutions • Positive solution has form of SHM, but • Negative solution has form of rotation expressed as complex number • Only discrete solutions are allowed • Schrödinger equation is QM version of conservation of energy

19. Lessons Learned (cont) • Gauge principle (Yang-Mills) is key to making principle of energy conservation work using wave equation. • Provides a way to change phase (rotate system) without rotating every point in the universe; that is, it allows replacing global invariance (symmetry) with local invariance (symmetry). z There is an infinite number of points on the unit circle. bi z = a + bi There is a complex number, z = a + bi, for every point between 1 and –1 on unit circle. a There is an infinite number of unit lines, (a2 + b2)1/2 = 1, corresponding to the z points. These rotations form a vector space, using one, complex, dimension, forming the basis of the U(1) Lie group, corresponding to 2D geometric rotations.

20. Lessons Learned (cont) • In two, complex, dimensions, the vector space forms the basis for the SU(2) Lie group, which corresponds to 3D geometric rotations. z z = a + bi There is an infinite number of points on the unit circle. bi z’ z’ = a’ + b’i There is a complex number, z = a + bi, for every point between 1 and –1 on unit circle. a There is an infinite number of unit lines, (a2 + b2)1/2 = 1, corresponding to the z points. These rotations form a vector space, using two, complex, dimensions, forming the basis of the SU(2) Lie group, corresponding to 3D geometric rotations.

21. Lessons Learned (cont) • SU(2) vis-a-vis R(3) • The SU(2) Lie group and Lie algebra correspond to the 3D geometric rotation group • Except it takes 720 degree rotation (4pi) to return to starting point, not 360 degrees (2pi)! (The story of spin) • Nobody knows why (do we?) • Hint: M3 cycle is a 720 degree cycle! Bruce Schumm writes (Deep Down Things, 2004): “What is spin and this oddly construed spin-space in which it lives? On the one hand it is quite real [corresponds to angular momentum]. On the other hand, a particle with no spatial extent [electron is point particle] shouldn’t possess angular momentum [or] have to be rotated through 720 degrees to return the particle to its original state. We don’t really have a clue about the physical origin of spin...”

22. Lessons Learned (cont) • The use of phase change in one, complex, dimension, enables application of conservation of energy in terms of phase changes (rotations) in U(1) group with reference to the electromagnetic interaction. • The isospin concept extends the idea of phase changes and conservation in SU(2) to the weak nuclear force, where phase changes in two, complex, dimensions leads to the weak interaction, which is a short-range force permitting the prediction of radioactive decay events.

23. Conclusions from Lessons Learned • The concept of M2, (change of position) motion, has been utilized to • attempt the description of magnitudes of M3 (change of interval) and M4 (change of scale) motion • The attempt has been only partially successful • As far as it goes, it’s extremely accurate, but it’s incomplete without the Higgs potential and Higgs boson • There are many flags to alert us that chart of motion will shed light on these problems • Natural explanation of spin • Insight into rotation – change of interval correspondence • Clarification of point particles and charges (distribution of charge on electron)

24. Predictions • We will be able to bridge to Newtonian science, from Larsonian science. It’s only a matter of time and resources • Once there, LST physicists will take serious look at our new science • Those who jump on board now will be glad they did later.