Summary

Classical Mechanics
    Newton's 2nd Law in an Inertial Reference Frame, Cartesian Coordinates
    Newton's 2nd law in an Inertial Reference Frame, Polar Coordinates, for a Central Force
    Newton's 2nd Law in a Noninertial Rotating Reference Frame
    The Simple Harmonic Oscillator
    The Damped SHO
    The Wave Equation

Special Relativity
    Rotations and Dilations Create a Representation of the Lorentz Group
    An Alternative Algebra for the Lorentz Group
    
Electromagnetism
    The Maxwell Equations
    Maxwell Written With Potentials
    The Lorentz Force
    Conservation Laws
    The Field Tensor F in Different Gauges
    The Maxwell Equations in the Light Gauge (QED?)
    The Stress Tensor of teh Electromagnetic Field
    
Quantum Mechanics
    Quaternions in Polar Cordinate Form
    Multiplying Quaternion Exponentials
    Commutators of Observable Operators
    The Uncertainty Principle
    Automorphic Commutator Identities
    The Schrödinger Equation
    The Klein-Gordon Equation
    Time Reversal Transformations for Intervals
    
Gravity
    The 3 Fields: g, E & B
    Field Equations
    Recreating Maxwell
    Unified Field Equations
    Conservation Laws
    Gauge Transformations
    Equations of Motion
    Unified Equations of Motion
    Strings
    Dimensionless Strings
    Behaving Like a Relativistic Quantum Gravity Theory

Each of the following laws of physics are generated by quaternion operators acting on the appropriate quaternion-valued functions.  The generators of these common laws often provide insight.

Classical Mechanics

Newton's 2nd Law for an Inertial Reference Frame in Cartesian Coordinates
A = (d by dt, 0) squared acting on R = (0, R double dot)

Newton's 2nd Law in Polar Coordinates for a Central Force in a Plane
A = (cosine theta, 0, 0, - sine theta) times (d by dt,0) squared acting on R =

= (0, -L squared over m squared R cubed + R double dot, 2 L R dot over m R squared , 0)

Newton's 2nd Law in a Noninertial, Rotating Frame
A = (d by dt, Omega) acting on (- omega dot R, R dot + Omega Cross R)

= (- Omega dot dot R, R double dot + 2 Omega Cross R dot + Omega dot Cross R - Omega dot R Omega )

The Simple Harmonic Oscillator (SHO)
(d by dt, 0) squared acting on (0, x, 0, 0) + (0, k x over m, 0, 0) = (0,x double dot + k x/m, 0, 0) = 0

The Damped Simple Harmonic Oscillator
(d by dt,0) squared acting on (0,x, 0, 0) + (d by dt,0) acting on (0, b x, 0, 0) + (0, k x over m,0,0) =

= (0, x double dot + b x dot + k x over m, 0, 0) = 0

The Wave Equation
(1/v d by dt,d by dx, 0, 0) squared acting on (0, 0, f(t v + x), 0) =

=(0, 0, (-d squared by d x squared +1/v squared d squared by d t squared ) f(t v+x), 2 d squared f(t v + x)/v dt dx)

The third term is the one dimensional wave equation.  The forth term is the instantaneous power transmitted by the wave.

A Force Is Conservative If The Curl Is Zero
The commutator of the operator(d by dt,Del) acting on a function F = 0

A Force Is Conservative If There Exists a Potential Function for the Force
F = (d by dt,Del) acting on phi

A Force Is Conservative If the Line Integral of Any Closed Loop Is Zero
The closed loop integral of F dt = 0

A Force Is Conservative If the Line Integral Along Different Paths Is the Same
The closed loop integral of F dt = 0

Special Relativity

Rotations and Dilations Create the Lorentz Group
q prime = q + (gamma - 1) even(even (V conjugate,q),V) over absolute value of V squared + gamma even(V conjugate, q conjugate)=

An Alternative Algebra for Lorentz Boosts
(L times (t, x, y, z)) squared =

For boosts along the x axis...

If t = 0, then

L = gamma(1, beta, 0, 0)

If x = 0, then

L= gamma (1, -beta, 0, 0)

If t = x, then for blueshifts

L = gamma (1 - beta, 0, 0, 0)

For general boosts along the x axis

= (gamma t squared +gamma x squared - 2 beta gamma t x + y squared + z squared, beta gamma (-t squared +x squared ),over (t squared + x squared + y squared + z squared )

Electromagnetism

The Maxwell Equations
even((d by dt, Del), (0, B)) + odd((d by dt, Del), (0, E)) = (- div B, Curl E + B dot) = (0, 0)

odd((d by dt, Del), (0, B)) - even((d by dt, Del), (0,E)) = (div E , Curl B - E dot) = 4 pi (rho, J)

Maxwell Written with Potentials

The fields

E = vector(even((d by dt, - Del),(phi, - A))) = (0, - A dot - Grad phi)

B = odd((d by dt, - Del),(phi, - A)) = (0, Curl A )

The field equations

+odd((d by dt, Del), vector(even((d by dt, - Del), (phi, - A)))) =

= (- div Curl A , Curl A dot - Curl A dot - Curl Grad phi) = (- div B, B dot + Curl E) = (0, 0)

odd((d by dt, Del),odd((d by dt, - Del),(phi,A))) - even((d by dt, Del), vector(even((d by dt, - Del), (phi, -A)))) =

= (- Laplacian phi - div A dot, Curl Curl A + A double dot + Grad phi dot) = (Del · E, Curl B - E dot) = 4 pi (rho, J)

The Lorentz Force
odd((gamma, gamma Beta),(0, B)) - even((-gamma, gamma Beta), (0,E)) = (gamma Beta dotE, gamma E + gamma Beta Cross B)

Conservation Laws

The continuity equation

scalar((d by dt, - Del) acting on (div E, Curl B - E dot)) = (the time derivative of div E - div E dot + div Curl B, 0) =

= scalar((d by dt, - Del), 4 pi (rho, J)) = 4 pi (E dot j +d rho by dt, 0)

Poynting's theorem for energy conservation.

scalar((0, -E)(div E, Curl B - E dot)) = (E dot Curl B - E dot E dot, 0)

= scalar((0, -E), 4 pi (rho, J)) = 4 pi(E dot J, 0)

The Field Tensor F in Different Gauges

The anti-symmetric 2-rank electromagnetic field tensor F

(d by dt, - Del) acting on (phi, - A ) -(phi, A) acted on by (d by dt, Del) = (0, - A dot - Grad phi + Curl A)

F in the Lorenz gauge.

((d by dt, - Del) acting on ((phi,A)+(phi, -A)) - ((phi, A)-(phi, - A)) acted on by (d by dt, Del)) over 2 =

= (phi dot+ divA, - A dot - Grad phi + Curl A )

F in the Coulomb gauge

(d by dt, - Del) acting on (phi, -A) + (d by dt, - Del) acting on ((phi, A)-(phi, -A)) over 4 + ((phi, -A)-(phi, A)) acted on by (d by dt, Del) over 4 =

= (phi dot, - A dot - Grad phi + Curl A)

F in the temporal gauge.

(d by dt, - Del) acting on (phi, -A) - (d by dt, - Del) acting on ((phi, A) + (phi, -A)) over 4 - ((phi, -A) + (phi, A)) acted on by (d by dt, Del) over 4 =

= (- divA, - A dot - Grad phi + Curl A)

F in the light gauge.

(d by dt, - Del) acting on (phi, - A) = (phi dot- divA, - A dot - Grad phi + Curl A)

The light gauge is one sign different from the Lorenz gauge, but its generator is a simple as it gets.

The Maxwell Equations in the Light Gauge

Note: subsequent work has suggested that the scalar in these equations is part of a unified field theory.

even((d by dt, Del), odd((d by dt, Del), (phi, A))) + odd((d by dt, Del), even((d by dt, - Del), (phi, -A))) =

= (- div Curl A , - Curl Grad phi) = (0, 0)

odd((d by dt, Del),odd((d by dt, Del), (phi, A))) - even((d by dt, Del), even((d by dt, - Del), (phi, -A))) =

= (phi double dot + Laplacian phi , - A double dot + Curl (Curl A) - Del divA) = (phi double dot + laplacian phi , - A double dot - Laplacian A) = 4 pi (rho, J)

The Stress Tensor of the Electromagnetic Field
T sup ik = Sum of a from x thru z Sum of b from x thru z of (({Ua,Ub} over 3 - 1) times ((0,E) squared + (0,B) squared ) over 2 -

- even(E, Ua) even(E, Ub) - even(B, Ua) even(B, Ub) -

- even(odd(E, B), Ua) - even(odd(E, B), Ub)) over 4pi =

= (-Ex Ey - Ex Ez - Ey Ez - Bx By - Bx Bz - By Bz + Ey Bz - Ez By + Ez Bx - Ex Bz + Ex By - Ey Bx, 0) over 2 pi

Quantum Mechanics

Quaternions in Polar Coordinate Form
q = the norm of q times e to the (theta i) = q conjugate times q times (cosine (theta) + I sine (theta))

Multiplying Quaternion Exponentials
q times q' = the even conjugator of q, q prime + the absolute value of the odd conjugator of q, q' times e to the pi over 2 times the odd conjugator of q, q prime over its absolute value ...

Commutators of Observable Operators
The commutator of A hat, B hat acting on q = (A hat times B hat - B hat times A hat) acting on q = -a I d q by d a + I d a q by d a

= -a I d q by d a + a I d q by d a + I q d a by d a = I q

The Uncertainty Principle
The commutator of A, B over 2 = I over 2 is less than or equal to DA squared DB squared

Unifying the Representation of Spin and Angular Momentum

For small rotations:

The commutator of a rotation around e1, rotation around e2 = 2 (Re3=0 (theta squared) - R(0))

Automorphic Commutator Identities
The commutator of q, q prime = the commutator of q conjugate, q prime conjugated = the first conjugate of the commutator of q first conjugated, q prime first conjugated = the second conjugate of the commutator of q second conjugated, q prime second conjugated

The anti-commutator of q, q prime = the conjugate of the anti-commutator of q conjugated, q prime conjugated = - the first conjugate of the anti-commutator of q first conjugated, q prime first conjugated = - the anti-commutator of q second conjugated, q prime second conjugated

The Schrödinger Equation
psi = e to the (V normalized times (omega t - K dot X))

The hamiltonian operator acting on psi = -i h bar phi dot = -h bar squared over 2 m Laplacian psi + the potential V(0, X) psi

The Klein-Gordon Equation
the sum from n = 0 to infinity of ((d by dt, Del) squared + (d by dt, - Del) squared + (E sub n, P sub n) squared + (E sub n, - P sub n) squared) acting on (E sub n, P sub n) over 2 =

=the sum from n = 0 to infinity of (- div (Curl P sub n)- div Grad E sub n - P sub n·dot (P sub n Curl P sub n) - (P sub n·dot P sub n) E sub n + E sub n⁣ cubed + E sub n double dot,

Curl(Curl P sub n) + Curl(Grad E sub n) + P sub n Cross(P sub Cross P sub n) + (P sub Cross P sub n) E sub n - Grad⁣ (div P sub n) +⁣ P sub n E sub n squared - P sub n ⁣(P sub n·dot P sub n) + P sub n double dot)

It takes some skilled staring to assure that this equation contains the Klein-Gordon equation along with vector identities.

Time Reversal Transformations for Intervals
(t, X) goes to (-t, X) = R times (t, X)

R = (-t, X)times(t, X)inverse= (-t squared + X dot X, 2 t X) over (t squared + X dot X)

Classically

if beta is much much less than 1 then R is approximately equal to (-1, 2 t Beta)

R = (- epsilon over T, 1, 0, 0)

Gravity

The 3 Fields: g, E & B
(d by dt, - Del) acting on (phi, - A) = (phi dot - div A, - A dot - Grad phi + Curl A) = (g, E + B)

Field Equations: Almost Maxwell and a Dynamic g
(d by dt, Del) acting on (g, E + B) = (g dot - div E - div B, E dot + Curl B + B dot + Curl E + Grad g) = 4 pi (rho sub g + rho sub e, J + J sub e)

(d by dt, Del) acting on (g, E + B) = (g dot + div E + div B, E dot - Curl B + B dot - Curl E - Grad g) = 4 pi(rho sub g + rho sub e, J sub g + J sub e)

Recreating Maxwell
Let U = (g dot - div E - div B, E dot + Curl B + B dot + Curl E + Grad g)

W = (g dot + div E + div B, E dot - Curl B + B dot - Curl E - Grad g)

Mirror((U + W) over 2) + (W - U) conjugated over 2 = (g dot + div E + div B, grad g - E dot + Curl B + B dot + Curl E )

Unified Field Equations
(d by dt, Del) acting on (d by dt, - Del) acting on (phi, - A) =

= (phi double dot - div A dot + div A dot + div Grad phi - div Curl A, - A double dot - Grad phi dot + Curl A dot + Grad phi dot - Grad div A - Curl A dot - Curl Grad phi + Curl Curl A)

= (phi double dot + Laplacian phi, - A double dot - Laplacian A) = 4 pi (rho sub u, J sub u)

(d by dt, - Del) acting on (d by dt, - Del) acting on (phi, - A) =

= (phi double dot - div A dot - div A dot + div Grad phi + div Curl A, - A double dot - Grad phi dot + d Curl A by dt - Grad phi dot + Grad div A + Curl A dot + Curl Grad phi - Curl Curl A) =

= (phi double dot - Laplacian phi - 2 div A dot, - A double dot - Laplacian A - 2 Grad phi dot + 2 Curl A dot) =

= 4 pi (rho sub u, J sub u)

Conservation Laws
(d by dt, - Del) acting on (d by dt, Del) acting on (g, E + B) =

= (g double dot + Laplacian g, E double dot+ d Curl B by dt + Grad div E - Curl E dot - Curl Curl B) =

= (d by dt, - Del) acting on 4 pi (rho sub g + rho sub e, J sub g + J sub e) =

= 4 pi(rho sub g dot + rho sub e dot + div J sub g + div J sub e, J sub g dot + J sub e dot - Grad rho sub g - Grad rho sub e - Curl J sub g - Curl J sub e)

rho sub e dot + div J sub e = 0

J sub g dot - Grad rho sub g - Curl J sub g = 0

If the differential operator acts on the hyperbolic equation,  analogous results are obtained:

(d by dt, Del) acting on (d by dt, - Del) acting on (g, E + B) =

= (g double dot + Laplacian g, E double dot + B double dot + Grad div E - Curl Curl B - Curl Curl E) =

= (d by dt, - Del) acting on 4 pi (rho sub g + rho sub e, J sub g + J sub e) =

= 4 pi(rho sub g dot + rho sub e dot - div J sub g - div J sub e, J sub g dot + J sub e dot + Grad rho sub g + Grad rho sub e + Curl J sub g + Curl J sub e)

There are two conservation laws here, charge conservation for electromagnetism in the scalar, and a vector conservation for gravity.

rho sub e dot - div J sub e = 0 is charge

J sub g dot + Grad rho sub g + Curl J sub g = 0

Gauge Transformations
(phi, A) gets transformed to (phi prime, A prime) = (phi -lambda dot - div L, A + Grad lambda - L dot + Curl L)

Equations of Motion
(gamma, gamma Beta) times (g, E + B) =

= (gamma g - gamma Beta.E - gamma Beta.B, gamma E + gamma Beta Cross B + gamma B + gamma Beta Cross E + gamma Beta g) =

= (W dot over m + W dot over e, P dot over m + P dot over e)

Unified Equations of Motion

Repeat the exercise from above, but this time, look to the potentials.

(gamma, gamma Beta) times (d by dt, - Del) acting on (phi, A) = (gamma, gamma Beta) times (phi dot - div A, - A dot - Grad phi + Curl A) =

= (gamma phi dot - gamma div A dot + gamma Beta . Grad phi - gamma Beta . Curl A, - gamma A double dot - gamma Grad phi dot + gamma Curl A dot + phi dot gamma Beta - div A gamma Beta - gamma Beta Cross A dot - gamma Beta Cross Grad phi + gamma Beta Cross Curl A)

That is pretty complicated!  The key to simplifying this equation is to see what happens for light, where dt/dx = dx/dt.  Gamma blows up, but if the equation is over gamma, that problem becomes a scaling factor.  With beta equal to one, a number of terms cancel, which can be seen more clearly if the terms are written out explicitly.

= (phi dot - d by d X A dot + X dot dot A dot + d by d X dot d phi by d X - X dot dot d by d X Cross A, - A dot - d by d X phi dot + d by d X Cross A dot + phi dot X dot - d by d X dot A X dot - d by d X Cross A dot - d by d X Cross d phi/d X + d by d X Cross d by d X Cross A)

It would take a real mathematician to state the proper constraints on the three pairs of cancellations that happen when velocities get flipped.  There are also a pair of vector identities, presuming simple connectedness.  This leads to the following equation:

= (2 phi dot, - A dot - d by d X dot A X dot + d by d X Cross d by d X Cross A)

The scalar change in energy depends only on the scalar potential, and the 3-vector change in momentum only depends on the 3-vector A.

Strings
dq squared = (da0 squared e0 squared + da1 squared e1 squared over 9 + da2 squared e2 squared over 9 + da3 squared e3 squared over 9, 2 da0 da1 e0 e1 over 3, 2 da0 da2 e0 e2 over 3, 2 da0 da3 e0 e3 over 3) =

= (interval squared, 3-string)

Dimensionless Strings

As far as the units are concerned, this is relativistic (c) quantum (h) gravity (G).   Take this constants to zero or infinity, and the difference of a quaternion blows up or disappears.

Behaving Like a Relativistic Quantum Gravity Theory
Case 1:  Constant Intervals and Strings
A transformation T from dq to dq prime such that the scalar part of dq squared equals the scalar part of dq prime squared and the vector part of dq squared is equal to the vector part of dq prime squared

Case 2:  Constant Intervals
A transformation T from dq to dq prime such that the scalar part of dq squared equals the scalar part of dq prime squared and the vector part of dq squared is not equal to the vector part of dq prime squared

Case 3:  Constant Strings
A transformation T from dq to dq prime such that the scalar part of dq squared does not equal the scalar part of dq prime squared and the vector part of dq squared equals the vector part of dq prime squared

Case 4:  No Constants
A transformation T from dq to dq prime such that the scalar part of dq squared does not equal the scalar part of dq prime squared and the vector part of dq squared does not equal to the vector part of dq prime squared

In this proposal, changes in the reference frame of an inertial observer are logically independent from changing the mass density.  The two effects can be measured separately.  The change in the length-time of the string will involve the inertial reference frame, and the change in the interval will involve changes in the mass density.

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