Local gauge invariance, action-reaction in electromagnetism, weak, strong & gravity interactions

On Jan 26, 2014, at 9:02 PM, JACK SARFATTI <jacksarfatti@gmail.com> wrote:

G is the local transformation of COINCIDENT LNIF <-> LNIF’ at a fixed small region of space-time.

The Levi-Civita Christoffel connection transforms as

{LC} -> {LC}’ = {LC}GG^-1G^-1 - G^-1G^-1dG   (eq. 2.39 in I p.73, )

dV/ds -> dV’/ds  = GdV/ds + dGVV (eq. 2.40 in I p. 74)

V -> V’ = GV

DV/ds -> DV’/ds = dV’/ds + {LC}’V’V’

{LC}’V’V’ = ({LC}GG^-1G^-1 - G^-1G^-1dG)GGVV = G{LC}VV - dGVV

Therefore the acceleration gauge shift terms dGVV cancel out of

DV’/ds = GDV/ds

That is, the T4(x) gravity analog of virtual momentum hdS in U1 EM is the acceleration  c^2dGVV.

Putting in dimensions, V is dimensionless dX/ds

d has dimension 1/Length

G is dimensionless

c^2dGVV is the shift in acceleration g i.e.

g(LNIF) - g’(LNIF’) = c^2dGVV

in time &t

c^2dGVV/&t = exchange of “jerk" between test particle acceleration dV/ds and gravity field {LC} equivalence principle acceleration.

P in U1 EM is replaced by DV/ds

| is “A short course in general relativity” J Forster, J D Nightingale, Springer 1995

On Jan 26, 2014, at 2:23 PM, JACK SARFATTI <jacksarfatti@gmail.com> wrote:

I correct a previous error from some days ago

P = mV + @A

A = AaT^a = connection “potential” field

T^a are the elements of the internal symmetry Lie algebra - they are matrices in practical calculations

mV -> mV + hGdG^-1

@A -> @A - hGdG^-1

So P is still gauge invariant even in Yang-Mills.

Note that the Yang-Mills internal fiber curvature is

F -> F’ = GFG^-1

G is a matrix representation of the internal symmetry Lie group

The matrix representation is that for the multiplet structure of the charges

http://en.wikipedia.org/wiki/Hypercharge

e.g. quarks, leptons, W-bosons, gluons

GdG^-1 is the generalized dS of simple U1 theory.

On Jan 26, 2014, at 1:38 PM, JACK SARFATTI <jacksarfatti@gmail.com> wrote:

the pope has spoken 

divine revelation

It’s now clear to me that neither Jim nor Z understand the physical meaning of Einstein’s GR.

Jim has still never defined precisely with the relevant MAINSTREAM literature math what he means by

“inertia”

“inertial force”

“inertial reaction force”

“phi”

Ag

The only role of the gravitational field in Einstein’s GR is to provide the global objective structure of real-force free timelike geodesics for rest massive test particles and the null geodesics for zero rest mass test particles in terms of source fields in Tuv.

Real forces push rest-massive test charges off timelike geodesics.

m the rest mass of a test particle cancels out of the geodesic equation completely ( dm/ds = 0).

We are only sure of three real forces

EM U1

Weak SU2

Strong SU3

Real forces come from internal symmetry fiber spaces beyond base spacetime (fiber bundle)

Fictitious forces come from spacetime symmetries.

Gravity is a local gauge field from spacetime symmetries.

Real forces and fictitious forces intersect in the local gauge theory minimal coupling.

For U1 simplest case 3-vector

P =  mV + (e/c)A    at a point event

The U1 gauge transformations are

mV -> mV’ = mV + hdS

S = classical gauge function = quantum phase of test particle

h = Planck’s quantum of action

(e/c)A -> (e/c)A’ = (e/c)A - hdS

Therefore 

P -> P’ = P is a U1 gauge invariant local observable canonical momentum of charge + field.

hdS is the momentum exchange of a virtual longitudinal polarized photon between the coincident charge and EM field where A is the Glauber macro-quantum coherent order parameter that is the classical EM field.

Newton’s 3rd law of action-reaction comes from momentum conservation which, in turn, comes from translational symmetry

mV and (e/c)A form an effective CLOSED SYSTEM that is LOCALLY TRANSLATIONALLY INVARIANT

the exchange of hdS guarantees local momentum conservation

force = hdS/&t

where

&E&t < h (virtual particle)

therefore, Newton’s 3rd law is obeyed LOCALLY using gauge invariance

>—|

> is the test charge (e,m)

— is the virtual momentum exchange hdS

| is the Glauber coherent state order parameter A representing the classical EM field.

Similar arguments for SU2 and SU3

also for gravity, which is a T4(x) local gauge theory