Saturday, October 29, 2005

Vacuum Instability for any Lie Group of Internal Symmetries

n components of macro-quantum ODLRO order parameter are each local complex functions of Einstein's "local coincidences" P (in general). However, for now I only use Minkowski space-time without gravity and we can be sloppy and use "x" instead of "P".

We use Mexican Hat quartic potential because no other seems to permit renormalizable quantum field theories sans gravity of the SSB scheme.

Using the previous notation of n scalar fields for the order parameter in fiber space V over physical base space M of dimension d, with a defect of dimension d' surrounded by space S of dim r in d.

1 + r + d' = d

Topological stability means homotopy group PIr(V) more then identity group generally implies

d' = d - n

1. SImplest global U(1) spin 0 complex scalar field SSB no gauge field.

n = 2 real scalar fields = 1 complex field. In spacelike slice of Minkowski space-time d = 3, therefore the only stable defects are d' = 3 - 2 = 1 line vortex Goldstone phase defects where the Higgs amplitude vanishes surrounded by non-bounding closed loops giving quantized Bohm-Aharonov "Flux without flux" in stable "stationary states" (Bohr-Sommerfeld).

Small vibrations (real quanta on-mass-shell) of the local zero entropy order parameter Goldstone phase along the minimum circle of the Mexican Hat Potential are massless - no energy gap at infinite wavelength. In contrast the small vibrations of the Higgs amplitude up the slope of the potential have a mass gap m ~ Higgs amplitude itself, i.e.

Let (0|phi|0) be the order parameter (VEV = Vacuum Expectation Value), the Mexican Hat Potential of the dynamical complex scalar field phi is

V(phi) = (k/4)(|phi|^2 - |(0|phi|0)|^2)^2

m(Higgs) = k^1/2|(0|phi|0)| real and positive

2. Same spin 0 n = 2 field above with a massless Abelian local gauge field. That is we locally gauged the global n= 2 U(1) above with minimal coupling.

Now, m(Higgs) = k^1/2|(0|phi|0)|

same as before, but now no massless Goldstone phase quantum. It is absorbed into the gauge force field giving it a mass

m(gauge boson) = 2^1/2e|(0|phi|0)

Similarly if we put in a MASSLESS Dirac spinor with a Yukawa coupling @ to the scalar field

m(fermion) ~ @^1/2(0|phi|0)

so that failing to find Higgs boson has NOTHING to do with the origin of inertia of the fermion leptons & quarks in the standard model that uses the more complicated Yang-Mills theory, but this basic idea remains the same.

The objections of James Woodward & Eric Davis about this have no foundation in fact in terms of the actual mainstream theory for the origin of inertia of the basic particles. Hadrons are trapped bags of light quarks and most of the hadronic mass is trapped kinetic energy (e.g. F. Wilczek).

Yang-Mills Nonabelian Gauge Theories beyond n = 2.

Let there be n-complex scalar fields, i.e. n here is not same as in the above notation for real fields. One can see that stable topological defects may require extra space dimensions. For now we will not worry about the stability of the topological defects.

Let G be a Lie group of RIGID global internal symmetry transformations on the n complex scalar field components. "Scalar" under the Poincare space-time group. Let g be an element of G. The nxn matrix representation of G is D(g).

phij --> phi'j = Dj^j'(g)phij'

g = e^-i@aL^a

@a are "phases" (not Goldstone phases as yet) conjugate to the Lie algebra generators L^a where the commutators are

[L^a,L^b] = f^a^bcL^c

fab^c are the "structure constants"

(range of a = N, which need not be same as n)

The N-dim rep is the "adjoint rep" in which

(L^a)bc = -if^abc

Tr[L^aL^b] = (1)^a^b (Identity NxN matrix)

More generally for any n that need not = N but can of course:

T^a = Dnxn(L^a)

a = 1 ... N

The Vacuum Manifold V = G/H.

SSB of G to normal subgroup H.

H is a normal AKA "invariant" subgroup because

H = {g< G|D(g)(0|phi|0) = (0|phi|0)}

Therefore since

g = e^-i@aT^a ~ 1 - i@aT^a + powers of T^a

T^a(0|phi|0) = 0

note this is a nxn matrix eigenvalue equation with ZERO EIGENVALUE with the n-component order parameter as the singular eigenvector.


Do not confuse the above with the mass matrix.

Invariance of the ACTION under the full G implies

V,iTij^aphi^j = 0

V,i is functional derivative of V with respect to phii.

Again do small vibrations about the equilibrium order parameter minima to get real quanta excited states.

There is a squared mass matrix

m^2ij = V,i,j > 0 at the minima on the landscape

where V,i = 0

m^2ijT^a^j^k(0|phi|0)k = 0

Since T^a(0|phi|0) =/= 0 for the BROKEN GENERATORS of G not in H. They are linearly independent and therefore have ZERO MASSES.

That is, the BROKEN GENERATORS of G (non-zero eigenvalues in previous sense) have MASSLESS Goldstone phase excitations on mass shell outside the vacuum. The unbroken generators of H CAN have massive Goldstone phase excitations. All of this BEFORE putting in any gauge fields mind you!

The degenerate Vacuum Manifold of order parameters is the coset space

V = G/H

Next step Yang-Mills gauge fields.

to be continued.

On Oct 29, 2005, at 12:34 PM, Jack Sarfatti wrote:

PS Cartan form generalization of GR to torsion.

B the curved tetrad 1-form comes from local gauging of T4 to Diff(4)

And, in addition, from Spontaneous Broken Vacuum ODLRO of standard model Higgs in U(1)xSU(2)

B = (hG/c^3)^1/2'd'(Effective Goldstone Phase from Higgs Field in Standard Model)

Therefore gravity & inertia together in accord with equivalence principle
i.e. origin of inertia of leptons, quarks, W bosons & Higgs itself seamlessly integrated with emergence of gravity without gravitons, without quantum foam, but with classical gravity waves (LIGO, LISA)

To get torsion 2-form locally gauge O(1,3) in addition to T4

This gives the torsion tetrad 1-form that is also equal to

T' = (hG/c^3)d'd'(Goldstone Phase) = Torsion 2-form

in sense of Bohm-Aharonov "Flux-without-flux", i.e. T = 0 locally on the surrounding non-bounding 1-cycle, but not nonlocally in the sense to the total 2-form integral over the interior of the non-bounding 1-cycle including the Goldstone Phase singularity. That is, the Goldstone Phase Singularities have Bohm-Aharonov NONLOCAL "Flux-without flux" actions-at-a-distance.

Furthermore, in the global deRham integral sense:

T' = dS + W/\S + S/\(1 + B + S) = (hG/c^3)d'd'(Goldstone Phase) =/= 0 when S =/= 0

Note that S/\S means SK^I/\S^KJ

Both S & W are 1-forms with 2 tangent space indices.

D' = d + W/\ + S/
The extended torsion field equations I postulate are:

R' = D'(W + S) = dW + W/\W + S/\W = R + S/\W + S/\S

D'R' = 0 "Bianchi identity"

D'*R' = *J(4D Translational Sources)

D*J(4D Translational Sources) = 0

D'T' = 0

D'*T' = *J(4D Rotating Sources)

D*J(4D Rotating Sources)

ANSATZ: Hypothesis
All local gauge theories (using Einstein's "local coincidences" P MOD Diff(4)) obey the SAME UNIVERSAL
Cartan-form TEMPLATE

A = potential 1-form

F = DA field 2-form

DF = 0 Bianchi identity

D*F = *J Source Equation

D*J = 0 Local covariant conservation of Source Current Densities

Lagrangian Density ~ F/\F

In the case of 1915 GR we start with B and from there get spin connection W where

T = De = dB + W/\(1 + B) = 0

Determines W from B and

D = d + W/\ = Diff(4) covariant exterior derivative

in 1915 GR

I have proved that GR is a local gauge theory
R = DW

D = d + W/
DR = 0

D*R = *J

D*J = 0

R = Curvature 2-form

above is only for 1915 GR limit of zero torsion 2-form

T = De = D(1 + B) = 0

R is like EM field F in

F = dA

dF = 0

d*F = *j

dj* = 0

Where B(P), W(P), R(P)

P is local coincidence not a bare manifold point x

Technically P = {x} = Coset Orbit of All x in manifold mod Diff(4).

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