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Spinor Qubits in Curved Space-Time



 
 
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Old December 29th 06, 02:30 AM posted to sci.math,sci.physics.relativity,sci.astro,sci.philosophy.tech,sci.physics.particle
Jack Sarfatti
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Default Spinor Qubits in Curved Space-Time

The basic spinor qubits here that are square roots of null world vectors
are not directly in the quantum Hilbert space of Bohm's pilot waves
guiding extended test particles that can be thought of as little
wormholes complementary to strings and membranes. Consult R. Wald for
the complex details of getting the Hilbert space irreducible
representations of the globally flat Poincare group for quantum theory
in any interpretation.

The Penrose spinors rely on the fact that the Lie algebra of the
globally rigid 10-parameter Poincare group specify 10 Killing vector
field isometries - not so simple in general relativity's fields of
curvature (Ricci contraction/expansion from matter/ZPF & Weyl vacuum
tidal stretch-squeeze) and beyond to torsionic fields where this
Poincare group is locally gauged to get 16 non-trivial tetrads and 24
non-trivial spin connections as dynamical compensating gauge potentials.

The spinors in curved base space-time live in the local tangent fibers
not in the base space-time. That is, given the 4 Einstein-Cartan 1-forms e^a

e^A^A' = e^a(Sigma)a^A^A' all in the (co)tangent fiber.

The topology of spacetime has profound restrictions on the definition of
global spinor field in curved and possibly torsioned spacetime - a
non-trivial complicated subject. 13.2 The parallel transport of spinors
is defined in terms of its null field flag tensor F. Suppressing these
subtle points on topology for now, one can construct the spinor
covariant derivative DAA' corresponding to Einstein's 1915 torsion-free
theory. Given any Penrose spinor qubit |C

(DAA'DBB' - DBB'DAA')|C = ChiAA'BB'C^J|J

The spinor square root of Einstein's 1915 GR curvature tensor is then
the 4-qubit string

RAA'BB'CC'^JJ' = ChiAA'BB'C^JE*C'^J' + Chi*AA'BB'C'^J'EmC^J

Chi splits into 3 pieces (13.2.24) p. 371

Similarly R splits into a Weyl vacuum + Ricci contracted source + Ricci
scalar.

The Ricci contraction spinor is independent of the Weyl vacuum spinor.

Newman-Penrose do not use the usual Einstein-Cartan tetrad orthonormal
basis with 1 timelike and 3 spacelike tetrads. They use this Penrose
spinor basis where

A+|A- = 0

A+|A+ = A-|A- = 0

They replace the 24 real spin connections by 12 complex ones. Local
gauging of O(1,3) implicit, i.e. Shipov's torsionic field.

They have a special "complex null tetrad" basis

|l^A^A'* = |A+|A'+* retarded future light cone

l^A^A'*|l^A^A'* = 0 null tetrad

similarly for null

|n^A^A* = |A-|A'-* advanced RETRO-CAUSAL past light cone

with two spacelike tetrads

|m^A^A' = |A+|A'-*

|m^A^A'* = |A'+*|A-

|x^A^A' = (1/2)^1/2[|A'+*|A- + |A+|A'-*] spin triplet S = 1 Sz = 0
analog

|y^A^A' = i(1/2)^1/2[|A'+*|A- = |A+|A'-*] spin singlet S = 0 Sz =
0 analog

The Weyl vacuum stretch-squeeze gravity wave spinor is the symmetrized
product of 4 principal null spinors

|A|B|C|J

|A can repeat 2,3,4 times.

What about fields with a given rest mass m and spin in globally flat
Minkowski spacetime with zero real intrinsic curvature?
The STRICT EQUIVALENCE PRINCIPLE demands MINIMAL COUPLING i.e. use
covariant DAA'.
The local spinor currents are no longer conserved - same problem of the
stress-energy of the pure gravity field that must be nonlocal. There is
no well-posed initial value problem for spin 1 p. 375. This is why
it's better to use renormalizable spin 1 curved tetrad fields and then
get spin 2, 1, 0 effects from entangled 1 + 1 = 2,1,0 pairs. For zero
rest mass, initial value problem is well posed only for spin 1/2 & spin
1 in curved 3+1 spacetime. This is fine for my theory of gravity
emergent from vacuum ODLRO which provides m =/= 0 from ODLRO itself. All
basic fields are massless.

On Dec 28, 2006, at 12:23 PM, Jack Sarfatti wrote:


http://www.people.cornell.edu/pages/gnl2/cave.htm

13.1 on Robert Wald's "General Relativity" has a nice discussion of
measurement theory in general relativity in terms of "families of
observers" that agrees with what I say below on the distinction between
local objective invariants (Platonic forms, Jungian Archetypes) and
their observer-dependent subjective faithful representations and
provides more details.

Diffeomorphisms are not physical, they are too general with tremendous
gauge freedom redundancy. We only want their sub Lie group of isometries
("Killing" (man's name) vector fields) so that different observers can
meaningfully compare their subjective data, put it into the machine and
crank out the same local invariants from the theory. Their goal is each
to get the same set of real numbers for observations of equivalent
happenings.

"when (and only when) Diff(4) is an isometry, we can use [it] ... to map
our original family of physical observers associated with the [tetrads
e^a] into a new family of observers associated with [tetrads e^a'] ..."
p. 343.

"Here upon we're all agreed all that we two will agree to. To entrust
you in The Art, elemental, fundamental ..." W.S. Gilbert, Yeoman of the
Guard

Given a set of tetrad 1-forms e^a then as spinors we have

e^AA' = e^a(Newman-Penrose)a^AA'

Each A index = 0,1 is 1 qubit

for this

IT FROM QUBIT

ureye.gif

e^BB' = I^BB(flat)' + A(warped)^BB'

Now we come to a strange miracle - the Bell quantum teleportation pair
states appear.

Use qubit |A+ of Wald type (0,1;0,0) & qubit |A'-of Wald type
(0,0;0,1) (p. 347) each in 2 complex dimensions.

Then, for special relativity, space-time AKA emerges from 2-qubit
strings |A & |A'.

First take the local diagonal at the same local objective IT coincidence
P = P', the space-time operators are entangled 2-qubit strings in 4
complex dimensions, i.e., spinor tensors of type (0,1;0,1) in Wald's
notation Ch 13). Thus, we have the pair entangled qubit states

|t^AA'(P) = (2)^-1/2[|A(P)+|A'(P)+ + |A(P)-|A'(P)-]

Note |t^0^0', |t^0^1', |t^1^0' |t^1^1' span 4 complex dimensions.
Similarly for

|x^AA'(P) = (2)^-1/2[|A(P)+|A'(P)- + |A(P)-|A'(P)+]

|y^AA'(P) = (2)^-1/2[|A(P)+|A'(P)- - |A(P)-|A'(P)+]

|z^AA'(P) = (2)^-1/2[|A(P)+|A'(P)+ - |A(P)-|A'(P)-]

We can generalize the above to the non-local pair states

|t^AA'(P,P') = (2)^-1/2[|A(P)+|A'(P')+ + |A(P)-|A'(P')-]

|x^AA'(P,P') = (2)^-1/2[|A(P)+|A'(P')- + |A(P)-|A'(P')+]

|y^AA'(P,P') = (2)^-1/2[|A(P)+|A'(P')- - |A(P)-|A'(P')+]

|z^AA'(P,P') = (2)^-1/2[|A(P)+|A'(P')+ - |A(P)-|A'(P')-]

We may need to put a path-dependent anholonomic propagator U(P---P')
here, one for each homotopy equivalence class of paths --- homotopic
to each other from topological obstructions like black hole event
horizons and observer dependent horizons in dark energy dominates
deSitter space-time.

Below everything is local on the diagonal P = P' for Einstein's "local
coincidences" defined by Rovelli in Ch II of his 'Quantum Gravity."

guv --- gAA'BB' = EABE*A'B" (13.1.15) p. 349 Wald's GR

|EAB = |A+|B- - |A-|B+ = -|EBA

where A+|A- = EAB|A+B- = 1

A+|A+ = A-|A- = 0 i.e. these qubits have a null inner product that
cannot be interpreted as in quantum Hilbert space. That is a differently
defined product with complex conjugates. My use of the Dirac bra-ket for
e-mail convenience may not be the best here because it can lead to that
confusion.

Choose the following canonical basis with components (A), (B) etc. The
2x2 "basis" matrices are

E(A)(B)11 = 0

E(A)(B)12 = 1

E(A)(B)21 = -1

E(A)(B)22 = 0

t^(A)^(A') = Pauli Spin Matrix(t) (with added factor -1/2^1/2)

x^(A)^(A') = Pauli Spin Matrix(x)

y^(A)^(A') = Pauli Spin Matrix(y)

z^(A)^(A') = Pauli Spin Matrix(z)

These 2x2 matrices do not commute, so we have a non-commutative geometry
in spinor tensor space. They have the same formal Lie algebra of
internal SU(2) of the weak force fiber even though they represent
Minkowski spacetime base space.

Any globally flat Minkowski world vector first rank tensor can be
represented as the second rank spinor (mod - 1/2^1/2) in a 2x2 matrix
representation

V^A^A' = V^t Pauli Spin Matrix(t) + Vx Pauli Spin Matrix(t) + (13.1.29)
p. 351 Wald

Lorentz transformations are SL(2C) similarity transformations in this
notation.

These single qubit "potential" spinors map to null vectors on the light
cone in globally flat Minkowski spacetime. The qubit is the square root
of a null world vector V^^a^a', i.e.

|V^A^A'* = |A+|A'+*

V^A^A'*|V^A^A'* = 0

One can also make 2nd rank antisymmetric null world field tensors (AKA
null bivectors).

F^A^A'^B^B' = |A||BE*^A'^B' + |A'*||B'*E^A^B

F^2 = 0

FV = 0

(indices understood) p. 352 Wald.

This null F defines the Penrose "null flag".

To include T4 translations we must extend SL(2,C) to ISL(2,C) covering
the 10-parameter Poincare group whose local gauging in all non-gravity
field actions i.e. equivalence principle gives curvature + torsion i.e.
independent 16 non-trivial tetrad components and 24 antisymmetric spin
connection components packaged into 4 tetrad 1-forms e^a and 6
spin-connection 1-forms W^a^b in tangent vector/cotangent form fibers.

Lie derivatives of spinor fields are only definable for Killing
isometries. You cannot do it for a general Diff(4) p. 353 Wald.

Newman-Penrose components are using the 16 tetrad components

(Sigma)^aAA' = e^a0tAA' - e^a1xAA' - e^a2yAA' - e^a3zAA'

so this shows how to generalize to curved and torsioned spacetime!

to be continued.


Jack Sarfatti

"If we knew what it was we were doing, it would not be called research,
would it?"
- Albert Einstein
http://lifeboat.com/ex/bios.jack.sarfatti
http://qedcorp.com/APS/Dec122006.ppt
http://video.google.com/videoplay?do...ausation&hl=en
http://www.flickr.com/photos/lub/sets/72157594439814784
On Dec 26, 2006, at 1:00 PM, Jack Sarfatti wrote:

"The Question is: What is The Question?" John Archibald Wheeler

12essa.1.jpg
http://cecelia.physics.indiana.edu/p...m/wheeler.html

kolb1new_Page_05.jpg

http://www-conf.slac.stanford.edu/ss...age_12_jpg.htm

kolb1new_Page_51.jpg
http://www-conf.slac.stanford.edu/ss...age_51_jpg.htm

Kolb2_Page_46.jpg
http://www-conf.slac.stanford.edu/ss...age_46_jpg.htm

The total energy of the universe is obviously not conserved in the
standard model that fits observations to ~ 1% precision.

kolb1new_Page_08.jpg
http://www-conf.slac.stanford.edu/ss...age_08_jpg.htm

The total w = -1 dark energy repulsive cosmic antigravity field
accelerating the 3D space expansion of the universe is obviously not
conserved. The total w = -1 dark energy content of our universe from
Rocky Kolb's first slide above scales as a(t)^3.

The total w = 0 on-mass shell finite rest mass matter-energy is
conserved scaling as a(t)^0 = 1.

The total zero rest mass w = +1/3 on light cone radiation energy is not
conserved, it scales as a(t)^-1 - zero from the cosmic redshift. Note
that the chemical potential of radiation is zero, therefore there is no
general global conservation law for radiation.

The general point here is that global space-time conservation laws are
not fundamental unlike the local versions, which are satisfied e.g.

Tuv(matter)^;v = 0

locally in 1915 GR with the (LC) connection covariant partial derivative ;v.

Global conservation laws require

[Tuv(Matter) + tuv(Matter-Gravity)]^,v = 0

where ,v is the ordinary partial derivative.

This cannot be done in general in curved space-time with
tuv(Matter-Gravity) as a kosher localized T4 tensor. tuv(Matter-Gravity)
is a pseudo-tensor because in LNIF's the energy-momentum of the
non-geodesic detectors powered by non-gravity forces makes a
contribution to the vacuum gravity field that, by the equivalence
principle, cannot be locally distinguished from the "real gravity
field." Garbage in - garbage out. Ask a stupid question, get a stupid
answer. Trying to globally conserve total energy, trying to conserve
total linear and angular momentum in a generally curved and torsioned
spacetime is a stupid thing to try to do. We already know this from
Noether's theorem. This is simply because the global Poincare group is
locally gauged and all we can hope for is to locally conserve the total
stress-energy current densities, which in fact is the case.

((-detguv)^1/2Tuv(matter))^,v +
(-detguv)^1/2(LC(observer))u^v^wTvw(matter) = 0

Actual observations given above show that the total energy of the
universe is not conserved. It's time to slay that Sacred Cow.

What the pure mathematicians, who lack physical understanding do not
get, is that any representation of curved space-time guv is observer
dependent i.e. relative to any conceivable network of ideal local
observers on arbitrary worldlines inside the local light cone field. For
example, in the non-rotating black hole outside the event horizon rs/r 1

g00 = -1/g11 = 1 - rs/r

is only for that special class of static LNIF "shell" (J.A. Wheeler's
term) observers at fixed r without orbital angular momentum. They need
to fire rockets to stay in place. Note that warp drive "UFO" observers
see a different metric field representation. What "Diff(4)" (local T4)
frame shifts do is to connect different networks of local observers.
That's the physical meaning of the abstract math missed by many of the
formalists in the field. The local relation between objective reality
and the observer's experience is

ds^2(objective observer invariant) = guv(observer dependent)dx^udx^v

= (Tetrad)^a(Tetrad)a

Space-time physics is local only because curved space-time is emergent
from a local vacuum ODLRO world hologram Higgsian field with several
coherent Goldstone phases that encode the 10^122 bits of our universe
retrocausally from far future Omega to past Alpha at The Creation in our
deSitter Universe.

http://qedcorp.com/APS/Adam.jpg

http://qedcorp.com/APS/desitter.jpg

Jack Sarfatti

"If we knew what it was we were doing, it would not be called research,
would it?"
- Albert Einstein
http://lifeboat.com/ex/bios.jack.sarfatti
http://video.google.com/videoplay?do...ausation&hl=en
 




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